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.
Highlights
Several recent research has investigated the integration of a ‘convex optimization layer’ within the computational graph of machine learning architectures in applications such as optimal control (de Avila Belbute-Peres et al 2018; Amos et al 2018), computer vision (Bertinetto et al 2019; Lee et al 2019) or filtering (Barratt and Boyd 2019)
We introduced in this paper an interpolation-based projection onto a convex set that can be readily computed for any convex domain defining function
We derived a descent algorithm based on the composition of the objective and the projection and showed that this surprisingly yields a convergent algorithm when the objective is linear, despite the ‘sub-optimality’ of the projection
Summary
Several recent research has investigated the integration of a ‘convex optimization layer’ within the computational graph of machine learning architectures in applications such as optimal control (de Avila Belbute-Peres et al 2018; Amos et al 2018), computer vision (Bertinetto et al 2019; Lee et al 2019) or filtering (Barratt and Boyd 2019). We proposed in Akrour et al (2019) differentiable policy parameterizations that comply with these constraints by construction, allowing the policy optimization problem to be solved by standard gradient descent algorithms These parameterizations were based on interpolating any input parameterization of a distribution with a constraint satisfying parameterization. Interpolating an input discrete distribution with the uniform distribution, that satisfies any reasonable minimal entropy constraint These projections were not ‘optimal’ in the sense that they do not minimize a distance to the admissible set, we noted empirically (see Akrour et al (2019), Fig. 1 and surrounding text) that such parameterization would always drive the descent algorithm to an optimum on a toy problem with a linear objective and a convex, entropy constraint. The practical implications being a cheap way of adding convex constraints to machine learning models as shown in the experimental validation section
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