Almost sure convergence rates of stochastic proximal gradient descent algorithm

Stochastic proximal gradient descent (Prox-SGD) is a standard optimization algorithm for solving stochastic composite opti-mization problems in machine learning. However, the existing convergence rate analysis of Prox-SGD is mostly focused on convergence in expectation. To this end, we provide an almost sure convergence rate analysis of Prox-SGD in two optimiza-tion settings: convexity and strong convexity, showing that any implementation of the stochastic algorithm will converge with probability one. Specifically, we show that the average-iterates almost sure convergence rate of the function values v for Prox-SGD is arbitrarily close to the optimal rate o(1/v T) in the convex setting. For strongly convex objective functions, we obtain the almost sure convergence rate of the iterative sequence, which is also arbitrarily close to the optimal rate o(1/T). Last but not least, we establish the more challeng-ing last-iterate convergence rate of the function values on convex problems, which contrasts with existing results on convergence for a weighted average.