Quantitative Central Limit Theorems for Discrete Stochastic Processes

Many randomized algorithms in machine learning can be analyzed as some kind of stochastic process. For example, MCMC algorithms intentionally inject carefully designed randomness in order to sample from a desired target distribution. There is a second category of randomized algorithms for which the for which the goal is optimization rather than sampling, and the randomness is viewed as a price to pay for computational tractability. For example, stochastic gradient methods for large scale optimization use noisy estimates of a gradient because they are cheap. While such algorithms are not designed with the goal of sampling from a target distribution, an algorithm of this kind has random outputs, and its behavior is determined by the distribution of its output. Results in this paper provide tools for analyzing the convergence of such algorithms as stochastic processes.