An Incremental Gradient Method for Large-scale Distributed Nonlinearly Constrained Optimization

TL;DR

This paper introduces a new incremental gradient method tailored for large-scale distributed optimization problems with nonlinear constraints, avoiding costly projections and improving efficiency in sensor network and machine learning applications.

Contribution

The paper proposes the aIR-IG algorithm that handles nonlinear constraints without hard-to-project steps, providing convergence guarantees and outperforming existing methods in distributed SVM tasks.

Findings

Outperforms standard projected IG methods in distributed SVM problems

Provides non-asymptotic convergence rates for suboptimality and infeasibility

Effectively handles nonlinear and large linear constraints without costly projections

Abstract

Motivated by applications arising from sensor networks and machine learning, we consider the problem of minimizing a finite sum of nondifferentiable convex functions where each component function is associated with an agent and a hard-to-project constraint set. Among well-known avenues to address finite sum problems is the class of incremental gradient (IG) methods where a single component function is selected at each iteration in a cyclic or randomized manner. When the problem is constrained, the existing IG schemes (including projected IG, proximal IAG, and SAGA) require a projection step onto the feasible set at each iteration. Consequently, the performance of these schemes is afflicted with costly projections when the problem includes: (1) nonlinear constraints, or (2) a large number of linear constraints. Our focus in this paper lies in addressing both of these challenges. We develop an algorithm called averaged iteratively regularized incremental gradient (aIR-IG) that does not involve any hard-to-project computation. Under mild assumptions, we derive non-asymptotic rates of convergence for both suboptimality and infeasibility metrics. Numerically, we show that the proposed scheme outperforms the standard projected IG methods on distributed soft-margin support vector machine problems.