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

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.