Online Sampling from Log-Concave Distributions
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Given a sequence of convex functions f_0, f_1, ..., f_T, we study the problem of sampling from the Gibbs distribution π_t ∝ e^-∑_k=0^t f_k for each epoch t in an online manner. This problem occurs in applications to machine learning, Bayesian statistics, and optimization where one constantly acquires new data, and must continuously update the distribution. Our main result is an algorithm that generates independent samples from a distribution that is a fixed ε TV-distance from π_t for every t and, under mild assumptions on the functions, makes poly(T) gradient evaluations per epoch. All previous results for this problem imply a bound on the number of gradient or function evaluations which is at least linear in T. While we assume the functions have bounded second moment, we do not assume strong convexity. In particular, we show that our assumptions hold for online Bayesian logistic regression, when the data satisfy natural regularity properties. In simulations, our algorithm achieves accuracy comparable to that of a Markov chain specialized to logistic regression. Our main result also implies the first algorithm to sample from a d-dimensional log-concave distribution π_T ∝ e^-∑_k=0^T f_k where the f_k's are not assumed to be strongly convex and the total number of gradient evaluations is roughly T(T)+poly(d), as opposed to T·poly(d) implied by prior works. Key to our algorithm is a novel stochastic gradient Langevin dynamics Markov chain that has a carefully designed variance reduction step built-in with fixed constant batch size. Technically, lack of strong convexity is a significant barrier to the analysis, and, here, our main contribution is a martingale exit time argument showing the chain is constrained to a ball of radius roughly poly(T) for the duration of the algorithm.
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