Self-Repellent Random Walks on General Graphs - Achieving Minimal Sampling Variance via Nonlinear Markov Chains (Extended Abstract)

We consider random walks on discrete state spaces, such as general undirected graphs, where the random walkers are designed to approximate a target quantity over the network topology via sampling and neighborhood exploration in the form of Markov chain Monte Carlo (MCMC) procedures. Given any Markov chain corresponding to a target probability distribution, we design a self-repellent random walk (SRRW) which is less likely to transition to nodes that were highly visited in the past, and more likely to transition to seldom visited nodes. For a class of SRRWs parameterized by a positive real α, we prove that the empirical distribution of the process converges almost surely to the target (stationary) distribution of the underlying Markov chain kernel. We then provide a central limit theorem and derive the exact form of the arising asymptotic co-variance matrix, which allows us to show that the SRRW with stronger repellence (larger α) always achieves a smaller asymptotic covariance, in the sense of Loewner ordering of co-variance matrices. Especially for SRRW-driven MCMC algorithms, we show that the decrease in the asymptotic sampling variance is of the order O(1/α), eventually going down to zero. After generalizing these results for a class of weighted empirical measures, we use them as a stepping stone to show that a similar performance ordering can also be obtained for distributed stochastic optimization tasks using token algorithms. More explicitly, by replacing a Markovian token by a SRRW version with the same target distribution, we show that the asymptotic co-variance of the optimization iterates decreases at rate O(1/α^2) - the performance benefit of using SRRW thereby amplified in the stochastic optimization context. Empirical results support our theoretical findings.