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Gradient Flossing: Improving Gradient Descent through Dynamic Control of Jacobians (2312.17306v1)

Published 28 Dec 2023 in cs.LG, cs.AI, nlin.CD, q-bio.NC, and stat.ML

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 demonstrate 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 prior to 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 gradient problem.

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Authors (1)
  1. Rainer Engelken (5 papers)
Citations (3)
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Summary

  • The paper introduces gradient flossing as a novel technique to regulate Lyapunov exponents and improve long-term error propagation in RNNs.
  • It demonstrates that gradient flossing accelerates convergence and boosts success rates on synthetic tasks by stabilizing training in various RNN architectures.
  • The method’s compatibility with advanced initialization techniques highlights its potential to mitigate exploding and vanishing gradients in deep learning.

Introduction

Training recurrent neural networks (RNNs) poses significant challenges due to the potential instability of gradients when information is propagated across many time steps. These unstable gradients can either explode or diminish sharply, resulting in poor network training performance. To address this, researchers have explored a variety of methods aimed at stabilizing gradients, including the use of specialized units like LSTMs and GRUs, gradient clipping, normalization techniques, and specialized network architectures.

Understanding Gradient Instability

When RNNs are trained on tasks with long time dependencies, the chain of recursive derivatives can lead to exponentially amplified or reduced gradients, known as exploding and vanishing gradients. This gradient instability is closely related to the singular value spectrum of long-term Jacobians and the Lyapunov exponents from dynamical systems theory. The latter describes how infinitesimal perturbations either diverge or converge over time, affecting the network's ability to learn across extensive temporal intervals.

Gradient Flossing: A Novel Approach

A new technique called gradient flossing aims to address this instability by regulating Lyapunov exponents. By keeping these exponents near zero, gradient flossing improves the condition number of the long-term Jacobian, enabling error signals to propagate over longer time horizons without destabilizing. The effectiveness of gradient flossing is demonstrated through its application to various RNN architectures and tasks with differing temporal complexity.

Benefits of Gradient Flossing During Training

Empirical tests on synthetic tasks indicate that gradient flossing not only enhances the success rate and speed of convergence, but also remains beneficial when applied during training. By providing regular stabilization to the training process, it allows networks to maintain robustness in learning over extended time sequences. Additionally, when combined with other strategies such as dynamic mean-field theory for initialization and orthogonal initialization, the efficacy of gradient flossing in mitigating gradient instabilities is further showcased.

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