Online Sampling from LogConcave Distributions
Abstract
Given a sequence of convex functions $f_0, f_1, \ldots, f_T$, we study the problem of sampling from the Gibbs distribution $\pi_t \propto e^{\sum_{k=0}^tf_k}$ for each epoch $t$ in an 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 realworld 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. In particular, our assumptions hold for online Bayesian logistic regression, when the data satisfy natural regularity properties, giving a sampling algorithm with updates that are polylogarithmic in $T$. In simulations, our algorithm achieves accuracy comparable to an algorithm specialized to logistic regression. Key to our algorithm is a novel stochastic gradient Langevin dynamics Markov chain with a carefully designed variance reduction step and constant batch size. Technically, lack of strong convexity is a significant barrier to analysis and, here, our main contribution is a martingale exit time argument that shows our Markov chain remains in a ball of radius roughly polylogarithmic in $T$ for enough time to reach within $\varepsilon$ of $\pi_t$.
 Publication:

arXiv eprints
 Pub Date:
 February 2019
 arXiv:
 arXiv:1902.08179
 Bibcode:
 2019arXiv190208179L
 Keywords:

 Computer Science  Machine Learning;
 Computer Science  Data Structures and Algorithms;
 Mathematics  Probability;
 Statistics  Computation;
 Statistics  Machine Learning
 EPrint:
 42 pages