Minibatch stochastic subgradient-based projection algorithms for solving convex inequalities

TL;DR

This paper introduces minibatch stochastic subgradient projection algorithms for convex inequalities, providing convergence analysis and conditions under which minibatch methods outperform single-sample approaches.

Contribution

It is the first to establish conditions where minibatch stochastic subgradient projection methods have better complexity than single-sample methods.

Findings

Proved sublinear convergence of the algorithms.

Established linear convergence under regularity conditions.

Derived explicit dependence of convergence rates on minibatch size.

Abstract

This paper deals with the convex feasibility problem, where the feasible set is given as the intersection of a (possibly infinite) number of closed convex sets. We assume that each set is specified algebraically as a convex inequality, where the associated convex function is general (possibly non-differentiable). For finding a point satisfying all the convex inequalities we design and analyze random projection algorithms using special subgradient iterations and extrapolated stepsizes. Moreover, the iterate updates are performed based on parallel random observations of several constraint components. For these minibatch stochastic subgradient-based projection methods we prove sublinear convergence results and, under some linear regularity condition for the functional constraints, we prove linear convergence rates. We also derive conditions under which these rates depend explicitly on the minibatch size. To the best of our knowledge, this work is the first deriving conditions that show when minibatch stochastic subgradient-based projection updates have a better complexity than their single-sample variants.