A regularization interpretation of the proximal point method for weakly convex functions

Abstract

Empirical evidence and theoretical results suggest that the proximal point method can be computed approximately and still converge faster than the corresponding gradient descent method, in both the stochastic and exact gradient case. In this article we provide a perspective on this result by interpreting the method as gradient descent on a regularized function. This perspective applies in the case of weakly convex functions where proofs of the faster rates are not available. Using this analysis we find the optimal value of the regularization parameter in terms of the weak convexity.