Georgia Institute of Technology College of Sciences

MCMC and variational inference are two competing paradigms for the problem of sampling from a given probability distribution. In this talk, I'll show how they can work together to give the first polynomial-time sampling algorithm for approximately low-rank Ising models. Sampling was previously known when all eigenvalues of the interaction matrix fit in an interval of length 1; however, a single outlier can cause Glauber dynamics to mix torpidly. Our result covers the case when all but O(1) eigenvalues lie in an interval of length 1. To deal with positive eigenvalues, we use a temperature-based heuristic for MCMC called simulated tempering, while to deal with negative eigenvalues, we define a nonconvex variational problem over Ising models, solved using SGD. Our result has applications to sampling Hopfield networks with a fixed number of patterns, Bayesian clustering models with low-dimensional contexts, and antiferromagnetic/ferromagnetic Ising model on expander graphs.

In this talk, we will look into the two most widely studied settings of the stochastic multi-armed bandit problems - regret minimization and pure exploration. The algorithm is presented with a finite set of unknown distributions from which it can generate samples. In the regret-minimization setting, its aim is to sample sequentially so as to maximize the total average reward accumulated. In the pure exploration setting, we are interested in algorithms that identify the arm with the maximum mean in a small number of samples on an average while keeping the probability of false selection to at most a pre-specified and small value. Both of these problems are well studied in literature and tight lower bounds and optimal algorithms exist when the arm distributions are known to belong to simple classes of distributions such as single-parameter exponential family, distributions that have bounded support, etc. However, in practice, the distributions may not satisfy these assumptions and may even be heavy-tailed. In this talk, we will look at techniques and algorithms for optimally solving these two problems with minimal assumptions on the arm distributions. These ideas can be extended to a more general objective of identifying the distribution with the minimum linear combination of risk and reward, which captures the risk-reward trade-off that is popular in many practical settings, including in finance.

In 1939, Wolfgang Gröbner proposed using differential operators to represent ideals in a polynomial ring. Using Macaulay inverse systems, he showed a one-to-one correspondence between primary ideals whose variety is a rational point, and finite dimensional vector spaces of differential operators with constant coefficients. The question for general ideals was left open. Significant progress was made in the 1960's by analysts, culminating in a deep result known as the Ehrenpreis-Palamodov fundamental principle, connecting polynomial ideals and modules to solution sets of linear, homogeneous partial differential equations with constant coefficients. 

This talk aims to survey classical results, and provide new constructions, applications, and insights, merging concepts from analysis and nonlinear algebra. We offer a new formulation generalizing Gröbner's duality for arbitrary polynomial ideals and modules and connect it to the analysis of PDEs. This framework is amenable to the development of symbolic and numerical algorithms. We also study some applications of algebraic methods in problems from analysis.

Link: https://gatech.zoom.us/j/95997197594?pwd=RDN2T01oR2JlaEcyQXJCN1c4dnZaUT09