Efficient Sampling from Time-Varying Log-Concave Distributions

TL;DR

This paper introduces an efficient random walk algorithm for sampling from time-varying log-concave distributions, with applications in streaming data, optimization, and online learning, providing theoretical guarantees and practical examples.

Contribution

It develops a rapid mixing random walk method that adapts to changing distributions with theoretical guarantees, enabling efficient sampling in dynamic settings.

Findings

Rapid mixing and tracking of time-varying distributions

Applications to streaming posterior sampling and optimization

Achieves low oracle complexity and one-step tracking in some cases

Abstract

We propose a computationally efficient random walk on a convex body which rapidly mixes and closely tracks a time-varying log-concave distribution. We develop general theoretical guarantees on the required number of steps; this number can be calculated on the fly according to the distance from and the shape of the next distribution. We then illustrate the technique on several examples. Within the context of exponential families, the proposed method produces samples from a posterior distribution which is updated as data arrive in a streaming fashion. The sampling technique can be used to track time-varying truncated distributions, as well as to obtain samples from a changing mixture model, fitted in a streaming fashion to data. In the setting of linear optimization, the proposed method has oracle complexity with best known dependence on the dimension for certain geometries. In the context of online learning and repeated games, the algorithm is an efficient method for implementing no-regret mixture forecasting strategies. Remarkably, in some of these examples, only one step of the random walk is needed to track the next distribution.