Gradient Flossing: Improving Gradient Descent through Dynamic Control of Jacobians

Rainer Engelken

Keywords: exploding/vanishing gradients, Lyapunov exponents, Lyapunov spectrum, chaos, RNN, condition number, Jacobian

TL;DR: We propose a novel method to analyze and ameliorate the vanishing/exploding gradient problem based on concepts form dynamical systems theory (Lyapunov exponents)

Abstract:

Training recurrent neural networks (RNNs) remains a challenge due to the instability of gradients across long time horizons, which can lead to exploding and vanishing gradients. Recent research has linked these problems to the values of Lyapunov exponents for the forward-dynamics, which describe the growth or shrinkage of infinitesimal perturbations. Here, we propose gradient flossing, a novel approach to tackling gradient instability by pushing Lyapunov exponents of the forward dynamics toward zero during learning. We achieve this by regularizing Lyapunov exponents through backpropagation using differentiable linear algebra. This enables us to "floss" the gradients, stabilizing them and thus improving network training. We show that gradient flossing controls not only the gradient norm but also the condition number of the long-term Jacobian, facilitating multidimensional error feedback propagation. We find that applying gradient flossing before training enhances both the success rate and convergence speed for tasks involving long time horizons.
For challenging tasks, we show that gradient flossing during training can further increase the time horizon that can be bridged by backpropagation through time. Moreover, we demonstrate the effectiveness of our approach on various RNN architectures and tasks of variable temporal complexity. Additionally, we provide a simple implementation of our gradient flossing algorithm that can be used in practice. Our results indicate that gradient flossing via regularizing Lyapunov exponents can significantly enhance the effectiveness of RNN training and mitigate the exploding and vanishing gradients problem.

Submission Number: 1463