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1. Faster Algorithms for High-Dimensional Robust Covariance Estimation.

   Cheng, Yu ; Diakonikolas, Ilias ; Ge, Rong ; Woodruff, David P. ( June 2019 , The 32’nd Annual Conference on Learning Theory (COLT 2019))

   We study the problem of estimating the covariance matrix of a high-dimensional distribution when a small constant fraction of the samples can be arbitrarily corrupted. Recent work gave the first polynomial time algorithms for this problem with near-optimal error guarantees for several natural structured distributions. Our main contribution is to develop faster algorithms for this problem whose running time nearly matches that of computing the empirical covariance. Given N = Ω(d^2/\eps^2) samples from a d-dimensional Gaussian distribution, an \eps-fraction of which may be arbitrarily corrupted, our algorithm runs in time O(d^{3.26}/ poly(\eps)) and approximates the unknown covariance matrix to optimal error up to a logarithmic factor. Previous robust algorithms with comparable error guarantees all have runtimes Ω(d^{2ω}) when \eps = Ω(1), where ω is the exponent of matrix multiplication. We also provide evidence that improving the running time of our algorithm may require new algorithmic techniques. 

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2. Resource-Efficient Common Randomness and Secret-Key Schemes

   Ghazi, Badih ; Jayram, T.S. ( January 2018 , Proceedings of the annual ACM-SIAM Symposium on Discrete Algorithms)

   We study common randomness where two parties have access to i.i.d. samples from a known random source, and wish to generate a shared random key using limited (or no) communication with the largest possible probability of agreement. This problem is at the core of secret key generation in cryptography, with connections to communication under uncertainty and locality sensitive hashing. We take the approach of treating correlated sources as a critical resource, and ask whether common randomness can be generated resource-efficiently. We consider two notable sources in this setup arising from correlated bits and correlated Gaussians. We design the first explicit schemes that use only a polynomial number of samples (in the key length) so that the players can generate shared keys that agree with constant probability using optimal communication. The best previously known schemes were both non-constructive and used an exponential number of samples. In the amortized setting, we characterize the largest achievable ratio of key length to communication in terms of the external and internal information costs, two well-studied quantities in theoretical computer science. In the relaxed setting where the two parties merely wish to improve the correlation between the generated keys of length k, we show that there are no interactive protocols using o(k) bits of communication having agreement probability even as small as 2–o(k). For the related communication problem where the players wish to compute a joint function f of their inputs using i.i.d samples from a known source, we give a simultaneous message passing protocol using 2O(c) bits where c is the interactive randomized public-coin communication complexity of f. This matches the lower bound shown previously while the best previously known upper bound was doubly exponential in c. Our schemes reveal a new connection between common randomness and unbiased error-correcting codes, e.g., dual-BCH codes and their analogues in Euclidean space. Read More: https://epubs.siam.org/doi/10.1137/1.9781611975031.120 

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3. High-Dimensional Robust Mean Estimation in Nearly-Linear Time

   Cheng, Yu ; Diakonikolas, Ilias ; Ge, Rong ( April 2019 , Proceedings of the annual ACM-SIAM Symposium on Discrete Algorithms)

   We study the fundamental problem of high-dimensional mean estimation in a robust model where a constant fraction of the samples are adversarially corrupted. Recent work gave the first polynomial time algorithms for this problem with dimension-independent error guarantees for several families of structured distributions. In this work, we give the first nearly-linear time algorithms for high-dimensional robust mean estimation. Specifically, we focus on distributions with (i) known covariance and sub-gaussian tails, and (ii) unknown bounded covariance. Given N samples on R^d, an \eps-fraction of which may be arbitrarily corrupted, our algorithms run in time eO(Nd)/poly(\eps) and approximate the true mean within the information-theoretically optimal error, up to constant factors. Previous robust algorithms with comparable error guarantees have running times \Omega(Nd^2), for \eps= O(1) Our algorithms rely on a natural family of SDPs parameterized by our current guess ν for the unknown mean μ. We give a win-win analysis establishing the following: either a near-optimal solution to the primal SDP yields a good candidate for μ — independent of our current guess ν — or a near-optimal solution to the dual SDP yields a new guess ν0 whose distance from μ is smaller by a constant factor. We exploit the special structure of the corresponding SDPs to show that they are approximately solvable in nearly-linear time. Our approach is quite general, and we believe it can also be applied to obtain nearly-linear time algorithms for other high-dimensional robust learning problems. 

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4. Decidability of Non-Interactive Simulation of Joint Distributions

   Ghazi, Badih ; Kamath, Pritish ; Sudan, Madhu ( October 2016 , Annual Symposium on Foundations of Computer Science)

   We present decidability results for a sub-class of “non-interactive” simulation problems, a well-studied class of problems in information theory. A non-interactive simulation problem is specified by two distributions P(x, y) and Q(u, v): The goal is to determine if two players, Alice and Bob, that observe sequences Xn and Y n respectively where {(Xi, Yi)}n i=1 are drawn i.i.d. from P(x, y) can generate pairs U and V respectively (without communicating with each other) with a joint distribution that is arbitrarily close in total variation to Q(u, v). Even when P and Q are extremely simple: e.g., P is uniform on the triples {(0, 0), (0, 1), (1, 0)} and Q is a “doubly symmetric binary source”, i.e., U and V are uniform ±1 variables with correlation say 0.49, it is open if P can simulate Q. In this work, we show that whenever P is a distribution on a finite domain and Q is a 2 × 2 distribution, then the non-interactive simulation problem is decidable: specifically, given δ > 0 the algorithm runs in time bounded by some function of P and δ and either gives a non-interactive simulation protocol that is δ-close to Q or asserts that no protocol gets O(δ)-close to Q. The main challenge to such a result is determining explicit (computable) convergence bounds on the number n of samples that need to be drawn from P(x, y) to get δ-close to Q. We invoke contemporary results from the analysis of Boolean functions such as the invariance principle and a regularity lemma to obtain such explicit bounds. 

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5. Local Access to Huge Random Objects Through Partial Sampling

   Biswas, Amartya Shankha ; Rubinfeld, Ronitt ; Yodpinyanee, Anak ( January 2020 , 11th Innovations in Theoretical Computer Science Conference, ITCS 2020)

   null (Ed.)

   Consider an algorithm performing a computation on a huge random object (for example a random graph or a "long" random walk). Is it necessary to generate the entire object prior to the computation, or is it possible to provide query access to the object and sample it incrementally "on-the-fly" (as requested by the algorithm)? Such an implementation should emulate the random object by answering queries in a manner consistent with an instance of the random object sampled from the true distribution (or close to it). This paradigm is useful when the algorithm is sub-linear and thus, sampling the entire object up front would ruin its efficiency. Our first set of results focus on undirected graphs with independent edge probabilities, i.e. each edge is chosen as an independent Bernoulli random variable. We provide a general implementation for this model under certain assumptions. Then, we use this to obtain the first efficient local implementations for the Erdös-Rényi G(n,p) model for all values of p, and the Stochastic Block model. As in previous local-access implementations for random graphs, we support Vertex-Pair and Next-Neighbor queries. In addition, we introduce a new Random-Neighbor query. Next, we give the first local-access implementation for All-Neighbors queries in the (sparse and directed) Kleinberg’s Small-World model. Our implementations require no pre-processing time, and answer each query using O(poly(log n)) time, random bits, and additional space. Next, we show how to implement random Catalan objects, specifically focusing on Dyck paths (balanced random walks on the integer line that are always non-negative). Here, we support Height queries to find the location of the walk, and First-Return queries to find the time when the walk returns to a specified location. This in turn can be used to implement Next-Neighbor queries on random rooted ordered trees, and Matching-Bracket queries on random well bracketed expressions (the Dyck language). Finally, we introduce two features to define a new model that: (1) allows multiple independent (and even simultaneous) instantiations of the same implementation, to be consistent with each other without the need for communication, (2) allows us to generate a richer class of random objects that do not have a succinct description. Specifically, we study uniformly random valid q-colorings of an input graph G with maximum degree Δ. This is in contrast to prior work in the area, where the relevant random objects are defined as a distribution with O(1) parameters (for example, n and p in the G(n,p) model). The distribution over valid colorings is instead specified via a "huge" input (the underlying graph G), that is far too large to be read by a sub-linear time algorithm. Instead, our implementation accesses G through local neighborhood probes, and is able to answer queries to the color of any given vertex in sub-linear time for q ≥ 9Δ, in a manner that is consistent with a specific random valid coloring of G. Furthermore, the implementation is memory-less, and can maintain consistency with non-communicating copies of itself. 

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