﻿ 【创源大讲堂】 Variance Reduced Random Relaxed Projection Method for Constrained Finite-sum Minimization-西南交通大学新闻网

# 【创源大讲堂】 Variance Reduced Random Relaxed Projection Method for Constrained Finite-sum Minimization

Title: Variance Reduced Random Relaxed Projection Method for Constrained Finite-sum Minimization Problems

(Joint work with Zhichun Yang, Kai Tu and Man-Chung Yue)

Abstract. We consider the problem of minimizing a large sum of smooth convex functions subject to a large number of constraints defined by convex functions that are possibly non-smooth. Such a problem finds a wide range of applications in many areas, such as machine learning and signal processing. In this paper, we devise a new random projection method (RPM) for efficiently solving this problem. Compared with existing RPMs, our proposed algorithm features two useful algorithmic ideas. First, at each iteration, instead of projection onto a subset of the feasible region, our algorithm requires only a relaxed projection in the sense that only the projection onto a half-space approximation of the subset is needed. This significantly reduces the per iteration computational cost as the relaxed projection admits a closed-form formula. Second, to exploit the structure that the objective function is a large sum of convex functions, the variance reduction technique is incorporated into our algorithm to improve the empirical convergence behaviour. As our theoretical contributions, under an error bound-type condition and some other standard conditions, we prove that almost surely the proposed algorithm converges to an optimal solution and that both the optimality and feasibility gaps decrease to zero, with rates $O(1/\sqrt(K))$ and $O(log(K)/K)$, respectively. As a side contribution, we also show that the said error bound-type condition holds some mild assumptions, which could be of independent interest. Numerical results on synthetic problem instances are also presented to demonstrate the practical effectiveness of the variance reduction technique and the superiority of our proposed RPM as compared with two existing ones.

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