  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.
