Georgia Institute of Technology College of Sciences

In recent years, the few classical results in large deviations for random matrices have been complemented by a variety of new ones, in both the math and physics literatures, whose proofs leverage connections with Harish-Chandra/Itzykson/Zuber integrals. We present one such result, focusing on extreme eigenvalues of deformed sample-covariance and Wigner random matrices. This confirms recent formulas of Maillard (2020) in the physics literature, precisely locating a transition point whose analogue in non-deformed models is not yet fully understood. Joint work with Jonathan Husson.

Empirical Bayes provides a powerful approach to learning and adapting to latent structure in data. Theory and algorithms for empirical Bayes have a rich literature for sequence models, but are less understood in settings where latent variables and data interact through more complex designs.

In this work, we study empirical Bayes estimation of an i.i.d. prior in Bayesian linear models, via the nonparametric maximum likelihood estimator (NPMLE). We introduce and study a system of gradient flow equations for optimizing the marginal log-likelihood, jointly over the prior and posterior measures in its Gibbs variational representation using a smoothed reparametrization of the regression coefficients. A diffusion-based implementation yields a Langevin dynamics MCEM algorithm, where the prior law evolves continuously over time to optimize a sequence-model log-likelihood defined by the coordinates of the current Langevin iterate.

We show consistency of the NPMLE under mild conditions, including settings of random sub-Gaussian designs under high-dimensional asymptotics. In high noise, we prove a uniform log-Sobolev inequality for the mixing of Langevin dynamics, for possibly misspecified priors and non-log-concave posteriors. We then establish polynomial-time convergence of the joint gradient flow to a near-NPMLE if the marginal negative log-likelihood is convex in a sub-level set of the initialization.

This is joint work with Leying Guan, Yandi Shen, and Yihong Wu.

Higher-order multiway data is ubiquitous in machine learning and statistics and often exhibits community-like structures, where each component (node) along each different mode has a community membership associated with it. In this talk we propose the tensor mixed-membership blockmodel, a generalization of the tensor blockmodel positing that memberships need not be discrete, but instead are convex combinations of latent communities. We first study the problem of estimating community memberships, and we show that a tensor generalization of a matrix algorithm can consistently estimate communities at a rate that improves relative to the matrix setting, provided one takes the tensor structure into account. Next, we study the problem of testing whether two nodes have the same community memberships, and we show that a tensor analogue of a matrix test statistic can yield consistent testing with a tighter local power guarantee relative to the matrix setting. If time permits we will also examine the performance of our estimation procedure on flight data. This talk is based on two recent works with Anru Zhang.

Abstract: Motivated by applications in Reinforcement Learning (RL), this talk focuses on the Stochastic Appproximation (SA) method to find fixed points of a contractive operator. First proposed by Robins and Monro, SA is a popular approach for solving fixed point equations when the information is corrupted by noise. We consider the SA algorithm for operators that are contractive under arbitrary norms (especially the l-infinity norm). We present finite sample bounds on the mean square error, which are established using a Lyapunov framework based on infimal convolution and generalized Moreau envelope. We then present our more recent result on concentration of the tail error, even when the iterates are not bounded by a constant. These tail bounds are obtained using exponential supermartingales in conjunction with the Moreau envelop and a novel bootstrapping approach. Our results immediately imply the state-of-the-art sample complexity results for a large class of RL algorithms.

Bio: Siva Theja Maguluri is Fouts Family Early Career Professor and Associate Professor in the H. Milton Stewart School of Industrial and Systems Engineering at Georgia Tech. He obtained his Ph.D. and MS in ECE as well as MS in Applied Math from UIUC, and B.Tech in Electrical Engineering from IIT Madras. His research interests span the areas of Control, Optimization, Algorithms and Applied Probability and include Reinforcement Learning theory and Stochastic Networks. His research and teaching are recognized through several awards including the  Best Publication in Applied Probability award, NSF CAREER award, second place award at INFORMS JFIG best paper competition, Student best paper award at IFIP Performance, CTL/BP Junior Faculty Teaching Excellence Award, and Student Recognition of Excellence in Teaching: Class of 1934 CIOS Award.