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# On stochastic proximal gradient algorithms

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### Abstract

We study a perturbed version of the proximal gradient algorithm for which the gradient is not known in closed form and should be approximated. We address the convergence and derive a non-asymptotic bound on the convergence rate for the perturbed proximal gradient, a perturbed averaged version of the proximal gradient algorithm and a perturbed version of the fast iterative shrinkage-thresholding (FISTA) of \cite{becketteboulle09}. When the approximation is achieved by using Monte Carlo methods, we derive conditions involving the Monte Carlo batch-size and the step-size of the algorithm under which convergence is guaranteed. In particular, we show that the Monte Carlo approximations of some averaged proximal gradient algorithms and a Monte Carlo approximation of FISTA achieve the same convergence rates as their deterministic counterparts. To illustrate, we apply the algorithms to high-dimensional generalized linear mixed models using $$\ell_1$$-penalization.

### Author and article information

###### Journal
2014-02-10
2015-01-29
1402.2365

60F15, 60G42
35 pages, 4 figures
math.ST math.OC stat.TH

Numerical methods, Statistics theory