For Spring 2016 the usual time for the meetings will be Fridays at 12pm in 417 Mudd (DSI). Abstracts for talks are given below the talk schedule.

Sepehr Asadi

University of Pennsylvania

**Abstract:**
We consider the problem of approximating a maximum matching in dynamic graph streams where the stream may include both edge insertions and deletions. Our main result is a resolution of the space complexity of linear sketches for approximating the maximum matching in this model.

Specifically, we show that for any $\eps > 0$, there exists a single-pass streaming algorithm, which only maintains a linear sketch of size roughly $n^{2-3\eps}$ bits and recovers an $n^\epsilon$-approximate maximum matching in dynamic graph streams, where $n$ is the number of vertices in the graph. We complement this result with the following lower bound result: any linear sketch for approximating the maximum matching to within a factor of $n^\eps$ has to be of size at least $n^{2-3\eps -o(1)}$ bits.

This is based on joint work with Sanjeev Khanna, Yang Li, and Grigory Yaroslavtsev.

Li-Yang Tan

Toyota Technological Institute at Chicago

**Abstract:**
The well-known method of random restrictions (Subbotovskaya, Furst--Saxe--Sipser, Yao, Håstad) lies at the heart of numerous landmark lower bounds in Boolean circuit complexity. I will present three recent circuit lower bounds that are obtained via random projections, a generalization of random restrictions:

1. We prove an average-case depth hierarchy theorem for Boolean circuits, an average-case extension of the classical worst-case depth hierarchy theorems of Sipser, Yao, and Håstad. This resolves a main open problem in Håstad’s 1986 Ph.D. thesis. Via a classical connection between Boolean circuits and structural complexity, our circuit lower bound implies that the polynomial hierarchy is infinite relative to a random oracle, resolving a similarly longstanding conjecture of Håstad, Cai, and Babai from the 1980s.

2. We prove the first super-polynomial lower bounds against the "depth-d Frege" propositional proof system for d = \Theta(\sqrt{\log n}). Previous work of Pitassi--Beame--Impagliazzo (1993) and Krajicek--Pudlak--Woods (1995) gave super-polynomial lower bounds against depth-d Frege for d = \Theta(\log log n).

3. We obtain near-optimal small-depth circuit lower bounds for the distance-k connectivity problem: given an n-node graph G and vertices s and t, is there an s-t path of length at most k in G? Our n^{k^{1/d}/d} lower bound almost matches the folklore n^{k^{2/d}} upper bound based on Savitch's "repeated squaring" algorithm; improving our lower bound even to n^{k^{\Omega(1/d)}} would separate P from NC1.

Based on joint works with Xi Chen, Igor Oliveira, Toniann Pitassi, Benjamin Rossman, and Rocco Servedio.

Ilias Diakonikolas

University of Southern California

**Abstract:**
We study problems in distribution property testing: Given sample access to one or more unknown discrete distributions, we want to determine whether they have some global property or are $\eps$-far from having the property in $\ell_1$ distance. We provide a simple and general approach to obtain upper bounds in this setting, by reducing $\ell_1$-testing to $\ell_2$-testing. Our reduction yields optimal $\ell_1$-testers, by using a standard $\ell_2$-tester as a black-box.

Using our framework, we obtain sample-optimal and computationally efficient estimators for a wide variety of $\ell_1$ distribution testing problems, including the following: identity testing to a fixed distribution, closeness testing between two unknown distributions (with equal/unequal sample sizes), independence testing (in any number of dimensions), closeness testing for collections of distributions, and testing $k$-flatness. For several of these problems, we give the first optimal tester in the literature. Moreover, our estimators are significantly simpler to state and analyze compared to previous approaches.

As our second main contribution, we provide a direct general approach for proving distribution testing lower bounds, by bounding the mutual information. Our lower bound approach is not restricted to symmetric properties, and we use it to prove tight lower bounds for all the aforementioned problems.

Joint work with Daniel Kane.

Mark Bun

Harvard University

**Abstract:**
The line of work on differential privacy is aimed at enabling rich statistical analyses on data while provably protecting individual-level privacy. The last decade of research has shown that, at least in principle, a number of fundamental statistical tasks are compatible with differential privacy. However, privacy-preserving analyses often require additional complexity over their non-private counterparts: for instance, in terms of the number of data samples one needs to collect in order to get accurate results.

In this talk, we will examine the price of privacy for several basic tasks involving one-dimensional threshold functions. We give the first non-trivial lower bounds on the sample complexity of performing these tasks with (approximate) differential privacy. Surprisingly, our techniques seem closely related to classic results in distributed computing, and bolster connections between privacy and combinatorics.

Joint work with Kobbi Nissim, Uri Stemmer, and Salil Vadhan.

Yaron Singer

Harvard University

**Abstract:**
As we grow highly dependent on data for making predictions, we translate these predictions into models that help us make informed decisions. But how do the guarantees we have on predictions translate to guarantees on decisions? In many cases, we learn models from sampled data and then aim to use these models to make decisions. This intuitive approach turns out to have non-trivial limitations. In some cases, despite having access to large data sets, the current frameworks we have for learnability do not suffice to guarantee desirable outcomes. In other cases, the learning techniques we have introduce estimation errors which can result in poor outcomes and stark impossibility results. In this talk we will formalize some of these ideas using convex and combinatorial optimization and discuss some possibility and impossibility results of this agenda.

Aaron Bernstein

Columbia University

**Abstract:**
Computing shortest paths in a graph is one of the fundamental problems of graph algorithms. The goal of dynamic single source shortest paths (SSSP) is to maintain a shortest path tree from a fixed source s as the edges of the graph change over time. The most general case is the fully dynamic one, where each adversarial update inserts or deletes an arbitrary edge. The trivial algorithm is to recompute SSSP after ever update in O(m) time. For the fully dynamic case, no non-trivial algorithm is known.

We can, however, improve upon the trivial algorithm by restricting the update sequence to be partially dynamic: only insertions, or only deletions. For example, in the only-insertions case, we start with an empty graph and the adversary inserts one edge at at a time until we end with some final graph with m edges; the goal is to maintain SSSP throughout this entire sequence of insertions. The trivial algorithm takes O(m) time per update, but as far back as 1981 Even and Shiloach showed how to achieve O(n) time per update. There are conditional lower bounds showing that this O(n) is optimal.

In 2011 Bernstein and Roditty showed that with a (1+eps) approximation we can break through this lower bound: they achieved ~O(n^2/m) update time. This was successively improved, with the state of the art achieving ~O(m^{1/\sqrt{log(n)}}) = m^o(1) update time. All of these algorithms, however, are randomized. Moreover, the randomization forces them to adopt a weaker model with a non-adaptive adversary, which makes these algorithms unsuitable to many settings.

We present a (1+eps)-approximate deterministic algorithm with update time ~O(n^2/m). This does not match the best randomized algorithm, but it is the first deterministic algorithm to beyond O(n). It is also much simpler than all of the above randomized algorithms. Our main tool is an entirely new deterministic sparsification technique, which we believe could be applicable to many other deterministic shortest path algorithms.

Accepted to STOC 2016. Joint work with Shiri Chechik.

Dimitris Paparas

Columbia University

**Abstract:**
We consider the problem of designing Revenue Maximizing Bayesian Auctions in both the Deterministic and the Randomized setting.

In the main part of the talk, we will focus on a Unit-demand buyer. For the Deterministic setting, we obtain a polynomial time algorithm for the case that the valuations of the buyer are drawn from a discrete product distribution of support size 2. On the negative side, we prove that the problem is NP-complete in the general case. For the Randomized setting, we study the menu-size complexity of the problem, presenting cases were a small menu suffices as well as cases that require an exponentially large menu. Regarding the computational complexity of the problem, we prove that the problem is #P-hard.

In the remaining part, we will focus on an Additive buyer and briefly discuss some recent results in this setting.

Based on joint work with Xi Chen, Ilias Diakonikolas, George Matikas, Anthi Orfanou, Xiaorui Sun, and Mihalis Yannakakis.

Jing Chen

Stony Brook University

**Abstract:**
Interactive proofs model a world where a verifier delegates computation to
an untrustworthy prover, verifying the prover's claims before accepting
them. These proofs have applications to delegation of computation,
probabilistically checkable proofs, crowdsourcing, and more.

In some of these applications, the verifier may pay the prover based on the quality of his work. Rational proofs, introduced by Azar and Micali (2012), are an interactive proof model in which the prover is rational rather than untrustworthy---he may lie, but only to increase his payment. This allows the verifier to leverage the greed of the prover to obtain better protocols: while rational proofs are no more powerful than interactive proofs, the protocols are simpler and more efficient. Azar and Micali posed as an open problem whether multiple provers are more powerful than one for rational proofs.

We provide a model that extends rational proofs to allow multiple provers. In this model, a verifier can cross-check the answers received by asking several provers. The verifier can pay the provers according to the quality of their work, incentivizing them to provide correct information.

We analyze rational proofs with multiple provers from a complexity-theoretic point of view. We fully characterize this model by giving tight upper and lower bounds on its power. On the way, we resolve Azar and Micali's open problem in the affirmative, showing that multiple rational provers are strictly more powerful than one (under standard complexity-theoretic assumptions). We further show that the full power of rational proofs with multiple provers can be achieved using only two provers and five rounds of interaction. Finally, we consider more demanding models where the verifier wants the provers' payment to decrease significantly when they are lying, and fully characterize the power of the model when the payment gap must be noticeable (i.e., at least 1/p where p is a polynomial).

Periklis Papakonstantinou

Rutgers University

**Abstract:**
In 1989 it was shown by Allender and Hertrampf that every circuit of depth d and gates AND,OR,NOT, and MODp, for prime p, can be reduced to a depth 3 circuit of size 2^( (logn)^O(d) ). The question about MODm, for composite m, was handled a year later by Yao (randomized circuits) and by Beigel and Tarui (derandomizing Yao's), but with triple-exponential size construction, i.e. 2^((logn)^2^O(d)).

We obtain for composites the same asymptotic result as for primes.

I will outline this exponentially-improved depth reduction and discuss the most significant among its implications. For example, one consequence is an (almost) exponentially better depth lower bound (from loglogn to logn/loglogn) in William's program for NEXP. If time permits I may also discuss various consequences in the interactive compression settings.

This is joint work with Shiteng Chen.

Xiaorui Sun

Columbia University

**Abstract:**
We study the graph isomorphism problem in property testing framework: Given edge queries to two unknown graphs, how many edge queries are required to distinguish between the case that two graphs are isomorphic, and the case that they are \epsilon-far from being isomorphic.

Fischer and Matsliah (SICOMP 2008) showed that the number of queries necessary to (approximately) test isomorphism is at least Omega(n) and at most O(n^(5/4)). We close this gap by showing an algorithm that makes n*2^O(sqrt(log n)) queries.

Joint work with Krzysztof Onak (IBM TJ Watson Research Center).

Greg Valiant

Stanford University

**Abstract:**
We discuss two problems related to the general challenge of making
accurate inferences about a complex distribution, in the regime in
which the amount of data (i.e the sample size) is too small for the
empirical distribution of the samples to be an accurate representation
of the underlying distribution. The first problem we consider is the
following basic learning task: given independent draws from an unknown
distribution over a discrete support, output an approximation of the
distribution that is as accurate as possible in L1 distance (ie total
variation distance). Perhaps surprisingly, it is
often possible to "de-noise" the empirical distribution of the samples
to return an approximation of the true distribution that is
significantly more accurate than the empirical distribution, without
relying on any prior assumptions on the distribution. We present an
instance optimal learning algorithm which optimally performs this
de-noising for every distribution for which such a de-noising is
possible. One curious implication of our techniques is an algorithm
for accurately estimating the number of new domain elements that would
be seen given a new larger sample, of size up to n*log n.
(Extrapolation beyond this sample size is provable information
theoretically impossible, without additional assumptions on the
distribution.) While these results are applicable generally, we
highlight an adaptation of this general approach to some problems in
genomics (e.g. quantifying the number of unobserved protein coding
variants).

The second problem we discuss is the problem of recovering a low-rank approximation to a matrix of probabilities P, given access to an observed matrix of "counts" obtained via independent samples from the distribution defined by P. This problem can be viewed as a generalization of "community detection", and is relevant to several recent machine learning efforts, including the work on constructing "word embeddings", and recommendation systems derived from sparse data.

This talk is based on three papers, which are joint works with Paul Valiant, with Paul Valiant and James Zou, and with Qingqing Huang, Sham Kakade, and Weihao Kong.

Noga Ron-Zewi

Princeton University and Rutgers University

**Abstract:**
We show an efficient, deterministic interactive coding scheme that simulates any interactive protocol under random errors with nearly optimal parameters.

Specifically, our coding scheme achieves a communication rate of 1 – õ(\sqrt{epsilon}) and a failure probability of exp(-n/ \log n), where n is the protocol length and each bit is flipped independently with constant probability epsilon. Prior to our work, all nontrivial deterministic schemes (either efficient or not) had a rate bounded away from 1. Furthermore, the best failure probability achievable by an efficient deterministic scheme with constant rate was only quasi-polynomial, i.e., of the form exp(-polylog(n)) (Braverman, ITCS 2012). The rate of our scheme is essentially optimal (up to poly-logarithmic factors) by a result of Kol and Raz (STOC 2013).

Joint work with Ran Gelles, Bernhard Haeupler, Gillat Kol and Avi Wigderson.

Contact Alexandr Andoni if you want to

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