OptNet: Differentiable Optimization as a Layer in Neural Networks

TL;DR

OptNet introduces a novel neural network layer that incorporates quadratic programming optimization problems, enabling the encoding of complex constraints and dependencies within end-to-end trainable models, demonstrated through learning to play mini-Sudoku.

Contribution

This work develops a differentiable optimization layer using quadratic programs, with efficient GPU-based solvers and applications in constraint learning and game playing.

Findings

Successfully differentiates through quadratic programming layers

Efficient GPU-based solver with minimal additional cost

Learned to play mini-Sudoku without prior rule knowledge

Abstract

This paper presents OptNet, a network architecture that integrates optimization problems (here, specifically in the form of quadratic programs) as individual layers in larger end-to-end trainable deep networks. These layers encode constraints and complex dependencies between the hidden states that traditional convolutional and fully-connected layers often cannot capture. We explore the foundations for such an architecture: we show how techniques from sensitivity analysis, bilevel optimization, and implicit differentiation can be used to exactly differentiate through these layers and with respect to layer parameters; we develop a highly efficient solver for these layers that exploits fast GPU-based batch solves within a primal-dual interior point method, and which provides backpropagation gradients with virtually no additional cost on top of the solve; and we highlight the application of these approaches in several problems. In one notable example, the method is learns to play mini-Sudoku (4x4) given just input and output games, with no a-priori information about the rules of the game; this highlights the ability of OptNet to learn hard constraints better than other neural architectures.