<?xml version="1.0" encoding="utf-8"?><rss version="2.0" xmlns:atom="http://www.w3.org/2005/Atom" xmlns:dc="http://purl.org/dc/elements/1.1/"><channel><title>Kamathematics</title><link>https://blog.gautamkamath.com</link><description>Essays and notes by Gautam Kamath</description><language>en</language><atom:link href="https://blog.gautamkamath.com/feed.xml" rel="self" type="application/rss+xml"/><item><title>Theory Jobs 2026</title><link>https://blog.gautamkamath.com/2026/06/01/theory-jobs-2026/</link><guid isPermaLink="true">https://blog.gautamkamath.com/2026/06/01/theory-jobs-2026/</guid><pubDate>Mon, 01 Jun 2026 14:59:45 +0000</pubDate><dc:creator>Gautam Kamath</dc:creator><description>
&lt;p class="wp-block-paragraph"&gt;The Theory community has a tradition of a crowdsourced spreadsheet of sharing who has accepted which jobs (see, e.g., &lt;a href="https://blog.gautamkamath.com/2025/06/03/theory-jobs-2025/"&gt;2025&lt;/a&gt;).&lt;br&gt;&lt;br&gt;You can report jobs accepted via &lt;a href="https://docs.google.com/forms/d/1QcnDP4qyRyT954RmCZ-4cHxi829sSWfTSAIurby7HTI/viewform?edit_requested=true"&gt;this form&lt;/a&gt; (anonymous), and view the results on &lt;a href="https://docs.google.com/spreadsheets/d/1Cem8vwVE2lD3xhuPFM_S6dEYbGyTixtNs-Wilr8_4MU/edit?usp=sharing"&gt;this sheet&lt;/a&gt;. If you feel a change needs to be made to the results reported in the form, please &lt;a href="mailto:g@csail.mit.edu"&gt;email me&lt;/a&gt; or fill out this other &lt;a href="https://docs.google.com/forms/d/e/1FAIpQLSeaGCkEdvrnEobtQuj7zlyjdkV4dKKHiNdioen2X0BhtSZL0w/viewform"&gt;form&lt;/a&gt; (also anonymous, unless you choose to identify yourself).&lt;br&gt;&lt;br&gt;The rules are as follows:&lt;/p&gt;



&lt;ul class="wp-block-list"&gt;
&lt;li&gt;You are welcome to add yourself, or people your department has hired.&lt;/li&gt;



&lt;li&gt;Hires should be connected to theoretical computer science, broadly defined.&lt;/li&gt;



&lt;li&gt;Only add jobs that you are absolutely sure have been offered and accepted. This is not the place for speculation and rumors. Please, be particularly careful when adding senior hires (people who already have an academic or industrial job) &amp;#8212; end dates of their current positions might be still in the future.&lt;/li&gt;
&lt;/ul&gt;



&lt;p class="wp-block-paragraph"&gt;To keep this list focused around theory, mild curation will be applied: individuals should have publications at theory-centric conferences such as STOC, FOCS, SODA, ICALP, STACS, COLT, CRYPTO, CCC, ITCS, etc., or publications of a similar style. If you don&amp;#8217;t meet this criterion, but still identify as a theorist/theoretician (at least partially), please email me (ideally, with a sentence or two of explanation) at g@csail.mit.edu and I&amp;#8217;ll include you, no further questions asked.&lt;/p&gt;
</description></item><item><title>Making a talk, without and with AI</title><link>https://blog.gautamkamath.com/2026/05/27/making-a-talk-without-and-with-ai/</link><guid isPermaLink="true">https://blog.gautamkamath.com/2026/05/27/making-a-talk-without-and-with-ai/</guid><pubDate>Wed, 27 May 2026 20:09:26 +0000</pubDate><dc:creator>Gautam Kamath</dc:creator><description>
&lt;p class="wp-block-paragraph"&gt;Some of the discussion online has been about how not to use AI in making academic talks (see, e.g., &lt;a href="https://x.com/JessicaHullman/status/2059003724677226653"&gt;this post&lt;/a&gt; by &lt;a href="https://users.eecs.northwestern.edu/~jhullman/"&gt;Jessica Hullman&lt;/a&gt;). A junior researcher asked my opinion on using AI to help make slides and posters. I shared my thoughts with them, and I thought it was worth writing down in a more organized fashion.&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;I&amp;#8217;ll first give my take on the value of making talks by hand (which I don&amp;#8217;t think I&amp;#8217;m terrible at). I&amp;#8217;ll then comment on how AI could fit into the process well or poorly. I will say that I&amp;#8217;m not (yet?) a wizard at using AI to enhance my talks, so don&amp;#8217;t come here expecting pro tips (except, spoilers, that the process of writing itself is immensely important). Since AI is a fast-moving topic, note that this post was originally written in May 2026, but I don&amp;#8217;t think that the general principles here are very dependent on current AI capabilities.&lt;/p&gt;



&lt;h2 class="wp-block-heading"&gt;Making talks by hand&lt;/h2&gt;



&lt;p class="wp-block-paragraph"&gt;Roughly speaking, the contents of a talk can be decomposed into three levels. From highest to lowest level, they are:&lt;/p&gt;



&lt;ul class="wp-block-list"&gt;
&lt;li&gt;The story you&amp;#8217;re telling the audience&lt;/li&gt;



&lt;li&gt;The contents of each slide&lt;/li&gt;



&lt;li&gt;The words you actually say&lt;/li&gt;
&lt;/ul&gt;



&lt;p class="wp-block-paragraph"&gt;Before you even make a single slide, you should already have a good idea of the &lt;em&gt;story&lt;/em&gt; of the talk, which is your overarching narrative. This includes things like, what is the motivation for studying this problem, which are the important related works to mention, what technical results to highlight, etc.&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;&lt;strong&gt;Designing the story is the single most valuable part in preparing any talk&lt;/strong&gt;, both in terms of your intellectual engagement with your own research and making sure you give a coherent and educational talk. Every one of these details requires thought and consideration. Why should people care about this problem? What examples illustrate the problem cleanly and intuitively? Which technical results are the most interesting, and how should they be simplified for the purpose of a talk? &lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;Note that all of these story design choices are highly subjective. There is no &amp;#8220;one talk&amp;#8221; to give about a particular result, and two co-authors of the same paper may prepare and give totally different talks. That type of variance is fine and by design. A talk is meant to give your own perspective on the research results, and that should come through in the story you choose to tell. This is why you shouldn&amp;#8217;t rely too heavily on slides from talk already given by a co-author: it&amp;#8217;s their story, it doesn&amp;#8217;t have to be yours.&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;The process of designing your story will also help you understand your own results better. Thinking about how to motivate a problem will help you discover new connections. Thinking about a simple example will help you understand the nature of the phenomenon you&amp;#8217;re demonstrating. Thinking about which results to present and how to do it (e.g., what to simplify) will help you distill the core technical ideas in your work. The important part is not only the final content on your slides, but the thinking and your process to get there.&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;Once you design and internalize your story, it will be useful in other settings as well. For example, if you haven&amp;#8217;t written the paper yet, it will make the introduction and technical overview much easier to write. And if someone asks you to explain your result impromptu, you&amp;#8217;ll be able to do so more clearly.&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;Thus far, our discussion has been about designing the story. I often find that, when you have a sufficiently clear story in your head, writing it down is very easy. But there is another disconnect, between the words and pictures on the slides and the actual words you say. This is where practice and rehearsal comes in: you&amp;#8217;ll frequently find that the most natural way to present a slide is entirely different than what you put down in the first place, and you have to go back and refine the slide afterwards (and potentially iterate). This is another reason why using someone else&amp;#8217;s slides is a common pitfall: the way that is natural for you to present a result is not the same way that is natural for them.&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;It&amp;#8217;s not the main point of this post, but a brief comment on delivery. It should be clear to the audience that you care about what you&amp;#8217;re presenting about. After all, if you don&amp;#8217;t care, why should they? This is of course challenging for some people, who are not naturally charismatic or are afraid of public speaking. But this is a core part of academic research, so it will benefit you to improve at it.&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;As a final point here, in case it&amp;#8217;s not obvious, why is giving good talks an important thing to do? In some sense, being a researcher is being an influencer. Research is inherently an attentionally-driven field. You can have the most brilliant ideas and results, but if no one knows about or understands them, then the value of the research is a fraction of what it could be. A talk focuses all the attention on you and your results, so make the most of the spotlight when you have it. &lt;/p&gt;



&lt;h2 class="wp-block-heading"&gt;How can AI help? &lt;/h2&gt;



&lt;p class="wp-block-paragraph"&gt;I&amp;#8217;ll start by outlining ways that AI can hurt you when preparing talks, but end with some suggestions on how it can help you make better talks than you made before. &lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;Using AI too broadly can be disastrous. Suppose you use it to design your &lt;em&gt;entire talk&lt;/em&gt;. Then you are completely skipping the sometimes-painful but essential process of coming up with the story. You won&amp;#8217;t understand your research as deeply. You won&amp;#8217;t have internalized it as you would have if you sweated through it, and you won&amp;#8217;t be able to explain the research as clearly in settings when you don&amp;#8217;t have AI at your fingertips. &lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;I&amp;#8217;ve talked with people who are very anti-AI for writing technical content, due to the value of thinking through the details, but are OK with using AI for helping make talks, since (in their opinion) it&amp;#8217;s not the &amp;#8220;real&amp;#8221; part of research. I disagree strongly with the latter: good communication is an essential part of research, and (hopefully as I&amp;#8217;ve argued above) thinking about how to communicate something well is an essential part of the technical component of research. &lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;Using AI too heavily also discards a lot of &lt;em&gt;you&lt;/em&gt; from your talks. People came to the talk not only to hear about the research, but for &lt;em&gt;your perspective&lt;/em&gt; on the research. The authenticity is very important (and, I believe, will only become more important over time). &lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;This ties into another point, about delivery, where it is important to show that you care about your research. The easiest way to show you &lt;em&gt;don&amp;#8217;t&lt;/em&gt; care about your research is to dump slop slides (i.e., only lightly edited AI outputs) at your audience. You expect me to believe this is something you care at all about if you can&amp;#8217;t spend just a few hours preparing a decent talk? &lt;strong&gt;It also signals that you have no respect for your audience.&lt;/strong&gt; If I catch you presenting slop slides, you will correspondingly lose a lot of my respect for a long time.&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;AI can also result in slides that have a big disconnect in the content on the slides versus what you would actually say. It&amp;#8217;s effectively like having someone else give you their slides. The &amp;#8220;AI&amp;#8217;s voice&amp;#8221; is likely to be entirely different from yours. You can of course refine and iterate on them, but at that point, you might as well just make the slides yourself.&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;So with this said, what can AI be useful for? AI can be help by giving a second opinion on things. Maybe you&amp;#8217;ve done the thinking yourself already (which, as mentioned before, is the important part), and you want to see if there&amp;#8217;s any other perspectives you didn&amp;#8217;t consider. It might suggest some other ideas that you already thought of and discarded, or other new ones that you hadn&amp;#8217;t considered. But you should really think of it as being a &lt;em&gt;second&lt;/em&gt; opinion, your own perspective and preferences should still dominate. &lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;AI can also help make content that you wouldn&amp;#8217;t have the time or the skills to make before. For example, AI can help make high-quality visualizations, animations, and interactive demos. These tools can help communicate an underlying technical idea much better than just words, but could previously have been too difficult to actually create. EDIT: After I released this blog post, &lt;a href="https://www.neeldhara.com/"&gt;Neeldhara Misra&lt;/a&gt; linked a &lt;a href="https://www.youtube.com/watch?v=qe3TzBkm9ic"&gt;talk&lt;/a&gt; by &lt;a href="https://www.imsc.res.in/s_viswanath"&gt;S Viswanath&lt;/a&gt;, which (at a glance, I only skimmed) appears to use AI quite well to improve the quality of the talk.&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;AI models are not going away. You should use them to make your talks better: communicate and express your underlying  message in a way you would not have been able to do before. But you should not use them in a way that makes your thinking or your talks worse: stealing away your valuable thoughts, removing what makes you &lt;em&gt;you&lt;/em&gt; from your talks, or embarrassing yourself in front of your professional colleagues. &lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;EDIT: see also Jessica Hullman&amp;#8217;s independent follow-up &lt;a href="https://statmodeling.stat.columbia.edu/2026/05/28/what-if-scientists-really-were-dispassionate-observers-communicating-ideas-without-irrational-commitment-look-here-says-ai/"&gt;post&lt;/a&gt; about AI use in talks.&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;&lt;/p&gt;
</description></item><item><title>How to turn down an offer</title><link>https://blog.gautamkamath.com/2025/06/18/how-to-turn-down-an-offer/</link><guid isPermaLink="true">https://blog.gautamkamath.com/2025/06/18/how-to-turn-down-an-offer/</guid><pubDate>Wed, 18 Jun 2025 14:47:28 +0000</pubDate><dc:creator>Gautam Kamath</dc:creator><description>
&lt;p class="wp-block-paragraph"&gt;If you are incredibly lucky at some point in your life, you will be blessed with the choice of multiple opportunities, of which you can only accept one. This might be for graduate school, an internship, or a full-time job. This post is how to decline such an offer with grace and poise.&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;Why is this important to do? Getting an offer in front of someone can be an enormous amount of work, and often requires the stars to align (particularly thinking about faculty job offers). There&amp;#8217;s at least one person (and likely multiple people) highly invested in your accepting, professionally and emotionally. Maybe they went out of their way to answer your questions and help you make a decision. And behind the scenes, a lot might have gone on. They could have fought on your behalf and done a lot of work to make that offer come through. The goal is to communicate your decision to these people in a polite and thoughtful way. Besides being a kind thing to do, these individuals are also likely to continue to be relevant in your research career, so best to not leave them with a bad feeling about you.&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;The worst thing to do: ghost your contacts. That is, after you receive an offer, ignore any and all messages from your contacts and hope they get the picture. They eventually will, but this is quite rude.&amp;nbsp;&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;Perhaps par for the course: officially decline the offer, by clicking a button in a system, or emailing the person who sent you the offer (e.g., the department chair, the HR rep, etc.). This may be sufficient, but given the context above, I would recommend a bit more. Imagine you were highly invested in a candidate, you really went out of your way to try to convince them to come, and you learned about their decision only through an impersonal online system, where a field changed to &amp;#8220;offer declined.&amp;#8221;&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;A general principle: the more warmth and kindness someone shows you, the more you should give them in return. This principle holds much more broadly than the present context.&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;Correspondingly, try to identify all the people who you think spent a lot of time/energy/emotions on you and/or would have been excited to see you accept the offer. In the context of a faculty job, this could include the department chair, the host for your visit, and members of the group you&amp;#8217;d be joining. If you&amp;#8217;re not sure about someone, it doesn&amp;#8217;t hurt to include them. And send them a kind email, thanking them, telling them you&amp;#8217;re declining the offer, and, if it has been decided, what you&amp;#8217;ll be doing instead. In a pinch, even a one sentence email thanking them for their time but you&amp;#8217;ve decided to pursue other opportunities is better than nothing. One tricky point is whether to include a reason: if you can do it in a non-offensive way that leaves the door open for further interactions, go for it (e.g., &lt;em&gt;don&amp;#8217;t &lt;/em&gt;tell them that the people there bored you to tears with their research). If you can&amp;#8217;t, feel free to leave it blank. The recipients understand you can only pursue one opportunity, and will appreciate you telling them personally. Sending such emails will of course take some time, but it&amp;#8217;s a lower order term compared to the time invested by both parties to reach this point, so it&amp;#8217;s worth doing. You can send just one email to all such individuals at an institution, though if you feel like you had a closer connection with someone, you can send them a separate personal message.&amp;nbsp;&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;The above may feel difficult to some. After all, sending someone negative news is never pleasant, and one may prefer to just avoid it. But as you progress in your career, uncomfortable situations like this will arise again and again, so it&amp;#8217;s worth developing the skills to handle them.&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;&lt;strong&gt;Acknowledgments&lt;/strong&gt;&lt;br&gt;Thanks to Bailey Kacsmar, Sandeep Silwal, Adam Smith, and Thomas Steinke for comments on this blog post.&lt;/p&gt;
</description></item><item><title>Guest Post: Margins of My Dissertation : Life Lessons that My PhD Taught Me</title><link>https://blog.gautamkamath.com/2025/06/08/guest-post-margins-of-my-dissertation-life-lessons-that-my-phd-taught-me/</link><guid isPermaLink="true">https://blog.gautamkamath.com/2025/06/08/guest-post-margins-of-my-dissertation-life-lessons-that-my-phd-taught-me/</guid><pubDate>Sun, 08 Jun 2025 01:26:15 +0000</pubDate><dc:creator>Chhavi Yadav</dc:creator><description>
&lt;p class="wp-block-paragraph"&gt;This is a guest post by &lt;a href="https://www.chhaviyadav.org/"&gt;Chhavi Yadav&lt;/a&gt; &amp;#8212; a personal and thoughtful retrospective on her PhD experience and lessons learned therein. Also posted on &lt;a href="https://www.chhaviyadav.org/blogs#h.v6iryjtvrbwj"&gt;her own blog&lt;/a&gt;. &lt;/p&gt;



&lt;figure class="wp-block-image size-large"&gt;&lt;img loading="lazy" width="683" height="1024" data-attachment-id="400" data-permalink="https://kamathematics.wordpress.com/2025/06/08/guest-post-margins-of-my-dissertation-life-lessons-that-my-phd-taught-me/image-2/" data-orig-file="https://kamathematics.wordpress.com/wp-content/uploads/2025/06/image.png" data-orig-size="1024,1536" data-comments-opened="1" data-image-title="image" data-image-description="" data-image-caption="" data-large-file="https://kamathematics.wordpress.com/wp-content/uploads/2025/06/image.png?w=683" src="https://blog.gautamkamath.com/assets/3278b85cfd20f0bf5ab5.png" alt="" class="wp-image-400" loading="lazy" decoding="async"&gt;&lt;/img&gt;&lt;/figure&gt;



&lt;p class="wp-block-paragraph"&gt;I expected a spark of joy upon graduating from my PhD. Instead, there was stillness. A quiet that didn’t celebrate the finish, but asked: &lt;em&gt;What did all of this truly mean?&lt;/em&gt; Was it just about writing papers, answering reviewer comments, and collecting citations? Or had this long, often messy journey transformed me in deeper, more personal ways? Once the noise quieted, reflection surfaced. Not just on my research, but on who I had become in the process.&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;The answers, I realized, lived not in the main chapters of my dissertation, but in the margins — those blank spaces that don’t carry equations or arguments — but carry what truly changed. The breakdowns, the risings, the softening, the unlearning, the relearning, the pauses, the unexpected kindness. These reflections became this piece. This isn’t a success story or a guide to academic glory. It’s a deeply personal account of growth, written for anyone who’s been through—or is about to begin—the long haul of a PhD.&amp;nbsp;&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;&lt;em&gt;Are we forgetting the human behind the author?&lt;/em&gt;&amp;nbsp;&lt;em&gt;Do I recommend doing a PhD? What parts of me did it sharpen&lt;/em&gt;&lt;em&gt;&amp;nbsp;&lt;/em&gt;&lt;em&gt;and what did it soften? What did I have to unlearn and relearn? Read on to find out.&lt;/em&gt;&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;&lt;strong&gt;Kindness&amp;nbsp;&lt;/strong&gt;&lt;strong&gt;i&lt;/strong&gt;&lt;strong&gt;s Not a Footnote — It&amp;#8217;s a Foundation&lt;/strong&gt;&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;I’d heard that “words can heal,” but it wasn’t until I sat across from people on hard days—mine and theirs—that I understood how powerful gentle words can be. Not hollow flattery, not surface-level positivity, but real, sincere warm words. Many think that to be taken seriously, one must be harsh and emotionless. But some of the most thoughtful and intellectually rigorous people I&amp;#8217;ve met are also deeply kind. And yes, many of them are men, despite the stereotype that emotional warmth is a feminine trait. I’ve learned that choosing compassion doesn’t diminish your intellect, it deepens it. Kindness doesn’t take away rigor, it humanizes it. You don’t need to be harsh to be right and you don’t need to wound to make a point.&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;A kind word after a failed submission that soothes an aching heart, a moment of listening during burnout, the simple act of checking in &amp;#8212; these don’t show up in your CV, but they’re what keep people going. Especially newcomers, who&amp;#8217;re still building their sense of self and in professions where people suffer silently and often feel alone.&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;&lt;strong&gt;The Human Behind the Author&lt;/strong&gt;&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;Guilty as charged, the naive me used to subconsciously judge others—and myself—by citation counts and the shine of CVs. But slowly, I saw how misguided—and frankly, how pitiable and almost inhumane—this mindset was. Even the most decorated scholars are just people. There can be brilliant minds who can leave one feeling cold, and soft-spoken researchers whose presence can bring warmth and ease. What matters to me now is the character of the person. How do we treats others. Do we listen. Do we uplift. Do we rise after falling. This shift feels even more urgent today, when paper acceptances are often random, culture is fueled by hype, and visibility can overshadow thoughtful work. Tying worth to these moving targets is a guaranteed path to exhaustion as one will always be chasing. Now, I care utmost about the human behind the author. Everything else is secondary.&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;&lt;strong&gt;Becoming THE Author&lt;/strong&gt;&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;There was a time when I did research to prove something, I cannot pinpoint exactly what I was trying to prove, perhaps seeking validation as a serious researcher?? Nevermind, that part of me feels distant now. I started working not to prove anything, but to discover what I actually wanted to say. And slowly, I moved towards becoming THE author — not just of papers, but of my own questions, voice and direction. I still care deeply about research, but I do it with a different intent : to feel at peace and for clarity. Because these questions keep tugging at me, and answering them feels meaningful, even when no one is watching. This resonance is far more grounding I must say.&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;&lt;strong&gt;Keep Moving—You Never Know When It Clicks&lt;/strong&gt;&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;One of the most profound lessons of my PhD is: things shift when you keep moving. There were months when nothing worked, when rejections piled up, writing stalled, or I doubted the point of it all. But something always changed when I stayed in motion. An idea clicked. Stuff clarified. Help arrived unexpectedly. It didn’t mean pushing relentlessly or ignoring rest, it meant not abandoning everything completely. So much of progress is not about brilliance, but about staying in the game just long enough for the tide to turn.&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;Confusion, I’ve learned, isn’t a sign to stop. It’s a sign to lean in. There’s a quote I’ve come to live by: &lt;em&gt;If you feel lost, study harder. If you feel behind, move with discipline. If you feel unsure, create something anyway.&lt;/em&gt; Clarity comes through motion and self-authority is earned in small, quiet ways.&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;&lt;strong&gt;Real Friends, Real Breaks, Real Joy&lt;/strong&gt;&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;One of the most unexpected lessons of this journey was about rest and friendship. Burnout doesn’t knock politely—it arrives unannounced. The only way I survived was by building rhythms of recovery: real vacations, long walks, weekend hikes, moving my body, eating food I enjoyed, talking to people I like. Some of my best thinking didn’t happen at my desk—it happened while traveling, laughing, swimming, or just being deeply present with life. And through this, I found my people. Friends who were honest and real. Who celebrated my wins and held me during losses. Who reminded me that I am more than my research — why limit yourself to research anyway when you could be soooooo much more??&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;&lt;strong&gt;Do I Recommend doing a PhD?&lt;/strong&gt;&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;Only if you know it’s going to break you—and rebuild you—in ways you didn’t expect. If you come in with a romantic and rozy idea of research, it will shatter extremely quickly. But if you’re curious, willing to unlearn, and open to being reshaped by failure and growth—it can be profound. Just don’t go in expecting only answers. You’ll come out with better questions.&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;So yes, my PhD didn’t just teach me how to publish,&amp;nbsp;present or argue a thesis. It taught me how to be a better human. One who is gentler, wiser, and a little more peaceful. These are the margins of my dissertation and they hold more than any chapter ever could.&lt;/p&gt;
</description></item><item><title>Theory Jobs 2025</title><link>https://blog.gautamkamath.com/2025/06/03/theory-jobs-2025/</link><guid isPermaLink="true">https://blog.gautamkamath.com/2025/06/03/theory-jobs-2025/</guid><pubDate>Tue, 03 Jun 2025 13:50:29 +0000</pubDate><dc:creator>Gautam Kamath</dc:creator><description>
&lt;p class="wp-block-paragraph"&gt;The Theory community has a tradition of a crowdsourced spreadsheet of sharing who has accepted which jobs, previously hosted by &lt;a href="https://grigory.us/blog/theory-jobs-2024/"&gt;Grigory Yaroslavtsev&lt;/a&gt; and &lt;a href="https://blog.computationalcomplexity.org/2017/06/theory-jobs-2016.html"&gt;Lance Fortnow&lt;/a&gt;.  &lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;Past years have been slightly chaotic, due to the sheet being an anonymous free-for-all in terms of edits. This year, you can report jobs accepted via &lt;a href="https://docs.google.com/forms/d/1Q2s21yCWvculAzuFkpas3T9eBikSdH7F-6iaRXcQPQ4/edit"&gt;this form&lt;/a&gt; (which is still anonymous), and view the results on &lt;a href="https://docs.google.com/spreadsheets/d/1fyVE8l7FVPM5NQye5xFGOWQdgGPco2yXlJ-Zzy5ZJaU/"&gt;this sheet&lt;/a&gt;. If you feel a change needs to be made to the results reported in the form, please &lt;a href="mailto:g@csail.mit.edu"&gt;email me&lt;/a&gt; or fill out this other &lt;a href="https://docs.google.com/forms/d/1HoH9X_rlFrn18Ids9lysyi9aJo1g1dDe_ohjFktSS2c/edit"&gt;form&lt;/a&gt; (also anonymous, unless you choose to identify yourself).&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;The rules, from past years, are as follows:&lt;/p&gt;



&lt;ul class="wp-block-list"&gt;
&lt;li&gt;You are welcome to add yourself, or people your department has hired.&lt;/li&gt;



&lt;li&gt;Hires should be connected to theoretical computer science, broadly defined.&lt;/li&gt;



&lt;li&gt;Only add jobs that you are absolutely sure have been offered and accepted. This is not the place for speculation and rumors. Please, be particularly careful when adding senior hires (people who already have an academic or industrial job) &amp;#8212; end dates of their current positions might be still in the future.&lt;/li&gt;
&lt;/ul&gt;



&lt;p class="wp-block-paragraph"&gt;To keep this list focused around theory (and particularly theoretical computer science), mild curation will be applied: individuals should have publications at theory-centric conferences such as STOC, FOCS, SODA, ICALP, STACS, COLT, CRYPTO, CCC, ITCS, etc. If you don&amp;#8217;t meet this criterion, but still identify as a theorist/theoretician (at least partially), please email me (ideally, with a sentence or two of explanation) at g@csail.mit.edu and I&amp;#8217;ll include you, no further questions asked.&lt;/p&gt;
</description></item><item><title>Tips on How to Connect at Academic Conferences</title><link>https://blog.gautamkamath.com/2025/05/01/tips-on-how-to-connect-at-academic-conferences/</link><guid isPermaLink="true">https://blog.gautamkamath.com/2025/05/01/tips-on-how-to-connect-at-academic-conferences/</guid><pubDate>Thu, 01 May 2025 08:47:23 +0000</pubDate><dc:creator>Gautam Kamath</dc:creator><description>
&lt;p class="wp-block-paragraph"&gt;I was a kinda awkward teenager. If you are a CS researcher reading this post, then chances are, you were too. How to navigate social situations and make friends is not always intuitive, and has to be learnt. This is particularly true at a conference, where the event is short (just a few days) and the number of people may be intimidatingly large (in the thousands of people). So I wrote this post to give some pointers on how I might go about socially navigating such events, particularly as a newcomer.&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;I hope to do the following two things:&lt;/p&gt;



&lt;ul class="wp-block-list"&gt;
&lt;li&gt;Convince you that connecting with other researchers in your area is worthwhile; and&lt;/li&gt;



&lt;li&gt;Give concrete advice on &lt;em&gt;how&lt;/em&gt; to connect with people at conferences.&lt;/li&gt;
&lt;/ul&gt;



&lt;p class="wp-block-paragraph"&gt;Throughout this post, I use the term &amp;#8220;connect.&amp;#8221; You can think of it as meaning something in the space of &amp;#8220;befriend,&amp;#8221; &amp;#8220;make acquaintance,&amp;#8221; &amp;#8220;develop a rapport,&amp;#8221; or &amp;#8220;establish commonalities.&amp;#8221; The meaning of the term has been somewhat cheapened by professional connotations and, e.g., LinkedIn.&amp;nbsp;&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;Finally, this post is primarily written with large-scale machine learning conferences (e.g., NeurIPS, ICML, ICLR) in mind. Vibes and strategies may differ at other styles of conferences. For example, theoretical computer science conferences don&amp;#8217;t usually have poster sessions. And at a smaller conference, depending on how fitting the topic is, you might have an easier or harder time finding someone you can connect with. So your mileage may vary, but I think the overarching principles remain sound across a variety of conferences (and social settings more generally).&amp;nbsp;&lt;/p&gt;



&lt;h2 class="wp-block-heading"&gt;Why connect with other researchers?&lt;/h2&gt;



&lt;p class="wp-block-paragraph"&gt;Primarily: because it&amp;#8217;s fun! For better or worse, researchers often tend to be friends with other researchers. This is perhaps unsurprising, as we befriend people with whom we have something in common, and research tends to be a large part of researchers&amp;#8217; lives. I have made many friends through the research world, who I look forward to seeing every time I go to a conference or workshop. It is generally fun to have friends all around the world: case in point, I started drafting this post while waiting for a researcher friend to arrive, as we were going sightseeing in Hong Kong. I talk daily with many friends from the research community, who are based in a variety of locations, on a plethora of different topics (research and non-research related).&amp;nbsp;&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;Secondarily, it can be useful for professional reasons. People don&amp;#8217;t generally enter and leave the research community multiple times: the people you will meet at conferences may be your professional colleagues for your entire career. You will read each other&amp;#8217;s papers. You will do service together. You will meet again and again at conferences. You may even go to each other&amp;#8217;s 65th birthday parties (a common academic tradition). So it can be worthwhile to meet and connect with the relevant people early on. It will give you some familiarity of who&amp;#8217;s doing work in your research area and facilitate more natural interactions with them in the future.&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;This is not (directly) a guide on how to &amp;#8220;network.&amp;#8221; Networking connotes a primarily professional motive: for example, you&amp;#8217;re seeking to find a job, or to feel important because you&amp;#8217;re interacting with (supposedly) important people. I just like making friends and connecting with other human beings. Part of why I do research is because of the community. It feels good to be part of something bigger than just myself. I find it a bit off-putting and transparent when people are interacting to network while pretending to be trying to connect. Nonetheless, like I said, connecting with people can be professionally valuable.&lt;/p&gt;



&lt;h2 class="wp-block-heading"&gt;How to connect with other researchers?&lt;/h2&gt;



&lt;p class="wp-block-paragraph"&gt;Now that we&amp;#8217;ve established that it&amp;#8217;s worthwhile to connect with people, how do you do it? Note that the advice I&amp;#8217;ll give here is fairly general purpose (broadly speaking, &amp;#8220;find things you have in common&amp;#8221;) but specialized for academic settings.&lt;/p&gt;



&lt;h3 class="wp-block-heading"&gt;Connecting at the conference&lt;/h3&gt;



&lt;p class="wp-block-paragraph"&gt;Easy mode: find a socially capable senior mentor in your area (e.g., your advisor) and tag along with them. They will know people, they will know the people you should know, and they probably have enough confidence and social capital to be able to introduce you to relevant people. I have done this for some of my students and mentees before. A caricature of such an interaction: &amp;#8220;Hey [student X], remember we read this paper by [student Y] together, let&amp;#8217;s go check out their poster! Hey [student Y], good to see you again after [conference Z], this is my student [student X]. Can you tell us about your work? (&amp;#8230;conversation about their work&amp;#8230;) [Student X] has some interesting results on [topic W] that you&amp;#8217;ve been working on, maybe you two can schedule some time to chat later?&amp;#8221;&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;But maybe your advisor couldn&amp;#8217;t attend the conference, or they&amp;#8217;re busy, and you&amp;#8217;re left to fend for yourself. What can you do? Again: try your best to find things you have in common.&amp;nbsp;&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;The most natural commonality: research interests. This can manifest in a few different ways. Suppose you see someone whose paper you have read. If you&amp;#8217;ve done related work in the area, even better. You can literally walk up to them and say &amp;#8220;Hey, you&amp;#8217;re [person X], right? My name is [person Y]. I read your paper on [topic W] and thought it was really cool!&amp;#8221; You can then ask more questions &amp;#8212; maybe something about their paper, whether they&amp;#8217;re still working on that problem, etc. You might also tell them about your work in the area.&amp;nbsp;&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;Note that this type of interaction will work much better for folks at (or below) your seniority rather than above. For example, it would be very memorable and validating for a junior grad student to hear that someone else read and liked their paper, whereas this might be more common for senior researchers. This is OK. Your goal here is to make friends (likely to be people of comparable seniority), and not &amp;#8220;network&amp;#8221; with &amp;#8220;celebrities.&amp;#8221; Besides, there is probably more scientific and professional value to connecting with people at a similar seniority to you, as they will be your contemporaries throughout your career.&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;Another approach for connecting with people is discovering mutual friends. Perhaps you and the other person have a mutual co-author. Maybe they&amp;#8217;ve hung out with your advisor before. If you can mention this in a natural way, it could build an instant connection and some rapport. I&amp;#8217;m advocating for, e.g., if you have a suspicion that the person is close with your advisor,&amp;nbsp; &amp;#8220;Hey, I&amp;#8217;m [person X], and I&amp;#8217;m advised by [person Y]&amp;#8221; and see if they know each other. This is more natural than, e.g., &amp;#8220;I know [people A, B, and C]. Do you know any of these people?&amp;#8221; (though if you have a suspicion they do know someone, e.g., they&amp;#8217;re in the same group, that could be less awkward). Mutual friends will generally not be obvious in advance, so a bit of trial and error may be needed.&amp;nbsp;&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;This strategy could be described using a loaded term: &amp;#8220;name dropping.&amp;#8221; Indeed, it is a type of name dropping. But that term usually implies the goal of impressing someone with how famous your advisor is or what elite school you go to. Don&amp;#8217;t try to do that. It will only make people like you less. Our goal is to make friends. We&amp;#8217;re name dropping to connect with people, not to impress people.&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;Courtesy &lt;a href="https://sites.google.com/bu.edu/marika/home"&gt;Marika Swanberg&lt;/a&gt;, here are some other helpful icebreakers and conversation topics:&lt;/p&gt;



&lt;ul class="wp-block-list"&gt;
&lt;li&gt;&amp;#8220;Hi, I don’t think we’ve met,&amp;#8221; if it&amp;#8217;s the first time you&amp;#8217;re meeting someone. This will generally lead them to introducing themselves, which can be less intimidating than introducing yourself to someone else. In the case you have met before, then they&amp;#8217;ll tell you where, and that&amp;#8217;s another easy start to the conversation.&lt;/li&gt;



&lt;li&gt;&amp;#8220;Are there any sessions/posters you&amp;#8217;re looking forward to this afternoon?&amp;#8221; This has a dual purpose. You&amp;#8217;ll learn more about them and what they&amp;#8217;re interested in. And second, if their research is adjacent to yours, you&amp;#8217;ll get some information about what work in the field has got the attention of others. &lt;/li&gt;
&lt;/ul&gt;



&lt;h3 class="wp-block-heading"&gt;When to connect?&lt;/h3&gt;



&lt;p class="wp-block-paragraph"&gt;We&amp;#8217;ve talked about strategies to connect with people. But when do you do it?&amp;nbsp;&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;Coffee breaks are, in one sense, the ideal time: the whole point is to grab snacks and socialize. In theory, people are happy to be approached and to talk to people. On the other hand, I know from first-hand experience it can be quite intimidating, especially if you don&amp;#8217;t know anyone yet. In practice, you&amp;#8217;ll see people talking in small groups like they&amp;#8217;re best friends, and it feels like everyone already knows each other (hint: in many cases, they just met within the last day or two). You can squeeze into most such circles and people will generally be courteous and make space for you. Again, this can be scary if you&amp;#8217;re not a social butterfly (and if you&amp;#8217;re a CS researcher reading this post, you probably aren&amp;#8217;t), especially since you have to work your way into the conversation after that point. Slightly easier is to join a group with someone you already know somehow (e.g., co-author, mentor, advisor, someone you already met at the conference) &amp;#8212; you don&amp;#8217;t have to know them &lt;em&gt;well&lt;/em&gt;, even a brief chat at a poster earlier makes it feel a lot less awkward. As a general throughline of this entire post, &amp;#8220;friends beget more friends.&amp;#8221; But again, this still leaves open how to approach someone for the first time.&amp;nbsp;&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;Poster sessions can be a better time to approach people. If someone is presenting their poster, they &lt;em&gt;want&lt;/em&gt; to be approached, and you should do so. The caveat is that they are also likely to be busy, with other people wanting to learn about their work, and you should be mindful to not take up too much time. You can build up some rapport with them, and then schedule some time to meet with them later. You may also find people you want to talk to wandering around the poster session (e.g., in the area with posters related to your research). This may actually be the ideal way to approach someone. It has all the social benefits of the coffee break (people want to talk to people), but without some of the drawbacks (people are less likely to be entrenched in conversation groups).&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;You can also chat with people during conference talks! I don&amp;#8217;t mean literally &lt;em&gt;in&lt;/em&gt; a talk (that&amp;#8217;s just rude), but in the hallway, simultaneous to when talks are going on (sometimes called &amp;#8220;the hallway track&amp;#8221;). When you first go to a conference, you might think that your job is to attend literally every talk. This can be exhausting and you might not get much out of it. Some senior researchers I know are able to appreciate one or two talks per day, but not much more. Don&amp;#8217;t be afraid to skip talks if something else interesting is going on (like a good hallway conversation with someone you&amp;#8217;re really getting along with).&amp;nbsp;&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;Another common strategy is to email people ahead of the conference, asking if they&amp;#8217;ll be there and if they want to meet. This can be tricky, since you might not know who&amp;#8217;s at the conference, and even if they are, they might not be so organized as to have planned out their entire schedule. Again, you should try to email people who you&amp;#8217;re most likely to have a real connection with, and not just for the sake of networking.&amp;nbsp;&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;Conference lunch can have a unique dynamic. If the conference is catered, don&amp;#8217;t be shy and sit at a table with literally anyone. This is easier than breaking into conversation circles, because everyone needs to sit on a chair and eat food. If people have to go out for lunch, you&amp;#8217;ll usually see groups forming in the lobby. It is not weird to ask to join a random group, even if you don&amp;#8217;t know anyone, though once again, you&amp;#8217;ll probably feel more comfortable if you do. These groups often merge with others, and then eventually split when you can&amp;#8217;t find a place that can seat everyone. Tagging along with a senior mentor can also be very helpful. As an advanced tip: I would suggest trying to eat meals with different groups of people throughout a conference. After finding one group, it&amp;#8217;s very easy to just stick with them the whole conference. And if this is your first conference and you&amp;#8217;re able to do that, that&amp;#8217;s great! But if you can step outside your comfort zone, you might benefit from meeting even more people.&amp;nbsp;&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;There are times when you should not approach people at a conference. If they seem in a hurry and walking somewhere. If they&amp;#8217;re in a call or a research meeting. If they&amp;#8217;re feeding their child. Use your judgement if people look focused or busy with a task &amp;#8212; but don&amp;#8217;t mistake this with being busy socializing, because that&amp;#8217;s a signal that they&amp;#8217;re in the socializing headspace.&lt;/p&gt;



&lt;h3 class="wp-block-heading"&gt;What next?&lt;/h3&gt;



&lt;p class="wp-block-paragraph"&gt;After you&amp;#8217;ve connected with someone, what do you do next? At the conference itself, there might be plans for the evenings (e.g., dinner) or sightseeing before/after the conference (or even skipping days of the conference itself). It&amp;#8217;s easy enough to ask questions like &amp;#8220;What are you doing for dinner?&amp;#8221; or &amp;#8220;Did you have any plans to do stuff in the area?&amp;#8221; Either they don&amp;#8217;t, and they might be happy to plan something with you, or they do have plans, and might invite you along (if it&amp;#8217;s not something small/exclusive). You might end up in a group chat on WhatsApp etc., which will facilitate other conference socializing. There&amp;#8217;s often a person or two who takes the lead by suggesting plans &amp;#8212; you will naturally find that person. Advanced tip: be that person. People like to do things but don&amp;#8217;t always like to organize/lead things, so they appreciate such a person.&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;Remember that these connections can last beyond the conference, and potentially even a lifetime. If you had a good research conversation with someone, you can send them an email to talk more about it afterwards, potentially starting a collaboration. If you&amp;#8217;re traveling through the city where they&amp;#8217;re based, you could meet and grab coffee or a meal. You can even ask if you can give a research talk to their group/department.&amp;nbsp;&lt;/p&gt;



&lt;h3 class="wp-block-heading"&gt;Some caveats&lt;/h3&gt;



&lt;p class="wp-block-paragraph"&gt;This document is written by someone who considers himself relatively extroverted and is already established in the community. I tried to put myself in the shoes of someone who differs in both aspects, but let me caveat the advice I give above.&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;To address an elephant in the room: many people will be at the conference trying to network. This is not inherently a bad thing, it may be something you are interested in doing as well (though as mentioned before, this is not directly the guide for that). There are however some negative consequences. Interactions with others can feel &amp;#8220;fake&amp;#8221; or transactional, especially if people network under the guise of connecting. You may feel like you&amp;#8217;re being judged constantly, on things ranging from your institutional affiliation to your race or gender. Unfortunately, these elements are present to some degree at pretty much every conference. It can be quite mentally and emotionally draining, and naturally make you want to retreat. But in the context of this post, people who are fake, transactional, elitist, or judging: they&amp;#8217;re not really people you want to connect with anyway. If you find yourself in an interaction that&amp;#8217;s draining your energy or otherwise making you feel bad, politely excuse yourself and find yourself a way to recover, which may involve hanging out with your closest friends at the conference (co-authors, mentors, colleagues) or decompressing by yourself.&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;On that note, even absent awkward interactions, conferences can be extremely tiring. There&amp;#8217;s a lot of stuff going on, with a huge number of people. On top of this, you might have jet lag. You don&amp;#8217;t have to succumb to your FOMO, and it&amp;#8217;s perfectly normal to take some time for yourself. This is probably not your last conference, you don&amp;#8217;t have to do absolutely everything the first time. I usually try to book accommodation as close to the venue as possible in case I need to take a breather.&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;You might think that you are not suited for socializing at a conference because you are an awkward CS nerd. But you are at a CS conference: pretty much everyone is an awkward CS nerd. Stated differently, most people are in the same boat that you are, at a big event with lots of people they don&amp;#8217;t know, but cautiously eager to talk to people. So don&amp;#8217;t think you&amp;#8217;re the odd one out who doesn&amp;#8217;t belong: we&amp;#8217;re all drawn together by our mutual research interests, so everyone belongs.&lt;/p&gt;



&lt;h3 class="wp-block-heading"&gt;Other perspectives&lt;/h3&gt;



&lt;p class="wp-block-paragraph"&gt;While I was drafting this post, someone shared the following article &amp;#8220;&lt;a href="https://virtual.aistats.org/Conferences/2025/BlogPost_1"&gt;New in ML: A Guide for Navigating the ML Conference Scene&lt;/a&gt;,&amp;#8221; by &lt;a href="https://kkhetarpal.github.io/"&gt;Khimya Khetarpal&lt;/a&gt; and &lt;a href="https://cheng-soon.ong-home.my/"&gt;Cheng Soon Ong&lt;/a&gt;. This is a fantastic guide for a bunch of things you can do at a machine learning conference. It is more professionally-focused than (and thus complementary to) my post. Some folks I talked to found it a bit stress-inducing, so take it more as a collection of suggestions and advice, rather than a must-do guide for a conference. You might vibe more with the strategies in that post (and this one) if you are a &lt;a href="https://en.wikipedia.org/wiki/Type_A_and_Type_B_personality_theory"&gt;Type A&lt;/a&gt; personality.&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;On that note, remember that everyone experiences and enjoys a conference differently! Some people really like interacting with people, others prefer to keep to themselves. Some folks get a lot out of conference talks, while others are spent after just a talk or two. And obviously, everyone has different strategies on how to socialize and connect with people. What I suggest in this guide might not vibe with you. It might even make you think twice about going to a conference! But if that&amp;#8217;s the case for you, ignore everything I wrote, go to the conference, and tell me what worked for you. Send me an email! Your strategies or suggestions might even end up in this post (with attribution).&lt;/p&gt;



&lt;h2 class="wp-block-heading"&gt;Acknowledgements&lt;/h2&gt;



&lt;p class="wp-block-paragraph"&gt;Thanks to &lt;a href="https://www.cs.helsinki.fi/u/ahonkela/"&gt;Antti Honkela&lt;/a&gt;, &lt;a href="https://sandeepsilwal.com/"&gt;Sandeep Silwal&lt;/a&gt;, &lt;a href="https://cs-people.bu.edu/ads22/"&gt;Adam Smith&lt;/a&gt;, &lt;a href="https://thomas-steinke.net/"&gt;Thomas Steinke&lt;/a&gt;, &lt;a href="https://sites.google.com/bu.edu/marika/home"&gt;Marika Swanberg&lt;/a&gt;, &lt;a href="https://jonathan-ullman.github.io/"&gt;Jonathan Ullman&lt;/a&gt; for comments and feedback on this post.&lt;/p&gt;
</description></item><item><title>Guest Post: Advice for Faculty and Industry Interviews</title><link>https://blog.gautamkamath.com/2025/01/02/guest-post-advice-for-faculty-and-industry-interviews/</link><guid isPermaLink="true">https://blog.gautamkamath.com/2025/01/02/guest-post-advice-for-faculty-and-industry-interviews/</guid><pubDate>Thu, 02 Jan 2025 15:18:50 +0000</pubDate><dc:creator>Abhradeep Guha Thakurta, with commentary by Gautam Kamath</dc:creator><description>
&lt;p class="wp-block-paragraph"&gt;The following is a guest post by &lt;a href="https://athakurta.squarespace.com/"&gt;Abhradeep Guha Thakurta&lt;/a&gt;, on differences between the interview process for faculty and industry research positions in computer science. What follows is his post, and I&amp;#8217;ll add a bit of my own perspective at the end.&lt;/p&gt;



&lt;hr class="wp-block-separator has-alpha-channel-opacity"&gt;&lt;/hr&gt;



&lt;p class="wp-block-paragraph"&gt;By Abhradeep Guha Thakurta&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;&lt;strong&gt;My background&lt;/strong&gt;&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;I have spent significant time both in industry and academia, and have been on hiring/interviewing panels on numerous occasions in both. I have noticed striking yet subtle differences in the expectations of the interviewers which tend to make considerable differences in the outcome. In the following, I tried to summarize (in my opinion) a few of the critical differences. These thoughts initially originated from a chat thread with a few of my colleagues, which I translated into a more stand-alone document on the advice of Prof. Gautam Kamath.&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;To bring some credibility to the points I raise, here is my professional background: I have been in the following industry research labs: Microsoft Research (post-doc), Yahoo Labs, Apple, Google Research &amp;#8211; Brain Team, and Google DeepMind. Additionally, I have been a tenure-track faculty at the UC Santa Cruz Computer Science and Engineering Department. I have more than 50+ publications, 10+ patents, and 700K+ worth of NSF Grants as PI.&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;In the following, I will contrast the academic and industry research interviews.&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;&lt;strong&gt;Disclaimer:&lt;/strong&gt; These are all my personal opinions, and may grossly underestimate the nuances in the process.&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;&lt;strong&gt;Interviews in academia&lt;/strong&gt;&lt;/p&gt;



&lt;ol class="wp-block-list"&gt;
&lt;li&gt;&lt;strong&gt;Qualities on which a candidate is judged: &lt;/strong&gt;The interviews in academia typically judge a candidate on the following two aspects.&lt;/li&gt;
&lt;/ol&gt;



&lt;ul class="wp-block-list"&gt;
&lt;li&gt;&lt;strong&gt;Research:&lt;/strong&gt; The candidate is judged on whether the candidate does interesting research which the peers will appreciate. What appreciation means here are a) Can the colleagues at the university write papers/grants with the candidate, and b) Will hiring this person bring repute to the university. (a) mainly translates to NSF/NIH style grants with dollar amounts associated, and (b) mainly translates to how many top quality research papers the person produces.&lt;/li&gt;



&lt;li&gt;&lt;strong&gt;Teaching: &lt;/strong&gt;It is a very important part of hiring. Can the candidate articulate their research (to students). This thoroughly should come out from the job talk. The candidate has to articulate the importance of their research to folks who are completely out of domain expertise. As a result, in academia bombing a job talk is mostly the end of the world.&lt;/li&gt;
&lt;/ul&gt;



&lt;ol start="2" class="wp-block-list"&gt;
&lt;li&gt;&lt;strong&gt;Research statement and teaching statements: &lt;/strong&gt;They are super lengthy. Most people do not read it. What hiring folks do is skim over it. I got brilliant advice from a mentor of mine, who said &amp;#8220;If you want anything to be read by the hiring committee, write it in the first page, and write it in bold. Do not waste space to introduce your research area. Since, if someone does not know your area, they are anyway not going to fight for you.&lt;br&gt;&lt;/li&gt;



&lt;li&gt;&lt;strong&gt;Job-talk vs 1:1 interviews: &lt;/strong&gt;Job talk plays a crucial role in the interview process, possibly much more than the 1:1s. The purpose of the talk is to demonstrate three things simultaneously: a) Can the candidate teach, b) Can the candidate explain the research to faculty in allied areas, and c) Does the research have real substance. (a) is judged by whether the talk (at least a half of it) is understandable to a first-year grad student. (b) is judged by conversations during the talk with colleagues from allied areas. Again the objective here is to figure out whether the candidate can articulate their research to a NSF panel to get grants. Finally, (c) is judged by one expert in the area, which typically is the candidate’s host.&lt;br&gt;1:1’s are fluid, and the candidate is mostly judged subjectively. However, even under subjectivity, it is necessary that the candidate creates a synergy with at least two senior colleagues who would fight for the candidate in the hiring committee meeting.&amp;nbsp; One uncomfortable fact is that because of the subjectivity, this is where most of the variance in the hiring process comes from. It is almost impossible to optimize for it beyond getting a few of the basics right. For example, doing research on the interviewer (including asking folks about the personality), being polite and respectful, and trying to relate the research to the interviewer’s work in a genuine way.&lt;/li&gt;
&lt;/ol&gt;



&lt;ol start="4" class="wp-block-list"&gt;
&lt;li&gt;&lt;strong&gt;Timing for application: &lt;/strong&gt;Mostly synchronized. Happens at the end of the year, with interviews in Jan to April.&lt;br&gt;&lt;/li&gt;



&lt;li&gt;&amp;nbsp;&lt;strong&gt;Repute of the candidate: &lt;/strong&gt;The hiring has a strong correlation with the repute of the candidate, and how famous the letter writers (and of course what they have written &lt;img loading="lazy" width="23" height="23" src="https://blog.gautamkamath.com/assets/700aafb2d0e1683ac633.png" alt=":slightly_smiling_face:" loading="lazy" decoding="async"&gt;). Best letters are the ones which make comparison statements. (E.g., The candidate is comparable to researcher A, who the writer has mentored over the years.) Telling a candidate is a great person is not super important.&lt;/li&gt;
&lt;/ol&gt;



&lt;p class="wp-block-paragraph"&gt;&lt;strong&gt;Interviews in industry research labs&lt;/strong&gt;&lt;/p&gt;



&lt;ol class="wp-block-list"&gt;
&lt;li&gt;&lt;strong&gt;Qualities on which a candidate is judged: &lt;/strong&gt;The interviews in industry labs typically judge a candidate on the following aspect.&lt;/li&gt;
&lt;/ol&gt;



&lt;ul class="wp-block-list"&gt;
&lt;li&gt;&lt;strong&gt;Research:&lt;/strong&gt; The hiring team wants to see the depth of the candidate&amp;#8217;s research and not the breadth. The candidate will be mostly talking to domain experts from very aligned areas of the candidate. The job talk is of a very different flavor than academic one. The best industry job talk is the one that focuses on one problem and goes arbitrarily deep into it.
&lt;ul class="wp-block-list"&gt;
&lt;li&gt;&lt;strong&gt;Uncomfortable fact 1: &lt;/strong&gt;Industry job talks are just conversation starters. Unless the candidate manages to annoy an interviewer present in the talk, it does not count much even if the candidate semi-bombs the talk.&lt;/li&gt;



&lt;li&gt;&lt;strong&gt;Uncomfortable fact 2: &lt;/strong&gt;Most industry labs interviews nowadays have a coding component (LeetCode type or otherwise). If the candidate fails in those, the chances of getting an offer is fairly low, even if the candidate is stellar researchwise. The reason is labs typically want folks who are generally strong, and can pivot to different areas as needed, and basic general purpose skills are mandatory requirements.&lt;/li&gt;
&lt;/ul&gt;
&lt;/li&gt;
&lt;/ul&gt;



&lt;ol start="2" class="wp-block-list"&gt;
&lt;li&gt;&lt;strong&gt;Research statements: &lt;/strong&gt;Many research labs require research statements and letters of references. They are much less valued in the industry than in academia. In many places they are there for legacy reasons, since the nature of research labs have changed significantly over the years. Of course, a bad letter of reference strongly counts against the candidate.&lt;br&gt;&lt;/li&gt;



&lt;li&gt;&lt;strong&gt;Job-talk vs 1:1 interviews: &lt;/strong&gt;In industry job talks are conversation starters, and the focus of the job talk is much narrower. Unlike academic job talks, the sole purpose of the talk for the candidate is to demonstrate the depth of research, which can “wow” an expert in the field. Talking about a problem at great depth is more advantageous than demonstrating breadth. Think of giving an invited talk at a workshop only attended by experts in the field. If a candidate is preparing a job talk both for academia and industry, it is worth preparing two different talks.&amp;nbsp;&lt;br&gt;1:1 interviews are much more structured than academia. There are usually two sets of interviews. One that figures out research depth, and the other that figures out actual hands-on experience with coding (extending from LeetCode to PyTorch debugging.) Research interviews typically are general conversations on research, with a specific problem to be solved at the end (usually in the last 15 mins). Getting a satisfactory answer to this question decides the result of that interview. Coding interviews are standard with correct/incorrect answers. So, there is no subjectivity. Also, they hold a much higher value than a research interview. In both research and coding interviews it is usually safe to be grounded, and not making broad statements. The interviewers that are assigned to Ph.D.’s or post-docs are usually senior people, who can easily tease apart truth from fake.&lt;/li&gt;
&lt;/ol&gt;



&lt;p class="wp-block-paragraph"&gt;3. &lt;strong&gt;Repute of the candidate: &lt;/strong&gt;While the repute of the candidate may be important towards securing an interview. It hardly has any impact on the outcome of the interviews. Mostly the outcome depends on the performance of the candidate in the interviews. One crucial aspect of industry research interviews is how the candidate interacts with engineers in the group and in the rest of the organization. Meaning, does the candidate have the appetite and skill to sit down and solve a problem end-to-end, or will just provide high-level advice.&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;4. &lt;strong&gt;Timing: &lt;/strong&gt;Industry lab hiring happens all the time, mainly based on the requirements. However, once an offer is received it can be delayed pretty much. The reason is both parties (the candidate and the company) know that the employment is at-will. So, it does not make sense for someone to come and leave in a few months with monetary damages to the company. Rather it is best to give the candidate time to choose. There are of course exceptions.&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;5. &lt;strong&gt;Post-doc versus full-time: &lt;/strong&gt;There is not much of a difference in the industry in terms of job description. Of course one is term limited, no stocks, and lesser pay. If one is looking for an industry post-doc, it is sometimes worth taking a full-time position, and deferring the decision to later.&lt;/p&gt;



&lt;hr class="wp-block-separator has-alpha-channel-opacity"&gt;&lt;/hr&gt;



&lt;p class="wp-block-paragraph"&gt;I (Gautam Kamath) will add some of my own commentary on the academic side of things, given my experience as an assistant professor of computer science at the University of Waterloo in Canada.&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;I agree with most of what Abhradeep wrote. I diverge slightly when it comes to judging a candidate&amp;#8217;s research. To remind, the two conditions that he talked about were the following: a) can the colleagues at the university write papers/grants with the candidate, and b) will hiring this person bring repute to the university. In my experience, the former has not been very important. I personally have never heard discussion on whether a candidate would be able to win grants. This is likely due to a combination of a) being at a Canadian university rather than a US university, and b) being in a computer science department inside a school of mathematics, rather than a computer science and engineering department inside a school of engineering. We also don&amp;#8217;t explicitly look for collaborators. On the contrary, if a candidate&amp;#8217;s expertise is too similar to others in the department, this may be a reason to pass them over. While collaboration is of course welcome, in the end, faculty are expected to lead their own independent research program. Overall, our goal is to identify and recruit researchers doing field-leading research.&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;&lt;br&gt;Abhradeep also comments that, in letters, &amp;#8220;telling a candidate is a great person is not super important.&amp;#8221; I agree, but for me personally, it is valuable that a candidate &lt;em&gt;is&lt;/em&gt; a great person. That is to say, besides hiring a researcher, you are also hiring someone who might be your colleague for the next 50 years. I would prefer to hire someone who is willing to help out around the department or group, chime in with their expertise at times when it is needed, and generally work towards making their departmental community a better place. This is always a lower order term compared to research, but being a great person can make me extra excited about a candidate. And certainly, on the other hand, if a candidate has a negative reputation about their personality, this can outweigh the merits of their research.&lt;/p&gt;
</description></item><item><title>Foundations of Responsible Computing Job Market Profiles 2024</title><link>https://blog.gautamkamath.com/2024/12/24/foundations-of-responsible-computing-job-market-profiles-2024/</link><guid isPermaLink="true">https://blog.gautamkamath.com/2024/12/24/foundations-of-responsible-computing-job-market-profiles-2024/</guid><pubDate>Tue, 24 Dec 2024 18:47:44 +0000</pubDate><dc:creator>Gautam Kamath</dc:creator><description>
&lt;p class="wp-block-paragraph"&gt;&lt;a href="https://cs-people.bu.edu/ads22/"&gt;Adam Smith&lt;/a&gt;, &lt;a href="https://jonathan-ullman.github.io/"&gt;Jon Ullman&lt;/a&gt;, and I compiled a collection of profiles of &amp;#8220;FORC-y&amp;#8221; people on the faculty job market this season. FORC stands for foundations of responsible computing, which means &amp;#8220;mathematical research in computation and society writ large&amp;#8221;, including work on private data analysis, fairness, robustness, and mathematical approaches bridging computer science, law, and ethics. We had 29 people who chose to participate, check them out!&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;You can find the PDF with profiles &lt;a href="https://drive.google.com/file/d/1HyvgLfbzpfVT_WE6nhHhafdhwzke2hMZ/view?usp=sharing"&gt;here&lt;/a&gt;.&lt;/p&gt;
</description></item><item><title>Guest Post: ISAAC’24 in Sydney: Registration deadline soon!</title><link>https://blog.gautamkamath.com/2024/11/11/guest-post-isaac24-in-sydney-registration-deadline-soon/</link><guid isPermaLink="true">https://blog.gautamkamath.com/2024/11/11/guest-post-isaac24-in-sydney-registration-deadline-soon/</guid><pubDate>Mon, 11 Nov 2024 03:40:26 +0000</pubDate><dc:creator>Shared by Clément Canonne, with an introduction by Gautam Kamath</dc:creator><description>
&lt;p class="wp-block-paragraph"&gt;Clément Canonne asked to pass along the following registration info for ISAAC 2024, which will be in Sydney in early December. I was lucky to visit Sydney in July 2017 for ICML, and it was one of the most fun conference trips I&amp;#8217;ve had! (Painfully long flight from Boston aside&amp;#8230;) Definitely attend if you can!&lt;/p&gt;



&lt;hr class="wp-block-separator has-alpha-channel-opacity"&gt;&lt;/hr&gt;



&lt;p class="wp-block-paragraph"&gt;&lt;br&gt;The registration for the 35th International Symposium on Algorithms and Computation (ISAAC 2024), to be held in Sydney (Australia) on December 8-11, is open for only a few more weeks!&lt;br&gt;Besides the conference itself, the registration includes lunches as well as a ticket to the conference banquet: &lt;a href="https://www.trybooking.com/events/landing/1249687" target="_blank" rel="noreferrer noopener"&gt;https://www.trybooking.com/events/landing/1249687&lt;/a&gt;&lt;br&gt;Registrations are open until November 29, AoE:&lt;br&gt;Student: AUD 480 (~USD 325)&lt;br&gt;Regular: AUD 960 (~USD 650)&lt;br&gt;(Note: There will be no on-site registration.)&lt;br&gt;For more details, see &lt;a href="https://sites.google.com/view/isaac2024/registration?authuser=0" target="_blank" rel="noreferrer noopener"&gt;https://sites.google.com/view/isaac2024/registration?authuser=0&lt;/a&gt;&lt;br&gt;Looking forward to seeing you at ISAAC in Sydney!&lt;/p&gt;
</description></item><item><title>Principles for Planning an Academic Workshop</title><link>https://blog.gautamkamath.com/2023/08/08/principles-for-planning-an-academic-workshop/</link><guid isPermaLink="true">https://blog.gautamkamath.com/2023/08/08/principles-for-planning-an-academic-workshop/</guid><pubDate>Tue, 08 Aug 2023 11:40:36 +0000</pubDate><dc:creator>Gautam Kamath</dc:creator><description>
&lt;p class="wp-block-paragraph"&gt;Workshops are my favourite type of academic meeting. Smaller than conferences, more focused on a particular area of interest, and featuring the latest results. Planning a workshop can be a challenging endeavour. As an organizer, it is hard to foresee what makes for a good experience for the audience. I have co-organized a number of workshops of different shapes and sizes (e.g.,&lt;a href="https://tpdp.journalprivacyconfidentiality.org/2021/"&gt; 1&lt;/a&gt;, &lt;a href="https://tpdp.journalprivacyconfidentiality.org/2022/"&gt;2&lt;/a&gt;, &lt;a href="https://upml2022.github.io/"&gt;3&lt;/a&gt;, &lt;a href="http://www.fields.utoronto.ca/activities/22-23/privacy"&gt;4&lt;/a&gt;, &lt;a href="http://www.cs.columbia.edu/~ccanonne/workshop-focs2019/"&gt;5&lt;/a&gt;), and put a lot of thought into what makes for the best audience experience. I will share some of my thoughts and opinions on how to plan an effective and enjoyable workshop.&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;This post will focus on single-day workshops with a combination of invited talks and contributed papers, which in particular are common at machine learning conferences. However, the same basic principles should still hold for other formats, e.g., week-long workshops.&lt;/p&gt;



&lt;h2 class="wp-block-heading"&gt;Principles&lt;/h2&gt;



&lt;p class="wp-block-paragraph"&gt;Let&amp;#8217;s start from the basics. What are the goals of an academic workshop? Why are we even doing this? Here are some reasons, listed in roughly decreasing order of importance.&lt;/p&gt;



&lt;ul class="wp-block-list"&gt;
&lt;li&gt;&lt;strong&gt;Facilitating meaningful interactions between participants.&lt;/strong&gt; While somewhat abstract, this is by far the most important goal of any workshop. Connections between individuals lead to sharing of ideas, new collaborations, and help develop a sense of community. Through a combination of these factors, I find myself most fulfilled and inspired after a workshop with a lot of individual or small group interactions. While the importance of the first two factors may be self-explanatory, I really want to emphasize the last one. Research can be &lt;a href="https://twitter.com/thegautamkamath/status/1647262686080651264"&gt;lonely&lt;/a&gt;, and it helps to recognize that you&amp;#8217;re part of something bigger than yourself, and that someone else actually cares about the same things you do. Some ways to facilitate interactions include breaks, poster sessions, breakout groups, and social events.&lt;/li&gt;



&lt;li&gt;&lt;strong&gt;Sharing the latest work in the area.&lt;/strong&gt; Research moves fast. By the time things are published in conferences, the work and ideas may be a year (or more) old. Participants can learn what others are &lt;em&gt;actually&lt;/em&gt; working on right now, and those with work-in-progress can get thoughtful feedback from experts in the field. This is typically done via poster presentations and contributed talks.&lt;/li&gt;



&lt;li&gt;&lt;strong&gt;Highlighting significant lines of research in the field, and next directions.&lt;/strong&gt; Research areas can be broad, and it is easy to myopically focus on one&amp;#8217;s smaller sub-area. It is important to &amp;#8220;zoom out&amp;#8221; and see what&amp;#8217;s been going on elsewhere in the field. This is typically done via invited plenary talks.&lt;/li&gt;
&lt;/ul&gt;



&lt;p class="wp-block-paragraph"&gt;Based on these principles, from most to least important, the schedule should feature breaks, poster sessions, contributed talks, and invited talks. However, I&amp;#8217;ve noticed workshop organizers often prioritize these in the opposite order.&lt;/p&gt;



&lt;h2 class="wp-block-heading"&gt;Concrete Suggestions&lt;/h2&gt;



&lt;p class="wp-block-paragraph"&gt;Here are some suggestions when planning the schedule for your own workshop.&lt;/p&gt;



&lt;ul class="wp-block-list"&gt;
&lt;li&gt;&lt;strong&gt;Have plenty of breaks.&lt;/strong&gt; Workshops can be tiring, and few have the endurance to maintain their attention for hours at a time. Have plenty of breaks to keep people fresh. This also gives people the opportunity to mingle and discuss the ideas presented so far. If your workshop is part of a bigger conference featuring set coffee breaks, make sure to sync with those breaks.&lt;/li&gt;



&lt;li&gt;&lt;strong&gt;Schedule for lunch.&lt;/strong&gt; Everyone needs to go for lunch. If your workshop has lunch provided on-site, an hour should be enough time. If folks have to leave the venue, budget &lt;em&gt;at least&lt;/em&gt; 90 minutes. If you leave less time, you risk the poor post-lunch presenter(s) having no audience. And it should go without saying, but do &lt;em&gt;not&lt;/em&gt; schedule other things overlapping lunch (except maybe something social, see below). &lt;/li&gt;



&lt;li&gt;&lt;strong&gt;Poster sessions are the most valuable part of any workshop.&lt;/strong&gt; At pretty much every event I&amp;#8217;ve gone to, the poster session has been the most enriching part of the program. Make sure it is scheduled at a time when the most people will attend. If there are many posters, consider having multiple disjoint sessions. I&amp;#8217;ve found that 90 minutes is a sweet spot for the length of a poster session. And if you have control over the logistics, make sure the posters are well spaced and easy to gather around. If poster sessions are too crowded or difficult to navigate, people will do something else instead.&lt;/li&gt;



&lt;li&gt;&lt;strong&gt;Throw in some social events.&lt;/strong&gt; While you&amp;#8217;ve got everyone in one place, consider scheduling some non-technical components. One favourite of mine is the &amp;#8220;junior-senior lunch&amp;#8221;: organize so that a &amp;#8220;senior&amp;#8221; person (i.e., someone with a permanent job) go for lunch with a group of &amp;#8220;junior&amp;#8221; people (i.e., students and postdocs). This mixes people who might not otherwise interact, and gives the opportunity for senior folks to share some valuable mentorship with the juniors. Another idea is to organize an optional and informal post-workshop outing for food, drinks, etc.: a great chance to casually unwind after a long day.&lt;/li&gt;



&lt;li&gt;&lt;strong&gt;Encourage intellectual diversity.&lt;/strong&gt; When curating the program, make sure you represent a variety of different perspectives and topics. For example, when planning the talks for &lt;a href="https://tpdp.journalprivacyconfidentiality.org/2022/"&gt;TPDP 2022&lt;/a&gt;, we made sure to have a mix of theory and practice, academics and industry, etc. The audience shouldn&amp;#8217;t leave the workshop with the impression that only one sub-area is important.&lt;/li&gt;



&lt;li&gt;&lt;strong&gt;Prompt your invited speakers.&lt;/strong&gt; Invited speakers are often established researchers, who have broad research agendas, a deep understanding of a sub-area of the field, and meta-perspectives that go beyond a single paper. Thus, it&amp;#8217;s a bit disappointing if an invited speaker just chooses to focus narrowly on their most recent paper. Try to guide them to think bigger, potentially speaking about a collection of works, or a research direction that they think is particularly interesting or important. This distinguishes them from any contributed/spotlight talks, which are more focused on a single work.&lt;/li&gt;



&lt;li&gt;&lt;strong&gt;Accept generously, spotlight judiciously.&lt;/strong&gt; Space restrictions aside, there is little harm to accepting many papers as posters. As long as the results are relevant to the theme of the workshop, seem sound/correct, have at least something new/interesting, and don&amp;#8217;t &amp;#8220;do harm&amp;#8221; (say, the paper is clearly misleading or poorly framed), I tend towards acceptance. That said, time on the schedule is at a premium (see the final point in this list), so be conservative when choosing which submissions to spotlight with a talk. Try to select things that you believe &amp;#8220;everyone must know.&amp;#8221; Otherwise, the interested parties can connect during the poster session. Another benefit to fewer spotlight talks is that the talks can be longer: many find it difficult to get something meaningful out of very short lightning talks, and the rapid switching is very tiring.&lt;/li&gt;



&lt;li&gt;&lt;strong&gt;Dialogue is better than monologue.&lt;/strong&gt; Many people enjoy and get a lot more out of interactivity. It can get boring and hard to follow if one just sits around all day watching a bunch of talks (invited or contributed). Instead, consider ways to have participants engage on technical topics one-on-one or in small groups. The obvious way to do this is via poster sessions, but be creative! Other options include breakout groups or open problem sessions.&lt;/li&gt;



&lt;li&gt;&lt;strong&gt;Fewer invited talks.&lt;/strong&gt; Invited talks take up a lot of time. It is common to schedule these for at least 30 minutes, and having multiple such talks adds up fast. At the same time, workshop organizers are often incentivized into inviting tons of big-name speakers: it may help your workshop get accepted, and attract attendees to your workshop. But this backs you into a corner after you have to design a schedule around them. Consider having only a small number of invited speakers (I think two to three per day is ideal). This features some big picture talks and attracts an audience, while still leaving space for more interactive exchange of ideas. I consider a full day of invited speakers to be a missed opportunity: the vast majority of attendees would end up not saying a single word. &lt;/li&gt;



&lt;li&gt;&lt;strong&gt;Less is more.&lt;/strong&gt; More generally, the most common pitfall is succumbing to the urge to cram everything into a single day. Many organizers first commit to all the incredible content they want to host, and only after do they figure out what can actually fit in the schedule. These two steps need to be done simultaneously. Otherwise, the schedule becomes far too packed, and the day becomes a marathon of back-to-back talks with few or no breaks. It&amp;#8217;s important to remember that things will also inevitably fall behind schedule, so leave some breathing room to keep things comfortable for the attendees and less stressful for the organizers. I once attended a workshop intended as a &amp;#8220;mixer&amp;#8221; between two groups, where the organizers stuffed the schedule to be way too full. I left the workshop without the chance to talk to anyone from the other group!&lt;/li&gt;
&lt;/ul&gt;



&lt;p class="wp-block-paragraph"&gt;For a concrete example of what I think was a well-scheduled workshop, take a look at &lt;a href="https://tpdp.journalprivacyconfidentiality.org/2022/"&gt;TPDP 2022&lt;/a&gt;. We had frequent breaks, multiple long poster sessions, and &amp;#8220;only&amp;#8221; six spotlight talks and two invited talks. This might seem like a sparse program, but it gave lots of room for people to interact, and the talks were all high quality and long enough for people to really get something out of. If I could change anything, I would have tried to coordinate junior-senior groups for the lunch break, and organize a social outing after the workshop. But folks at the time were quite cautious about COVID, so we decided against it.&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;&lt;strong&gt;Acknowledgments.&lt;/strong&gt; Thanks to Ferdinando Fioretto, Thomas Steinke, and Florian Tramèr for comments and feedback on previous versions of this post.&lt;/p&gt;
</description></item><item><title>Guest post: COLT 2022 Call for Open Problems</title><link>https://blog.gautamkamath.com/2022/06/17/guest-post-colt-2022-call-for-open-problems/</link><guid isPermaLink="true">https://blog.gautamkamath.com/2022/06/17/guest-post-colt-2022-call-for-open-problems/</guid><pubDate>Fri, 17 Jun 2022 16:46:29 +0000</pubDate><dc:creator>Shared by Clément Canonne, with an introduction by Gautam Kamath</dc:creator><description>
&lt;p class="wp-block-paragraph"&gt;The tireless &lt;a href="https://ccanonne.github.io/"&gt;Clément Canonne&lt;/a&gt; is the open problem chair for COLT 2022. He asked to share this call for open problems. Please submit your best!&lt;/p&gt;



&lt;hr class="wp-block-separator has-alpha-channel-opacity"&gt;&lt;/hr&gt;



&lt;p class="wp-block-paragraph"&gt;&lt;a href="https://learningtheory.org/colt2022/index.html"&gt;The 35th Annual Conference on Learning Theory&lt;/a&gt; (COLT 2022), to be held in London on July 2-5 and remotely, will follow in the footsteps of previous editions and feature an &lt;em&gt;Open Problems session&lt;/em&gt;, where attendees can present their open problems and suggest them to the learning community &amp;#8212; and possibly offer prizes for their resolution! (After all, a little incentive goes a long way…)&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;The deadline to submit an open problem has been extended to &lt;em&gt;Monday June 20, 4pm PDT.&lt;/em&gt; If you have any nagging question or stubborn problem, please submit them!&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;More information and CfP: &lt;a href="https://learningtheory.org/colt2022/cfp.html#openproblems" rel="nofollow"&gt;https://learningtheory.org/colt2022/cfp.html#openproblems&lt;/a&gt;&lt;/p&gt;
</description></item><item><title>How to Ask for a Letter of Recommendation</title><link>https://blog.gautamkamath.com/2021/08/18/how-to-ask-for-a-letter-of-recommendation/</link><guid isPermaLink="true">https://blog.gautamkamath.com/2021/08/18/how-to-ask-for-a-letter-of-recommendation/</guid><pubDate>Wed, 18 Aug 2021 16:35:21 +0000</pubDate><dc:creator>Gautam Kamath</dc:creator><description>
&lt;p class="wp-block-paragraph"&gt;The end of the year is approaching at an alarming rate, which means that many students will require letters of recommendations for graduate school or jobs. These letters are written by more senior academics and researchers at the busiest time of the year, when they&amp;#8217;re handling other responsibilities like research, teaching, grant writing, and reviewing, and may be writing a dozen letters on top of it all. The purpose of this post is to guide applicants on how to ask for a letter of recommendation, simultaneously strengthening the applicant&amp;#8217;s package, and making the letter writing process as painless as possible for their writers. Feel free to share this with anyone applying for grad school or jobs. If someone is asking you for a letter, you could share this post with them, so they know what you need.&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;Note that this document is written by a computer scientist and intended for applications to computer science things, and may or may not carry over to other fields.&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;&lt;strong&gt;Whom to ask&lt;/strong&gt;&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;Ask people who know you and your work well. Ideally people who you&amp;#8217;ve worked on research with. Their words will carry the most influence for whatever position you&amp;#8217;re applying to. On the other hand, if you don&amp;#8217;t have a letter from someone you did a major research project with (especially when applying for grad school), this is a red flag.&amp;nbsp;Aim to have at least one such letter. While grad school applications typically solicit three letters of recommendation, very few people have all focused on research (I personally had two, from co-advisors on one project).&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;Some have suggested having someone &amp;#8220;at arm&amp;#8217;s length&amp;#8221; is beneficial for faculty applications (and anything that comes later, including awards and fellowships), in order to show that your work is known more broadly in the community.&amp;nbsp;&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;Letters from people who did nothing more than teach you a course are typically valued much less than research-based letters. Nonetheless, most grad school applicants will have at least one or two such letters, since it is rare to have the opportunity to do research projects with three different advisors. If you must, try to request letters from people who you left an impression on, or otherwise had meaningful interactions with. Try to avoid letters from people who can do nothing more than repeat information on your CV/transcript: these letters are often very short and uninformative, and as with any such letter, are unlikely to help your case (and may even hurt it).&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;&lt;strong&gt;When to ask&lt;/strong&gt;&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;Please ask &lt;strong&gt;well &lt;/strong&gt;in advance. As a rough rule of thumb, by late October is reasonable for &amp;#8220;standard&amp;#8221; deadlines which are in December, which is common for CS grad school, postdoc, and faculty applications. You might have to ask earlier if there are earlier deadlines (which is the case for some postdoc applications), but give at least a 6 week lead time. Try to mention the earliest deadline in your email request.&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;Asking early also benefits the applicant. You should have a solid picture about these key details of your application well in advance.&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;I may still agree to write a letter for you even after the end of October, but chances decrease the longer you wait. On the other hand, I am more likely to agree to late requests if I know you very well or we have worked together very closely. However, I would still be annoyed, and you probably want to avoid annoying your allies in this process.&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;&lt;strong&gt;How many places to apply?&lt;/strong&gt;&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;A common pitfall I&amp;#8217;ve observed: people apply for only a few positions, with the justification that they &amp;#8220;don&amp;#8217;t want to waste their letter writers time.&amp;#8221; Don&amp;#8217;t do this. 99% of the time and effort goes into writing the letter, it&amp;#8217;s easy submit to arbitrarily many places. Be selfish: once they have agreed to write a letter, they&amp;#8217;ve committed to supporting you, and you should put your own preferences first. And think of what&amp;#8217;s a better use of their time: if you apply to 2 places and get 0 offers, or if you apply to 30 places and get an offer that you accept?&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;&lt;strong&gt;What to provide&lt;/strong&gt;&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;You should give your letter writers as much information as possible. They might not use all of it, but it is good to have on hand when they are inevitably writing letters at 4 AM the day &lt;s&gt;before&lt;/s&gt; after the deadline. I&amp;#8217;m writing this as a catch-all list, so some points might not be relevant for the position you&amp;#8217;re applying to. Say, when you&amp;#8217;re applying for faculty positions, they don&amp;#8217;t need to know your undergraduate grades. The goal is to give your writers the gist of who you&amp;#8217;re trying to market yourself as, so they can support your story as best they can.&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;This doesn&amp;#8217;t all need to be provided when you first ask, but please share it as early as possible. Make it as easy to find as you can. For example, a single email with everything as an attachment, and a clear title to the email such as &amp;#8220;X&amp;#8217;s application materials&amp;#8221;. You could also share a link to a Dropbox or Google Drive folder with all the materials. This has the benefit that you can include preliminary versions of your materials, and update them as you refine them.&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;Mandatory:&lt;/p&gt;



&lt;ul class="wp-block-list"&gt;
&lt;li&gt;A shared spreadsheet with a list of all the places you&amp;#8217;re applying to, the deadlines, any special instructions for letter submission, and a space to mark when the letter is submitted.&lt;/li&gt;



&lt;li&gt;Your research statement/statement of purpose&lt;/li&gt;



&lt;li&gt;Your CV&lt;/li&gt;
&lt;/ul&gt;



&lt;p class="wp-block-paragraph"&gt;Optional but recommended:&lt;/p&gt;



&lt;ul class="wp-block-list"&gt;
&lt;li&gt;Anything else that the application asks you for, e.g. your cover letter, teaching statement,&amp;nbsp; diversity statement, transcript.&lt;/li&gt;



&lt;li&gt;Who else is writing letters for you, and what your relationship to them is.&lt;/li&gt;



&lt;li&gt;A brief summary of the work you’ve done or the contributions you’ve made with the recommender, to refresh their memory.&lt;/li&gt;
&lt;/ul&gt;



&lt;p class="wp-block-paragraph"&gt;The last one is helpful since I like to know what &amp;#8220;role&amp;#8221; I&amp;#8217;m playing. Am I your lead letter writer who people will look to first? Or am I the third letter, mostly confirming what other people already know? There may also be some unique perspective I can give, based on who your other reviewers are.&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;Finally, be sure to waive your right to view the letter when requesting it. I personally will not write a letter if this right has not been waived (and I would also be annoyed to receive such a request). It&amp;#8217;s also not to your advantage to have such a letter, since a reader wouldn&amp;#8217;t be able to trust what the writer wrote.&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;Some additional tips, which may be helpful for slightly older readers of this post (e.g., junior faculty):&lt;/p&gt;



&lt;ul class="wp-block-list"&gt;
&lt;li&gt;Consider providing an &amp;#8220;annotated bibliography.&amp;#8221; This isn&amp;#8217;t a well-defined object, but I have a document describing (in plain English) some of my most important works and their significance/impact.&lt;/li&gt;



&lt;li&gt;If you ask the same person for a letter multiple times, they will probably update their previous letter. Consequently, it is helpful if you provide them with materials that highlight (e.g., in colour) the differences with the previous materials. &lt;/li&gt;
&lt;/ul&gt;



&lt;p class="wp-block-paragraph"&gt;&lt;strong&gt;Afterwards?&lt;/strong&gt;&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;To the best of my knowledge, it is not customary in Computer Science to send gifts to your letter writers. You should just thank them genuinely, and pass on the favour when you are in a similar position of power.&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;&lt;strong&gt;What if&amp;#8230;&lt;/strong&gt;&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;It has been brought to my attention that, unfortunately, many letter writers ask the applicant for either a draft or full letter, which they simply sign. I could write a whole post on this topic, but in short, I consider this unacceptable behaviour from the perspective of the letter writer, and I urge them to rethink this practice.&amp;nbsp;&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;But what should the applicant do when put in this position? It helps if you can see what typical academic recommendation letters look like, since they have a certain tone, but access to such letters is generally out of the reach of most applicants. My recommendation is to be positive (now is not the time for modesty), but don&amp;#8217;t lie (these things can follow you around for longer than you might think). Try to make the first paragraph a brief summary that conveys the overall sentiment of the letter. In subsequent paragraphs, you should describe the research you worked on with the letter writer (don&amp;#8217;t assume the reader is highly familiar with the topic, but be succinct in terms of background), as well as your specific contributions to the project. Include anecdotes, if appropriate. This may be awkward, but you have been put into an awkward position.&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;&lt;strong&gt;Statement of Purpose?&lt;/strong&gt;&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;You probably haven&amp;#8217;t written anything like a Statement of Purpose before. Here&amp;#8217;s two resources (&lt;a href="https://let-all.com/assets/slides/How-to-COLT-Boaz.pdf"&gt;1&lt;/a&gt;, &lt;a href="https://let-all.com/assets/slides/How-to-COLT-Yisong.pdf"&gt;2&lt;/a&gt;), by &lt;a href="https://www.boazbarak.org/"&gt;Boaz Barak&lt;/a&gt; and &lt;a href="http://www.yisongyue.com/"&gt;Yisong Yue&lt;/a&gt;, where they describe what goes into a Statement of Purpose. &lt;a href="https://sites.google.com/view/naama-ben-david/home"&gt;Naama Ben-David&lt;/a&gt;&amp;#8216;s statement is used as an example.&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;&lt;strong&gt;Acknowledgments&lt;/strong&gt;&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;Thanks to Anand Sarwate, Thomas Steinke, and Jon Ullman for helpful comments.&lt;/p&gt;
</description></item><item><title>Social at STOC 2021</title><link>https://blog.gautamkamath.com/2021/06/21/social-at-stoc-2021/</link><guid isPermaLink="true">https://blog.gautamkamath.com/2021/06/21/social-at-stoc-2021/</guid><pubDate>Mon, 21 Jun 2021 02:16:07 +0000</pubDate><dc:creator>Gautam Kamath</dc:creator><description>
&lt;p class="wp-block-paragraph"&gt;I&amp;#8217;ve been asked to pass along a message by &lt;a href="https://ccanonne.github.io/"&gt;Clément Canonne&lt;/a&gt;, social chair for STOC 2021. This STOC might have the best social program of any I&amp;#8217;ve ever seen, either virtual or in-person, so be sure to check it out!&lt;/p&gt;



&lt;hr class="wp-block-separator"&gt;&lt;/hr&gt;



&lt;p class="wp-block-paragraph"&gt;STOC&amp;#8217;21 is around the corner, starting tomorrow; [don&amp;#8217;t forget to register](&lt;a href="http://acm-stoc.org/stoc2021/"&gt;http://acm-stoc.org/stoc2021/&lt;/a&gt;), if you haven&amp;#8217;t yet! This year, the (virtual) conference will include several social activities (games, TCS trivia, mystery hunt&amp;#8230;); among which, two &amp;#8220;junior/senior lunches,&amp;#8221; on Monday and Friday.&lt;br&gt;&lt;br&gt;Those both will be held in the Gather space for STOC (&lt;a href="http://acm-stoc.org/stoc2021/venue.html"&gt;http://acm-stoc.org/stoc2021/venue.html&lt;/a&gt;), and &amp;#8212; as in previous years &amp;#8212; are the occasion for senior researchers in the field, broadly construed, to have an informal chat with students, postdocs, and junior faculty, answer their questions, discuss their research, and generally have a nice conversation.&lt;br&gt;&lt;br&gt;If you are interested, don&amp;#8217;t forget to sign up! This is done through the &amp;#8220;feedback box&amp;#8221; placed on the Information Desk in the Gather space&amp;#8217;s Lobby, which gives access to a spreadsheet.&lt;br&gt;&lt;br&gt;Hoping to see you at STOC!&lt;/p&gt;



&lt;figure class="wp-block-image size-large"&gt;&lt;img loading="lazy" width="778" height="453" data-attachment-id="325" data-permalink="https://kamathematics.wordpress.com/image/" data-orig-file="https://kamathematics.wordpress.com/wp-content/uploads/2021/06/image.png" data-orig-size="778,453" data-comments-opened="1" data-image-title="image" data-image-description="" data-image-caption="" data-large-file="https://kamathematics.wordpress.com/wp-content/uploads/2021/06/image.png?w=778" src="https://blog.gautamkamath.com/assets/f0779ade3a59d6f4724e.png" alt="" class="wp-image-325" loading="lazy" decoding="async"&gt;&lt;/img&gt;&lt;/figure&gt;
</description></item><item><title>Learning Theory Alliance and Mentoring Workshop</title><link>https://blog.gautamkamath.com/2021/02/02/learning-theory-alliance-and-mentoring-workshop/</link><guid isPermaLink="true">https://blog.gautamkamath.com/2021/02/02/learning-theory-alliance-and-mentoring-workshop/</guid><pubDate>Tue, 02 Feb 2021 00:00:39 +0000</pubDate><dc:creator>Gautam Kamath</dc:creator><description>
&lt;p class="wp-block-paragraph"&gt;&lt;a href="https://www.cs.utexas.edu/~surbhi/"&gt;Surbhi Goel&lt;/a&gt;, &lt;a href="https://people.eecs.berkeley.edu/~nika/"&gt;Nika Haghtalab&lt;/a&gt;, and &lt;a href="https://vitercik.github.io/"&gt;Ellen Vitercik&lt;/a&gt; are the organizers of an excellent new initiative called the &lt;a href="https://www.let-all.com/"&gt;Learning Theory Alliance&lt;/a&gt;. They have the following inspiring mission statement:&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;&lt;em&gt;Our mission is to develop a strong, supportive learning theory community and ensure its healthy growth by fostering inclusive community engagement and encouraging active contributions from researchers at all stages of their careers.&lt;/em&gt;&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;Their first event is a mentoring workshop, to be held at ALT 2021. I&amp;#8217;ll be helping out by mentoring the creation of some written ALT highlights. Read on for more details from the organizers.&lt;/p&gt;



&lt;hr class="wp-block-separator"&gt;&lt;/hr&gt;



&lt;p class="wp-block-paragraph"&gt;We are pleased to announce the first&lt;strong&gt;&amp;nbsp;&lt;a href="https://www.let-all.com/alt.html"&gt;Learning Theory Mentorship Workshop&lt;/a&gt;&lt;/strong&gt;&amp;nbsp;in collaboration with the &lt;a href="http://algorithmiclearningtheory.org/alt2021/"&gt;Conference on Algorithmic Learning Theory (ALT) 2021&lt;/a&gt; to be held virtually on&amp;nbsp;&lt;strong&gt;March 4-5, 2021&lt;/strong&gt;. The workshop will focus on building technical and networking skills while giving participants an opportunity to interact with fellow researchers in the field.&amp;nbsp;&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;The workshop is intended for upper-level undergraduate and all-level graduate students as well as postdoctoral researchers who are excited about the possibility of learning theory research. No prior research experience in the field is expected.&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;We have several planned events including:&lt;/p&gt;



&lt;ul class="wp-block-list"&gt;&lt;li&gt;&lt;strong&gt;How-to talks&amp;nbsp;&lt;/strong&gt;which will provide general advice about giving talks, structuring papers, writing reviews, networking, and attending conferences.&lt;/li&gt;&lt;li&gt;A small group discussion&amp;nbsp;&lt;strong&gt;dissecting a short talk&lt;/strong&gt;&amp;nbsp;with feedback from a senior researcher.&lt;/li&gt;&lt;li&gt;An informal and interactive&lt;strong&gt;&amp;nbsp;“Ask Me Anything”&amp;nbsp;&lt;/strong&gt;sessionwith a senior member of the learning theory community.&lt;/li&gt;&lt;li&gt;&lt;strong&gt;General audience talks&lt;/strong&gt;&amp;nbsp;about recent learning theory research which will be accessible to new researchers.&lt;/li&gt;&lt;li&gt;&lt;strong&gt;Social events&amp;nbsp;&lt;/strong&gt;such as board games.&lt;/li&gt;&lt;/ul&gt;



&lt;p class="wp-block-paragraph"&gt;Our lineup includes Jacob Abernethy, Kamalika Chaudhuri, Nadav Cohen, Rafael Frongillo, Shafi Goldwasser, Zhiyi Huang, Robert Kleinberg, Pravesh Kothari, Po-Ling Loh, Lester Mackey, Jamie Morgenstern, Praneeth Netrapalli, Vatsal Sharan and Mary Wootters.&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;Together with Gautam Kamath, we will also organize a written account of ALT titled&amp;nbsp;&lt;strong&gt;“ALT Highlights”&lt;/strong&gt;&amp;nbsp;which will summarize the research presented at ALT. We will assist students and postdocs to set up interviews with presenters and keynote speakers as part of the highlights.&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;A short application&lt;a rel="noreferrer noopener" target="_blank" href="https://forms.gle/v8b8aeJMgWxJ1Bbx9"&gt;&amp;nbsp;form&lt;/a&gt;&amp;nbsp;is required to participate with an&amp;nbsp;&lt;strong&gt;application deadline of Friday, Feb. 19, 2021&lt;/strong&gt;. Students with backgrounds that are underrepresented or underserved in related fields are especially encouraged to apply.&amp;nbsp;&lt;strong&gt;We will be accommodating all time zones.&lt;/strong&gt;&amp;nbsp;More information can be found on the event’s website:&lt;a rel="noreferrer noopener" target="_blank" href="http://let-all.com/alt.html"&gt;&amp;nbsp;http://let-all.com/alt.html&lt;/a&gt;.&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;This workshop is part of our broader community building initiative called the Learning Theory Alliance (advised by Peter Bartlett, Avrim Blum, Stefanie Jegelka, Po-Ling Loh and Jenn Wortman Vaughan). Check out &lt;a rel="noreferrer noopener" target="_blank" href="http://let-all.com/"&gt;http://let-all.com/&lt;/a&gt;&amp;nbsp;for more details and to sign up to volunteer.&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;Best,&lt;br&gt;Surbhi Goel, Nika Haghtalab and Ellen Vitercik&lt;/p&gt;
</description></item><item><title>SODA 2021: Funds for Student Registration Fees</title><link>https://blog.gautamkamath.com/2021/01/05/soda-2021-funds-for-student-registration-fees/</link><guid isPermaLink="true">https://blog.gautamkamath.com/2021/01/05/soda-2021-funds-for-student-registration-fees/</guid><pubDate>Tue, 05 Jan 2021 19:25:00 +0000</pubDate><dc:creator>Gautam Kamath</dc:creator><description>
&lt;p class="wp-block-paragraph"&gt;UPDATE: An update from Shang-Hua Teng:&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;Dear TCS Students:&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;SIAM has today reopened the application portal for SODA21 Student Registration Waiver. The new deadline will be this Friday, January 8 at 11:59pm EST.&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;SODA21 will be held virtually on Jan. 10 &amp;#8211; Jan. 13, 2021.&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;Please note that having paper(s) in SODA and its associated Symposiums is not a requirement. For this year&amp;#8217;s application, the letter of recommendation from advisors is not required as well. The goal of the support from Google, Microsoft, and ACM is to increase opportunity for students to attend this premier TCS conference.&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;Looking forward to seeing you in SODA&amp;#8217;21.&lt;/p&gt;



&lt;hr class="wp-block-separator"&gt;&lt;/hr&gt;



&lt;p class="wp-block-paragraph"&gt;ORIGINAL MESSAGE: Please see below for a message from Shang-Hua Teng, regarding the possibility of waivers for SODA 2021 registration for students.&lt;/p&gt;



&lt;hr class="wp-block-separator"&gt;&lt;/hr&gt;



&lt;p class="wp-block-paragraph"&gt;Dear TCS students:&lt;br&gt;&lt;br&gt;By now, it is hard to overestimate the impact of the COVID19 pandemic to society. However, like every challenge, it has created some opportunities. For example, essentially all major conferences in TCS this year have been transformed into virtual ones, making them more accessible to scholars/students across the world (of course at the expense of traditional&amp;nbsp;interactions).&amp;nbsp;&lt;br&gt;&lt;br&gt;ACM-SIAM Symposium on Discrete Algorithms (SODA21) will be held virtually this year, on Jan. 10 &amp;#8211; 13, 2021.&amp;nbsp;As you may know, this is the premier conference on algorithms .&lt;br&gt;&lt;br&gt;See &lt;a rel="noreferrer noopener" href="https://www.siam.org/conferences/cm/conference/soda21" target="_blank"&gt;https://www.siam.org/conferences/cm/conference/soda21&lt;/a&gt;&lt;br&gt;&lt;br&gt;Thanks to our industry partners and ACM SIGACT group, SODA has some funds for covering student registrations. I am writing to informing you this opportunity and encourage you to apply:&lt;br&gt; See: &lt;br&gt;1.&amp;nbsp;&lt;a rel="noreferrer noopener" href="https://awards.siam.org/" target="_blank"&gt;https://awards.siam.org/&lt;/a&gt; &lt;br&gt;2.&amp;nbsp;&lt;a rel="noreferrer noopener" href="https://www.siam.org/conferences/cm/lodging-and-support/travel-support/soda21-travel-support" target="_blank"&gt;https://www.siam.org/conferences/cm/lodging-and-support/travel-support/soda21-travel-support&lt;/a&gt;&lt;br&gt;That deadline is Dec. 27, 2020. Like before, having papers in SODA&amp;nbsp;is not prerequisite.&lt;br&gt;&lt;br&gt;Shang-Hua Teng&lt;br&gt;On Behalf of SODA Steering Committee&lt;br&gt;&lt;/p&gt;
</description></item><item><title>Virtual STOC 2020 – Behind the Screens</title><link>https://blog.gautamkamath.com/2020/06/30/virtual-stoc-2020-behind-the-screens/</link><guid isPermaLink="true">https://blog.gautamkamath.com/2020/06/30/virtual-stoc-2020-behind-the-screens/</guid><pubDate>Tue, 30 Jun 2020 13:13:16 +0000</pubDate><dc:creator>Gautam Kamath</dc:creator><description>
&lt;p class="wp-block-paragraph"&gt;In order to assist organizers of other virtual conferences, the general chairs of STOC 2020 (myself, Konstantin Makarychev, Yury Makarychev and Madhur Tulsiani, with input from PC chair Julia Chuzhoy) wrote a detailed document describing the design and execution of the conference. I personally felt the conference went about as well as it could have gone, and despite many moving parts, there were minimal technical difficulties.&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;The guide is available here: &lt;a rel="noreferrer noopener" href="https://docs.google.com/document/d/1nzyvfdsXLzqYXxxdjw1y_OHAYwGolHCZUkRVmlxG9BE/edit?ts=5efa758c#" target="_blank"&gt;Virtual STOC 2020 &amp;#8211; Behind the Screens&lt;/a&gt;.&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;If you have any questions or comments, feel free to comment below, or join in the conversation on &lt;a href="https://twitter.com/thegautamkamath/status/1277959908168695808"&gt;Twitter&lt;/a&gt;.&lt;/p&gt;
</description></item><item><title>STOC 2020 Goes Virtual!</title><link>https://blog.gautamkamath.com/2020/06/19/stoc-2020-goes-virtual/</link><guid isPermaLink="true">https://blog.gautamkamath.com/2020/06/19/stoc-2020-goes-virtual/</guid><pubDate>Fri, 19 Jun 2020 20:20:16 +0000</pubDate><dc:creator>Gautam Kamath</dc:creator><description>
&lt;p class="wp-block-paragraph"&gt;Starting on Monday, STOC is joining the trend of conferences going online, I believe the biggest theory conference to do so thus far. Given my experience with &lt;a href="https://sites.google.com/site/plustcs/"&gt;TCS+&lt;/a&gt;, I volunteered to lend a hand with the organization and logistics. It&amp;#8217;s been a journey (with some &lt;a href="https://twitter.com/thegautamkamath/status/1273055827092549634"&gt;unusual technical challenges&lt;/a&gt;), but I think we have something which I hope will be engaging and generally a lot of fun. In addition to the typical academic component, we also have a social component planned as well. We learnt from the work of others, including the &lt;a href="https://www.acm.org/virtual-conferences"&gt;ACM virtual conferences guide&lt;/a&gt;, &lt;a href="https://iclr.cc/Conferences/2020"&gt;ICLR 2020&lt;/a&gt;, and &lt;a href="https://www.daniellitt.com/blog/2020/4/20/wagon-lessons-learned"&gt;WAGON&lt;/a&gt;. I may make some version of our logistics docs available to others after the conference, so others can learn from our experience as well. Anyway, read on for an announcement from me and the other General Chairs, Konstantin Makarychev, Yury Makarychev, and Madhur Tulsiani. See also the &lt;a href="http://acm-stoc.org/stoc2020/"&gt;main STOC page&lt;/a&gt; for a more complete list of credits.&lt;/p&gt;



&lt;hr class="wp-block-separator"&gt;&lt;/hr&gt;



&lt;p class="wp-block-paragraph"&gt;Dear fellow theorists,&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;As you already know, STOC 2020 this year will be a virtual conference. If you are interested in attending the conference, but haven’t registered yet, &lt;a href="http://www.cvent.com/events/52nd-annual-acm-symposium-on-theory-of-computing-stoc-2020-/event-summary-ea5fa7861d1a476d82bc10f667a1c0f4.aspx"&gt;please do so soon&lt;/a&gt; (students: $25, regular: $50). This will help us ensure we have capacity for various online events.&amp;nbsp;&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;Upon registration, you should receive a confirmation email from CVENT, also containing access information for various conference events. Also, if you are a student looking to register for STOC but the cost is a burden, please email us at &lt;a href="mailto:stoc2020@ttic.edu"&gt;stoc2020@ttic.edu&lt;/a&gt;.&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;&lt;strong&gt;How will the conference work?&lt;/strong&gt;&lt;/p&gt;



&lt;ul class="wp-block-list"&gt;&lt;li&gt;&lt;strong&gt;Videos&lt;/strong&gt;: The videos for all conference talks are now available on YouTube, and can be accessed through the links in the &lt;a href="http://acm-stoc.org/stoc2020/STOCprogram.html"&gt;conference program&lt;/a&gt;. Registration is &lt;em&gt;not required&lt;/em&gt; to view the talks on Youtube.&lt;/li&gt;&lt;/ul&gt;



&lt;ul class="wp-block-list"&gt;&lt;li&gt;&lt;strong&gt;Slack&lt;/strong&gt;: The conference has a Slack workspace, with one channel for every paper and workshop, and additional channels for information, announcements, social events, help, etc. The invitations for the Slack workspace will be sent to registered participants. Authors are also encouraged to monitor the channels for their papers. All access information for the conference will also be available here. The workspace is currently active, and will remain active for at least one week after the conference.&lt;/li&gt;&lt;/ul&gt;



&lt;ul class="wp-block-list"&gt;&lt;li&gt;&lt;strong&gt;Zoom sessions&lt;/strong&gt;: The conference will feature Zoom sessions with short presentations by the speakers. The total time for each paper is 10 minutes. Given that participants have access to the full talks by the speakers on Youtube, these can be thought of as being analogues of poster sessions. The workshops will also be held as separate sessions. The links for the Zoom session via information in the confirmation email.&lt;/li&gt;&lt;/ul&gt;



&lt;ul class="wp-block-list"&gt;&lt;li&gt;&lt;strong&gt;Social events&lt;/strong&gt;: The conference will include junior/senior “lunches”, breakout tables for impromptu and scheduled hangouts, and a group event using &lt;a href="https://gather.town"&gt;gather.town&lt;/a&gt;. The timings for the events can be found in the conference program. Sign-up links for various events will be sent to all registered participants &amp;#8211; please do sign-up soon!&lt;/li&gt;&lt;/ul&gt;



&lt;p class="wp-block-paragraph"&gt;See you all at (virtual) STOC 2020. Please do let us know if you have any questions or suggestions.&lt;/p&gt;
</description></item><item><title>A Primer on Private Statistics – Part II</title><link>https://blog.gautamkamath.com/2020/04/21/a-primer-on-private-statistics-part-ii/</link><guid isPermaLink="true">https://blog.gautamkamath.com/2020/04/21/a-primer-on-private-statistics-part-ii/</guid><pubDate>Tue, 21 Apr 2020 13:36:53 +0000</pubDate><dc:creator>Gautam Kamath</dc:creator><description>&lt;p&gt;By&amp;nbsp;&lt;a href="http://www.gautamkamath.com/"&gt;Gautam Kamath&lt;/a&gt;&amp;nbsp;and&amp;nbsp;&lt;a href="http://www.ccs.neu.edu/home/jullman/"&gt;Jonathan Ullman&lt;/a&gt;&lt;/p&gt;
&lt;p&gt;The second part of our brief survey of differentially private statistics. This time, we show how to privately estimate the CDF of a distribution (i.e., estimate the distribution in Kolmogorov distance), and conclude with pointers to some other work in the space.&lt;/p&gt;
&lt;p&gt;The first part of this series is &lt;a href="https://blog.gautamkamath.com/2020/04/14/a-primer-on-private-statistics-part-i/"&gt;here&lt;/a&gt;, and you can download both parts in PDF form &lt;a href="http://www.gautamkamath.com/writings/primer.pdf"&gt;here&lt;/a&gt;.&lt;/p&gt;
&lt;p&gt;&lt;b&gt;1. CDF Estimation for Discrete, Univariate Distributions &lt;/b&gt;&lt;/p&gt;
&lt;p&gt;Suppose we have a distribution &lt;img src="https://blog.gautamkamath.com/assets/4f5401ebbe0c17f4eccf.png" alt="{P}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; over the ordered, discrete domain &lt;img src="https://blog.gautamkamath.com/assets/4407c017d265b4669c20.png" alt="{\{1,\dots,D\}}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; and let &lt;img src="https://blog.gautamkamath.com/assets/294e1e8b0355c1aba775.png" alt="{\mathcal{P}}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; be the family of all such distributions. The CDF of the distribution is the function &lt;img src="https://blog.gautamkamath.com/assets/85bf3adb548649da45f6.png" alt="{\Phi_{P} : \{1,\dots,D\} \rightarrow [0,1]}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; given by&lt;/p&gt;
&lt;p align="center"&gt;&lt;img src="https://blog.gautamkamath.com/assets/d1e543ac5464c6e40219.png" alt="\displaystyle \Phi_{P}(j) = \mathop{\mathbb P}(P \leq j). \ \ \ \ \ (1)" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;&lt;/p&gt;
&lt;p&gt;A natural measure of distance between CDFs is the &lt;img src="https://blog.gautamkamath.com/assets/88a76707a42e3635a91d.png" alt="{\ell_\infty}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; distance, as this is the sort of convergence guarantee that the empirical CDF satisfies. That is, in the non-private setting, the empirical CDF will achieve the minimax rate, which it known by [&lt;a href="#DKW56"&gt;DKW56&lt;/a&gt;, &lt;a href="#Mas90"&gt;Mas90&lt;/a&gt;] to be &lt;a name="eqdkw"&gt;&lt;/a&gt;&lt;/p&gt;
&lt;p&gt;&lt;a name="eqdkw"&gt;&lt;/a&gt;&lt;/p&gt;
&lt;p&gt;&lt;a name="eqdkw"&gt;&lt;/a&gt;&lt;/p&gt;
&lt;p align="center"&gt;&lt;img src="https://blog.gautamkamath.com/assets/741f1091d27da4408d00.png" alt="\displaystyle \max_{P \in \mathcal{P}} \mathop{\mathbb E}_{X_{1 \cdots n} \sim P}(\| \Phi_{X} - \Phi_{P} \|_{\infty}) = O\left(\sqrt{\frac{1}{n}} \right). \ \ \ \ \ (2)" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;&lt;/p&gt;
&lt;p&gt;&lt;a name="eqdkw"&gt;&lt;/a&gt;&lt;a name="eqdkw"&gt;&lt;/a&gt;&lt;a name="eqdkw"&gt;&lt;/a&gt;&lt;/p&gt;
&lt;p&gt;&lt;b&gt; 1.1. Private CDF Estimation &lt;/b&gt;&lt;/p&gt;
&lt;blockquote&gt;&lt;p&gt;&lt;b&gt;Theorem 1&lt;/b&gt; &lt;em&gt; &lt;a name="thmcdf-ub"&gt;&lt;/a&gt; For every &lt;img src="https://blog.gautamkamath.com/assets/b4d4518917a0cd03094d.png" alt="{n \in {\mathbb N}}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; and every &lt;img src="https://blog.gautamkamath.com/assets/32f811523253e1b85605.png" alt="{\epsilon,\delta &amp;gt; 0}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;, there exists an &lt;img src="https://blog.gautamkamath.com/assets/22aed5ab6dec88b9744b.png" alt="{(\epsilon,\delta)}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;-differentially private mechanism &lt;img src="https://blog.gautamkamath.com/assets/03e84b366e4aa0c348a3.png" alt="{M}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; such that &lt;/em&gt;&lt;/p&gt;
&lt;p align="center"&gt;&lt;img src="https://blog.gautamkamath.com/assets/373e34942deab6b417a0.png" alt="\displaystyle \max_{P \in \mathcal{P}} \mathop{\mathbb E}_{X_{1 \cdots n} \sim P}(\| M(X_{1 \cdots n}) - \Phi_{P} \|_{\infty}) = O\left(\sqrt{\frac{1}{n}} + \frac{\log^{3/2}(D) \log^{1/2}(1/\delta)}{\epsilon n} \right). \ \ \ \ \ (3)" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;&lt;/p&gt;
&lt;/blockquote&gt;
&lt;p&gt;&lt;em&gt;Proof:&lt;/em&gt; Assume without loss of generality that &lt;img src="https://blog.gautamkamath.com/assets/37fb0011daf34eb6e047.png" alt="{D = 2^{d}}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; for an integer &lt;img src="https://blog.gautamkamath.com/assets/5f5e0e55877b97523735.png" alt="{d \geq 1}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;. Let &lt;img src="https://blog.gautamkamath.com/assets/4598a97e5fe0289a2985.png" alt="{X_{1 \cdots n} \sim P}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; be a sample. By the triangle inequality, we have&lt;/p&gt;
&lt;p align="center"&gt;&lt;img src="https://blog.gautamkamath.com/assets/738d2ba469c20c9efe30.png" alt="\displaystyle \begin{array}{rll} \mathop{\mathbb E}_{X_{1 \cdots n} \sim P}{\| M(X_{1 \cdots n}) - \Phi_{P} \|_{\infty}} &amp;amp;\leq{} \mathop{\mathbb E}_{X_{1 \cdots n} \sim P}(\| \Phi_{X} - \Phi_{P} \|_{\infty} + \| M(X_{1 \cdots n}) - \Phi_{X} \|_{\infty}) \\ &amp;amp;\leq{} O(\sqrt{1/n}) + \mathop{\mathbb E}_{X_{1 \cdots n} \sim P}(\| M(X_{1 \cdots n}) - \Phi_{X} \|_{\infty}), \end{array} " class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;&lt;/p&gt;
&lt;p&gt;so we will focus on constructing &lt;img src="https://blog.gautamkamath.com/assets/03e84b366e4aa0c348a3.png" alt="{M}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; to approximate &lt;img src="https://blog.gautamkamath.com/assets/0efc169d465fb20f7192.png" alt="{\Phi_{X}}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;.&lt;/p&gt;
&lt;p&gt;For any &lt;img src="https://blog.gautamkamath.com/assets/bd645a37e27d852a00fb.png" alt="{\ell = 0,\dots,d-1}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; and &lt;img src="https://blog.gautamkamath.com/assets/b9ee1cd92965f3ccfc24.png" alt="{j = 1,\dots,2^{d - \ell}}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;, consider the statistics&lt;/p&gt;
&lt;p align="center"&gt;&lt;img src="https://blog.gautamkamath.com/assets/1feed0f976784dd743be.png" alt="\displaystyle f_{\ell,j}(X_{1 \cdots n}) = \frac{1}{n} \sum_{i=1}^{n} {\bf 1}\{ (j-1)2^{\ell} + 1 \leq X_i \leq j 2^{\ell} \}. \ \ \ \ \ (4)" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;&lt;/p&gt;
&lt;p&gt;Let &lt;img src="https://blog.gautamkamath.com/assets/674284633fbed85ef5a7.png" alt="{f : \{1,\dots,D\}^n \rightarrow [0,1]^{2D - 2}}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; be the function whose output consists of all &lt;img src="https://blog.gautamkamath.com/assets/9b2bd7828480c2ba0689.png" alt="{2D-2}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; such counts. To decipher this notation, for a given &lt;img src="https://blog.gautamkamath.com/assets/3edd496692f37353df29.png" alt="{\ell}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;, the counts &lt;img src="https://blog.gautamkamath.com/assets/942690fb6210e4276c46.png" alt="{f_{\ell,\cdot}}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; form a histogram of &lt;img src="https://blog.gautamkamath.com/assets/5ac0839addd4ec5b132d.png" alt="{X_{1 \cdots n}}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; using consecutive bins of width &lt;img src="https://blog.gautamkamath.com/assets/ba06986a33005b605776.png" alt="{2^{\ell}}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;, and we consider the &lt;img src="https://blog.gautamkamath.com/assets/3c1cdc77d583c5fb53f2.png" alt="{\log(D)}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; histograms of geometrically increasing width &lt;img src="https://blog.gautamkamath.com/assets/8cb3c7588705b5828a99.png" alt="{1,2,4,\dots,D}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;. First, we claim that the function &lt;img src="https://blog.gautamkamath.com/assets/d4f19f783c6310f0e53d.png" alt="{f}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; has low sensitivity&amp;#8212;for adjacent samples &lt;img src="https://blog.gautamkamath.com/assets/80bfa982f798c252f90b.png" alt="{X}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; and &lt;img src="https://blog.gautamkamath.com/assets/d471ab4b17655d60fd1a.png" alt="{X&amp;#x27;}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;,&lt;/p&gt;
&lt;p align="center"&gt;&lt;img src="https://blog.gautamkamath.com/assets/8c6d8c4c1f3df4719413.png" alt="\displaystyle \| f(X) - f(X&amp;#x27;) \|_2^2 \leq \frac{2 \log(D)}{n^2}. \ \ \ \ \ (5)" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;&lt;/p&gt;
&lt;p&gt;Thus, we can use the Gaussian mechanism:&lt;/p&gt;
&lt;p align="center"&gt;&lt;img src="https://blog.gautamkamath.com/assets/d13f331b6c62380ddd3d.png" alt="\displaystyle M&amp;#x27;(X_{1 \cdots n}) = f(X_{1 \cdots n}) + \mathcal{N}\left(0, \frac{2 \log(D) \log(1/\delta)}{\epsilon^2 n^2} \cdot \mathbb{I}_{2D \times 2D}\right). \ \ \ \ \ (6)" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;&lt;/p&gt;
&lt;p&gt;As we will argue, there exists a matrix &lt;img src="https://blog.gautamkamath.com/assets/d515d695cd8402a90609.png" alt="{A \in {\mathbb R}^{2D \times 2D}}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; such that &lt;img src="https://blog.gautamkamath.com/assets/427ff00b4fbb3fc4262f.png" alt="{\Phi_{X} = A \cdot f(X_{1 \cdots n})}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;. We will let &lt;img src="https://blog.gautamkamath.com/assets/e70c1aed33b95386e0ac.png" alt="{M(X_{1 \cdots n}) = A \cdot M&amp;#x27;(X_{1 \cdots n})}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;. Since differential privacy is closed under post-processing, &lt;img src="https://blog.gautamkamath.com/assets/03e84b366e4aa0c348a3.png" alt="{M}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; inherits the privacy of &lt;img src="https://blog.gautamkamath.com/assets/96d7c14781768ac52b44.png" alt="{M&amp;#x27;}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;.&lt;/p&gt;
&lt;p&gt;We will now show how to construct the matrix &lt;img src="https://blog.gautamkamath.com/assets/b516c6332b0f158eefc6.png" alt="{A}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; and analyze the error of &lt;img src="https://blog.gautamkamath.com/assets/03e84b366e4aa0c348a3.png" alt="{M}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;. For any &lt;img src="https://blog.gautamkamath.com/assets/0c9c29351436eaaf3f5f.png" alt="{j = 1,\dots,D}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;, we can form the interval &lt;img src="https://blog.gautamkamath.com/assets/7825dc7a26ad4d2b1edb.png" alt="{\{1,\dots,j\}}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; as the union of at most &lt;img src="https://blog.gautamkamath.com/assets/887aedb9d834cc20ed9d.png" alt="{\log D}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; disjoint intervals of the form we&amp;#8217;ve computed, and therefore we can obtain &lt;img src="https://blog.gautamkamath.com/assets/218a2c4f063b258463a4.png" alt="{\Phi_{X}(j)}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; as the sum of at most &lt;img src="https://blog.gautamkamath.com/assets/887aedb9d834cc20ed9d.png" alt="{\log D}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; of the entries of &lt;img src="https://blog.gautamkamath.com/assets/7557004afc1e2b7e2a03.png" alt="{f(X)}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;. For example, if &lt;img src="https://blog.gautamkamath.com/assets/6a6069b2fe19bbd9d09d.png" alt="{j = 5}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; then we can write&lt;/p&gt;
&lt;p align="center"&gt;&lt;img src="https://blog.gautamkamath.com/assets/1ed80c31a8e87c4d1375.png" alt="\displaystyle \{1,\dots,7\} = \{1,\dots,4\} \cup \{5,6\} \cup \{7\} \ \ \ \ \ (7)" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;&lt;/p&gt;
&lt;p&gt;and&lt;/p&gt;
&lt;p align="center"&gt;&lt;img src="https://blog.gautamkamath.com/assets/4a417fcc1b734a4c45cd.png" alt="\displaystyle \Phi_{X}(5) = f_{2,1} + f_{1,3} + f_{0,7}. \ \ \ \ \ (8)" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;&lt;/p&gt;
&lt;p&gt;See the following diagram for a visual representation of the decomposition.&lt;/p&gt;
&lt;p&gt;&lt;img loading="lazy" data-attachment-id="142" data-permalink="https://kamathematics.wordpress.com/2020/04/21/a-primer-on-private-statistics-part-ii/bin-tree-mech/" data-orig-file="https://kamathematics.wordpress.com/wp-content/uploads/2020/04/bin-tree-mech.png" data-orig-size="1903,1045" data-comments-opened="1" data-image-title="bin-tree-mech" data-image-description="" data-image-caption="" data-large-file="https://kamathematics.wordpress.com/wp-content/uploads/2020/04/bin-tree-mech.png?w=1024" class=" wp-image-142 aligncenter" src="https://blog.gautamkamath.com/assets/5cc23523ce794fd59957.png" alt="bin-tree-mech" width="547" height="300" loading="lazy" decoding="async"&gt;&lt;/p&gt;
&lt;p&gt;This shows hierarchical decomposition of the domain &lt;img src="https://blog.gautamkamath.com/assets/e89815cbd4f3103eb860.png" alt="{\{1,\dots,8\}}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; using 14 intervals. The highlighted squares represent the interval &lt;img src="https://blog.gautamkamath.com/assets/f11b6999baf8a512e532.png" alt="{\{1,\dots,7\}}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; and the highlighted circles show the decomposition of this interval into a union of &lt;img src="https://blog.gautamkamath.com/assets/27768e2012e6e42cc1c0.png" alt="{3}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; intervals in the tree.&lt;/p&gt;
&lt;p&gt;Thus we can construct the matrix &lt;img src="https://blog.gautamkamath.com/assets/b516c6332b0f158eefc6.png" alt="{A}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; using this information. Note that each entry of &lt;img src="https://blog.gautamkamath.com/assets/5e6f8114eb2731df6a48.png" alt="{A f(X)}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; is the sum of at most &lt;img src="https://blog.gautamkamath.com/assets/3c1cdc77d583c5fb53f2.png" alt="{\log(D)}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; entries of &lt;img src="https://blog.gautamkamath.com/assets/7557004afc1e2b7e2a03.png" alt="{f(X)}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;. Thus, if we use the output of &lt;img src="https://blog.gautamkamath.com/assets/c5ebb6c6934aba55b6ca.png" alt="{M&amp;#x27;(X_{1 \cdots n})}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; in place of &lt;img src="https://blog.gautamkamath.com/assets/31b314747e568fb8243b.png" alt="{f(X_{1 \cdots n})}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;, for every &lt;img src="https://blog.gautamkamath.com/assets/6d65a15f265336d360c4.png" alt="{j}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; we obtain&lt;/p&gt;
&lt;p align="center"&gt;&lt;img src="https://blog.gautamkamath.com/assets/9621b605adf021be43fe.png" alt="\displaystyle \Phi_{X}(j) + \mathcal{N}(0, \sigma^2) \quad \textrm{for} \quad \sigma^2 = \frac{ 2 \log^2(D) \log(1/\delta)}{\epsilon^2 n^2}. \ \ \ \ \ (9)" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;&lt;/p&gt;
&lt;p&gt;Applying standard bounds on the expected supremum of a Gaussian process, we have&lt;/p&gt;
&lt;p align="center"&gt;&lt;img src="https://blog.gautamkamath.com/assets/9a8363a7dd1dc9fa587d.png" alt="\displaystyle \mathop{\mathbb E}(\| M(X_{1 \cdots n}) - \Phi_{X} \|_{\infty}) = O( \sigma \sqrt{\log D}) = O\left(\frac{\log^{3/2}(D) \log^{1/2}(1/\delta)}{\epsilon n} \right). \ \ \ \ \ (10)" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;&lt;/p&gt;
&lt;p&gt;&lt;img src="https://blog.gautamkamath.com/assets/b7d45ccd758dec885fd2.png" alt="\Box" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;&lt;/p&gt;
&lt;p&gt;&lt;b&gt; 1.2. Why Restrict the Domain? &lt;/b&gt;&lt;/p&gt;
&lt;p&gt;A drawback of the estimator we constructed is that it only applies to distributions of finite support &lt;img src="https://blog.gautamkamath.com/assets/74b16fbd2bc093769eb0.png" alt="{\{1,2,\dots,D\}}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;, albeit with a relatively mild dependence on the support size. If privacy isn&amp;#8217;t a concern, then no such restriction is necessary, as the bound &lt;a href="#eqdkw"&gt;(2)&lt;/a&gt; applies equally well to any distribution over &lt;img src="https://blog.gautamkamath.com/assets/4d6d4e4b8e9e89081970.png" alt="{{\mathbb R}}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;. Can we construct a differentially private estimator for distributions with infinite support?&lt;/p&gt;
&lt;p&gt;Perhaps surprisingly, the answer to this question is no! Any differentially private estimator for the CDF of the distribution has to have a rate that depends on the support size, and cannot give non-trivial rates for distributions with infinite support.&lt;/p&gt;
&lt;blockquote&gt;&lt;p&gt;&lt;b&gt;Theorem 2 ([&lt;a href="#BNSV15"&gt;BNSV15&lt;/a&gt;])&lt;/b&gt; &lt;em&gt; &lt;a name="thmcdf-lb"&gt;&lt;/a&gt; If &lt;img src="https://blog.gautamkamath.com/assets/294e1e8b0355c1aba775.png" alt="{\mathcal{P}}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; consists of all distributions on &lt;img src="https://blog.gautamkamath.com/assets/4407c017d265b4669c20.png" alt="{\{1,\dots,D\}}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;, then &lt;/em&gt;&lt;/p&gt;
&lt;p align="center"&gt;&lt;img src="https://blog.gautamkamath.com/assets/4dd3f71ebc3769c1808f.png" alt="\displaystyle \min_{M \in \mathcal{M}_{1, \frac{1}{n}}} \max_{P \in \mathcal{P}} \mathop{\mathbb E}_{X_{1 \cdots n} \sim P}(\| M(X_{1 \cdots n}) - \Phi_{P} \|_{\infty}) = \Omega\left(\frac{\log^* D}{n} \right). \ \ \ \ \ (11)" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;&lt;/p&gt;
&lt;/blockquote&gt;
&lt;p&gt;The notation &lt;img src="https://blog.gautamkamath.com/assets/f38e1d31bd6860c933ed.png" alt="{\log^* D}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; refers to the &lt;a href="https://en.wikipedia.org/wiki/Iterated_logarithm"&gt;iterated logarithm&lt;/a&gt;.&lt;/p&gt;
&lt;p&gt;We emphasize that this theorem shouldn&amp;#8217;t meet with too much alarm, as &lt;img src="https://blog.gautamkamath.com/assets/f38e1d31bd6860c933ed.png" alt="{\log^* D}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; grows remarkably slowly with &lt;img src="https://blog.gautamkamath.com/assets/fdeab5d00a28ef8fd381.png" alt="{D}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;. There are differentially private CDF estimators that achieve very mild dependence on &lt;img src="https://blog.gautamkamath.com/assets/fdeab5d00a28ef8fd381.png" alt="{D}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; [&lt;a href="#BNS13"&gt;BNS13&lt;/a&gt;, &lt;a href="#BNSV15"&gt;BNSV15&lt;/a&gt;], including one nearly matching the lower bound in Theorem &lt;a href="#thmcdf-lb"&gt;2&lt;/a&gt;. Moreover, if we want to estimate a distribution over &lt;img src="https://blog.gautamkamath.com/assets/4d6d4e4b8e9e89081970.png" alt="{{\mathbb R}}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;, and are willing to make some mild regularity conditions on the distribution, then we can approximate it by a distribution with finite support and only increase the rate slightly. However, what Theorem &lt;a href="#thmcdf-lb"&gt;2&lt;/a&gt; shows is that there is no &amp;#8220;one-size-fits-all&amp;#8221; solution to private CDF estimation that achieves similar guarantees to the empirical CDF. That is, the right algorithm has to be tailored somewhat to the application and the assumptions we can make about the distribution.&lt;/p&gt;
&lt;p&gt;&lt;b&gt;2. More Private Statistics &lt;/b&gt;&lt;/p&gt;
&lt;p&gt;Of course, the story doesn&amp;#8217;t end here! There&amp;#8217;s a whole wide world of differentially private statistics beyond what we&amp;#8217;ve mentioned already. We proceed to survey just a few other directions of study in private statistics.&lt;/p&gt;
&lt;p&gt;&lt;b&gt; 2.1. Parameter and Distribution Estimation &lt;/b&gt;&lt;/p&gt;
&lt;p&gt;A number of the early works in differential privacy give methods for differentially private statistical estimation for i.i.d. data. The earliest works [&lt;a href="#DN03"&gt;DN03&lt;/a&gt;, &lt;a href="#DN04"&gt;DN04&lt;/a&gt;, &lt;a href="#BDMN05"&gt;BDMN05&lt;/a&gt;, &lt;a href="#DMNS06"&gt;DMNS06&lt;/a&gt;], which introduced the Gaussian mechanism, among other foundational results, can be thought of as methods for estimating the mean of a distribution over the hypercube &lt;img src="https://blog.gautamkamath.com/assets/1d4be3da5e423617b749.png" alt="{\{0,1\}^d}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; in the &lt;img src="https://blog.gautamkamath.com/assets/88a76707a42e3635a91d.png" alt="{\ell_\infty}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; norm. Tight lower bounds for this problem follow from the tracing attacks introduced in [&lt;a href="#BUV14"&gt;BUV14&lt;/a&gt;, &lt;a href="#DSSUV15"&gt;DSSUV15&lt;/a&gt;, &lt;a href="#BSU17"&gt;BSU17&lt;/a&gt;, &lt;a href="#SU17a"&gt;SU17a&lt;/a&gt;, &lt;a href="#SU17b"&gt;SU17b&lt;/a&gt;]. A very recent work of Acharya, Sun, and Zhang [&lt;a href="#ASZ20"&gt;ASZ20&lt;/a&gt;] adapts classical tools for proving estimation and testing lower bounds (lemmata of Assouad, Fano, and Le Cam) to the differentially private setting. Steinke and Ullman [&lt;a href="#SU17b"&gt;SU17b&lt;/a&gt;] give tight minimax lower bounds for the weaker guarantee of selecting the largest coordinates of the mean, which were refined by Cai, Wang, and Zhang [&lt;a href="#CWZ19"&gt;CWZ19&lt;/a&gt;] to give lower bounds for sparse mean-estimation problems.&lt;/p&gt;
&lt;p&gt;Nissim, Raskhodnikova, and Smith introduced the highly general sample-and-aggregate paradigm, which they apply to several learning problems (e.g., learning mixtures of Gaussians) [&lt;a href="#NRS07"&gt;NRS07&lt;/a&gt;]. Later, Smith [&lt;a href="#Smi11"&gt;Smi11&lt;/a&gt;] showed that this paradigm can be used to transform any estimator for any asymptotically normal, univariate statistic over a bounded data domain into a differentially private one with the same asymptotic convergence rate.&lt;/p&gt;
&lt;p&gt;Subsequent work has focused on both relaxing the assumptions in [&lt;a href="#Smi11"&gt;Smi11&lt;/a&gt;], particularly boundedness, and on giving finite-sample guarantees. Karwa and Vadhan investigated the problem of Gaussian mean estimation, proving the first near-optimal bounds for this setting [&lt;a href="#KV18"&gt;KV18&lt;/a&gt;]. In particular, exploiting concentration properties of Gaussian data allows us to achieve non-trivial results even with unbounded data, which is impossible in general. Following this, Kamath, Li, Singhal, and Ullman moved to the multivariate setting, investigating the estimation of Gaussians and binary product distributions in total variation distance [&lt;a href="#KLSU19"&gt;KLSU19&lt;/a&gt;]. In certain cases (i.e., Gaussians with identity covariance), this is equivalent to mean estimation in &lt;img src="https://blog.gautamkamath.com/assets/63aec118fb6952ceb9ea.png" alt="{\ell_2}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;-distance, though not always. For example, for binary product distribution, one must estimate the mean in a type of &lt;img src="https://blog.gautamkamath.com/assets/2295a500f8cd5173566c.png" alt="{\chi^2}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;-distance instead. The perspective of distribution estimation rather than parameter estimation can be valuable. Bun, Kamath, Steinke, and Wu [&lt;a href="#BKSW19"&gt;BKSW19&lt;/a&gt;] develop a primitive for private hypothesis selection, which they apply to learn any coverable class of distributions under pure differential privacy. Through the lens of distribution estimation, their work implies an upper bound for mean estimation of binary product distributions that bypasses lower bounds for the same problem in the empirical setting. In addition to work on mean estimation in the sub-Gaussian setting, such as the results discussed earlier, mean estimation has also been studied under weaker moment conditions [&lt;a href="#BS19"&gt;BS19&lt;/a&gt;, &lt;a href="#KSU20"&gt;KSU20&lt;/a&gt;]. Beyond these settings, there has also been study of estimation of discrete multinomials, including estimation in Kolmogorov distance [&lt;a href="#BNSV15"&gt;BNSV15&lt;/a&gt;] and in total variation distance for structured distributions [&lt;a href="#DHS15"&gt;DHS15&lt;/a&gt;], and parameter estimation of Markov Random Fields [&lt;a href="#ZKKW20"&gt;ZKKW20&lt;/a&gt;].&lt;/p&gt;
&lt;p&gt;A different approach to constructing differentially private estimators is based on robust statistics. This approah begins with the influential work of Dwork and Lei [&lt;a href="#DL09"&gt;DL09&lt;/a&gt;], which introduced the propose-test-release framework, and applied to estimating robust statistics such as the median and interquartile range. While the definitions in robust statistics and differential privacy are semantically similar, formal connections between the two remain relatively scant, which suggests a productive area for future study.&lt;/p&gt;
&lt;p&gt;&lt;b&gt; 2.2. Hypothesis Testing &lt;/b&gt;&lt;/p&gt;
&lt;p&gt;An influential work of Homer et al. [&lt;a href="#HSRDTMPSNC08"&gt;HSRDTMPSNC08&lt;/a&gt;] demonstrated the vulnerability of classical statistics in a genomic setting, showing that certain &lt;img src="https://blog.gautamkamath.com/assets/2295a500f8cd5173566c.png" alt="{\chi^2}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;-statistics on many different variables could allow an attacker to determine the presence of an individual in a genome-wide association study (GWAS). Motivated by these concerns, an early line of work from the statistics community focused on addressing these issues [&lt;a href="#VS09"&gt;VS09&lt;/a&gt;, &lt;a href="#USF13"&gt;USF13&lt;/a&gt;, &lt;a href="#YFSU14"&gt;YFSU14&lt;/a&gt;].&lt;/p&gt;
&lt;p&gt;More recently, work on private hypothesis testing can be divided roughly into two lines. The first focuses on the minimax sample complexity, in a line initiated by Cai, Daskalakis, and Kamath [&lt;a href="#CDK17"&gt;CDK17&lt;/a&gt;], who give an algorithm for privately testing goodness-of-fit (more precisely, a statistician might refer to this problem as one-sample testing of multinomial data). A number of subsequent works have essentially settled the complexity of this problem [&lt;a href="#ASZ18"&gt;ASZ18&lt;/a&gt;, &lt;a href="#ADR18"&gt;ADR18&lt;/a&gt;], giving tight upper and lower bounds. Other papers in this line study related problems, including the two-sample version of the problem, independence testing, and goodness-of-fit testing for multivariate product distributions [&lt;a href="#ASZ18"&gt;ASZ18&lt;/a&gt;, &lt;a href="#ADR18"&gt;ADR18&lt;/a&gt;, &lt;a href="#ADKR19"&gt;ADKR19&lt;/a&gt;, &lt;a href="#CKMUZ19"&gt;CKMUZ19&lt;/a&gt;]. A related paper studies the minimax sample complexity of property &lt;em&gt;estimation&lt;/em&gt;, rather than testing of discrete distributions, including support size and entropy [&lt;a href="#AKSZ18"&gt;AKSZ18&lt;/a&gt;]. Other recent works in this vein focus on testing of simple hypotheses [&lt;a href="#CKMTZ18"&gt;CKMTZ18&lt;/a&gt;, &lt;a href="#CKMSU19"&gt;CKMSU19&lt;/a&gt;]. In particular [&lt;a href="#CKMSU19"&gt;CKMSU19&lt;/a&gt;] proves an analogue of the Neyman-Pearson Lemma for differentially private testing of simple hypotheses. A paper of Awan and Slavkovic [&lt;a href="#AS18"&gt;AS18&lt;/a&gt;] gives a universally optimal test when the domain size is two, however Brenner and Nissim [&lt;a href="#BN14"&gt;BN14&lt;/a&gt;] shows that such universally optimal tests cannot exist when the domain has more than two elements. A related problem in this space is private change-point detection [&lt;a href="#CKMTZ18"&gt;CKMTZ18&lt;/a&gt;, &lt;a href="#CKMSU19"&gt;CKMSU19&lt;/a&gt;, &lt;a href="#CKLZ19"&gt;CKLZ19&lt;/a&gt;] &amp;#8212; in this setting, we are given a time series of datapoints which are sampled from a distribution, which at some point, changes to a different distribution. The goal is to (privately) determine when this point occurs.&lt;/p&gt;
&lt;p&gt;Complementary to minimax hypothesis testing, a line of work [&lt;a href="#WLK15"&gt;WLK15&lt;/a&gt;, &lt;a href="#GLRV16"&gt;GLRV16&lt;/a&gt;, &lt;a href="#KR17"&gt;KR17&lt;/a&gt;, &lt;a href="#KSF17"&gt;KSF17&lt;/a&gt;, &lt;a href="#CBRG18"&gt;CBRG18&lt;/a&gt;, &lt;a href="#SGGRGB19"&gt;SGGRGB19&lt;/a&gt;, &lt;a href="#CKSBG19"&gt;CKSBG19&lt;/a&gt;] designs differentially private versions of popular test statistics for testing goodness-of-fit, closeness, and independence, as well as private ANOVA, focusing on the performance at small sample sizes. Work by Wang et al. [&lt;a href="#WKLK18"&gt;WKLK18&lt;/a&gt;] focuses on generating statistical approximating distributions for differentially private statistics, which they apply to hypothesis testing problems.&lt;/p&gt;
&lt;p&gt;&lt;b&gt; 2.3. Differential Privacy on Graphs &lt;/b&gt;&lt;/p&gt;
&lt;p&gt;There is a significant amount of work on differentially private analysis of graphs. We remark that these algorithms can satisfy either edge or node differential privacy. The former (easier) guarantee defines a neighboring graph to be one obtained by adding or removing a single edge, while in the latter (harder) setting, a neighboring graph is one that can be obtained by modifying the set of edges connected to a single node. The main challenge in this area is that most graph statistics can have high sensitivity in the worst-case.&lt;/p&gt;
&lt;p&gt;The initial works in this area focused on the empirical setting, and goals range from counting subgraphs [&lt;a href="#KRSY11"&gt;KRSY11&lt;/a&gt;, &lt;a href="#BBDS13"&gt;BBDS13&lt;/a&gt;, &lt;a href="#KNRS13"&gt;KNRS13&lt;/a&gt;, &lt;a href="#CZ13"&gt;CZ13&lt;/a&gt;, &lt;a href="#RS16"&gt;RS16&lt;/a&gt;] to outputting a privatized graph which approximates the original [&lt;a href="#GRU12"&gt;GRU12&lt;/a&gt;, &lt;a href="#BBDS12"&gt;BBDS12&lt;/a&gt;, &lt;a href="#Upa13"&gt;Upa13&lt;/a&gt;, &lt;a href="#AU19"&gt;AU19&lt;/a&gt;, &lt;a href="#EKKL20"&gt;EKKL20&lt;/a&gt;]. In contrast to the setting discussed in most of this series, it seems that there are larger qualitative differences between the study of empirical and population statistics due to the fact that many graph statistics have high worst-case sensitivity, but may have smaller sensitivity on typical graphs from many natural models.&lt;/p&gt;
&lt;p&gt;In the population statistics setting, recent work has focused on parameter estimation of the underlying random graph model. So far this work has given estimators for the &lt;img src="https://blog.gautamkamath.com/assets/233ae345edbcd14bde12.png" alt="{\beta}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;-model [&lt;a href="#KS16"&gt;KS16&lt;/a&gt;] and graphons [&lt;a href="#BCS15"&gt;BCS15&lt;/a&gt;,&lt;a href="#BCSZ18"&gt;BCSZ18&lt;/a&gt;]. Graphons are a generalization of the stochastic block model, which is, in turn, a generalization of the Erdös-Rényi model. Interestingly, the methods of Lipschitz-extensions introduced in the empirical setting by [&lt;a href="#BBDS13"&gt;BBDS13&lt;/a&gt;, &lt;a href="#KNRS13"&gt;KNRS13&lt;/a&gt;] are the main tool used in the statistical setting as well. While the first works on private graphon estimation were not computationally efficient, a recent focus has been on obviating these issues for certain important cases, such as the Erdös-Rényi setting [&lt;a href="#SU19"&gt;SU19&lt;/a&gt;].&lt;/p&gt;
&lt;p&gt;&lt;b&gt;Bibliography&lt;/b&gt;&lt;/p&gt;
&lt;p&gt;&lt;a name="ADKR19"&gt;&lt;/a&gt;[ADKR19] Maryam Aliakbarpour, Ilias Diakonikolas, Daniel M. Kane, and Ronitt Rubinfeld. Private testing of distributions via sample permutations. NeurIPS &amp;#8217;19.&lt;/p&gt;
&lt;p&gt;&lt;a name="ADR18"&gt;&lt;/a&gt;[ADR18] Maryam Aliakbarpour, Ilias Diakonikolas, and Ronitt Rubinfeld. Differentially private identity and closeness testing of discrete distributions. ICML &amp;#8217;18.&lt;/p&gt;
&lt;p&gt;&lt;a name="AKSZ18"&gt;&lt;/a&gt;[AKSZ18] Jayadev Acharya, Gautam Kamath, Ziteng Sun, and Huanyu Zhang. Inspectre: Privately estimating the unseen. ICML &amp;#8217;18.&lt;/p&gt;
&lt;p&gt;&lt;a name="AS18"&gt;&lt;/a&gt;[AS18] Jordan Awan and Aleksandra Slavković. Differentially private uniformly most powerful tests for binomial data. NeurIPS &amp;#8217;18.&lt;/p&gt;
&lt;p&gt;&lt;a name="ASZ18"&gt;&lt;/a&gt;[ASZ18] Jayadev Acharya, Ziteng Sun, and Huanyu Zhang. Differentially private testing of identity and closeness of discrete distributions. NeurIPS &amp;#8217;18.&lt;/p&gt;
&lt;p&gt;&lt;a name="ASZ20"&gt;&lt;/a&gt;[ASZ20] Jayadev Acharya, Ziteng Sun, and Huanyu Zhang. Differentially private Assouad, Fano, and Le Cam. arXiv, 2004.06830, 2020.&lt;/p&gt;
&lt;p&gt;&lt;a name="AU19"&gt;&lt;/a&gt;[AU19] Raman Arora and Jalaj Upadhyay. On differentially private graph sparsification and applications. NeurIPS &amp;#8217;19.&lt;/p&gt;
&lt;p&gt;&lt;a name="BBDS12"&gt;&lt;/a&gt;[BBDS12] Jeremiah Blocki, Avrim Blum, Anupam Datta, and Or Sheffet. The Johnson-Lindenstrauss transform itself preserves differential privacy. FOCS &amp;#8217;12.&lt;/p&gt;
&lt;p&gt;&lt;a name="BBDS13"&gt;&lt;/a&gt;[BBDS13] Jeremiah Blocki, Avrim Blum, Anupam Datta, and Or Sheffet. Differentially private data analysis of social networks via restricted sensitivity. ITCS &amp;#8217;13.&lt;/p&gt;
&lt;p&gt;&lt;a name="BCS15"&gt;&lt;/a&gt;[BCS15] Christian Borgs, Jennifer Chayes, and Adam Smith. Private graphon estimation for sparse graphs. NIPS &amp;#8217;15.&lt;/p&gt;
&lt;p&gt;&lt;a name="BCSZ18"&gt;&lt;/a&gt;[BCSZ18] Christian Borgs, Jennifer Chayes, Adam Smith, and Ilias Zadik. Revealing network structure, confidentially: Improved rates for node-private graphon estimation. FOCS &amp;#8217;18.&lt;/p&gt;
&lt;p&gt;&lt;a name="BDMN05"&gt;&lt;/a&gt;[BDMN05] Avrim Blum, Cynthia Dwork, Frank McSherry, and Kobbi Nissim. Practical privacy: The SuLQ framework. PODS &amp;#8217;05.&lt;/p&gt;
&lt;p&gt;&lt;a name="BKSW19"&gt;&lt;/a&gt;[BKSW19] Mark Bun, Gautam Kamath, Thomas Steinke, and Zhiwei Steven Wu. Private hypothesis selection. NeurIPS &amp;#8217;19.&lt;/p&gt;
&lt;p&gt;&lt;a name="BN14"&gt;&lt;/a&gt;[BN14] Hai Brenner and Kobbi Nissim. Impossibility of differentially private universally optimal mechanisms. SIAM Journal on Computing, 43(5), 2014.&lt;/p&gt;
&lt;p&gt;&lt;a name="BNS13"&gt;&lt;/a&gt;[BNS13] Amos Beimel, Kobbi Nissim, and Uri Stemmer. Private learning and sanitization: Pure vs. approximate differential privacy. APPROX-RANDOM &amp;#8217;13.&lt;/p&gt;
&lt;p&gt;&lt;a name="BNSV15"&gt;&lt;/a&gt;[BNSV15] Mark Bun, Kobbi Nissim, Uri Stemmer, and Salil Vadhan. Differentially private release and learning of threshold functions. FOCS &amp;#8217;15.&lt;/p&gt;
&lt;p&gt;&lt;a name="BS19"&gt;&lt;/a&gt;[BS19] Mark Bun and Thomas Steinke. Average-case averages: Private algorithms for smooth sensitivity and mean estimation. NeurIPS &amp;#8217;19.&lt;/p&gt;
&lt;p&gt;&lt;a name="BSU17"&gt;&lt;/a&gt;[BSU17] Mark Bun, Thomas Steinke, and Jonathan Ullman. Make up your mind: The price of online queries in differential privacy. SODA &amp;#8217;17.&lt;/p&gt;
&lt;p&gt;&lt;a name="BUV14"&gt;&lt;/a&gt;[BUV14] Mark Bun, Jonathan Ullman, and Salil Vadhan. Fingerprinting codes and the price of approximate differential privacy. STOC &amp;#8217;14.&lt;/p&gt;
&lt;p&gt;&lt;a name="CBRG18"&gt;&lt;/a&gt;[CBRG18] Zachary Campbell, Andrew Bray, Anna Ritz, and Adam Groce. Differentially private ANOVA testing. ICDIS &amp;#8217;18.&lt;/p&gt;
&lt;p&gt;&lt;a name="CDK17"&gt;&lt;/a&gt;[CDK17] Bryan Cai, Constantinos Daskalakis, and Gautam Kamath. Priv&amp;#8217;it: Private and sample efficient identity testing. ICML &amp;#8217;17.&lt;/p&gt;
&lt;p&gt;&lt;a name="CKLZ19"&gt;&lt;/a&gt;[CKLZ19] Rachel Cummings, Sara Krehbiel, Yuliia Lut, and Wanrong Zhang. Privately detecting changes in unknown distributions. arXiv, 1910.01327, 2019.&lt;/p&gt;
&lt;p&gt;&lt;a name="CKMSU19"&gt;&lt;/a&gt;[CKMSU19] Clément L. Canonne, Gautam Kamath, Audra McMillan, Adam Smith, and Jonathan Ullman. The structure of optimal private tests for simple hypotheses. STOC &amp;#8217;19.&lt;/p&gt;
&lt;p&gt;&lt;a name="CKMTZ18"&gt;&lt;/a&gt;[CKMTZ18] Rachel Cummings, Sara Krehbiel, Yajun Mei, Rui Tuo, and Wanrong Zhang. Differentially private change-point detection. NeurIPS &amp;#8217;18.&lt;/p&gt;
&lt;p&gt;&lt;a name="CKMUZ19"&gt;&lt;/a&gt;[CKMUZ19] Clément L. Canonne, Gautam Kamath, Audra McMillan, Jonathan Ullman, and Lydia Zakynthinou. Private identity testing for high-dimensional distributions. arXiv, 1905.11947, 2019.&lt;/p&gt;
&lt;p&gt;&lt;a name="CKSBG19"&gt;&lt;/a&gt;[CKSBG19] Simon Couch, Zeki Kazan, Kaiyan Shi, Andrew Bray, and Adam Groce. Differentially private nonparametric hypothesis testing. CCS &amp;#8217;19.&lt;/p&gt;
&lt;p&gt;&lt;a name="CWZ19"&gt;&lt;/a&gt;[CWZ19] T. Tony Cai, Yichen Wang, and Linjun Zhang. The cost of privacy: Optimal rates of convergence for parameter estimation with differential privacy. arXiv, 1902.04495, 2019.&lt;/p&gt;
&lt;p&gt;&lt;a name="CZ13"&gt;&lt;/a&gt;[CZ13] Shixi Chen and Shuigeng Zhou. Recursive mechanism: Towards node differential privacy and unrestricted joins. SIGMOD &amp;#8217;13.&lt;/p&gt;
&lt;p&gt;&lt;a name="DHS15"&gt;&lt;/a&gt;[DHS15] Ilias Diakonikolas, Moritz Hardt, and Ludwig Schmidt. Differentially private learning of structured discrete distributions. NIPS &amp;#8217;15.&lt;/p&gt;
&lt;p&gt;&lt;a name="DKW56"&gt;&lt;/a&gt;[DKW56] Aryeh Dvoretzky, Jack Kiefer, and Jacob Wolfowitz. Asymptotic minimax character of the sample distribution function and of the classical multinomial estimator. The Annals of Mathematical Statistics, 27(3), 1956.&lt;/p&gt;
&lt;p&gt;&lt;a name="DL09"&gt;&lt;/a&gt;[DL09] Cynthia Dwork and Jing Lei. Differential privacy and robust statistics. STOC &amp;#8217;09.&lt;/p&gt;
&lt;p&gt;&lt;a name="DMNS06"&gt;&lt;/a&gt;[DMNS06] Cynthia Dwork, Frank McSherry, Kobbi Nissim, and Adam Smith. Calibrating noise to sensitivity in private data analysis. TCC &amp;#8217;06.&lt;/p&gt;
&lt;p&gt;&lt;a name="DN03"&gt;&lt;/a&gt;[DN03] Irit Dinur and Kobbi Nissim. Revealing information while preserving privacy. PODS &amp;#8217;03.&lt;/p&gt;
&lt;p&gt;&lt;a name="DN04"&gt;&lt;/a&gt;[DN04] Cynthia Dwork and Kobbi Nissim. Privacy-preserving datamining on vertically partitioned databases. CRYPTO &amp;#8217;04.&lt;/p&gt;
&lt;p&gt;&lt;a name="DSSUV15"&gt;&lt;/a&gt;[DSSUV15] Cynthia Dwork, Adam Smith, Thomas Steinke, Jonathan Ullman, and Salil Vadhan. Robust traceability from trace amounts. FOCS &amp;#8217;15.&lt;/p&gt;
&lt;p&gt;&lt;a name="EKKL20"&gt;&lt;/a&gt;[EKKL20] Marek Eliáš, Michael Kapralov, Janardhan Kulkarni, and Yin Tat Lee. Differentially private release of synthetic graphs. SODA &amp;#8217;20.&lt;/p&gt;
&lt;p&gt;&lt;a name="GLRV16"&gt;&lt;/a&gt;[GLRV16] Marco Gaboardi, Hyun-Woo Lim, Ryan M. Rogers, and Salil P. Vadhan. Differentially private chi-squared hypothesis testing: Goodness of fit and independence testing. ICML &amp;#8217;16.&lt;/p&gt;
&lt;p&gt;&lt;a name="GRU12"&gt;&lt;/a&gt;[GRU12] Anupam Gupta, Aaron Roth, and Jonathan Ullman. Iterative constructions and private data release. TCC &amp;#8217;12.&lt;/p&gt;
&lt;p&gt;&lt;a name="HSRDTMPSNC08"&gt;&lt;/a&gt;[HSRDTMPSNC08] Nils Homer, Szabolcs Szelinger, Margot Redman, David Duggan, Waibhav Tembe, Jill Muehling, John V. Pearson, Dietrich A. Stephan, Stanley F. Nelson, and David W. Craig. PLoS Genetics, 4(8), 2008.&lt;/p&gt;
&lt;p&gt;&lt;a name="KLSU19"&gt;&lt;/a&gt;[KLSU19] Gautam Kamath, Jerry Li, Vikrant Singhal, and Jonathan Ullman. Privately learning high-dimensional distributions. COLT &amp;#8217;19.&lt;/p&gt;
&lt;p&gt;&lt;a name="KNRS13"&gt;&lt;/a&gt;[KNRS13] Shiva Prasad Kasiviswanathan, Kobbi Nissim, Sofya Raskhodnikova, and Adam Smith. Analyzing graphs with node differential privacy. TCC &amp;#8217;13.&lt;/p&gt;
&lt;p&gt;&lt;a name="KR17"&gt;&lt;/a&gt;[KR17] Daniel Kifer and Ryan M. Rogers. A new class of private chi-square tests. AISTATS &amp;#8217;17.&lt;/p&gt;
&lt;p&gt;&lt;a name="KRSY11"&gt;&lt;/a&gt;[KRSY11] Vishesh Karwa, Sofya Raskhodnikova, Adam Smith, and Grigory Yaroslavtsev. Private analysis of graph structure. VLDB &amp;#8217;11.&lt;/p&gt;
&lt;p&gt;&lt;a name="KS16"&gt;&lt;/a&gt;[KS16] Vishesh Karwa and Aleksandra Slavković. Inference using noisy degrees: Differentially private β-model and synthetic graphs. The Annals of Statistics, 44(1), 2016.&lt;/p&gt;
&lt;p&gt;&lt;a name="KSF17"&gt;&lt;/a&gt;[KSF17] Kazuya Kakizaki, Jun Sakuma, and Kazuto Fukuchi. Differentially private chi-squared test by unit circle mechanism. ICML &amp;#8217;17.&lt;/p&gt;
&lt;p&gt;&lt;a name="KSU20"&gt;&lt;/a&gt;[KSU20] Gautam Kamath, Vikrant Singhal, and Jonathan Ullman. Private mean estimation of heavy-tailed distributions. arXiv, 2002.09464, 2020.&lt;/p&gt;
&lt;p&gt;&lt;a name="KV18"&gt;&lt;/a&gt;[KV18] Vishesh Karwa and Salil Vadhan. Finite sample differentially private confidence intervals. ITCS &amp;#8217;18.&lt;/p&gt;
&lt;p&gt;&lt;a name="Mas90"&gt;&lt;/a&gt;[Mas90] Pascal Massart. The tight constant in the Dvoretzky-Kiefer-Wolfowitz inequality. The Annals of Probability, 18(3), 1990.&lt;/p&gt;
&lt;p&gt;&lt;a name="NRS07"&gt;&lt;/a&gt;[NRS07] Kobbi Nissim, Sofya Raskhodnikova, and Adam Smith. Smooth sensitivity and sampling in private data analysis. STOC &amp;#8217;07.&lt;/p&gt;
&lt;p&gt;&lt;a name="RS16"&gt;&lt;/a&gt;[RS16] Sofya Raskhodnikova and Adam D. Smith. Lipschitz extensions for node-private graph statistics and the generalized exponential mechanism. FOCS &amp;#8217;16.&lt;/p&gt;
&lt;p&gt;&lt;a name="Smi11"&gt;&lt;/a&gt;[Smi11] Adam Smith. Privacy-preserving statistical estimation with optimal convergence rates. STOC &amp;#8217;11.&lt;/p&gt;
&lt;p&gt;&lt;a name="SGGRGB19"&gt;&lt;/a&gt;[SGGRGB19] Marika Swanberg, Ira Globus-Harris, Iris Griffith, Anna Ritz, Adam Groce, and Andrew Bray. Improved differentially private analysis of variance. PETS &amp;#8217;19.&lt;/p&gt;
&lt;p&gt;&lt;a name="SU17a"&gt;&lt;/a&gt;[SU17a] Thomas Steinke and Jonathan Ullman. Between pure and approximate differential privacy. Journal of Privacy and Confidentiality, 7(2), 2017.&lt;/p&gt;
&lt;p&gt;&lt;a name="SU17b"&gt;&lt;/a&gt;[SU17b] Thomas Steinke and Jonathan Ullman. Tight lower bounds for differentially private selection. FOCS &amp;#8217;17.&lt;/p&gt;
&lt;p&gt;&lt;a name="SU19"&gt;&lt;/a&gt;[SU19] Adam Sealfon and Jonathan Ullman. Efficiently estimating Erdos-Renyi graphs with node differential privacy. NeurIPS &amp;#8217;19.&lt;/p&gt;
&lt;p&gt;&lt;a name="Upa13"&gt;&lt;/a&gt;[Upa13] Jalaj Upadhyay. Random projections, graph sparsification, and differential privacy. ASIACRYPT &amp;#8217;13.&lt;/p&gt;
&lt;p&gt;&lt;a name="USF13"&gt;&lt;/a&gt;[USF13] Caroline Uhler, Aleksandra Slavković, and Stephen E. Fienberg. Privacy-preserving data sharing for genome-wide association studies. The Journal of Privacy and Confidentiality, 5(1), 2013.&lt;/p&gt;
&lt;p&gt;&lt;a name="VS09"&gt;&lt;/a&gt;[VS09] Duy Vu and Aleksandra Slavković. Differential privacy for clinical trial data: Preliminary evaluations. ICDMW ’09.&lt;/p&gt;
&lt;p&gt;&lt;a name="WKLK18"&gt;&lt;/a&gt;[WKLK18] Yue Wang, Daniel Kifer, Jaewoo Lee, and Vishesh Karwa. Statistical approximating distributions under differential privacy. The Journal of Privacy and Confidentiality, 8(1), 2018.&lt;/p&gt;
&lt;p&gt;&lt;a name="WLK15"&gt;&lt;/a&gt;[WLK15] Yue Wang, Jaewoo Lee, and Daniel Kifer. Revisiting differentially private hypothesis tests for categorical data. arXiv, 1511.03376, 2015.&lt;/p&gt;
&lt;p&gt;&lt;a name="YFSU14"&gt;&lt;/a&gt;[YFSU14] Fei Yu, Stephen E. Fienberg, Aleksandra B. Slavković, and Caroline Uhler. Scalable privacy-preserving data sharing methodology for genome-wide association studies. Journal of Biomedical Informatics, 50, 2014.&lt;/p&gt;
&lt;p&gt;&lt;a name="ZKKW20"&gt;&lt;/a&gt;[ZKKW20] Huanyu Zhang, Gautam Kamath, Janardhan Kulkarni, and Zhiwei Steven Wu. Privately learning Markov random fields. arXiv, 2002.09463, 2020.&lt;/p&gt;
</description></item><item><title>A Primer on Private Statistics – Part I</title><link>https://blog.gautamkamath.com/2020/04/14/a-primer-on-private-statistics-part-i/</link><guid isPermaLink="true">https://blog.gautamkamath.com/2020/04/14/a-primer-on-private-statistics-part-i/</guid><pubDate>Tue, 14 Apr 2020 14:23:37 +0000</pubDate><dc:creator>Gautam Kamath</dc:creator><description>&lt;p&gt;By &lt;a href="http://www.gautamkamath.com/"&gt;Gautam Kamath&lt;/a&gt; and &lt;a href="http://www.ccs.neu.edu/home/jullman/"&gt;Jonathan Ullman&lt;/a&gt;&lt;/p&gt;
&lt;p&gt;Differentially private statistics is a very lively research area, and has seen a lot of activity in the last couple years. While the phrasing is a slight departure from previous work which focused on estimation with worst-case datasets, it turns out that the differences are often superficial. In a short series of blog posts, we hope to educate readers on some of the recent advancements in this area, as well as shed light on some of the connections between the old and the new. We&amp;#8217;ll describe the settings, cover a couple of technical examples, and give pointers to some other directions in the area. Thanks to &lt;a href="https://cs-people.bu.edu/ads22/"&gt;Adam Smith&lt;/a&gt; for helping kick off this project, &lt;a href="http://www.cs.columbia.edu/~ccanonne/"&gt;Clément Canonne&lt;/a&gt;, &lt;a href="https://www.cis.upenn.edu/~aaroth/"&gt;Aaron Roth&lt;/a&gt;, and &lt;a href="http://www.thomas-steinke.net/"&gt;Thomas Steinke&lt;/a&gt; for helpful comments, and &lt;a href="https://lucatrevisan.github.io/"&gt;Luca Trevisan&lt;/a&gt; for his &lt;a href="https://lucatrevisan.wordpress.com/latex-to-wordpress/"&gt;LaTeX2WP script&lt;/a&gt;.&lt;/p&gt;
&lt;p&gt;&lt;b&gt;1. Introduction &lt;/b&gt;&lt;/p&gt;
&lt;p&gt;Statistics and machine learning are now ubiquitous in data analysis. Given a dataset, one immediately wonders what it allows us to infer about the underlying population. However, modern datasets don&amp;#8217;t exist in a vacuum: they often contain sensitive information about the individuals they represent. Without proper care, statistical procedures will result in gross violations of privacy. Motivated by the shortcomings of ad hoc methods for data anonymization, Dwork, McSherry, Nissim, and Smith introduced the celebrated notion of differential privacy [&lt;a href="#DMNS06"&gt;DMNS06&lt;/a&gt;].&lt;/p&gt;
&lt;p&gt;From its inception, some of the driving motivations for differential privacy were applications in statistics and the social sciences, notably disclosure limitation for the US Census. And yet, the lion&amp;#8217;s share of differential privacy research has taken place within the computer science community. As a result, the specific applications being studied are often not formulated using statistical terminology, or even as statistical problems. Perhaps most significantly, much of the early work in computer science (though definitely not all) focus on estimating some property &lt;em&gt;of a dataset&lt;/em&gt; rather than estimating some property &lt;em&gt;of an underlying population&lt;/em&gt;.&lt;/p&gt;
&lt;p&gt;Although the earliest works exploring the interaction between differential privacy and classical statistics go back to at least 2009 [&lt;a href="#VS09"&gt;VS09&lt;/a&gt;,&lt;a href="#FRY10"&gt;FRY10&lt;/a&gt;], the emphasis on differentially private statistical inference in the computer science literature is somewhat more recent. However, while earlier results on differential privacy did not always formulate problems in a statistical language, statistical inference was a key motivation for most of this work. As a result many of the techniques that were developed have direct applications in statistics, for example establishing minimax rates for estimation problems.&lt;/p&gt;
&lt;p&gt;The purpose of this series of blog posts is to highlight some of those results in the computer science literature, and present them in a more statistical language. Specifically, we will discuss:&lt;/p&gt;
&lt;ul&gt;
&lt;li&gt;Tight minimax lower bounds for privately estimating the mean of a multivariate distribution over &lt;img src="https://blog.gautamkamath.com/assets/0cd21008a3d2da905e33.png" alt="{{\mathbb R}^d}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;, using the technique of &lt;em&gt;tracing attacks&lt;/em&gt; developed in [&lt;a href="#BUV14"&gt;BUV14&lt;/a&gt;,&lt;a href="#DSSUV15"&gt;DSSUV15&lt;/a&gt;, &lt;a href="#BSU17"&gt;BSU17&lt;/a&gt;, &lt;a href="#SU17a"&gt;SU17a&lt;/a&gt;, &lt;a href="#SU17b"&gt;SU17b&lt;/a&gt;, &lt;a href="#KLSU19"&gt;KLSU19&lt;/a&gt;].
&lt;p&gt; &lt;/p&gt;
&lt;/li&gt;
&lt;li&gt;Upper bounds for estimating a distribution in Kolmogorov distance, using the ubiquitous &lt;em&gt;binary-tree mechanism&lt;/em&gt; introduced in [&lt;a href="#DNPR10"&gt;DNPR10&lt;/a&gt;,&lt;a href="#CSS11"&gt;CSS11&lt;/a&gt;].&lt;/li&gt;
&lt;/ul&gt;
&lt;p&gt;In particular, we hope to encourage computer scientists working on differential privacy to pay more attention to the applications of their methods in statistics, and share with statisticians many of the powerful techniques that have been developed in the computer science literature.&lt;/p&gt;
&lt;p&gt; &lt;/p&gt;
&lt;p&gt;&lt;b&gt; 1.1. Formulating Private Statistical Inference &lt;/b&gt;&lt;/p&gt;
&lt;p&gt;Essentially every differentially private statistical estimation task can be phrased using the following setup. We are given a dataset &lt;img src="https://blog.gautamkamath.com/assets/c921ababa7e640330c4d.png" alt="{X = (X_1, \dots, X_n)}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; of size &lt;img src="https://blog.gautamkamath.com/assets/b0c4a337cdf1c86e0e02.png" alt="{n}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;, and we wish to design an algorithm &lt;img src="https://blog.gautamkamath.com/assets/dfe3271ecc25b62a6801.png" alt="{M \in \mathcal{M}}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; where &lt;img src="https://blog.gautamkamath.com/assets/94d92ffd79a4fe277ec2.png" alt="{\mathcal{M}}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; is the class of mechanisms that are both:&lt;/p&gt;
&lt;ol&gt;
&lt;li&gt;differentially private, and&lt;/li&gt;
&lt;li&gt;accurate, either in expectation or with high probability, according to some task-specific measure.&lt;/li&gt;
&lt;/ol&gt;
&lt;p&gt;A few comments about this framework are in order. First, although the accuracy requirement is stochastic in nature (i.e., an algorithm might not be accurate depending on the randomness of the algorithm and the data generation process), the privacy requirement is worst-case in nature. That is, the algorithm must protect privacy for every dataset &lt;img src="https://blog.gautamkamath.com/assets/80bfa982f798c252f90b.png" alt="{X}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;, even those we believe are very unlikely.&lt;/p&gt;
&lt;p&gt;Second, the accuracy requirement is stated rather vaguely. This is because the notion of accuracy of an algorithm is slightly more nuanced, depending on whether we are concerned with &lt;em&gt;empirical&lt;/em&gt; or &lt;em&gt;population&lt;/em&gt; statistics. A particular emphasis of these blog posts is to explore the difference (or, as we will see, the lack of a difference) between these two notions of accuracy. The former estimates a quantity of the observed dataset, while the latter estimates a quantity of an unobserved distribution which is assumed to have generated the dataset.&lt;/p&gt;
&lt;p&gt;More precisely, the former can be phrased in terms of empirical loss, of the form:&lt;/p&gt;
&lt;p align="center"&gt;&lt;img src="https://blog.gautamkamath.com/assets/eeb9ac15f8f50a9b0b39.png" alt="\displaystyle \min_{M \in \mathcal{M}}~\max_{X \in \mathcal{X}}~\mathop{\mathbb E}_M(\ell(M(X), f(X))), " class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;&lt;/p&gt;
&lt;p&gt;where &lt;img src="https://blog.gautamkamath.com/assets/94d92ffd79a4fe277ec2.png" alt="{\mathcal{M}}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; is some class of &lt;em&gt;randomized estimators&lt;/em&gt; (e.g., differentially private estimators), &lt;img src="https://blog.gautamkamath.com/assets/952184fc5a44d98d46e0.png" alt="{\mathcal{X}}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; is some class of &lt;em&gt;datasets&lt;/em&gt;, &lt;img src="https://blog.gautamkamath.com/assets/d4f19f783c6310f0e53d.png" alt="{f}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; is some quantity of interest, and &lt;img src="https://blog.gautamkamath.com/assets/3edd496692f37353df29.png" alt="{\ell}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; is some &lt;em&gt;loss function&lt;/em&gt;. That is, we&amp;#8217;re looking to find an estimator that has small expected loss on &lt;em&gt;any dataset&lt;/em&gt; in some class.&lt;/p&gt;
&lt;p&gt;In contrast, statistical minimax theory looks at statements about population loss, of the form:&lt;/p&gt;
&lt;p align="center"&gt;&lt;img src="https://blog.gautamkamath.com/assets/2e7a672727c8a15fd124.png" alt="\displaystyle \min_{M \in \mathcal{M}}~\max_{P \in \mathcal{P}}~\mathop{\mathbb E}_{X \sim P, M}(\ell(M(X),f(P))), " class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;&lt;/p&gt;
&lt;p&gt;where &lt;img src="https://blog.gautamkamath.com/assets/294e1e8b0355c1aba775.png" alt="{\mathcal{P}}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; is some family of &lt;em&gt;distributions&lt;/em&gt; over datasets (typically consisting of i.i.d. samples). That is, we&amp;#8217;re looking to find an estimator that has small expected loss on random data from &lt;em&gt;any distribution&lt;/em&gt; in some class. In particular, note that the randomness in this objective additionally includes the data generating procedure &lt;img src="https://blog.gautamkamath.com/assets/5848ed63823c1bc62f30.png" alt="{X \sim P}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;.&lt;/p&gt;
&lt;p&gt;These two formulations are formally very different in several ways. First, the empirical formulation requires an estimator to have small loss on &lt;em&gt;worst-case&lt;/em&gt; datasets, whereas the statistical formulation only requires the estimator to have small loss &lt;em&gt;on average&lt;/em&gt; over datasets drawn from certain distributions. Second, the statistical formulation requires that we estimate the unknown quantity &lt;img src="https://blog.gautamkamath.com/assets/3e4cc2cf5cfc987a4b44.png" alt="{f(P)}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;, and thus necessitates a solution to the non-private estimation problem. On the other hand, the empirical formulation only asks us to estimate the known quantity &lt;img src="https://blog.gautamkamath.com/assets/7557004afc1e2b7e2a03.png" alt="{f(X)}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;, and thus if there were no privacy constraint it would always be possible to compute &lt;img src="https://blog.gautamkamath.com/assets/7557004afc1e2b7e2a03.png" alt="{f(X)}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; exactly. Third, typically in the statistical formulation, we require that the dataset is drawn i.i.d., which means that we are more constrained when proving lower bounds for estimation than we are in the empirical problem.&lt;/p&gt;
&lt;p&gt;However, in practice (more precisely, in the practice of doing theoretical research), these two formulations are more alike than they are different, and results about one formulation often imply results about the other formulation. On the algorithmic side, classical statistical results will often tell us that &lt;img src="https://blog.gautamkamath.com/assets/f451bece928b3293d2ae.png" alt="{\ell(f(X),f(P))}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; is small, in which case algorithms that guarantee &lt;img src="https://blog.gautamkamath.com/assets/6a68035d8f56b9e1520a.png" alt="{\ell(M(X),f(X))}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; is small also guarantee &lt;img src="https://blog.gautamkamath.com/assets/6b660e71fae09cc573d8.png" alt="{\ell(M(X),f(P))}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; is small.&lt;/p&gt;
&lt;p&gt;Moreover, typical lower bound arguments for empirical quantities are often statistical in nature. These typically involving constructing some simple &amp;#8220;hard distribution&amp;#8221; over datasets such that no private algorithm can estimate well on average for this distribution, and thus these lower bound arguments also apply to estimating population statistics for some simple family of distributions. We will proceed to give some examples of estimation problems that were originally studied by computer scientists with the empirical formulation in mind. These results either implicitly or explicitly provide solutions to the corresponding population versions of the same problems&amp;#8212;our goal is to spell out and illustrate these connections.&lt;/p&gt;
&lt;p&gt;&lt;b&gt;2. DP Background &lt;/b&gt;&lt;/p&gt;
&lt;p&gt;Let &lt;img src="https://blog.gautamkamath.com/assets/ff0dd2d346a8c62b3d14.png" alt="{X = (X_1,X_2,\dots,X_n) \in \mathcal{X}^n}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; be a collection of &lt;img src="https://blog.gautamkamath.com/assets/b0c4a337cdf1c86e0e02.png" alt="{n}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; samples where each individual sample comes from the domain &lt;img src="https://blog.gautamkamath.com/assets/952184fc5a44d98d46e0.png" alt="{\mathcal{X}}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;. We say that two samples &lt;img src="https://blog.gautamkamath.com/assets/a4144f585930261d9ea8.png" alt="{X,X&amp;#x27; \in \mathcal{X}^*}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; are &lt;em&gt;adjacent&lt;/em&gt;, denoted &lt;img src="https://blog.gautamkamath.com/assets/d43fa12ae2a6c63ed516.png" alt="{X \sim X&amp;#x27;}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;, if they differ on at most one individual sample. Intuitively, a randomized algorithm &lt;img src="https://blog.gautamkamath.com/assets/03e84b366e4aa0c348a3.png" alt="{M}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;, which is often called a &lt;em&gt;mechanism&lt;/em&gt; for historical reasons, is &lt;em&gt;differentially private&lt;/em&gt; if the distribution of &lt;img src="https://blog.gautamkamath.com/assets/df5f1e3c67698162e30f.png" alt="{M(X)}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; and &lt;img src="https://blog.gautamkamath.com/assets/0a0f33ba928e3cab0159.png" alt="{M(X&amp;#x27;)}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; are similar for every pair of adjacent samples &lt;img src="https://blog.gautamkamath.com/assets/f715df54ae2e21533f4a.png" alt="{X,X&amp;#x27;}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;.&lt;/p&gt;
&lt;blockquote&gt;
&lt;p&gt;&lt;b&gt;Definition 1 ([&lt;a href="#DMNS06"&gt;DMNS06&lt;/a&gt;])&lt;/b&gt;&lt;em&gt; A mechanism &lt;img src="https://blog.gautamkamath.com/assets/76102597a55e137a85b6.png" alt="{M \colon \mathcal{X}^n \rightarrow \mathcal{R}}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; is &lt;em&gt;&lt;img src="https://blog.gautamkamath.com/assets/22aed5ab6dec88b9744b.png" alt="{(\epsilon,\delta)}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;-differentially private&lt;/em&gt; if for every pair of adjacent datasets &lt;img src="https://blog.gautamkamath.com/assets/d43fa12ae2a6c63ed516.png" alt="{X \sim X&amp;#x27;}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;, and every (measurable) &lt;img src="https://blog.gautamkamath.com/assets/bfe84339a211cfba5c7c.png" alt="{R \subseteq R}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; &lt;/em&gt;&lt;/p&gt;
&lt;p align="center"&gt;&lt;img src="https://blog.gautamkamath.com/assets/8987c5fde6103a9f49c3.png" alt="\displaystyle \mathop{\mathbb P}(M(X) \in R) \leq e^{\epsilon} \cdot \mathop{\mathbb P}(M(X&amp;#x27;) \in R) + \delta. " class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;&lt;/p&gt;
&lt;p&gt; &lt;/p&gt;
&lt;/blockquote&gt;
&lt;p&gt;We let &lt;img src="https://blog.gautamkamath.com/assets/156bd8629f505c50a69c.png" alt="{\mathcal{M}_{\epsilon,\delta}}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; denote the set of mechanisms that satisfy &lt;img src="https://blog.gautamkamath.com/assets/22aed5ab6dec88b9744b.png" alt="{(\epsilon,\delta)}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;-differential privacy.&lt;/p&gt;
&lt;blockquote&gt;
&lt;p&gt;&lt;b&gt;Remark 1&lt;/b&gt; &lt;em&gt; To simplify notation, and to maintain consistency with the literature, we adopt the convention of defining the mechanism only for a fixed sample size &lt;img src="https://blog.gautamkamath.com/assets/b0c4a337cdf1c86e0e02.png" alt="{n}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;. What this means in practice is that the mechanisms we describe treat the sample size &lt;img src="https://blog.gautamkamath.com/assets/b0c4a337cdf1c86e0e02.png" alt="{n}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; is &lt;em&gt;public information&lt;/em&gt; that need not be kept private. While one could define a more general model where &lt;img src="https://blog.gautamkamath.com/assets/b0c4a337cdf1c86e0e02.png" alt="{n}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; is not fixed, it wouldn&amp;#8217;t add anything to this discussion other than additional complexity. &lt;/em&gt;&lt;/p&gt;
&lt;/blockquote&gt;
&lt;blockquote&gt;
&lt;p&gt;&lt;b&gt;Remark 2&lt;/b&gt; &lt;em&gt; In these blog posts, we stick to the most general formulation of differential privacy, so-called &lt;em&gt;approximate differential privacy&lt;/em&gt;, i.e. &lt;img src="https://blog.gautamkamath.com/assets/22aed5ab6dec88b9744b.png" alt="{(\epsilon,\delta)}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;-differential privacy for &lt;img src="https://blog.gautamkamath.com/assets/a13440df3176a08d1130.png" alt="{\delta &amp;gt; 0}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; essentially because this is the notion that captures the widest variety of private mechanisms. Almost all of what follows would apply equally well, with minor technical modifications, to slightly stricter notions of &lt;em&gt;concentrated differential privacy [&lt;/em&gt;&lt;a href="#DR16"&gt;DR16&lt;/a&gt;, &lt;a href="#BS16"&gt;BS16&lt;/a&gt;, &lt;a href="#BDRS18"&gt;BDRS18&lt;/a&gt;], Rényi differential privacy [&lt;a href="#Mir17"&gt;Mir17&lt;/a&gt;], or &lt;em&gt;Gaussian differential privacy [&lt;a href="#DRS19"&gt;DRS19&lt;/a&gt;]&lt;/em&gt;. While so-called &lt;em&gt;pure differential privacy&lt;/em&gt;, i.e. &lt;img src="https://blog.gautamkamath.com/assets/0ba2a1742353e780078c.png" alt="{(\epsilon,0)}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;-differential privacy has also been studied extensively, this notion is artificially restrictive and excludes many differentially private mechanisms. &lt;/em&gt;&lt;/p&gt;
&lt;/blockquote&gt;
&lt;p&gt;A key property of differential privacy that helps when desinging efficient estimators is &lt;em&gt;closure under postprocessing&lt;/em&gt;:&lt;/p&gt;
&lt;blockquote&gt;
&lt;p&gt;&lt;b&gt;Lemma 2 (Post-Processing [&lt;a href="#DMNS06"&gt;DMNS06&lt;/a&gt;])&lt;/b&gt;&lt;em&gt; &lt;a name="lempost-processing"&gt;&lt;/a&gt; If &lt;img src="https://blog.gautamkamath.com/assets/76102597a55e137a85b6.png" alt="{M \colon \mathcal{X}^n \rightarrow \mathcal{R}}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; is &lt;img src="https://blog.gautamkamath.com/assets/22aed5ab6dec88b9744b.png" alt="{(\epsilon,\delta)}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;-differentially private and &lt;img src="https://blog.gautamkamath.com/assets/d2703324b57606788b43.png" alt="{M&amp;#x27; \colon \mathcal{R} \rightarrow \mathcal{R}&amp;#x27;}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; is any randomized algorithm, then &lt;img src="https://blog.gautamkamath.com/assets/104886056418b6c32d6d.png" alt="{M&amp;#x27; \circ M}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; is &lt;img src="https://blog.gautamkamath.com/assets/22aed5ab6dec88b9744b.png" alt="{(\epsilon,\delta)}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;-differentially private. &lt;/em&gt;&lt;/p&gt;
&lt;/blockquote&gt;
&lt;p&gt;The estimators we present in this work will use only one tool for achieving differential privacy, the &lt;em&gt;Gaussian Mechanism&lt;/em&gt;.&lt;/p&gt;
&lt;blockquote&gt;
&lt;p&gt;&lt;b&gt;Lemma 3 (Gaussian Mechanism)&lt;/b&gt; &lt;em&gt; &lt;a name="lemgauss-mech"&gt;&lt;/a&gt; Let &lt;img src="https://blog.gautamkamath.com/assets/32dd7221606fc3f6b77e.png" alt="{f \colon \mathcal{X}^n \rightarrow {\mathbb R}^d}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; be a function and let &lt;/em&gt;&lt;/p&gt;
&lt;p align="center"&gt;&lt;img src="https://blog.gautamkamath.com/assets/6ad125c9934245f198d5.png" alt="\displaystyle \Delta_{f} = \sup_{X\sim X&amp;#x27;} \| f(X) - f(X&amp;#x27;) \|_2 " class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;&lt;/p&gt;
&lt;p&gt;denote its &lt;em&gt;&lt;img src="https://blog.gautamkamath.com/assets/63aec118fb6952ceb9ea.png" alt="{\ell_2}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;-sensitivity&lt;/em&gt;. The &lt;em&gt;Gaussian mechanism&lt;/em&gt;&lt;/p&gt;
&lt;p align="center"&gt;&lt;img src="https://blog.gautamkamath.com/assets/56df99309d2f2b650f8c.png" alt="\displaystyle M(X) = f(X) + \mathcal{N}\left(0 , \frac{2 \log(2/\delta)}{\epsilon^2} \cdot \Delta_{f}^2 \cdot {\mathbb I}_{d \times d} \right) " class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;&lt;/p&gt;
&lt;p&gt;&lt;em&gt; satisfies &lt;img src="https://blog.gautamkamath.com/assets/22aed5ab6dec88b9744b.png" alt="{(\epsilon,\delta)}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;-differential privacy. &lt;/em&gt;&lt;/p&gt;
&lt;/blockquote&gt;
&lt;p&gt;&lt;b&gt;3. Mean Estimation in &lt;img src="https://blog.gautamkamath.com/assets/0cd21008a3d2da905e33.png" alt="{{\mathbb R}^d}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; &lt;/b&gt;&lt;/p&gt;
&lt;p&gt;Let&amp;#8217;s take a dive into the problem of &lt;em&gt;private mean estimation&lt;/em&gt; for some family &lt;img src="https://blog.gautamkamath.com/assets/294e1e8b0355c1aba775.png" alt="{\mathcal{P}}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; of multivariate distributions over &lt;img src="https://blog.gautamkamath.com/assets/0cd21008a3d2da905e33.png" alt="{{\mathbb R}^d}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;. This problem has been studied for various families &lt;img src="https://blog.gautamkamath.com/assets/294e1e8b0355c1aba775.png" alt="{\mathcal{P}}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; and various choices of loss function. Here we focus on perhaps the simplest variant of the problem, in which &lt;img src="https://blog.gautamkamath.com/assets/294e1e8b0355c1aba775.png" alt="{\mathcal{P}}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; contains distributions of bounded support &lt;img src="https://blog.gautamkamath.com/assets/cda29481bf817c69cb43.png" alt="{[\pm 1]^d}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; and the loss is the &lt;img src="https://blog.gautamkamath.com/assets/2a40b253579817c05188.png" alt="{\ell_2^2}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; error. We emphasize, however, that the methods we discuss here are quite versatile and can be used to derive minimax bounds for other variants of the mean-estimation problem.&lt;/p&gt;
&lt;p&gt;Note that, by a simple argument, the non-private minimax rate for this class is achieved by the empirical mean, and is&lt;/p&gt;
&lt;p align="center"&gt;&lt;img src="https://blog.gautamkamath.com/assets/214cd4de12075fe718c0.png" alt="\displaystyle \max_{P \in \mathcal{P}} \mathop{\mathbb E}_{X_{1 \cdots n} \sim P}(\| \overline{X} - \mu\|_2^2) = \frac{d}{n}. \ \ \ \ \ (1)" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;&lt;/p&gt;
&lt;p&gt;The main goal of this section is to derive the minimax bound &lt;a name="eqRd-minimax"&gt;&lt;/a&gt;&lt;/p&gt;
&lt;p align="center"&gt;&lt;img src="https://blog.gautamkamath.com/assets/b53cb8be5b023e5750b1.png" alt="\displaystyle \min_{M \in \mathcal{M}_{\epsilon,\frac{1}{n}}} \max_{P \in \mathcal{P}} \mathop{\mathbb E}_{X_{1 \cdots n} \sim P}(\| M(X_{1 \cdots n}) - \mu \|_2^2) = \frac{d}{n} + \tilde\Theta\left(\frac{d^2}{\epsilon^2 n^2}\right). \ \ \ \ \ (2)" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;&lt;/p&gt;
&lt;p&gt;&lt;a name="eqRd-minimax"&gt;&lt;/a&gt; Recall that &lt;img src="https://blog.gautamkamath.com/assets/ebdf1f4d3fd3051ab04b.png" alt="{\tilde \Theta(f(n))}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; refers to a function which is both &lt;img src="https://blog.gautamkamath.com/assets/5f9d2dd0aaa6cc3b3c7e.png" alt="{O(f(n) \log^{c_1} f(n))}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; and &lt;img src="https://blog.gautamkamath.com/assets/bc6efa206e8dceeb7448.png" alt="{\Omega(f(n) \log^{c_2} f(n))}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; for some constants &lt;img src="https://blog.gautamkamath.com/assets/382f304bc04f289cad2e.png" alt="{c_1, c_2}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;. The proof of this lower bound is based on &lt;em&gt;robust tracing attacks&lt;/em&gt;, also called &lt;em&gt;membership inference attacks&lt;/em&gt;, which were developed in a chain of papers [&lt;a href="#BUV14"&gt;BUV14&lt;/a&gt;, &lt;a href="#DSSUV15"&gt;DSSUV15&lt;/a&gt;, &lt;a href="#BSU17"&gt;BSU17&lt;/a&gt;, &lt;a href="#SU17a"&gt;SU17a&lt;/a&gt;, &lt;a href="#SU17b"&gt;SU17b&lt;/a&gt;, &lt;a href="#KLSU19"&gt;KLSU19&lt;/a&gt;]. We remark that this lower bound is almost identical to the minimax bound for mean estimation proven in the much more recent work of Cai, Wang, and Zhang [&lt;a href="#CWZ19"&gt;CWZ19&lt;/a&gt;], but it lacks tight dependence on the parameter &lt;img src="https://blog.gautamkamath.com/assets/02e2c691c45d8d91445b.png" alt="{\delta}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;, which we discuss in the following remark.&lt;/p&gt;
&lt;blockquote&gt;
&lt;p&gt;&lt;b&gt;Remark 3&lt;/b&gt; &lt;em&gt; The choice of &lt;img src="https://blog.gautamkamath.com/assets/febd5d80c4889ee340d6.png" alt="{\delta = 1/n}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; in &lt;a href="#eqRd-minimax"&gt;(2)&lt;/a&gt; may look strange at first. For the upper bound this choice is arbitrary&amp;#8212;as we will see, we can upper bound the rate for any &lt;img src="https://blog.gautamkamath.com/assets/a13440df3176a08d1130.png" alt="{\delta &amp;gt; 0}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; at a cost of a factor of &lt;img src="https://blog.gautamkamath.com/assets/30d4334a5095b55dcf1d.png" alt="{O(\log(1/\delta))}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;. The lower bound applies only when &lt;img src="https://blog.gautamkamath.com/assets/edfec914bfa50f9058d3.png" alt="{\delta \leq 1/n}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;. Note that the rate is qualitatively different when &lt;img src="https://blog.gautamkamath.com/assets/ef94e568e9b0dad6151b.png" alt="{\delta \gg 1/n}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;. However, we emphasize that &lt;img src="https://blog.gautamkamath.com/assets/22aed5ab6dec88b9744b.png" alt="{(\epsilon,\delta)}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;-differential privacy is not a meaningful privacy notion unless &lt;img src="https://blog.gautamkamath.com/assets/076163c552b37a070f64.png" alt="{\delta \ll 1/n}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;. In particular, the mechanism that randomly outputs &lt;img src="https://blog.gautamkamath.com/assets/a8554812e77665503711.png" alt="{\delta n}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; elements of the sample satisfies &lt;img src="https://blog.gautamkamath.com/assets/7219b429ecfc6ec69420.png" alt="{(0,\delta)}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;-differential privacy. However, when &lt;img src="https://blog.gautamkamath.com/assets/ef94e568e9b0dad6151b.png" alt="{\delta \gg 1/n}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;, this mechanism completely violates the privacy of &lt;img src="https://blog.gautamkamath.com/assets/b599071863f02f4b20f7.png" alt="{\gg 1}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; person in the dataset. Moreover, taking the empirical mean of these &lt;img src="https://blog.gautamkamath.com/assets/a8554812e77665503711.png" alt="{\delta n}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; samples gives rate &lt;img src="https://blog.gautamkamath.com/assets/a21f49ce82def3f70cfd.png" alt="{d/\delta n}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;, which would violate our lower bound when &lt;img src="https://blog.gautamkamath.com/assets/02e2c691c45d8d91445b.png" alt="{\delta}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; is large enough. On the other hand, we would expect the minimax rate to become slower when &lt;img src="https://blog.gautamkamath.com/assets/076163c552b37a070f64.png" alt="{\delta \ll 1/n}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;. This expectation is, in fact, correct, however the proof we present does not give the tight dependence on the parameter &lt;img src="https://blog.gautamkamath.com/assets/02e2c691c45d8d91445b.png" alt="{\delta}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;. See [&lt;a href="#SU17a"&gt;SU17a&lt;/a&gt;] for a refinement that can obtain the right dependence on &lt;img src="https://blog.gautamkamath.com/assets/02e2c691c45d8d91445b.png" alt="{\delta}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;, and [&lt;a href="#CWZ19"&gt;CWZ19&lt;/a&gt;] for the details of how to apply this refinement in the i.i.d. setting. &lt;/em&gt;&lt;/p&gt;
&lt;/blockquote&gt;
&lt;p&gt;&lt;b&gt; 3.1. A Simple Upper Bound &lt;/b&gt;&lt;/p&gt;
&lt;blockquote&gt;
&lt;p&gt;&lt;b&gt;Theorem 4&lt;/b&gt; &lt;em&gt; For every &lt;img src="https://blog.gautamkamath.com/assets/b4d4518917a0cd03094d.png" alt="{n \in {\mathbb N}}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;, and every &lt;img src="https://blog.gautamkamath.com/assets/32f811523253e1b85605.png" alt="{\epsilon,\delta &amp;gt; 0}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;, there exists an &lt;img src="https://blog.gautamkamath.com/assets/22aed5ab6dec88b9744b.png" alt="{(\epsilon,\delta)}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;-differentially private private mechanism &lt;img src="https://blog.gautamkamath.com/assets/03e84b366e4aa0c348a3.png" alt="{M}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; such that &lt;a name="eqmean-est-ub"&gt;&lt;/a&gt;&lt;/em&gt;&lt;/p&gt;
&lt;p&gt;&lt;em&gt;&lt;em&gt;&lt;a name="eqmean-est-ub"&gt;&lt;/a&gt;&lt;/em&gt;&lt;/em&gt;&lt;/p&gt;
&lt;p align="center"&gt;&lt;img src="https://blog.gautamkamath.com/assets/e1d925ccf6563bf6b1f8.png" alt="\displaystyle \max_{P \in \mathcal{P}} \mathop{\mathbb E}_{X_{1 \cdots n} \sim P}(\| M(X_{1 \cdots n}) - \mu \|_2^2) \leq \frac{d}{n} + \frac{2 d^2 \log(2/\delta)}{\epsilon^2 n^2}. \ \ \ \ \ (3)" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;&lt;/p&gt;
&lt;p&gt;&lt;em&gt;&lt;a name="eqmean-est-ub"&gt;&lt;/a&gt;&lt;/em&gt;&lt;/p&gt;
&lt;p&gt;&lt;em&gt;&lt;a name="eqmean-est-ub"&gt;&lt;/a&gt; &lt;/em&gt;&lt;/p&gt;
&lt;/blockquote&gt;
&lt;p&gt;&lt;em&gt;Proof:&lt;/em&gt; Define the mechanism&lt;/p&gt;
&lt;p align="center"&gt;&lt;img src="https://blog.gautamkamath.com/assets/aa9a72043980e92f0dfe.png" alt="\displaystyle M(X_{1 \cdots n}) = \overline{X} + \mathcal{N}\left(0, \frac{2 d \log(2/\delta)}{\varepsilon^2 n^2} \cdot \mathbb{I}_{d \times d} \right). \ \ \ \ \ (4)" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;&lt;/p&gt;
&lt;p&gt;This mechanism satisfies &lt;img src="https://blog.gautamkamath.com/assets/22aed5ab6dec88b9744b.png" alt="{(\epsilon,\delta)}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;-differential privacy by Lemma &lt;a href="#lemgauss-mech"&gt;3&lt;/a&gt;, noting that for any pair of adjacent samples &lt;img src="https://blog.gautamkamath.com/assets/5ac0839addd4ec5b132d.png" alt="{X_{1 \cdots n}}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; and &lt;img src="https://blog.gautamkamath.com/assets/0f209ec7551d936220f5.png" alt="{X&amp;#x27;_{1 \cdots n}}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;, &lt;img src="https://blog.gautamkamath.com/assets/7532c42e054c44488a50.png" alt="{\| \overline{X} - \overline{X}&amp;#x27;\|_2^2 \leq \frac{d}{n^2}}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;.&lt;/p&gt;
&lt;p&gt;Let &lt;img src="https://blog.gautamkamath.com/assets/1c45139abb578b5d62fd.png" alt="{\sigma^2 = \frac{2 d \log(2/\delta)}{\varepsilon^2 n^2}}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;. Note that since the Gaussian noise has mean &lt;img src="https://blog.gautamkamath.com/assets/473660f59b08ac5d8b7f.png" alt="{0}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; and is independent of &lt;img src="https://blog.gautamkamath.com/assets/a36dfc6e9765bf193eaa.png" alt="{\overline{X} - \mu}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;, we have&lt;/p&gt;
&lt;p align="center"&gt;&lt;img src="https://blog.gautamkamath.com/assets/8231a95b61171bbf5c69.png" alt="\displaystyle \begin{array}{rll} \mathop{\mathbb E}(\| M(X_{1 \cdots n}) - \mu \|_2^2) ={} &amp;amp;\mathop{\mathbb E}(\| \overline{X} - \mu \|_2^2) + \mathop{\mathbb E}(\| M(X_{1 \cdots n}) - \overline{X} \|_2^2 ) \\ \leq{} &amp;amp;\frac{d}{n} + \mathop{\mathbb E}(\| M(X_{1 \cdots n}) - \overline{X} \|_2^2 ) \\ ={} &amp;amp;\frac{d}{n} + \mathop{\mathbb E}(\| \mathcal{N}(0, \sigma^2 \mathbb{I}_{d \times d}) \|_2^2 ) \\ ={} &amp;amp;\frac{d}{n} + \sigma^2 d \\ ={} &amp;amp;\frac{d}{n} + \frac{2 d^2 \log(2/\delta)}{\epsilon^2 n^2}. \end{array} " class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;&lt;/p&gt;
&lt;p&gt;&lt;img src="https://blog.gautamkamath.com/assets/b7d45ccd758dec885fd2.png" alt="\Box" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;&lt;/p&gt;
&lt;p&gt; &lt;/p&gt;
&lt;p&gt;&lt;b&gt; 3.2. Minimax Lower Bounds via Tracing &lt;/b&gt;&lt;/p&gt;
&lt;blockquote&gt;
&lt;p&gt;&lt;b&gt;Theorem 5&lt;/b&gt; &lt;em&gt; &lt;a name="thmmean-lb"&gt;&lt;/a&gt; For every &lt;img src="https://blog.gautamkamath.com/assets/7f1bd6474895b14f0d50.png" alt="{n, d \in {\mathbb N}}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;, &lt;img src="https://blog.gautamkamath.com/assets/09fc50fb75768d21335a.png" alt="{\epsilon &amp;gt; 0}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;, and &lt;img src="https://blog.gautamkamath.com/assets/3154e2a59df72c7d1a0e.png" alt="{\delta &amp;lt; 1/96n}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;, if &lt;img src="https://blog.gautamkamath.com/assets/294e1e8b0355c1aba775.png" alt="{\mathcal{P}}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; is the class of all product distributions on &lt;img src="https://blog.gautamkamath.com/assets/5ff740012432555592f5.png" alt="{\{\pm 1\}^{d}}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;, then for some constant &lt;img src="https://blog.gautamkamath.com/assets/fed713cae7c7f126f8de.png" alt="{C &amp;gt; 0}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;, &lt;/em&gt;&lt;/p&gt;
&lt;p align="center"&gt;&lt;img src="https://blog.gautamkamath.com/assets/592f72b5e66cc80d77d6.png" alt="\displaystyle \min_{M \in \mathcal{M}_{\epsilon,\delta}} \max_{P \in \mathcal{P}} \mathop{\mathbb E}_{X_{1 \cdots n} \sim P,M}(\| M(X_{1 \cdots n}) - \mu \|_2^2) = \Omega\left(\min \left\{ \frac{d^2}{ \epsilon^2 n^2}, d \right\}\right). " class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;&lt;/p&gt;
&lt;p&gt; &lt;/p&gt;
&lt;/blockquote&gt;
&lt;p&gt;Note that it is trivial to achieve error &lt;img src="https://blog.gautamkamath.com/assets/5523e6609c58683631b8.png" alt="{d}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; for any distribution using the mechanism &lt;img src="https://blog.gautamkamath.com/assets/09363dc69dfe76e5d8d8.png" alt="{M(X_{1 \cdots n}) \equiv 0}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;, so the result says that the error must be &lt;img src="https://blog.gautamkamath.com/assets/c0d733aa6a71fb692c39.png" alt="{\Omega(d^2/\epsilon^2 n^2)}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; whenever this error is significantly smaller than the trivial error of &lt;img src="https://blog.gautamkamath.com/assets/5523e6609c58683631b8.png" alt="{d}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;.&lt;/p&gt;
&lt;p&gt;&lt;b&gt;Tracing Attacks.&lt;/b&gt;&lt;/p&gt;
&lt;p&gt;Before giving the formal proof, we will try to give some intuition for the high-level proof strategy. The proof can be viewed as constructing a &lt;em&gt;tracing attack &lt;/em&gt;[&lt;a href="#DSSU17"&gt;DSSU17&lt;/a&gt;] (sometimes called a &lt;em&gt;membership inference attack&lt;/em&gt;) of the following form. There is an attacker who has the data of some individual &lt;img src="https://blog.gautamkamath.com/assets/8a19517b5d3eaa4daa66.png" alt="{Y}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; chosen in one of the two ways: either &lt;img src="https://blog.gautamkamath.com/assets/8a19517b5d3eaa4daa66.png" alt="{Y}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; is a random element of the sample &lt;img src="https://blog.gautamkamath.com/assets/80bfa982f798c252f90b.png" alt="{X}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;, or &lt;img src="https://blog.gautamkamath.com/assets/8a19517b5d3eaa4daa66.png" alt="{Y}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; is an independent random sample from the population &lt;img src="https://blog.gautamkamath.com/assets/4f5401ebbe0c17f4eccf.png" alt="{P}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;. The attacker is given access to the true distribution &lt;img src="https://blog.gautamkamath.com/assets/4f5401ebbe0c17f4eccf.png" alt="{P}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; and the outcome of the mechanism &lt;img src="https://blog.gautamkamath.com/assets/df5f1e3c67698162e30f.png" alt="{M(X)}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;, and wants to determine which of the two is the case. If the attacker can succeed, then &lt;img src="https://blog.gautamkamath.com/assets/03e84b366e4aa0c348a3.png" alt="{M}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; cannot be differentially private. To understand why this is the case, if &lt;img src="https://blog.gautamkamath.com/assets/8a19517b5d3eaa4daa66.png" alt="{Y}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; is a member of the dataset, then the attacker should say &lt;img src="https://blog.gautamkamath.com/assets/8a19517b5d3eaa4daa66.png" alt="{Y}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; is in the dataset, but if we consider the adjacent dataset &lt;img src="https://blog.gautamkamath.com/assets/d471ab4b17655d60fd1a.png" alt="{X&amp;#x27;}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; where we replace &lt;img src="https://blog.gautamkamath.com/assets/8a19517b5d3eaa4daa66.png" alt="{Y}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; with some independent sample from &lt;img src="https://blog.gautamkamath.com/assets/4f5401ebbe0c17f4eccf.png" alt="{P}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;, then the attacker will now say &lt;img src="https://blog.gautamkamath.com/assets/8a19517b5d3eaa4daa66.png" alt="{Y}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; is independent of the dataset. Thus, &lt;img src="https://blog.gautamkamath.com/assets/df5f1e3c67698162e30f.png" alt="{M(X)}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; and &lt;img src="https://blog.gautamkamath.com/assets/0a0f33ba928e3cab0159.png" alt="{M(X&amp;#x27;)}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; cannot be close in the sense required by differential privacy.&lt;/p&gt;
&lt;p&gt;Thus, the proof works by constructing a test statistic &lt;img src="https://blog.gautamkamath.com/assets/d68d6b594083cb48de00.png" alt="{Z = Z(M(X),Y,P),}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; that the attacker can use to distinguish the two possibilities for &lt;img src="https://blog.gautamkamath.com/assets/8a19517b5d3eaa4daa66.png" alt="{Y}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;. In particular, we show that there is a distribution over populations &lt;img src="https://blog.gautamkamath.com/assets/4f5401ebbe0c17f4eccf.png" alt="{P}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; such that &lt;img src="https://blog.gautamkamath.com/assets/d52a308425be85c329d0.png" alt="{\mathop{\mathbb E}(Z)}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; is small when &lt;img src="https://blog.gautamkamath.com/assets/8a19517b5d3eaa4daa66.png" alt="{Y}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; is independent of &lt;img src="https://blog.gautamkamath.com/assets/80bfa982f798c252f90b.png" alt="{X}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;, but for &lt;em&gt;every&lt;/em&gt; sufficiently accurate mechanism &lt;img src="https://blog.gautamkamath.com/assets/03e84b366e4aa0c348a3.png" alt="{M}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;, &lt;img src="https://blog.gautamkamath.com/assets/d52a308425be85c329d0.png" alt="{\mathop{\mathbb E}(Z)}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; is large when &lt;img src="https://blog.gautamkamath.com/assets/8a19517b5d3eaa4daa66.png" alt="{Y}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; is a random element of &lt;img src="https://blog.gautamkamath.com/assets/80bfa982f798c252f90b.png" alt="{X}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;.&lt;/p&gt;
&lt;p&gt;&lt;b&gt;Proof of Theorem &lt;a href="#thmmean-lb"&gt;5&lt;/a&gt;.&lt;/b&gt;&lt;/p&gt;
&lt;p&gt;The proof that we present closely follows the one that appears in Thomas Steinke’s Ph.D. thesis [&lt;a href="#Ste16"&gt;Ste16&lt;/a&gt;].&lt;/p&gt;
&lt;p&gt;We start by constructing a &amp;#8220;hard distribution&amp;#8221; over the family of product distributions &lt;img src="https://blog.gautamkamath.com/assets/294e1e8b0355c1aba775.png" alt="{\mathcal{P}}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;. Let &lt;img src="https://blog.gautamkamath.com/assets/45b66696b9c88cf81ecf.png" alt="{\mu = (\mu^1,\dots,\mu^d) \in [-1,1]^d}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; consist of &lt;img src="https://blog.gautamkamath.com/assets/5523e6609c58683631b8.png" alt="{d}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; independent draws from the uniform distribution on &lt;img src="https://blog.gautamkamath.com/assets/9c87d3dc526bbd93ea57.png" alt="{[-1,1]}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; and let &lt;img src="https://blog.gautamkamath.com/assets/f661f821d1574d1e44f1.png" alt="{P_{\mu}}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; be the product distribution over &lt;img src="https://blog.gautamkamath.com/assets/5ff740012432555592f5.png" alt="{\{\pm 1\}^{d}}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; with mean &lt;img src="https://blog.gautamkamath.com/assets/2b2431897f982f8c084e.png" alt="{\mu}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;. Let &lt;img src="https://blog.gautamkamath.com/assets/4b33945a86fa3531f814.png" alt="{X_1,\dots,X_n \sim P_{\mu}}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; and &lt;img src="https://blog.gautamkamath.com/assets/2900104760846445364c.png" alt="{X = (X_1,\dots,X_n)}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;.&lt;/p&gt;
&lt;p&gt;Let &lt;img src="https://blog.gautamkamath.com/assets/71ed7757c90808b4125d.png" alt="{M \colon \{\pm 1\}^{n \times d} \rightarrow [\pm 1]^d}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; be any &lt;img src="https://blog.gautamkamath.com/assets/22aed5ab6dec88b9744b.png" alt="{(\epsilon,\delta)}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;-differentially private mechanism and let&lt;/p&gt;
&lt;p align="center"&gt;&lt;img src="https://blog.gautamkamath.com/assets/9ed51ceca2848a571044.png" alt="\displaystyle \alpha^2 = \mathop{\mathbb E}_{\mu,X,M}(\| M(X) - \mu\|_2^2 ) \ \ \ \ \ (5)" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;&lt;/p&gt;
&lt;p&gt;be its expected loss. We will prove the desired lower bound on &lt;img src="https://blog.gautamkamath.com/assets/fa5850a31d603387f7d6.png" alt="{\alpha^2}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;.&lt;/p&gt;
&lt;p&gt;For every element &lt;img src="https://blog.gautamkamath.com/assets/bde2ec82ce426934a048.png" alt="{i}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;, we define the random variables&lt;/p&gt;
&lt;p align="center"&gt;&lt;img src="https://blog.gautamkamath.com/assets/8807428f887f0a4c6c87.png" alt="\displaystyle Z_i = Z_i(M(X),X_i,\mu) = \left\langle M(X) - \mu, X_i - \mu \right\rangle \\ Z&amp;#x27;_{i} = Z&amp;#x27;_i(M(X_{\sim i}), X_i, \mu) = \left\langle M(X_{\sim i}) - \mu, X_i - \mu \right\rangle, " class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;&lt;/p&gt;
&lt;p&gt;where &lt;img src="https://blog.gautamkamath.com/assets/bc8fae26152c6456670b.png" alt="{X_{\sim i}}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; denotes &lt;img src="https://blog.gautamkamath.com/assets/a9bd3f5f9b700af20622.png" alt="{(X_1,\dots,X&amp;#x27;_i,\dots,X_n)}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; where &lt;img src="https://blog.gautamkamath.com/assets/ab3d34c248fea1d0675c.png" alt="{X&amp;#x27;_i}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; is an independent sample from &lt;img src="https://blog.gautamkamath.com/assets/5c29724a23a62ab92dc5.png" alt="{P_\mu}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;. Our goal will be to show that, privacy and accuracy imply both upper and lower bounds on &lt;img src="https://blog.gautamkamath.com/assets/c47afcc85af15b428faa.png" alt="{\mathop{\mathbb E}(\sum_i Z_i)}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; that depend on &lt;img src="https://blog.gautamkamath.com/assets/b2bbc350751ac4c8f523.png" alt="{\alpha}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;, and thereby obtain a bound on &lt;img src="https://blog.gautamkamath.com/assets/fa5850a31d603387f7d6.png" alt="{\alpha^2}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;.&lt;/p&gt;
&lt;p&gt;The first claim says that, when &lt;img src="https://blog.gautamkamath.com/assets/e3d0992002f2b19f272d.png" alt="{X_i}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; is &lt;em&gt;not&lt;/em&gt; in the sample, then the likelihood random variable has mean &lt;img src="https://blog.gautamkamath.com/assets/473660f59b08ac5d8b7f.png" alt="{0}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; and variance controlled by the expected &lt;img src="https://blog.gautamkamath.com/assets/2a40b253579817c05188.png" alt="{\ell_2^2}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; error of the mechanism.&lt;/p&gt;
&lt;blockquote&gt;
&lt;p&gt;&lt;b&gt;Claim 1&lt;/b&gt; &lt;em&gt; &lt;a name="clmmean-lb-1"&gt;&lt;/a&gt; For every &lt;img src="https://blog.gautamkamath.com/assets/bde2ec82ce426934a048.png" alt="{i}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;, &lt;img src="https://blog.gautamkamath.com/assets/ebd50dc6750ad837bfa7.png" alt="{\mathop{\mathbb E}(Z&amp;#x27;_i) = 0}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;, &lt;img src="https://blog.gautamkamath.com/assets/fbb55fbfdfd8b8dfacbb.png" alt="{\mathrm{Var}(Z&amp;#x27;_i) \leq 4\alpha^2}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;, and &lt;img src="https://blog.gautamkamath.com/assets/8ecadac948529dd61680.png" alt="{\|Z&amp;#x27;_i\|_\infty \leq 4d}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;. &lt;/em&gt;&lt;/p&gt;
&lt;/blockquote&gt;
&lt;p&gt;&lt;em&gt;Proof:&lt;/em&gt; Conditioned on any value of &lt;img src="https://blog.gautamkamath.com/assets/2b2431897f982f8c084e.png" alt="{\mu}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;, &lt;img src="https://blog.gautamkamath.com/assets/ba226feea7a3ee71089f.png" alt="{M(X_{\sim i})}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; is independent from &lt;img src="https://blog.gautamkamath.com/assets/e3d0992002f2b19f272d.png" alt="{X_i}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;. Moreover, &lt;img src="https://blog.gautamkamath.com/assets/74067b7358495f2a1630.png" alt="{\mathop{\mathbb E}(X_i - \mu) = 0}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;, so we have&lt;/p&gt;
&lt;p align="center"&gt;&lt;img src="https://blog.gautamkamath.com/assets/50bc545e00982abeff33.png" alt="\displaystyle \begin{array}{rll} &amp;amp;\mathop{\mathbb E}_{\mu,X,M}(\langle M(X_{\sim i}) - \mu, X_i - \mu \rangle) \\ = &amp;amp;\mathop{\mathbb E}_{\mu}(\mathop{\mathbb E}_{X,M}(\langle M(X_{\sim i}) - \mu, X_i - \mu \rangle)) \\ = &amp;amp;\mathop{\mathbb E}_{\mu}(\left\langle \mathop{\mathbb E}_{X,M}(M(X_{\sim i}) - \mu), \mathop{\mathbb E}_{X,M}(X_i - \mu) \right \rangle ) \\ = &amp;amp;\mathop{\mathbb E}_{\mu}(\left\langle \mathop{\mathbb E}_{X,M}(M(X_{\sim i}) - \mu), 0 \right \rangle ) \\ = &amp;amp;0. \end{array} " class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;&lt;/p&gt;
&lt;p&gt;For the second part of the claim, since &lt;img src="https://blog.gautamkamath.com/assets/a2bfcded8f8dc91cbacc.png" alt="{(X_i - \mu)^2 \leq 4}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;, we have &lt;img src="https://blog.gautamkamath.com/assets/fe55e0ff0fc8cf32fca4.png" alt="{\mathrm{Var}(Z&amp;#x27;_i) \leq 4 \cdot \mathop{\mathbb E}(\| M(X) - \mu \|_2^2) = 4\alpha^2}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;. The final part of the claim follows from the fact that every entry of &lt;img src="https://blog.gautamkamath.com/assets/d16c4b6303f21b68def9.png" alt="{M(X_{\sim i}) - \mu}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; and &lt;img src="https://blog.gautamkamath.com/assets/61d57cbd41401af02904.png" alt="{X_i - \mu}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; is bounded by &lt;img src="https://blog.gautamkamath.com/assets/25c270e45cbeab4171c7.png" alt="{2}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; in absolute value, and &lt;img src="https://blog.gautamkamath.com/assets/54e373a2df4079272a14.png" alt="{Z&amp;#x27;_i}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; is a sum of &lt;img src="https://blog.gautamkamath.com/assets/5523e6609c58683631b8.png" alt="{d}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; such entries, so its absolute value is always at most &lt;img src="https://blog.gautamkamath.com/assets/7d436da5262722f8d135.png" alt="{4d}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;. &lt;img src="https://blog.gautamkamath.com/assets/b7d45ccd758dec885fd2.png" alt="\Box" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;&lt;/p&gt;
&lt;p&gt;The next claim says that, because &lt;img src="https://blog.gautamkamath.com/assets/03e84b366e4aa0c348a3.png" alt="{M}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; is differentially private, &lt;img src="https://blog.gautamkamath.com/assets/3859f5b6a1f39c629345.png" alt="{Z_i}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; has similar expectation to &lt;img src="https://blog.gautamkamath.com/assets/54e373a2df4079272a14.png" alt="{Z&amp;#x27;_i}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;, and thus its expectation is also small.&lt;/p&gt;
&lt;blockquote&gt;
&lt;p&gt;&lt;b&gt;Claim 2&lt;/b&gt; &lt;em&gt;&lt;a name="clmmean-lb-2"&gt;&lt;/a&gt; &lt;img src="https://blog.gautamkamath.com/assets/d21610d989c8319e1e2d.png" alt="{\mathop{\mathbb E}(\sum_{i=1}^{n} Z_i) \leq 4n\alpha \epsilon + 8n \delta d.}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; &lt;/em&gt;&lt;/p&gt;
&lt;/blockquote&gt;
&lt;p&gt;&lt;em&gt;Proof:&lt;/em&gt; The proof is a direct calculation using the following inequality, whose proof is relatively simple using the definition of differential privacy:&lt;/p&gt;
&lt;p align="center"&gt;&lt;img src="https://blog.gautamkamath.com/assets/3931560230329df5bc8e.png" alt="\displaystyle \mathop{\mathbb E}(Z_i) \leq \mathop{\mathbb E}(Z&amp;#x27;_i) + 2\epsilon \sqrt{\mathrm{Var}(Z&amp;#x27;_i)} + 2\delta \| Z&amp;#x27;_i \|_\infty. " class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;&lt;/p&gt;
&lt;p&gt;Given the inequality and Claim &lt;a href="#clmmean-lb-1"&gt;1&lt;/a&gt;, we have&lt;/p&gt;
&lt;p align="center"&gt;&lt;img src="https://blog.gautamkamath.com/assets/d9d697c5a8685bad83f2.png" alt="\displaystyle \mathop{\mathbb E}(Z_i) \leq 0 + (2\epsilon)(2\alpha) + (2\delta)(2d) = 4\epsilon \alpha + 8 \delta d . " class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;&lt;/p&gt;
&lt;p&gt;The claim now follows by summing over all &lt;img src="https://blog.gautamkamath.com/assets/bde2ec82ce426934a048.png" alt="{i}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;. &lt;img src="https://blog.gautamkamath.com/assets/b7d45ccd758dec885fd2.png" alt="\Box" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;&lt;/p&gt;
&lt;p&gt;The final claim says that, because &lt;img src="https://blog.gautamkamath.com/assets/03e84b366e4aa0c348a3.png" alt="{M}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; is accurate, the expected sum of the random variables &lt;img src="https://blog.gautamkamath.com/assets/3859f5b6a1f39c629345.png" alt="{Z_i}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; is large.&lt;/p&gt;
&lt;blockquote&gt;
&lt;p&gt;&lt;b&gt;Claim 3&lt;/b&gt; &lt;em&gt; &lt;a name="clmmean-lb-3"&gt;&lt;/a&gt; &lt;img src="https://blog.gautamkamath.com/assets/00a079999a87daab7916.png" alt="{\mathop{\mathbb E}(\sum_{i=1}^{n} Z_i) \geq \frac{d}{3} - \alpha^2.}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; &lt;/em&gt;&lt;/p&gt;
&lt;/blockquote&gt;
&lt;p&gt;The proof relies on the following key lemma, whose proof we omit.&lt;/p&gt;
&lt;blockquote&gt;
&lt;p&gt;&lt;b&gt;Lemma 6 (Fingerprinting Lemma [&lt;a href="#BSU17"&gt;BSU17&lt;/a&gt;])&lt;/b&gt;&lt;em&gt; &lt;a name="lemfp"&gt;&lt;/a&gt; If &lt;img src="https://blog.gautamkamath.com/assets/bb6d6db29193d811e556.png" alt="{\mu \in [\pm 1]}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; is sampled uniformly, &lt;img src="https://blog.gautamkamath.com/assets/d7eb8674a96d1e770c1e.png" alt="{X_1,\dots,X_n \in \{\pm 1\}^{n}}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; are sampled independently with mean &lt;img src="https://blog.gautamkamath.com/assets/2b2431897f982f8c084e.png" alt="{\mu}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;, and &lt;img src="https://blog.gautamkamath.com/assets/7fbd59db405cc6da93be.png" alt="{f \colon \{\pm 1\}^n \rightarrow [\pm 1]}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; is any function, then &lt;/em&gt;&lt;/p&gt;
&lt;p align="center"&gt;&lt;img src="https://blog.gautamkamath.com/assets/d8185cf45dfb85daf8af.png" alt="\displaystyle \mathop{\mathbb E}_{\mu,X}((f(X) - \mu) \cdot \sum_{i=1}^{n} (X_i - \mu)) \geq \frac{1}{3} - \mathop{\mathbb E}_{\mu,X}((f(X) - \mu)^2). " class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;&lt;/p&gt;
&lt;p&gt; &lt;/p&gt;
&lt;/blockquote&gt;
&lt;p&gt;The lemma is somewhat technical, but for intuition, consider the case where &lt;img src="https://blog.gautamkamath.com/assets/6b3ae97bcf072fe477fc.png" alt="{f(X) = \frac{1}{n}\sum_{i} X_i}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; is the empirical mean. In this case we have&lt;/p&gt;
&lt;p align="center"&gt;&lt;img src="https://blog.gautamkamath.com/assets/25e3adede39cb6ed8d6f.png" alt="\displaystyle \begin{array}{rcl} \mathop{\mathbb E}_{\mu,X}((f(X) - \mu) \cdot \sum_{i=1}^n (X_i - \mu)) ={} \mathop{\mathbb E}_{\mu}(\frac{1}{n} \sum_i \mathop{\mathbb E}_{X}( (X_i - \mu)^2) ) ={} \mathop{\mathbb E}_{\mu}(\mathrm{Var}(X_i)) = \frac{1}{3}. \end{array} " class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;&lt;/p&gt;
&lt;p&gt;The lemma says that, when &lt;img src="https://blog.gautamkamath.com/assets/2b2431897f982f8c084e.png" alt="{\mu}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; is sampled this way, then any modification of &lt;img src="https://blog.gautamkamath.com/assets/d4f19f783c6310f0e53d.png" alt="{f}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; that reduces the correlation between &lt;img src="https://blog.gautamkamath.com/assets/7557004afc1e2b7e2a03.png" alt="{f(X)}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; and &lt;img src="https://blog.gautamkamath.com/assets/77c921db6538c1111fb8.png" alt="{\sum_i X_i}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; will increase the mean-squared-error of &lt;img src="https://blog.gautamkamath.com/assets/d4f19f783c6310f0e53d.png" alt="{f}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; proportionally.&lt;/p&gt;
&lt;p&gt;We now prove Claim &lt;a href="#clmmean-lb-3"&gt;3&lt;/a&gt;.&lt;/p&gt;
&lt;p&gt;&lt;em&gt;Proof:&lt;/em&gt; We can apply the lemma to each coordinate of the estimate &lt;img src="https://blog.gautamkamath.com/assets/df5f1e3c67698162e30f.png" alt="{M(X)}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;.&lt;/p&gt;
&lt;p align="center"&gt;&lt;img src="https://blog.gautamkamath.com/assets/a16027014b8f6c0877be.png" alt="\displaystyle \begin{array}{rll} \mathop{\mathbb E}(\sum_{i=1}^{n} Z_i) ={} &amp;amp;\mathop{\mathbb E}(\sum_{i=1}^{n} \left\langle M(X) - \mu, X_i - \mu \right\rangle) \\ ={} &amp;amp;\sum_{j=1}^{d} \mathop{\mathbb E}((M^j(X) - \mu^j)\cdot \sum_{i=1}^{n} (X_i^j - \mu^j)) \\ \geq{} &amp;amp;\sum_{j=1}^{d} \left( \frac{1}{3} - \mathop{\mathbb E}((M^j(X) - \mu^j)^2) \right) \\ ={} &amp;amp;\frac{d}{3} - \mathop{\mathbb E}(\| M(X) - \mu \|_2^2) ={} \frac{d}{3} - \alpha^2. \end{array} " class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;&lt;/p&gt;
&lt;p&gt;The inequality is Lemma &lt;a href="#lemfp"&gt;6&lt;/a&gt;. &lt;img src="https://blog.gautamkamath.com/assets/b7d45ccd758dec885fd2.png" alt="\Box" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;&lt;/p&gt;
&lt;p&gt;Combining Claims &lt;a href="#clmmean-lb-2"&gt;2&lt;/a&gt; and &lt;a href="#clmmean-lb-3"&gt;3&lt;/a&gt; gives&lt;/p&gt;
&lt;p align="center"&gt;&lt;img src="https://blog.gautamkamath.com/assets/36c5d57c2da76b59f4e3.png" alt="\displaystyle \frac{d}{3} - \alpha^2 \leq 4n\alpha \epsilon + 8n \delta d. \ \ \ \ \ (6)" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;&lt;/p&gt;
&lt;p&gt;Now, if &lt;img src="https://blog.gautamkamath.com/assets/ec99f04702da622d0ecd.png" alt="{\alpha^2 \geq \frac{d}{6}}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; then we&amp;#8217;re done, so we&amp;#8217;ll assume that &lt;img src="https://blog.gautamkamath.com/assets/1abd9b0d4d69d7f44285.png" alt="{\alpha^2 \leq \frac{d}{6}}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;. Further, by our assumption on the value of &lt;img src="https://blog.gautamkamath.com/assets/02e2c691c45d8d91445b.png" alt="{\delta}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;, &lt;img src="https://blog.gautamkamath.com/assets/5282e8fdb2ff72c7ad6b.png" alt="{8n \delta d \leq \frac{d}{12}}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;. In this case we can rearrange terms and square both sides to obtain&lt;/p&gt;
&lt;p align="center"&gt;&lt;img src="https://blog.gautamkamath.com/assets/26f435234d764d541ceb.png" alt="\displaystyle \alpha^2 \geq{} \frac{1}{16 \epsilon^2 n^2} \left(\frac{d}{3} - \alpha^2 - 8 n\delta d\right)^2 \geq \frac{1}{16 \epsilon^2 n^2} \left(\frac{d}{12}\right)^2 = \frac{d^2}{2304 \epsilon^2 n^2}. \ \ \ \ \ (7)" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;&lt;/p&gt;
&lt;p&gt;Combining the two cases for &lt;img src="https://blog.gautamkamath.com/assets/fa5850a31d603387f7d6.png" alt="{\alpha^2}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt; gives &lt;img src="https://blog.gautamkamath.com/assets/5a408d9d16fd85bcea21.png" alt="{\alpha^2 \geq \min\{ \frac{d}{6}, \frac{d^2}{2304 \epsilon^2 n^2} \}}" class="latex" loading="lazy" decoding="async"&gt;&lt;/img&gt;, as desired.&lt;/p&gt;
&lt;p&gt;&lt;b&gt;Bibliography&lt;/b&gt;&lt;/p&gt;
&lt;p&gt;[BDRS18]&lt;a name="BDRS18"&gt;&lt;/a&gt; Mark Bun, Cynthia Dwork, Guy N. Rothblum, and Thomas Steinke. Composable and versatile privacy via truncated CDP. STOC ’18.&lt;/p&gt;
&lt;p&gt;[BS16]&lt;a name="BS16"&gt;&lt;/a&gt; Mark Bun and Thomas Steinke. Concentrated differential privacy: Simplifications, extensions, and lower bounds. TCC ’16-B.&lt;/p&gt;
&lt;p&gt;[BSU17]&lt;a name="BSU17"&gt;&lt;/a&gt; Mark Bun, Thomas Steinke, and Jonathan Ullman. Make up your mind: The price of online queries in differential privacy. SODA ’17.&lt;/p&gt;
&lt;p&gt;[BUV14]&lt;a name="BUV14"&gt;&lt;/a&gt; Mark Bun, Jonathan Ullman, and Salil Vadhan. Fingerprinting codes and the price of approximate differential privacy. STOC ’14.&lt;/p&gt;
&lt;p&gt;[CSS11]&lt;a name="CSS11"&gt;&lt;/a&gt; T-H Hubert Chan, Elaine Shi, and Dawn Song. Private and continual release of statistics. ACM Transactions on Information and System Security, 14(3):26, 2011.&lt;/p&gt;
&lt;p&gt;[CWZ19]&lt;a name="CWZ19"&gt;&lt;/a&gt; T. Tony Cai, Yichen Wang, and Linjun Zhang. The cost of privacy: Optimal rates of convergence for parameter estimation with differential privacy. arXiv, 1902.04495, 2019.&lt;/p&gt;
&lt;p&gt;[DMNS06]&lt;a name="DMNS06"&gt;&lt;/a&gt; Cynthia Dwork, Frank McSherry, Kobbi Nissim, and Adam Smith. Calibrating noise to sensitivity in private data analysis. TCC ’06.&lt;/p&gt;
&lt;p&gt;[DNPR10]&lt;a name="DNPR10"&gt;&lt;/a&gt; Cynthia Dwork, Moni Naor, Toniann Pitassi, and Guy N. Rothblum. Differential privacy under continual observation. STOC &amp;#8217;10.&lt;/p&gt;
&lt;p&gt;[DR16]&lt;a name="DR16"&gt;&lt;/a&gt; Cynthia Dwork and Guy N. Rothblum. Concentrated differential privacy. arXiv, 1603.01887, 2016.&lt;/p&gt;
&lt;p&gt;[DRS19]&lt;a name="DRS19"&gt;&lt;/a&gt; Jinshuo Dong, Aaron Roth, and Weijie J. Su. Gaussian differential privacy. arXiv, 1905.02383, 2019.&lt;/p&gt;
&lt;p&gt;[DSSU17]&lt;a name="DSSU17"&gt;&lt;/a&gt; Cynthia Dwork, Adam Smith, Thomas Steinke, Jonathan Ullman, and Salil Vadhan. Robust traceability from trace amounts. FOCS ’15.&lt;/p&gt;
&lt;p&gt;[DSSUV15]&lt;a name="DSSUV15"&gt;&lt;/a&gt; Cynthia Dwork, Adam Smith, Thomas Steinke, and Jonathan Ullman. Exposed! a survey of attacks on private data. Annual Review of Statistics and Its Application, 4:61–84, 2017.&lt;/p&gt;
&lt;p&gt;[FRY10]&lt;a name="FRY10"&gt;&lt;/a&gt; Stephen E. Fienberg, Alessandro Rinaldo, and Xiaolin Yang. Differential privacy and the risk-utility tradeoff for multi-dimensional contingency tables. PSD ’10.&lt;/p&gt;
&lt;p&gt;[KLSU19]&lt;a name="KLSU19"&gt;&lt;/a&gt; Gautam Kamath, Jerry Li, Vikrant Singhal, and Jonathan Ullman. Privately learning high-dimensional distributions. COLT ’19.&lt;/p&gt;
&lt;p&gt;[Mir17]&lt;a name="Mir17"&gt;&lt;/a&gt; Ilya Mironov. Rényi differential privacy. CSF ’17.&lt;/p&gt;
&lt;p&gt;[Ste16]&lt;a name="Ste16"&gt;&lt;/a&gt; Thomas Alexander Steinke. Upper and Lower Bounds for Privacy and Adaptivity in Algorithmic Data Analysis. PhD thesis, 2016.&lt;/p&gt;
&lt;p&gt;[SU17a]&lt;a name="SU17a"&gt;&lt;/a&gt; Thomas Steinke and Jonathan Ullman. Between pure and approximate differential privacy. Journal of Privacy and Confidentiality, 7(2), 2017.&lt;/p&gt;
&lt;p&gt;[SU17b]&lt;a name="SU17b"&gt;&lt;/a&gt; Thomas Steinke and Jonathan Ullman. Tight lower bounds for differentially private selection. FOCS ’17.&lt;/p&gt;
&lt;p&gt;[VS09]&lt;a name="VS09"&gt;&lt;/a&gt; Duy Vu and Aleksandra Slavković. Differential privacy for clinical trial data: Preliminary evaluations. ICDMW ’09.&lt;/p&gt;


&lt;p class="wp-block-paragraph"&gt;&lt;/p&gt;
</description></item><item><title>ICALP (and LICS) 2020 – Relocation and Extended Deadline</title><link>https://blog.gautamkamath.com/2020/02/08/icalp-and-lics-2020-relocation-and-extended-deadline/</link><guid isPermaLink="true">https://blog.gautamkamath.com/2020/02/08/icalp-and-lics-2020-relocation-and-extended-deadline/</guid><pubDate>Sat, 08 Feb 2020 15:01:28 +0000</pubDate><dc:creator>Gautam Kamath</dc:creator><description>
&lt;p class="wp-block-paragraph"&gt;Due to the Wuhan coronavirus outbreak, the organizers of ICALP and LICS have made the difficult decision to relocate both (co-located) conferences from Beijing, China, to Saarbrücken, Germany. Speaking specifically about ICALP now (I do not have further information about LICS): As a result of previous uncertainty regarding the situation, the deadline has been extended by about six days, until Tuesday February 18, 2020, at 6 AM GMT. The dates of the conference remain (roughly) the same, July 8 &amp;#8211; 11, 2020. &lt;br&gt;The following is a more official message from ICALP Track A Chair, Artur Czumaj.&lt;/p&gt;



&lt;hr class="wp-block-separator"&gt;&lt;/hr&gt;



&lt;p class="wp-block-paragraph"&gt;The ICALP and the LICS steering committee have agreed together with the conference chairs in Beijing to relocate the two conferences.&lt;br&gt;ICALP and LICS 2020 will take place in &lt;strong&gt;Saarbrücken&lt;/strong&gt;, Germany, July 8 &amp;#8211; 11 2020 (with satellite workshops on July 6 &amp;#8211; 7 2020).&lt;br&gt;The deadline is extended, see below.&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;&lt;strong&gt;Call for Papers &amp;#8211; ICALP 2020&lt;/strong&gt;&lt;br&gt;&lt;strong&gt;July 8 &amp;#8211; 11 2020, Saarbrücken, Germany&lt;/strong&gt;&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;&lt;strong&gt;NEW Paper submission deadline: Tuesday February 18, 2020, 6am GMT&lt;/strong&gt;&lt;br&gt;&lt;a href="https://easychair.org/conferences/?conf=icalp2020"&gt;https://easychair.org/conferences/?conf=icalp2020&lt;/a&gt;&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;ICALP (International Colloquium on Automata, Languages and Programming) is the main European conference in Theoretical Computer Science and annual meeting of the European Association for Theoretical Computer Science (EATCS). ICALP 2020 will be hosted on the Saarland Informatics Campus in Saarbrücken, in co-location with LICS 2020 (ACM/IEEE Symposium on Logic in Computer Science).&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;&lt;strong&gt;Invited speakers:&lt;/strong&gt;&lt;br&gt;Track A: Virginia Vassilevska (MIT), Robert Krauthgamer (Weizmann)&lt;br&gt;Track B: Stefan Kiefer (Oxford)&lt;br&gt;Joint ICALP-LICS: Andrew Yao (Tsinghua), Jérôme Leroux (Bordeaux)&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;&lt;strong&gt;Submission Guidelines:&lt;/strong&gt; see &lt;a href="https://easychair.org/conferences/?conf=icalp2020"&gt;https://easychair.org/conferences/?conf=icalp2020&lt;/a&gt;&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;&lt;strong&gt;NEW Paper submission deadline: February 18&lt;/strong&gt;, 2020, 6am GMT&lt;br&gt;notifications: April 15, 2020&lt;br&gt;camera ready: April 28, 2020&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;Topics: ICALP 2020 will have the two traditional tracks&lt;br&gt;A (Algorithms, Complexity and Games &amp;#8211; including Algorithmic Game Theory, Distributed Algorithms and Parallel, Distributed and External Memory Computing) and&lt;br&gt;B (Automata, Logic, Semantics and Theory of Programming).&lt;br&gt;&lt;strong&gt;&lt;em&gt;&amp;nbsp;&amp;nbsp;&amp;nbsp; (Notice that the old tracks A and C have been merged into a single track A.)&lt;/em&gt;&lt;/strong&gt;&lt;br&gt;Papers presenting original, unpublished research on all aspects of theoretical computer science are sought.&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;Typical, but not exclusive topics are:&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;Track A &amp;#8212; Algorithmic Aspects of Networks and Networking, Algorithms for Computational Biology, Algorithmic Game Theory, Combinatorial Optimization, Combinatorics in Computer Science, Computational Complexity, Computational Geometry, Computational Learning Theory, Cryptography, Data Structures, Design and Analysis of Algorithms, Foundations of Machine Learning, Foundations of Privacy, Trust and Reputation in Network, Network Models for Distributed Computing, Network Economics and Incentive-Based Computing Related to Networks, Network Mining and Analysis, Parallel, Distributed and External Memory Computing, Quantum Computing, Randomness in Computation, Theory of Security in Networks&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;Track B &amp;#8212; Algebraic and Categorical Models, Automata, Games, and Formal Languages, Emerging and Non-standard Models of Computation, Databases, Semi-Structured Data and Finite Model Theory, Formal and Logical Aspects of Learning, Logic in Computer Science, Theorem Proving and Model Checking, Models of Concurrent, Distributed, and Mobile Systems, Models of Reactive, Hybrid and Stochastic Systems, Principles and Semantics of Programming Languages, Program Analysis and Transformation, Specification, Verification and Synthesis, Type Systems and Theory, Typed Calculi&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;&lt;strong&gt;PC Track A chair: Artur Czumaj&lt;/strong&gt; (University&amp;nbsp; of Warwick)&lt;br&gt;&lt;strong&gt;PC Track B chair: Anuj Dawar&lt;/strong&gt; (University of Cambridge)&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;Contact&lt;br&gt;All questions about submissions should be emailed to the PC Track chairs:&lt;br&gt;Artur Czumaj &lt;a href="mailto:A.Czumaj@warwick.ac.uk"&gt;A.Czumaj@warwick.ac.uk&amp;lt;mailto:A.Czumaj@warwick.ac.uk&amp;gt;&lt;/a&gt;&lt;br&gt;Anuj Dawar &lt;a href="mailto:Anuj.Dawar@cl.cam.ac.uk"&gt;Anuj.Dawar@cl.cam.ac.uk&amp;lt;mailto:Anuj.Dawar@cl.cam.ac.uk&amp;gt;&lt;/a&gt;&lt;/p&gt;
</description></item><item><title>Theory and Practice of Differential Privacy 2019</title><link>https://blog.gautamkamath.com/2019/06/20/theory-and-practice-of-differential-privacy-2019/</link><guid isPermaLink="true">https://blog.gautamkamath.com/2019/06/20/theory-and-practice-of-differential-privacy-2019/</guid><pubDate>Thu, 20 Jun 2019 18:05:19 +0000</pubDate><dc:creator>Gautam Kamath</dc:creator><description>
&lt;p class="wp-block-paragraph"&gt;While I&amp;#8217;m a relative newcomer to differential privacy (my &lt;a href="https://arxiv.org/abs/1703.10127"&gt;first paper&lt;/a&gt; on it was only in 2017), I&amp;#8217;ve found the community to be a pleasure to interact with: paradoxically, simultaneously tight-knit yet highly welcoming to newcomers. I partially credit this culture to the number of workshops and programs which bring people together, including, but not limited to, &lt;a href="https://www.birs.ca/events/2018/5-day-workshops/18w5189"&gt;a BIRS workshop&lt;/a&gt;, &lt;a href="https://privacytools.seas.harvard.edu/"&gt;the Privacy Tools project at Harvard&lt;/a&gt;, &lt;a href="https://simons.berkeley.edu/programs/privacy2019"&gt;a semester at the Simons Institute&lt;/a&gt;, &lt;a href="https://shonan.nii.ac.jp/seminars/164/"&gt;the forthcoming Shonan workshop&lt;/a&gt;, and &lt;a href="https://tpdp.cse.buffalo.edu/"&gt;the Theory and Practice of Differential Privacy (TPDP) Workshop&lt;/a&gt;.&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;I&amp;#8217;m writing this post to draw attention to the imminent deadline of &lt;a href="https://tpdp.cse.buffalo.edu/2019/"&gt;TPDP 2019&lt;/a&gt;, co-located with CCS 2019 in London. I&amp;#8217;ll spare you the full details (click the link for more information), but most pressing is the deadline tomorrow, June 21, 2019, anywhere on Earth (let me know if this presents hardship for you, and I can pass concerns on to the chair). Essentially anything related to the theory or practice of differential privacy is welcome. Submissions are limited to four pages in length and are lightly refereed, based on originality, relevance, interest, and clarity. There are no published proceedings, and previously published results are welcome. If you&amp;#8217;ve been looking to get to know the community, consider either submitting or attending the workshop! &lt;/p&gt;
</description></item><item><title>Hello World!</title><link>https://blog.gautamkamath.com/2019/06/17/hello-world/</link><guid isPermaLink="true">https://blog.gautamkamath.com/2019/06/17/hello-world/</guid><pubDate>Mon, 17 Jun 2019 23:42:13 +0000</pubDate><dc:creator>Gautam Kamath</dc:creator><description>
&lt;p class="wp-block-paragraph"&gt;Welcome to my blog! My name is Gautam Kamath, and I just started as an assistant professor in computer science at the University of Waterloo (for more info, see &lt;a href="https://blog.gautamkamath.com/about/"&gt;About&lt;/a&gt;).&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;This blog will, broadly speaking, be about topics relevant to those interested in the the theory of computer science, statistics, and machine learning. Posts will range from technical, to informational, to meta (read: basically whatever I want to write about, but I&amp;#8217;ll do my best to keep it topical). Stay tuned!&lt;/p&gt;



&lt;p class="wp-block-paragraph"&gt;The unofficial theme song of this blog is &lt;a href="https://www.youtube.com/watch?v=8Ir-zFC9nFE"&gt;&amp;#8220;Mathematics&amp;#8221; by Mos Def&lt;/a&gt;.&lt;/p&gt;
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