Newton Losses: Using Curvature Information for Learning with Differentiable Algorithms

TL;DR

Newton Losses leverage second-order curvature information of non-differentiable objectives to enhance neural network training, improving performance across various differentiable algorithms for sorting and shortest-path problems.

Contribution

The paper introduces Newton Losses, a novel method that uses second-order information of loss functions to improve training of neural networks with non-differentiable objectives.

Findings

Significant performance improvements on less-optimized algorithms.

Consistent improvements even on well-optimized algorithms.

Efficient use of second-order information without full second-order training.

Abstract

When training neural networks with custom objectives, such as ranking losses and shortest-path losses, a common problem is that they are, per se, non-differentiable. A popular approach is to continuously relax the objectives to provide gradients, enabling learning. However, such differentiable relaxations are often non-convex and can exhibit vanishing and exploding gradients, making them (already in isolation) hard to optimize. Here, the loss function poses the bottleneck when training a deep neural network. We present Newton Losses, a method for improving the performance of existing hard to optimize losses by exploiting their second-order information via their empirical Fisher and Hessian matrices. Instead of training the neural network with second-order techniques, we only utilize the loss function's second-order information to replace it by a Newton Loss, while training the network with gradient descent. This makes our method computationally efficient. We apply Newton Losses to eight differentiable algorithms for sorting and shortest-paths, achieving significant improvements for less-optimized differentiable algorithms, and consistent improvements, even for well-optimized differentiable algorithms.