Contextual Learning for Stochastic Optimization

TL;DR

This paper introduces a method for learning from samples of contextual value distributions in stochastic optimization, providing polynomial sample complexity bounds for various optimization problems.

Contribution

It proposes a novel approach to learn empirical distributions for contexts with guarantees on distributional proximity, enabling efficient policy learning.

Findings

Polynomial sample complexity for strongly monotone and stable problems

Effective learning of empirical distributions with small Lévý distance

Application to revenue maximization, Pandora's Box, and optimal stopping

Abstract

Motivated by stochastic optimization, we introduce the problem of learning from samples of contextual value distributions. A contextual value distribution can be understood as a family of real-valued distributions, where each sample consists of a context $x$ and a random variable drawn from the corresponding real-valued distribution $D_x$. By minimizing a convex surrogate loss, we learn an empirical distribution $D'_x$ for each context, ensuring a small L\'evy distance to $D_x$. We apply this result to obtain the sample complexity bounds for the learning of an $\epsilon$-optimal policy for stochastic optimization problems defined on an unknown contextual value distribution. The sample complexity is shown to be polynomial for the general case of strongly monotone and stable optimization problems, including Single-item Revenue Maximization, Pandora's Box and Optimal Stopping.