Convex Optimization with an Interpolation-based Projection and its Application to Deep Learning

TL;DR

This paper introduces a computationally efficient interpolation-based projection method for convex optimization, enabling its integration into deep learning models with theoretical convergence guarantees and practical benefits in reinforcement and supervised learning.

Contribution

It proposes a novel, cheap interpolation-based projection technique and an associated optimization algorithm with proven convergence for convex problems.

Findings

Interpolation projection is computationally cheaper than traditional convex projections.

The proposed algorithm converges for linear objectives with convex constraints.

Empirical results show improved efficiency and effectiveness in neural network applications.

Abstract

Convex optimizers have known many applications as differentiable layers within deep neural architectures. One application of these convex layers is to project points into a convex set. However, both forward and backward passes of these convex layers are significantly more expensive to compute than those of a typical neural network. We investigate in this paper whether an inexact, but cheaper projection, can drive a descent algorithm to an optimum. Specifically, we propose an interpolation-based projection that is computationally cheap and easy to compute given a convex, domain defining, function. We then propose an optimization algorithm that follows the gradient of the composition of the objective and the projection and prove its convergence for linear objectives and arbitrary convex and Lipschitz domain defining inequality constraints. In addition to the theoretical contributions, we demonstrate empirically the practical interest of the interpolation projection when used in conjunction with neural networks in a reinforcement learning and a supervised learning setting.