Online sampling from log-concave distributions

Given a sequence of convex functions f_0, f_1, \ldots, f_T, we study the problem of sampling from the Gibbs distribution \pi_t \propto e {-\sum_{k 0} t f_k} for each epoch t in an {\em online} manner. Interest in this problem derives from applications in machine learning, Bayesian statistics, and optimization where, rather than obtaining all the observations at once, one constantly acquires new data, and must continuously update the distribution. Our main result is an algorithm that generates roughly independent samples from \pi_t for every epoch t and, under mild assumptions, makes \mathrm{polylog}(T) gradient evaluations per epoch. All previous results imply a bound on the number of gradient or function evaluations which is at least linear in T . Motivated by real-world applications, we assume that functions are smooth, their associated distributions have a bounded second moment, and their minimizer drifts in a bounded manner, but do not assume they are strongly convex.