Ablation study wrt. the Tikhonov regularization strength hyperparameter λ. Displayed is the element-wise ranking accuracy (individual element ranks correctly identified), averaged over 10 seeds, and additionallyeach seed with low opacity in the background. Left: NeuralSort. Right: SoftSort. Each for n = 5. Newton Losses, and for both the Hessian and the Fisher variant, significantly improve over the baseline for up to (or beyond) 6 orders of magnitude in variation of its hyperparameter λ. Note the logarithmic horizontal axis.
Material
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