A Reliability Theory of Compromise Decisions for Large-Scale Stochastic Programs
Shuotao Diao, Suvrajeet Sen
TL;DR
This work addresses the reliability of decisions from large-scale stochastic programs solved via sampling-based methods. It introduces the compromise decision framework, where multiple SAA replications are aggregated with a proximal regularizer to reduce variance, and formalizes a pessimistic distance metric to quantify reliability. The authors derive finite-sample bounds on the expected error and variance of the compromise decision under convexity, via Rademacher complexity, and extend the results to exact and inexact replication settings as well as algorithmic implementations using cutting-plane and stochastic decomposition methods. A key contribution is showing that, under the proposed framework and standard regularity conditions, compromise decisions exhibit provable reliability improvements relative to conventional SAA-only solutions, with explicit guidance on parameter choices and replication structure. These results offer practical, theory-grounded means to enhance decision quality in large-scale stochastic optimization and point to fruitful avenues for future work in DRO connections and nonconvex settings.
Abstract
Stochastic programming models can lead to very large-scale optimization problems for which it may be impossible to enumerate all possible scenarios. In such cases, one adopts a sampling-based solution methodology in which case the reliability of the resulting decisions may be suspect. For such instances, it is advisable to adopt methodologies that promote variance reduction. One such approach goes under a framework known as "compromise decision", which requires multiple replications of the solution procedure. This paper studies the reliability of stochastic programming solutions resulting from the "compromise decision" process. This process is characterized by minimizing an aggregation of objective function approximations across replications, presumably conducted in parallel. We refer to the post-parallel-processing problem as the problem of "compromise decision". We quantify the reliability of compromise decisions by estimating the expectation and variance of the "pessimistic distance" of sampled instances from the set of true optimal decisions. Such pessimistic distance is defined as an estimate of the largest possible distance of the solution of the sampled instance from the "true" optimal solution set. The Rademacher average of instances is used to bound the sample complexity of the compromise decision.