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Geometric Dynamics of Signal Propagation Predict Trainability of Transformers (2403.02579v1)

Published 5 Mar 2024 in cond-mat.dis-nn and cs.LG

Abstract: We investigate forward signal propagation and gradient back propagation in deep, randomly initialized transformers, yielding simple necessary and sufficient conditions on initialization hyperparameters that ensure trainability of deep transformers. Our approach treats the evolution of the representations of $n$ tokens as they propagate through the transformer layers in terms of a discrete time dynamical system of $n$ interacting particles. We derive simple update equations for the evolving geometry of this particle system, starting from a permutation symmetric simplex. Our update equations show that without MLP layers, this system will collapse to a line, consistent with prior work on rank collapse in transformers. However, unlike prior work, our evolution equations can quantitatively track particle geometry in the additional presence of nonlinear MLP layers, and it reveals an order-chaos phase transition as a function of initialization hyperparameters, like the strength of attentional and MLP residual connections and weight variances. In the ordered phase the particles are attractive and collapse to a line, while in the chaotic phase the particles are repulsive and converge to a regular $n$-simplex. We analytically derive two Lyapunov exponents: an angle exponent that governs departures from the edge of chaos in this particle system, and a gradient exponent that governs the rate of exponential growth or decay of backpropagated gradients. We show through experiments that, remarkably, the final test loss at the end of training is well predicted just by these two exponents at the beginning of training, and that the simultaneous vanishing of these two exponents yields a simple necessary and sufficient condition to achieve minimal test loss.

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Citations (6)
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Summary

  • The paper demonstrates that analyzing the geometry of signal propagation reveals an order-chaos phase transition crucial for transformer trainability.
  • It identifies two critical Lyapunov exponents that correlate with final test loss and effectively regulate gradient behavior.
  • The study provides practical initialization hyperparameters that improve performance in applications like NLP and computer vision.

Geometric Dynamics of Signal Propagation and its Effect on Trainability in Deep Transformers

Overview

Recent research provides a comprehensive paper of the propagation of signals and gradients in deep transformers, revealing insights into the trainability of these models through a geometric lens. By analyzing the behavior of signal propagation, this paper identifies necessary and sufficient conditions for the initialization of transformer models that ensure effective training, demonstrating that the trainability of deep transformers can be predicted with high accuracy at the initialization stage.

Signal Propagation in Transformers

The paper begins by exploring the forward propagation of signals through transformers, revealing an order-chaos phase transition. The behavior of token representations through the layers can be classified into two distinct phases based on the strength of attentional and MLP (multilayer perceptron) residual connections and weight variances. In the ordered phase, token representations tend to collapse, converging to a linear configuration. Conversely, in the chaotic phase, representations disperse, converging to a more complex geometric configuration known as a regular n-simplex.

This dynamical system approach treats the evolution of token representations as interacting particles, with the geometry of these interactions revealing significant insights into the behavior of deep transformers. The paper extends existing work by providing quantitative descriptions of signal propagation that include nonlinear MLP layers, offering a more comprehensive understanding of transformer behavior at initialization.

Phase Transitions and Trainability

A key finding of this paper is the identification of two critical Lyapunov exponents that measure the departure from phase boundaries in initialization hyperparameter space. These exponents are pivotal in predicting the final test loss at the end of training, with their simultaneous vanishing providing a simple yet effective criterion for achieving minimal test loss.

The research uncovers two distinct phase transitions in transformers: one related to signal propagation leading to either a collapsed or a chaotic phase, and another associated with the backpropagation of gradients, indicating either vanishing or exploding gradients. Adjusting initialization hyperparameters to situate the model at the intersection of these phase boundaries ensures both optimal trainability and minimal test loss.

Practical Implications and Future Directions

The practical implications of this research are profound for the development and training of deep transformer models. By providing a set of initialization hyperparameters that ensure efficient training, the paper offers a pathway to improve the performance of transformers in various applications, from natural language processing to computer vision tasks.

Looking forward, this framework opens up new avenues for exploring other architectures and initialization schemes. The geometric perspective introduced could lead to a deeper understanding of model behavior across different domains, potentially unlocking more efficient and robust methods for training deep learning models.

Conclusion

This paper's geometric analysis of signal propagation in deep transformers unveils a necessary and sufficient condition for their trainability, rooted in the intricate dynamics of signal and gradient propagation. By bridging the gap between initialization and trainability, this work lays the groundwork for more predictable and efficient training of transformer models, marking a significant step forward in our understanding of deep learning systems.

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