Unbiased Gradients for a Class of Conditional Stochastic Optimization Problems
For researchers in stochastic optimization, this provides a method for CSO problems with inaccessible joint distributions, but the contribution is incremental as it builds on existing work (Goda & Kitade, 2023).
The paper tackles conditional stochastic optimization (CSO) problems where the joint distribution of random variables cannot be exactly sampled. It introduces a method combining Markovian stochastic approximation with unbiased approximation to find the optimizer, demonstrated on parameter estimation and portfolio selection.
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 $F(ξ) := \mathbb{E}\left[f\left(Z,\mathbb{E}[g(Z,X,ξ)|Z]\right)\right]$, 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 $F(ξ)$ 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.