Primal-dual algorithm for contextual stochastic combinatorial optimization

TL;DR

This paper presents a new primal-dual algorithm integrating neural networks and combinatorial optimization for contextual stochastic decision-making, demonstrating efficiency and scalability in complex problems.

Contribution

It introduces a novel primal-dual approach with a surrogate learning problem and regularization, extending Fenchel-Young loss and enabling tractable updates in stochastic combinatorial optimization.

Findings

Algorithm converges linearly under certain conditions.

Achieves performance comparable to expensive heuristics.

Scalable and efficient on complex stochastic problems.

Abstract

This paper introduces a novel approach to contextual stochastic optimization, integrating operations research and machine learning to address decision-making under uncertainty. Traditional methods often fail to leverage contextual information, which underscores the necessity for new algorithms. In this study, we utilize neural networks with combinatorial optimization layers to encode policies. Our goal is to minimize the empirical risk, which is estimated from past data on uncertain parameters and contexts. To that end, we present a surrogate learning problem and a generic primal-dual algorithm that is applicable to various combinatorial settings in stochastic optimization. Our approach extends classic Fenchel-Young loss results and introduces a new regularization method using sparse perturbations on the distribution simplex. This allows for tractable updates in the original space and can accommodate diverse objective functions. We demonstrate the linear convergence of our algorithm under certain conditions and provide a bound on the non-optimality of the resulting policy in terms of the empirical risk. Experiments on a contextual stochastic minimum weight spanning tree problem show that our algorithm is efficient and scalable, achieving performance comparable to imitation learning of solutions computed using an expensive Lagrangian-based heuristic.