Multistage Conditional Compositional Optimization

TL;DR

This paper introduces MCCO, a new decision-making framework that minimizes nested conditional expectations, and proposes multilevel Monte Carlo methods to address computational complexity issues.

Contribution

The paper develops a novel MCCO framework and introduces multilevel Monte Carlo techniques to reduce scenario complexity from exponential to polynomial growth.

Findings

MCCO unifies multistage stochastic programming and conditional optimization.

Naive nested sampling suffers from exponential complexity.

Multilevel Monte Carlo methods reduce complexity to polynomial growth.

Abstract

We introduce Multistage Conditional Compositional Optimization (MCCO) as a new paradigm for decision-making under uncertainty that combines aspects of multistage stochastic programming and conditional stochastic optimization. MCCO minimizes a nest of conditional expectations and nonlinear cost functions. It has numerous applications and arises, for example, in optimal stopping, linear-quadratic regulator problems, distributionally robust contextual bandits, as well as in problems involving dynamic risk measures. The na\"ive nested sampling approach for MCCO suffers from the curse of dimensionality familiar from scenario tree-based multistage stochastic programming, that is, its scenario complexity grows exponentially with the number of nests. We develop new multilevel Monte Carlo techniques for MCCO whose scenario complexity grows only polynomially with the desired accuracy.