Single-Loop Switching Subgradient Methods for Non-Smooth Weakly Convex Optimization with Non-Smooth Convex Constraints

In this paper, we consider a general non-convex constrained optimization problem, where the objective function is weakly convex and the constraint function is convex while they can both be non-smooth. This class of problems arises from many applications in machine learning such as fairness-aware supervised learning. To solve this problem, we consider the classical switching subgradient method by Polyak (1965), which is an intuitive and easily implementable first-order method. Before this work, its iteration complexity was only known for convex optimization. We prove its oracle complexity for finding a nearly stationary point when the objective function is non-convex. The analysis is derived separately when the constraint function is deterministic and stochastic. Compared to existing methods, especially the double-loop methods, the switching gradient method can be applied to non-smooth problems and only has a single loop, which saves the effort on tuning the number of inner iterations.