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

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.