Analysis of a Class of Stochastic Component-Wise Soft-Clipping Schemes

TL;DR

This paper provides a rigorous theoretical analysis of a broad class of stochastic soft-clipping schemes, establishing convergence guarantees and rates, thereby supporting their reliable use in machine learning optimization tasks.

Contribution

It introduces and analyzes a large class of stochastic soft-clipping schemes, providing convergence proofs and rates in both convex and non-convex settings.

Findings

Convergence in expectation under standard assumptions

Rates of convergence for convex and non-convex cases

Almost sure convergence to stationary points in non-convex scenarios

Abstract

Choosing the optimization algorithm that performs best on a given machine learning problem is often delicate, and there is no guarantee that current state-of-the-art algorithms will perform well across all tasks. Consequently, the more reliable methods that one has at hand, the larger the likelihood of a good end result. To this end, we introduce and analyze a large class of stochastic so-called soft-clipping schemes with a broad range of applications. Despite the wide adoption of clipping techniques in practice, soft-clipping methods have not been analyzed to a large extent in the literature. In particular, a rigorous mathematical analysis is lacking in the general, nonlinear case. Our analysis lays a theoretical foundation for a large class of such schemes, and motivates their usage. In particular, under standard assumptions such as Lipschitz continuous gradients of the objective function, we give rigorous proofs of convergence in expectation. These include rates in both the convex and the non-convex case, as well as almost sure convergence to a stationary point in the non-convex case. The computational cost of the analyzed schemes is essentially the same as that of stochastic gradient descent.