A carbon-free dissemination of ideas across the globe.

### TCS+ talk: Wednesday, June 9 — Pravesh Kothari (CMU) and Ankur Moitra (MIT)

The next TCS+ talk will take place this coming Wednesday, June 9th at 1:00 PM Eastern Time (10:00 AM Pacific Time, 19:00 Central European Time, 17:00 UTC). Pravesh Kothari  and Ankur Moitra from CMU and MIT will (jointly) speak about “Robustly Learning Mixtures of Gaussians” (abstract below).

Note that the seminar will be a bit longer than the usual: it’s a double feature!

You can reserve a spot as an individual or a group to join us live by signing up on the online form. Due to security concerns, registration is required to attend the interactive talk. (The recorded talk will also be posted on our website afterwards, so people who did not sign up will still be able to watch the talk) As usual, for more information about the TCS+ online seminar series and the upcoming talks, or to suggest a possible topic or speaker, please see the website.

Abstract: For a while now the problem of robustly learning a high-dimensional mixture of Gaussians has had a target on its back. The first works in algorithmic robust statistics gave provably robust algorithms for learning a single Gaussian. Since then there has been steady progress, including algorithms for robustly learning mixtures of spherical Gaussians, mixtures of Gaussians under separation conditions, and arbitrary mixtures of two Gaussians. In this talk we will discuss two recent works that essentially resolve the general problem. There are important differences in their techniques, setup, and overall quantitative guarantees, which we will discuss.

The talk will cover the following independent works:

• Liu, Moitra, “Settling the Robust Learnability of Mixtures of Gaussians”
• Bakshi, Diakonikolas, Jia, Kane, Kothari, Vempala, “Robustly Learning Mixtures of $k$ Arbitrary Gaussians”

### TCS+ talk: Wednesday, May 26 — Kira Goldner, Columbia University

The next TCS+ talk will take place this coming Wednesday, May 26th at 1:00 PM Eastern Time (10:00 AM Pacific Time, 19:00 Central European Time, 17:00 UTC). Kira Goldner from Columbia University will speak about “An Overview of Using Mechanism Design for Social Good” (abstract below).

You can reserve a spot as an individual or a group to join us live by signing up on the online form. Due to security concerns, registration is required to attend the interactive talk. (The recorded talk will also be posted on our website afterwards, so people who did not sign up will still be able to watch the talk) As usual, for more information about the TCS+ online seminar series and the upcoming talks, or to suggest a possible topic or speaker, please see the website.

Abstract: In order to accurately predict an algorithm’s outcome and quality when it interacts with participants who have a stake in the outcome, we must design it to be robust to strategic manipulation. This is the subject of algorithmic mechanism design, which borrows ideas from game theory and economics to design robust algorithms. In this talk, I will show how results from the theoretical foundations of algorithmic mechanism design can be used to solve problems of societal concern.

I will overview recent work in this area in many different applications — housing, labor markets, carbon license allocations, health insurance markets, and more — as well as discuss open problems and directions ripe for tools from both mechanism design and general TCS.

### TCS+ talk: Wednesday, May 12 — Santhoshini Velusamy, Harvard University

The next TCS+ talk will take place this coming Wednesday, May 12th at 1:00 PM Eastern Time (10:00 AM Pacific Time, 19:00 Central European Time, 17:00 UTC). Santhoshini Velusamy from Harvard University will speak about “Classification of the approximability of all finite Max-CSPs in the dynamic streaming setting” (abstract below).

You can reserve a spot as an individual or a group to join us live by signing up on the online form. Due to security concerns, registration is required to attend the interactive talk. (The recorded talk will also be posted on our website afterwards, so people who did not sign up will still be able to watch the talk)

As usual, for more information about the TCS+ online seminar series and the upcoming talks, or to suggest a possible topic or speaker, please see the website.

Abstract: A maximum constraint satisfaction problem, Max-CSP(F), is specified by a finite family of constraints F, where each constraint is of arity k. An instance of the problem on n variables is given by m applications of constraints from F to length-k subsequences of the n variables, and the goal is to find an assignment to the n variables that satisfies the maximum number of constraints. The class of Max-CSP(F) includes optimization problems such as Max-CUT, Max-DICUT, Max-3SAT, Max-q-Coloring, Unique Games, etc.

In this talk, I will present our recent dichotomy theorem on the approximability of Max-CSP(F) for every finite family F, in the single-pass dynamic streaming setting. In this setting, at each time step, a constraint is either added to or deleted from the stream. In the end, the streaming algorithm must estimate the maximum number of constraints that can be satisfied using space that is only polylogarithmic in n. No background in streaming algorithms or constraint satisfaction problems will be needed to enjoy this talk!

The talk will be based on this paper, and this paper with Chi-Ning Chou, Alexander Golovnev, and Madhu Sudan.

### TCS+ talk: Wednesday, April 28 — Ronen Eldan, Weizmann Institute

The next TCS+ talk will take place this coming Wednesday, April 28th at 1:00 PM Eastern Time (10:00 AM Pacific Time, 19:00 Central European Time, 17:00 UTC). Ronen Eldan from the Weizmann Institute will speak about “Localization, stochastic localization, and Chen’s recent breakthrough on the Kannan-Lovasz-Simonivits conjecture” (abstract below).

You can reserve a spot as an individual or a group to join us live by signing up on the online form. Due to security concerns, registration is required to attend the interactive talk. (The recorded talk will also be posted on our website afterwards, so people who did not sign up will still be able to watch the talk) As usual, for more information about the TCS+ online seminar series and the upcoming talks, or to suggest a possible topic or speaker, please see the website.

Abstract: The Kannan-Lovasz and Simonovits (KLS) conjecture considers the following isoperimetric problem on high-dimensional convex bodies: Given a convex body $K$, consider the optimal way to partition it into two pieces of equal volume so as to minimize their interface. Is it true that up to a universal constant, the minimal partition is attained via a hyperplane cut? Roughly speaking, this question can be thought of as asking “to what extent is a convex set a good expander”?

In analogy to expander graphs, such lower bounds on the capacity would imply bounds on mixing times of Markov chains associated with the convex set, and so this question has direct implications on the complexity of many computational problems on convex sets. Moreover, it was shown that a positive answer would imply Bourgain’s slicing conjecture.

Very recently, Yuansi Chen obtained a striking breakthrough, nearly solving this conjecture. In this talk, we will overview some of the central ideas used in the proof. We will start with the classical concept of “localization” (a very useful tool to prove concentration inequalities) and its extension, stochastic localization – the main technique used in the proof.

### TCS+ talk: Wednesday, April 14 — Andrea Lincoln, UC Berkeley

The next TCS+ talk will take place this coming Wednesday, April 14th at 1:00 PM Eastern Time (10:00 AM Pacific Time, 19:00 Central European Time, 17:00 UTC). Andrea Lincoln from UC Berkeley will speak about “New Techniques for Proving Fine-Grained Average-Case Hardness” (abstract below).

You can reserve a spot as an individual or a group to join us live by signing up on the online form. Due to security concerns, registration is required to attend the interactive talk. (The recorded talk will also be posted on our website afterwards, so people who did not sign up will still be able to watch the talk)

As usual, for more information about the TCS+ online seminar series and the upcoming talks, or to suggest a possible topic or speaker, please see the website.

Abstract: In this talk I will cover a new technique for worst-case to average-case reductions. There are two primary concepts introduced in this talk: “factored” problems and a framework for worst-case to average-case fine-grained (WCtoACFG) self reductions.

We will define new versions of OV, kSUM and zero-k-clique that are both worst-case and average-case fine-grained hard assuming the core hypotheses of fine-grained complexity. We then use these as a basis for fine-grained hardness and average-case hardness of other problems. Our hard factored problems are also simple enough that we can reduce them to many other problems, e.g. to edit distance, k-LCS and versions of Max-Flow. We further consider counting variants of the factored problems and give WCtoACFG reductions for them for a natural distribution.

To show hardness for these factored problems we formalize the framework of [Boix-Adsera et al. 2019] that was used to give a WCtoACFG reduction for counting k-cliques. We define an explicit property of problems such that if a problem has that property one can use the framework on the problem to get a WCtoACFG self reduction. In total these factored problems and the framework together give tight fine-grained average-case hardness for various problems including the counting variant of regular expression matching.

Based on joint work with Mina Dalirrooyfard and Virginia Vassilevska Williams.

### TCS+ talk: Wednesday, March 31 — Jasper Lee, Brown University

The next TCS+ talk will take place this coming Wednesday, March 31th at 1:00 PM Eastern Time (10:00 AM Pacific Time, 19:00 Central European Time, 17:00 UTC). Jasper Lee from Brown University will speak about “Optimal Sub-Gaussian Mean Estimation in $\mathbb{R}$” (abstract below).

You can reserve a spot as an individual or a group to join us live by signing up on the online form. Due to security concerns, registration is required to attend the interactive talk. (The recorded talk will also be posted on our website afterwards, so people who did not sign up will still be able to watch the talk)

As usual, for more information about the TCS+ online seminar series and the upcoming talks, or to suggest a possible topic or speaker, please see the website.

Abstract: We revisit and settle a fundamental problem in statistics: given access to independent samples from a 1D random variable (with finite but unknown mean and variance), what is the best way to estimate the mean in the high probability regime, in terms of error convergence with respect to sample size? The conventional wisdom is to use the empirical mean as our estimate. However, it is known that the empirical mean can in fact have exponentially sub-optimal convergence for certain heavy-tailed distributions. On the other hand, the median-of-means estimator (invented and reinvented in various literature) does have sub-Gaussian convergence for all finite-variance distributions, albeit only in the big-O sense with a sub-optimal multiplicative constant. The natural remaining question then, is whether it is possible to bridge the gap, and have an estimator that has optimal convergence with the right constant for all finite-variance distributions.

In this talk, we answer the question affirmatively by giving an estimator that converges with the optimal constant inside the big-O, up to a 1+o(1) multiplicative factor. The estimator is also easy to compute. The convergence analysis involves deriving tail bounds using linear and convex-concave programming dualities, which may be of independent interest.

Based on joint work with Paul Valiant.

### TCS+ talk: Wednesday, March 17 — Avishay Tal, UC Berkeley

The next TCS+ talk will take place this coming Wednesday, March 17th at 1:00 PM Eastern Time (10:00 AM Pacific Time, 18:00 Central European Time, 17:00 UTC). Avishay Tal from UC Berkeley will speak about “Junta Distance Approximation with Sub-Exponential Queries” (abstract below).

You can reserve a spot as an individual or a group to join us live by signing up on the online form. Due to security concerns, registration is required to attend the interactive talk. (The recorded talk will also be posted on our website afterwards, so people who did not sign up will still be able to watch the talk.) As usual, for more information about the TCS+ online seminar series and the upcoming talks, or to suggest a possible topic or speaker, please see the website.

Abstract: Joint Work with Vishnu Iyer and Michael Whitmeyer.

A Boolean function $f\colon \{0,1\}^n \to \{0,1\}$ is called a $k$-junta if it depends only on $k$ out of the $n$ input bits. Junta testing is the task of distinguishing between $k$-juntas and functions that are far from $k$-juntas. A long line of work settled the query complexity of testing $k$-juntas, which is $O(k log(k))$ [Blais, STOC 2009; Saglam, FOCS 2018]. Suppose, however, that $f$ is not a perfect $k$-junta but rather correlated with a $k$-junta. How well can we estimate this correlation? This question was asked by De, Mossel, and Neeman [FOCS 2019], who gave an algorithm for the task making $\sim\exp(k)$ queries. We present an improved algorithm that makes $\sim\exp(\sqrt{k})$ many queries. Along the way, we also give an algorithm, making $\mathrm{poly}(k)$ queries, that provides “implicit oracle access” to the underlying $k$-junta. Our techniques are Fourier analytical and introduce the notion of “normalized influences” that might be of independent interest.

### TCS+ talk: Wednesday, March 3 — Steve Hanneke, TTIC

The next TCS+ talk will take place this coming Wednesday, March 3th at 1:00 PM Eastern Time (10:00 AM Pacific Time, 19:00 Central European Time, 18:00 UTC). Steve Hanneke from TTIC will speak about “A Theory of Universal Learning” (abstract below).

You can reserve a spot as an individual or a group to join us live by signing up on the online form. Due to security concerns, registration is required to attend the interactive talk. (The recorded talk will also be posted on our website afterwards, so people who did not sign up will still be able to watch the talk.) As usual, for more information about the TCS+ online seminar series and the upcoming talks, or to suggest a possible topic or speaker, please see the website.

Abstract: How quickly can a given class of concepts be learned from examples? It is common to measure the performance of a supervised machine learning algorithm by plotting its “learning curve”, that is, the decay of the error rate as a function of the number of training examples. However, the classical theoretical framework for understanding learnability, the PAC model of Vapnik-Chervonenkis and Valiant, does not explain the behavior of learning curves: the distribution-free PAC model of learning can only bound the upper envelope of the learning curves over all possible data distributions. This does not match the practice of machine learning, where the data source is typically fixed in any given scenario, while the learner may choose the number of training examples on the basis of factors such as computational resources and desired accuracy.

In this work, we study an alternative learning model that better captures such practical aspects of machine learning, but still gives rise to a complete theory of the learnable in the spirit of the PAC model. More precisely, we consider the problem of universal learning, which aims to understand the performance of learning algorithms on every data distribution, but without requiring uniformity over the distribution. The main result of this work is a remarkable trichotomy: there are only three possible rates of universal learning. More precisely, we show that the learning curves of any given concept class decay either at an exponential, linear, or arbitrarily slow rates. Moreover, each of these cases is completely characterized by appropriate combinatorial parameters, and we exhibit optimal learning algorithms that achieve the best possible rate in each case.

Joint work with Olivier Bousquet, Shay Moran, Ramon van Handel, and Amir Yehudayoff.

### TCS+ talk: Wednesday, February 17 — William Hoza, UT Austin

Welcome back for a new season of TCS+! Our talks are back to a fortnightly schedule, with an exciting slate of speakers ahead of us.

The first TCS+ talk will take place in two weeks, Wednesday, February 17th at 1:00 PM Eastern Time (10:00 AM Pacific Time, 19:00 Central European Time, 18:00 UTC). William Hoza from UT Austin will speak about “Fooling Constant-Depth Threshold Circuits” (abstract below). You can reserve a spot as an individual or a group to join us live by signing up on the online form. Due to security concerns, registration is required to attend the interactive talk. (The talk will be recorded and posted on our website and YouTube channel afterwards, so people who did not sign up will still be able to watch the talk.) As usual, for more information about the TCS+ online seminar series and the upcoming talks, or to suggest a possible topic or speaker, please see the website.

Abstract: We present the first non-trivial pseudorandom generator (PRG) for linear threshold (LTF) circuits of arbitrary constant depth and super-linear size. This PRG fools circuits with depth $d$ and $n^{1 + \delta}$ wires, where $\delta = \exp(-O(d))$, using seed length $O(n^{1 - \delta})$ and with error $\exp(-n^{\delta})$. This tightly matches the best known lower bounds for this circuit class. As a consequence of our result, all the known hardness for LTF circuits has now effectively been translated into pseudorandomness. This brings the extensive effort in the last decade to construct PRGs and deterministic circuit-analysis algorithms for this class to the point where any subsequent improvement would yield breakthrough lower bounds.

A key ingredient in our construction is a pseudorandom restriction procedure that has tiny failure probability, but simplifies the function to a non-natural “hybrid computational model” that combines decision trees and LTFs. As part of our proof we also construct an “extremely low-error” PRG for the class of functions computable by an arbitrary function of s linear threshold functions that can handle even the extreme setting of parameters $s = n/\mathrm{polylog}(n)$ and $\epsilon = \exp(-n/\mathrm{polylog}(n))$.

Joint work with Pooya Hatami, Avishay Tal, and Roei Tell.

### TCS+ talk: Wednesday, December 2 — Yang Liu, Stanford University

We hope you are all doing well and, in the case of our US audience, are slowly emerging from a comfortable Thanksgiving food-induced stupor! Our last TCS+ talk of the semester will take place this coming Wednesday, December 2nd at 1:00 PM Eastern Time (10:00 AM Pacific Time, 19:00 Central European Time, 18:00 UTC). Yang Liu from Stanford University will speak about “Faster Algorithms for Unit Maximum Flow” (abstract below).

You can reserve a spot as an individual or a group to join us live by signing up on the online form. Due to security concerns, registration is required to attend the interactive talk. As usual, for more information about the TCS+ online seminar series and the upcoming talks, or to suggest a possible topic or speaker, please see the website.

Abstract: The maximum flow problem is one of the most well-studied problems in combinatorial optimization, encompassing a broad range of cut, matching, and scheduling problems. Here we present a recent line of work obtaining provably faster algorithms for solving the maximum flow problem using interior point methods. In particular, we show how to solve the maximum flow problem in $m$-edge unit capacity graphs in time almost $m^{4/3}$, improving over the breakthrough $m^{10/7}$ time algorithm of Mądry.

This is based on joint work with Aaron Sidford (arxiv.org/abs/1910.14276, arxiv.org/abs/2003.08929).