Data-Pooling in Stochastic Optimization

TL;DR

This paper introduces Shrunken-SAA, a novel data-pooling algorithm that combines data across unrelated stochastic optimization problems, outperforming decoupling even without structural links, especially when many small-data problems are involved.

Contribution

The paper proves that data pooling can outperform separate problem solving in stochastic optimization, even without prior problem structure, and introduces a practical algorithm with theoretical guarantees.

Findings

Data pooling can outperform decoupling in unrelated problems.

Shrunken-SAA effectively learns when pooling is beneficial.

Real-world application shows practical advantages of data pooling.

Abstract

Managing large-scale systems often involves simultaneously solving thousands of unrelated stochastic optimization problems, each with limited data. Intuition suggests one can decouple these unrelated problems and solve them separately without loss of generality. We propose a novel data-pooling algorithm called Shrunken-SAA that disproves this intuition. In particular, we prove that combining data across problems can outperform decoupling, even when there is no a priori structure linking the problems and data are drawn independently. Our approach does not require strong distributional assumptions and applies to constrained, possibly non-convex, non-smooth optimization problems such as vehicle-routing, economic lot-sizing or facility location. We compare and contrast our results to a similar phenomenon in statistics (Stein's Phenomenon), highlighting unique features that arise in the optimization setting that are not present in estimation. We further prove that as the number of problems grows large, Shrunken-SAA learns if pooling can improve upon decoupling and the optimal amount to pool, even if the average amount of data per problem is fixed and bounded. Importantly, we highlight a simple intuition based on stability that highlights when and why data-pooling offers a benefit, elucidating this perhaps surprising phenomenon. This intuition further suggests that data-pooling offers the most benefits when there are many problems, each of which has a small amount of relevant data. Finally, we demonstrate the practical benefits of data-pooling using real data from a chain of retail drug stores in the context of inventory management.