  Solving optimization problems with unknown parameters often requires learning
a predictive model to predict the values of the unknown parameters and then
solving the problem using these values. Recent work has shown that including
the optimization problem as a layer in the model training pipeline results in
predictions of the unobserved parameters that lead to higher decision quality.
Unfortunately, this process comes at a large computational cost because the
optimization problem must be solved and differentiated through in each training
iteration; furthermore, it may also sometimes fail to improve solution quality
due to non-smoothness issues that arise when training through a complex
optimization layer. To address these shortcomings, we learn a low-dimensional
surrogate model of a large optimization problem by representing the feasible
space in terms of meta-variables, each of which is a linear combination of the
original variables. By training a low-dimensional surrogate model end-to-end,
and jointly with the predictive model, we achieve: i) a large reduction in
training and inference time; and ii) improved performance by focusing attention
on the more important variables in the optimization and learning in a smoother
space. Empirically, we demonstrate these improvements on a non-convex adversary
modeling task, a submodular recommendation task and a convex portfolio
optimization task.
