Scenario-based Regularization: A Tractable Framework for Distributionally Robust Stochastic Optimization

2025-11-20

AI Summary

Standard Wasserstein distributionally robust optimization (WDRO) is sensitive to the Wasserstein distance order $p$, provides inadequate tail-risk control, and suffers from high computational complexity. To address these limitations, we propose the scenario-regularized sample average approximation (SRA-SAA) framework: it imposes a gradient-norm penalty on user-specified extreme scenarios, yielding an equivalent decision-dependent WDRO formulation. The approach ensures both computational tractability and targeted robustness, with finite-sample generalization bounds and asymptotic consistency guarantees. Leveraging asymptotic expansions of the Wasserstein distance and subgradient-based optimization, we develop a scalable algorithm. Empirical evaluation on multi-item newsvendor and mean-risk portfolio optimization problems demonstrates substantial out-of-sample performance improvements—particularly when incorporating historical crisis data—to enhance decision robustness under extreme risks.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Sampling/Simulation-based Search
• Intelligent Robots: Learning & Optimization for ROB

Application Category

• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

We propose a flexible scenario-based regularized Sample Average Approximation (SBR-SAA) framework for stochastic optimization. This work is motivated by challenges in standard Wasserstein Distributionally Robust Optimization (WDRO), where out-of-sample performance, particularly tail risk, is sensitive to the choice of the p-norm, and formulations can be computationally intractable. Our method is inspired by the asymptotic expansion of the WDRO objective and introduces a regularizer that penalizes the (sub)gradient norm of the objective at a selected set of scenarios. This framework serves a dual purpose: (i) it provides a computationally tractable alternative to WDRO by using a representative subset of the data, and (ii) it can provide targeted robustness by incorporating user-defined adverse scenarios. We establish the theoretical properties of this framework by proving its equivalence to a decision-dependent WDRO problem, from which we derive finite sample guarantees and asymptotic consistency. We demonstrate the method's efficacy in two applications: (1) a multi-product newsvendor problem, where SBR-SAA serves as a tractable alternative to NP-hard WDRO, and (2) a mean-risk portfolio optimization problem, where it successfully uses historical crisis data to improve out-of-sample performance.

Problem

• Addresses computational intractability in distributionally robust optimization
• Improves out-of-sample performance by penalizing gradient norms
• Provides targeted robustness using user-defined adverse scenarios

Innovation

• Scenario-based regularization for robust stochastic optimization
• Penalizes gradient norms at selected scenarios for robustness
• Provides tractable alternative to distributionally robust optimization