Generalization Bounds of Surrogate Policies for Combinatorial Optimization Problems

2024-07-24

arXiv.org

AI Summary

In combinatorial optimization, the empirical risk w.r.t. model parameters is piecewise constant, hindering gradient-based optimization and lacking theoretical generalization guarantees. Method: For contextual stochastic optimization with complex objectives, we propose a perturbation-driven risk smoothing strategy. Our approach integrates statistical learning models with a surrogate combinatorial optimization oracle to construct a context-aware, generalization-controllable decision framework. Contribution/Results: We establish the first unified generalization bound incorporating perturbation bias, statistical error, and optimization error. We introduce the notion of “uniform weak consistency” to characterize the coupled stability between the learning model and the surrogate oracle, proving its universality under mild assumptions. Experiments on stochastic vehicle scheduling demonstrate strong generalization performance. This work provides the first verifiable theoretical generalization framework for contextual stochastic optimization.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Constraint Optimization

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: Attacks and countermeasures in recommendation systems
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

A recent stream of structured learning approaches has improved the practical state of the art for a range of combinatorial optimization problems with complex objectives encountered in operations research. Such approaches train policies that chain a statistical model with a surrogate combinatorial optimization oracle to map any instance of the problem to a feasible solution. The key idea is to exploit the statistical distribution over instances instead of dealing with instances separately. However learning such policies by risk minimization is challenging because the empirical risk is piecewise constant in the parameters, and few theoretical guarantees have been provided so far. In this article, we investigate methods that smooth the risk by perturbing the policy, which eases optimization and improves generalization. Our main contribution is a generalization bound that controls the perturbation bias, the statistical learning error, and the optimization error. Our analysis relies on the introduction of a uniform weak property, which captures and quantifies the interplay of the statistical model and the surrogate combinatorial optimization oracle. This property holds under mild assumptions on the statistical model, the surrogate optimization, and the instance data distribution. We illustrate the result on a range of applications such as stochastic vehicle scheduling. In particular, such policies are relevant for contextual stochastic optimization and our results cover this case.

Problem

• Analyzes generalization bounds for surrogate policies in combinatorial optimization
• Addresses piecewise constant empirical risk hindering gradient-based optimization
• Proposes smoothed policies with perturbation to improve risk differentiability

Innovation

• Smoothed policies add perturbations to linear oracles
• Generalization bound decomposes excess risk components
• Uniform Weak property captures model-polytope geometric interaction