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.

凸优化器已经将许多应用称为深度神经架构中的可微层。这些凸层的一个应用是将点投影到凸集。然而，与典型的神经网络相比，这些凸层的前向和后向传递的计算成本要高得多。我们在本文中研究了不精确但更便宜的投影是否可以将下降算法推向最佳。具体来说，我们提出了一种基于插值的投影，在给定一个凸的域定义函数的情况下，它的计算成本低且易于计算。然后，我们提出了一种优化算法，该算法遵循目标和投影的组成梯度，并证明其收敛于线性目标和定义不等式约束的任意凸和 Lipschitz 域。