Sub-linear convergence of a tamed stochastic gradient descent method in Hilbert space

TL;DR

This paper introduces a tamed stochastic gradient descent method (TSGD) inspired by stochastic differential equations, proving its sub-linear convergence in Hilbert spaces with mild step size restrictions and demonstrating its practical utility in supervised learning.

Contribution

The paper presents a novel TSGD method with stability properties similar to implicit schemes, providing rigorous convergence analysis and practical validation in supervised learning.

Findings

Proves optimal sub-linear convergence of TSGD for strongly convex functions.

Shows TSGD has stability properties comparable to implicit schemes.

Demonstrates TSGD's effectiveness in a supervised learning problem.

Abstract

In this paper, we introduce the tamed stochastic gradient descent method (TSGD) for optimization problems. Inspired by the tamed Euler scheme, which is a commonly used method within the context of stochastic differential equations, TSGD is an explicit scheme that exhibits stability properties similar to those of implicit schemes. As its computational cost is essentially equivalent to that of the well-known stochastic gradient descent method (SGD), it constitutes a very competitive alternative to such methods. We rigorously prove (optimal) sub-linear convergence of the scheme for strongly convex objective functions on an abstract Hilbert space. The analysis only requires very mild step size restrictions, which illustrates the good stability properties. The analysis is based on a priori estimates more frequently encountered in a time integration context than in optimization, and this alternative approach provides a different perspective also on the convergence of SGD. Finally, we demonstrate the usability of the scheme on a problem arising in a context of supervised learning.