A carbon-free dissemination of ideas across the globe.

### 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).

### TCS+ talk: Wednesday, November 25 — The Coin Problem with Applications to Data Streams

The next TCS+ talk will take place this coming Wednesday, November 25th at 1:00 PM Eastern Time (10:00 AM Pacific Time, 19:00 Central European Time, 18:00 UTC). Sumegha Garg from Harvard University will speak about “The Coin Problem with Applications to Data Streams” (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: Consider the problem of computing the majority of a stream of $n$ i.i.d. uniformly random bits. This problem, known as the coin problem, is central to a number of counting problems in different data stream models. We show that any streaming algorithm for solving this problem with large constant advantage (over the uniform distribution) must use $\Omega(\log n)$ bits of space. Previously, it was known that computing the majority on every input with a constant probability takes $\Omega(\log n)$ space. We extend our lower bound to proving tight lower bounds for solving multiple, randomly interleaved copies of the coin problem, as well as for solving the OR of multiple copies of a variant of the coin problem. Our proofs involve new measures of information complexity that are well-suited for data streams.

We use these lower bounds to obtain a number of new results for data streams. In each case there is an underlying $d$-dimensional vector $x$ with additive updates to its coordinates given in a stream of length $m$. The input streams arising from our coin lower bound have nice distributional properties, and consequently for many problems for which we only had lower bounds in general turnstile streams, we now obtain the same lower bounds in more natural models, such as the bounded deletion model, in which $\|x\|_2$ never drops by a constant fraction of what it was earlier, or in the random order model, in which the updates are ordered randomly.

Based on joint work with Mark Braverman and David P. Woodruff.

### TCS+ talk: Wednesday, November 11 — Shuai Shao, UW-Madison

The next TCS+ talk will take place this coming Wednesday, November 11th at 1:00 PM Eastern Time (10:00 AM Pacific Time, 19:00 Central European Time, 18:00 UTC). Shuai Shao from UW-Madison will speak about “A Dichotomy for Real Boolean Holant Problems” (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: In this talk, we present a complexity dichotomy for Holant problems on the boolean domain with arbitrary sets of real-valued constraint functions. These constraint functions need not be symmetric nor do we assume any auxiliary functions as in previous results. It is proved that for every set $F$ of real-valued constraint functions, $\text{Holant}(F)$ is either $P$-time computable or $\#P$-hard. The classification has an explicit criterion. This is a culmination of much research on a decade-long classification program for Holant problems, and it uses previous results and techniques from many researchers. However, as it turned out, the journey to the present theorem has been arduous. Some particularly intriguing concrete functions f6, f8 and their associated families with extraordinary closure properties related to Bell states in quantum information theory play an important role in this proof.

Based on joint work with Jin-Yi Cai.

### TCS+ talk: Wednesday, November 4 — Shalev Ben-David, University of Waterloo

The next TCS+ talk will take place this coming Wednesday, November 4th at 1:00 PM Eastern Time (10:00 AM Pacific Time, 19:00 Central European Time, 18:00 UTC). Shalev Ben-David from University of Waterloo will speak about “Forecasting Algorithms, Minimax Theorems, and Randomized Lower Bounds” (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 link to the YouTube livestream will also be posted on our website on the day of the talk, so people who did not sign up will still be able to watch the talk live.) 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: I will present a new approach to randomized lower bounds, particularly in the setting where we wish to give a fine-grained analysis of randomized algorithms that achieve small bias. The approach is as follows: instead of considering ordinary randomized algorithms which give an output in $\{0,1\}$ and may err, we switch models to look at “forecasting” randomized algorithms which output a confidence in $[0,1]$ for whether they think the answer is 1. When scored by a proper scoring rule, the performance of the best forecasting algorithm is closely related to the bias of the best (ordinary) randomized algorithm, but is more amenable to analysis.
As an application, I’ll present a new minimax theorem for randomized algorithms, which can be viewed as a strengthening of Yao’s minimax theorem. Yao’s minimax theorem guarantees the existence of a hard distribution for a function $f$ such that solving $f$ against this distribution (to a desired error level) is as hard as solving $f$ in the worst case (to that same error level). However, the hard distribution provided by Yao’s theorem depends on the chosen error level. Our minimax theorem removes this dependence, giving a distribution which certifies the hardness of $f$ against all bias levels at once. In recent work, we used this minimax theorem to give a tight composition theorem for randomized query complexity.

Based on joint work with Eric Blais.

### TCS+ talk: Wednesday, October 28 — Omar Montasser, TTIC

The next TCS+ talk will take place this coming Wednesday, October 28th at 1:00 PM Eastern Time (10:00 AM Pacific Time, 18:00 Central European Time, 17:00 UTC). Omar Montasser from TTIC will speak about “Adversarially Robust Learnability: Characterization and Reductions” (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: We study the question of learning an adversarially robust predictor from uncorrupted samples. We show that any VC class $H$ is robustly PAC learnable, but we also show that such learning must sometimes be improper (i.e. use predictors from outside the class), as some VC classes are not robustly properly learnable. In particular, the popular robust empirical risk minimization approach (also known as adversarial training), which is proper, cannot robustly learn all VC classes. After establishing learnability, we turn to ask whether having a tractable non-robust learning algorithm is sufficient for tractable robust learnability and give a reduction algorithm for robustly learning any hypothesis class $H$ using a non-robust PAC learner for $H$, with nearly-optimal oracle complexity.
This is based on joint work with Steve Hanneke and Nati Srebro, available at https://arxiv.org/abs/1902.04217.

### TCS+ talk: Wednesday, October 21 — Aayush Jain, UCLA

The next TCS+ talk will take place this coming Wednesday, October 21th at 1:00 PM Eastern Time (10:00 AM Pacific Time, 19:00 Central European Time, 17:00 UTC). Aayush Jain from UCLA will speak about “Indistinguishability Obfuscation from Well-Founded Assumptions” (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 link to the YouTube livestream will also be posted on our website on the day of the talk, so people who did not sign up will still be able to watch the talk live.) 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 a construction of an indistinguishability obfuscation scheme, whose security rests on the subexponential hardness of four well-founded assumptions. We show the existence of an indistinguishability Obfuscation scheme for all circuits assuming sub-exponential security of the following assumptions:

• The Learning with Errors (LWE) assumption with arbitrarily small subexponential modulus-to-noise ratio,
• The SXDH assumption with respect to bilinear groups of prime order $p$,
• Existence of a Boolean Pseudorandom Generator (PRG) in $\mathsf{NC}^0$ with arbitrary polynomial stretch, that is, mapping $n$ bits to $n^{1+\tau}$ bits, for any constant \tau>0.
• The Learning Parity with Noise (LPN) assumption over $\mathbb{Z}_p$ with error-rate $\ell^{-\delta}$, where $\ell$ is the dimension of the secret and $\delta>0$ is an arbitrarily small constant.
Further, assuming only polynomial security of these assumptions, there exists a compact public-key functional encryption scheme for all circuits.

The main technical novelty is the introduction and construction of a cryptographic pseudorandom generator that we call a Structured-Seed PRG (sPRG), assuming LPN over $\mathbb{Z}_p$ and PRGs in $\mathsf{NC}^0$. During the talk, I will discuss how structured seed PRGs have evolved from different notions of novel pseudorandom generators proposed in the past few years, and how an interplay between different areas of theoretical computer science played a major role in providing valuable insights leading to this work. Time permitting, I will go into the details of how to construct sPRGs.

Joint work with Huijia (Rachel) Lin and Amit Sahai

### TCS+ talk: Wednesday, October 14 — Jayadev Acharya, Cornell University

The next TCS+ talk will take place this coming Wednesday, October 14th at 1:00 PM Eastern Time (10:00 AM Pacific Time, 19:00 Central European Time, 17:00 UTC). Jayadev Acharya from Cornell University will speak about “Distributed Statistical Inference under Local Information Constraints ” (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 link to the YouTube livestream will also be posted on our website on the day of the talk, so people who did not sign up will still be able to watch the talk live.) 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 consider statistical inference tasks in a distributed setting where access to data samples is subjected to strict “local constraints,” through a unified framework that captures communication limitations and (local) privacy constraints as special cases. We study estimation (learning) and goodness-of-fit (testing) for both discrete and high-dimensional distributions. Our goal is to understand how the sample complexity increases under the information constraints.

In this talk we will provide an overview of this field and a sample of some of our results. We will discuss the role of (public) randomness and interactivity in information-constrained inference, and make a case for thinking about randomness and interactivity as resources.

The work is part of a long-term ongoing collaboration with Clément Canonne (IBM Research) and Himanshu Tyagi (IISc), and includes works done with Cody Freitag (Cornell), Yanjun Han (Stanford), Yuhan Liu (Cornell), and Ziteng Sun (Cornell).

### TCS+ talk: Wednesday, October 7 — Susanna F. de Rezende, Mathematical Institute of the Czech Academy of Sciences

The next TCS+ talk will take place this coming Wednesday, October 7th at 1:00 PM Eastern Time (10:00 AM Pacific Time, 19:00 Central European Time, 17:00 UTC). Susanna F. de Rezende from Mathematical Institute of the Czech Academy of Sciences will speak about “Lifting with Simple Gadgets and Applications to Circuit and Proof Complexity” (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 link to the YouTube livestream will also be posted on our website on the day of the talk, so people who did not sign up will still be able to watch the talk live.) 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: Lifting theorems in complexity theory are a method of transferring lower bounds in a weak computational model into lower bounds for a more powerful computational model, via function composition. There has been an explosion of lifting theorems in the last ten years, essentially reducing communication lower bounds to query complexity lower bounds. These theorems only hold for composition with very specific “gadgets” such as indexing and inner product.

In this talk, we will present a generalization of the theorem lifting Nullstellensatz degree to monotone span program size by Pitassi and Robere (2018) so that it works for any gadget with high enough rank, in particular, for useful gadgets such as equality and greater-than. We will then explain how to apply our generalized theorem to solve three open problems:
– We present the first result that demonstrates a separation in proof power for cutting planes with unbounded versus polynomially bounded coefficients. Specifically, we exhibit CNF formulas that can be refuted in quadratic length and constant line space in cutting planes with unbounded coefficients, but for which there are no refutations in subexponential length and subpolynomial line space if coefficients are restricted to be of polynomial magnitude.
– We give the strongest separation to-date between monotone Boolean formulas and monotone Boolean circuits. Namely, we show that the classical GEN problem, which has polynomial-size monotone Boolean circuits, requires monotone Boolean formulas of size $2^{\Omega(n / \textrm{polylog} n)}$.
– We give the first explicit separation between monotone Boolean formulas and monotone real formulas. Namely, we give an explicit family of functions that can be computed with monotone real formulas of nearly linear size but require monotone Boolean formulas of exponential size. Previously only a non-explicit separation was known.

This talk is based on joint work with Or Meir, Jakob Nordström, Toniann Pitassi, Robert Robere, and Marc Vinyals, available at https://arxiv.org/abs/2001.02144

### TCS+ talk: Wednesday, September 30 — Alex Wein, NYU

The next TCS+ talk will take place this coming Wednesday, September 30th at 1:00 PM Eastern Time (10:00 AM Pacific Time, 19:00 Central European Time, 17:00 UTC). Alex Wein from NYU will speak about “Low-Degree Hardness of Random Optimization Problems” (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 link to the YouTube livestream will also be posted on our website on the day of the talk, so people who did not sign up will still be able to watch the talk live.) 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 high-dimensional statistical problems (including planted clique, sparse PCA, community detection, etc.), the class of “low-degree polynomial algorithms” captures many leading algorithmic paradigms such as spectral methods, approximate message passing, and local algorithms on sparse graphs. As such, lower bounds against low-degree algorithms constitute concrete evidence for average-case hardness of statistical problems. This method has been widely successful at explaining and predicting statistical-to-computational gaps in these settings.
While prior work has understood the power of low-degree algorithms for problems with a “planted” signal, we consider here the setting of “random optimization problems” (with no planted signal), including the problem of finding a large independent set in a random graph, as well as the problem of optimizing the Hamiltonian of mean-field spin glass models. I will define low-degree algorithms in this setting, argue that they capture the best known algorithms, and explain new proof techniques for giving lower bounds against low-degree algorithms in this setting. The proof involves a variant of the so-called “overlap gap property”, which is a structural property of the solution space.

Based on joint work with David Gamarnik and Aukosh Jagannath, available at arXiv:2004.12063.

### TCS+ talk: Wednesday, September 23 — Fotis Iliopoulos, Princeton and IAS

The next TCS+ talk will take place this coming Wednesday, September 23th at 1:00 PM Eastern Time (10:00 AM Pacific Time, 19:00 Central European Time, 17:00 UTC). Fotis Iliopoulos from Princeton and IAS will speak about “Stochastic Local Search and the Lovász Local Lemma” (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 link to the YouTube livestream will also be posted on our website on the day of the talk, so people who did not sign up will still be able to watch the talk live.) 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 Lovasz Local Lemma (LLL) is a powerful tool in probabilistic combinatorics which can be used to establish the existence of objects that satisfy certain properties. The breakthrough of Moser and Tardos (who recently received the Godel Prize for their work) and follow-up works revealed that the LLL has intimate connections with a class of stochastic local search algorithms for finding such desirable objects.

In this talk, I will survey this line of work through the perspective of recent unifying results, and also talk about recent applications to solving pseudo-random constraint satisfaction problems.