Folded Optimization for End-to-End Model-Based Learning

The integration of constrained optimization models as components in deep networks has led to promising advances in both these domains. A primary challenge in this setting is backpropagation through the optimization mapping, which typically lacks a closed form. A common approach is unrolling, which relies on automatic differentiation through the operations of an iterative solver. While flexible and general, unrolling can encounter accuracy and efficiency issues in practice. These issues can be avoided by differentiating the optimization mapping analytically, but current frameworks impose rigid requirements on the optimization problem's form. This paper provides theoretical insights into the backpropagation of unrolled optimizers, which lead to a system for generating equivalent but efficiently solvable analytical models. Additionally, it proposes a unifying view of unrolling and analytical differentiation through constrained optimization mappings. Experiments over various structured prediction and decision-focused learning tasks illustrate the potential of the approach both computationally and in terms of enhanced expressiveness.