Unified Analysis of Stochastic Gradient Projection Methods for Convex Optimization with Functional Constraints

This paper addresses a unified convergence analysis for a large family of stochastic gradient projection algorithms for dealing with constrained finite sum convex problems, with a smooth objective function satisfying a strong convexity condition and possibly nonsmooth functional constraints. At each iteration, the algorithm takes an optimal gradient step based on a stochastic unbiased estimate of the objective function aiming to minimize it and then a feasibility step reducing the infeasibility associated with randomly observed constraints. Our stochastic gradient estimate can take diverse forms (e.g., Stochastic Gradient Descent (SGD), Stochastic Average Gradient Acceleration (SAGA), Loopless-Stochastic Variance Reduced Gradient (L-SVRG). We conduct a convergence analysis of the proposed unified stochastic gradient projection algorithm under a diminishing stepsize, resulting in sublinear convergence rates, which are optimal for stochastic gradient methods within this problem class. Numerical evidence supports the effectiveness of our approach.