While we obtain those positive results for the stochastic setting, another main result of our work is an impossibility in the adversarial setting.

How to solve high-dimensional linear programs (LPs) efficiently is a fundamental question. Recently, there has been a surge of interest in reducing LP sizes using random projections, which can accelerate solving LPs independently of improving LP solvers. This paper explores a new direction of data-driven projections, which use projection matrices learned from data instead of random projection matrices. Given training data of n-dimensional LPs, we learn an n k projection matrix with n > k. When addressing a future LP instance, we reduce its dimensionality from n to k via the learned projection matrix, solve the resulting LP to obtain a k-dimensional solution, and apply the learned matrix to it to recover an n-dimensional solution. On the theoretical side, a natural question is: how much data is sufficient to ensure the quality of recovered solutions? We address this question based on the framework of data-driven algorithm design, which connects the amount of data sufficient for establishing generalization bounds to the pseudo-dimension of performance metrics.