Unbiased Gradients for a Class of Conditional Stochastic Optimization Problems

TL;DR

This paper introduces an unbiased gradient estimation method for a specific class of conditional stochastic optimization problems where joint distributions cannot be exactly sampled, demonstrated through applications in finance and parameter estimation.

Contribution

It develops a novel approach combining Markovian stochastic approximation with unbiased estimators for complex CSO problems where exact sampling is infeasible.

Findings

Method effectively finds optimizers in complex CSO scenarios.

Demonstrated on portfolio selection and parameter estimation examples.

Achieves unbiased gradient estimates in challenging sampling conditions.

Abstract

In this paper we consider the conditional stochastic optimization (CSO) problem. This consists of optimizing a function which can be written as the expectation of a function which is itself a function of a conditional expectation, i.e.~of the type $F(\xi) := \mathbb{E}\left[f\left(Z,\mathbb{E}[g(Z,X,\xi)|Z]\right)\right]$, where precise definitions are given in the main text. We address a particular class of CSO problems where the joint law of the random variables $X,Z$ cannot be exactly sampled; this case has been addressed in Goda & Kitade (2023). We introduce a method that combines Markovian stochastic approximation with unbiased approximation methods which allows one to find the optimizer of $F(\xi)$ in the context of interest. We illustrate our methodology on two examples associated to parameter estimation with model averaging and portfolio selection associated to high-dimensional full factor multivariate stochastic volatility models.