A Functional Model Method for Nonconvex Nonsmooth Conditional Stochastic Optimization

TL;DR

This paper introduces a novel single time-scale stochastic method for nonconvex, nonsmooth conditional stochastic optimization problems, enabling real-time learning with convergence guarantees.

Contribution

It develops a specialized stochastic algorithm for nonconvex conditional problems with Lipschitz smooth outer functions, using a rich parametric model for the inner expectation.

Findings

Convergence with probability one proved using differential inclusions.

Method applicable to real-time learning with one observation per iteration.

Numerical results demonstrate the approach's viability.

Abstract

We consider stochastic optimization problems involving an expected value of a nonlinear function of a base random vector and a conditional expectation of another function depending on the base random vector, a dependent random vector, and the decision variables. We call such problems conditional stochastic optimization problems. They arise in many applications, such as uplift modeling, reinforcement learning, and contextual optimization. We propose a specialized single time-scale stochastic method for nonconvex constrained conditional stochastic optimization problems with a Lipschitz smooth outer function and a generalized differentiable inner function. In the method, we approximate the inner conditional expectation with a rich parametric model whose mean squared error satisfies a stochastic version of a {\L}ojasiewicz condition. The model is used by an inner learning algorithm. The main feature of our approach is that unbiased stochastic estimates of the directions used by the method can be generated with one observation from the joint distribution per iteration, which makes it applicable to real-time learning. The directions, however, are not gradients or subgradients of any overall objective function. We prove the convergence of the method with probability one, using the method of differential inclusions and a specially designed Lyapunov function, involving a stochastic generalization of the Bregman distance. Finally, a numerical illustration demonstrates the viability of our approach.