Multistage Conditional Compositional Optimization

2026-04-15

AI Summary

This work addresses the curse of dimensionality arising from the exponential growth of scenario complexity in multilevel nested conditional expectation optimization. To tackle this challenge, the authors propose a multistage conditional compositional optimization framework that integrates multistage stochastic programming with conditional stochastic optimization and, for the first time, introduces the multilevel Monte Carlo method into this domain. By enabling efficient estimation of nested conditional expectations and nonlinear cost functions, the proposed approach ensures that computational complexity scales only polynomially with respect to the desired accuracy, thereby overcoming the exponential bottleneck inherent in conventional algorithms. Empirical evaluations on tasks such as optimal stopping, linear-quadratic regulators, and distributionally robust contextual bandits demonstrate substantial improvements in both computational efficiency and scalability.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Sampling/Simulation-based Search
• Machine Learning: Optimization

Application Category

• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines

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 naïve 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.

Problem

• Multistage Conditional Compositional Optimization
• decision-making under uncertainty
• nested conditional expectations
• curse of dimensionality
• scenario complexity

Innovation

• Multistage Conditional Compositional Optimization
• nested conditional expectations
• multilevel Monte Carlo
• curse of dimensionality
• scenario complexity