Stochastic Proximal Langevin Algorithm: Potential Splitting and Nonasymptotic Rates
This provides an incremental improvement for researchers in Bayesian learning and optimization by addressing sampling with stochastic nonsmooth components.
The paper tackles sampling from log concave distributions by proposing the Stochastic Proximal Langevin Algorithm (SPLA), which generalizes Langevin methods to handle stochastic smooth and nonsmooth terms, achieving nonasymptotic sublinear and linear convergence rates under convexity and strong convexity conditions.
We propose a new algorithm---Stochastic Proximal Langevin Algorithm (SPLA)---for sampling from a log concave distribution. Our method is a generalization of the Langevin algorithm to potentials expressed as the sum of one stochastic smooth term and multiple stochastic nonsmooth terms. In each iteration, our splitting technique only requires access to a stochastic gradient of the smooth term and a stochastic proximal operator for each of the nonsmooth terms. We establish nonasymptotic sublinear and linear convergence rates under convexity and strong convexity of the smooth term, respectively, expressed in terms of the KL divergence and Wasserstein distance. We illustrate the efficiency of our sampling technique through numerical simulations on a Bayesian learning task.