Convex Optimization with an Interpolation-based Projection and its Application to Deep Learning
This work addresses the efficiency bottleneck for convex layers in neural networks, offering a faster alternative with proven convergence, though it is incremental as it builds on existing convex optimization methods.
The paper tackles the computational expense of convex projection layers in deep learning by proposing a cheap interpolation-based projection and proves convergence for linear objectives with convex constraints, showing practical gains in reinforcement and supervised learning tasks.
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.