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

Convex optimizers have known many applications as differentiable layers\nwithin deep neural architectures. One application of these convex layers is to\nproject points into a convex set. However, both forward and backward passes of\nthese convex layers are significantly more expensive to compute than those of a\ntypical neural network. We investigate in this paper whether an inexact, but\ncheaper projection, can drive a descent algorithm to an optimum. Specifically,\nwe propose an interpolation-based projection that is computationally cheap and\neasy to compute given a convex, domain defining, function. We then propose an\noptimization algorithm that follows the gradient of the composition of the\nobjective and the projection and prove its convergence for linear objectives\nand arbitrary convex and Lipschitz domain defining inequality constraints. In\naddition to the theoretical contributions, we demonstrate empirically the\npractical interest of the interpolation projection when used in conjunction\nwith neural networks in a reinforcement learning and a supervised learning\nsetting.\n