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Convex optimizers have known many applications as differentiable layerswithin deep neural architectures. One application of these convex layers is toproject points into a convex set. However, both forward and backward passes ofthese convex layers are significantly more expensive to compute than those of atypical neural network. We investigate in this paper whether an inexact, butcheaper projection, can drive a descent algorithm to an optimum. Specifically,we propose an interpolation-based projection that is computationally cheap andeasy to compute given a convex, domain defining, function. We then propose anoptimization algorithm that follows the gradient of the composition of theobjective and the projection and prove its convergence for linear objectivesand arbitrary convex and Lipschitz domain defining inequality constraints. Inaddition to the theoretical contributions, we demonstrate empirically thepractical interest of the interpolation projection when used in conjunctionwith neural networks in a reinforcement learning and a supervised learningsetting.

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