Online Sampling from Log-Concave Distributions

Abstract

Given a sequence of convex functions $f_0,f_1,\dots ,f_T$, 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 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. 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 poly-logarithmic 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 poly-logarithmic in $T$ for enough time to reach within $\epsilon$ of ${\pi }_t$.

arXiv comment42 pages

References in corpus (11)

• Streaming Variational Bayes
• Coresets for Scalable Bayesian Logistic Regression
• Log-concave sampling: Metropolis-Hastings algorithms are fast
• Automated Scalable Bayesian Inference via Hilbert Coresets
• Leave Pima Indians alone: binary regression as a benchmark for Bayesian computation
• On the convergence of Hamiltonian Monte Carlo
• Logistic Regression: Tight Bounds for Stochastic and Online Optimization
• The True Cost of Stochastic Gradient Langevin Dynamics
• Logistic Regression: The Importance of Being Improper
• Enabling scalable stochastic gradient-based inference for Gaussian processes by employing the Unbiased LInear System SolvEr (ULISSE)
• Simulated Tempering Langevin Monte Carlo II: An Improved Proof using Soft Markov Chain Decomposition

Cited by in corpus (1)

• Nonconvex sampling with the Metropolis-adjusted Langevin algorithm