Sub-linear convergence of a stochastic proximal iteration method in Hilbert space

Abstract

We consider a stochastic version of the proximal point algorithm for optimization problems posed on a Hilbert space. A typical application of this is supervised learning. While the method is not new, it has not been extensively analyzed in this form. Indeed, most related results are confined to the finite-dimensional setting, where error bounds could depend on the dimension of the space. On the other hand, the few existing results in the infinitedimensional setting only prove very weak types of convergence, owing to weak assumptions on the problem. In particular, there are no results that show convergence with a rate. In this article, we bridge these two worlds by assuming more regularity of the optimization problem, which allows us to prove convergence with an (optimal) sub-linear rate also in an infinite-dimensional setting. We illustrate these results by discretizing a concrete infinite-dimensional classification problem with varying degrees of accuracy.

SRKCD: a stabilized Runge-Kutta method for stochastic optimization

“…This, however, requires that method is implicit. One such method would be implicit Euler, which, when applied to the gradient flow is equivalent to the proximal point method in the context of optimization [2,6]. While this can be applied in certain cases when F has a specific structure that allows the arising nonlinear equation systems to be solved efficiently, in general (usually) this is not feasible.…”

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

“…See, e.g. [3,7,10,20,22,28,29,30] for analyses of this setting. In general, however, it means that we have to solve an unfeasibly large system of nonlinear equations in each step.…”