On perturbed proximal gradient algorithms. (English) Zbl 1433.90199

Atchadé, Yves F.; Fort, Gersende; Moulines, Eric

J. Mach. Learn. Res. 18(2017-2018), Paper No. 10, 33 p. (2017).

Summary: We study a version of the proximal gradient algorithm for which the gradient is intractable and is approximated by Monte Carlo methods (and in particular Markov Chain Monte Carlo). We derive conditions on the step size and the Monte Carlo batch size under which convergence is guaranteed: both increasing batch size and constant batch size are considered. We also derive non-asymptotic bounds for an averaged version. Our results cover both the cases of biased and unbiased Monte Carlo approximation. To support our findings, we discuss the inference of a sparse generalized linear model with random effect and the problem of learning the edge structure and parameters of sparse undirected graphical models.

MSC:

90C59	Approximation methods and heuristics in mathematical programming
65C05	Monte Carlo methods

Keywords:

proximal gradient methods; stochastic optimization; Monte Carlo approximations; perturbed majorization-minimization algorithms