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Algorithmic Regularization in Learning Deep Homogeneous Models: Layers are Automatically Balanced (1806.00900v2)

Published 4 Jun 2018 in cs.LG, math.OC, and stat.ML

Abstract: We study the implicit regularization imposed by gradient descent for learning multi-layer homogeneous functions including feed-forward fully connected and convolutional deep neural networks with linear, ReLU or Leaky ReLU activation. We rigorously prove that gradient flow (i.e. gradient descent with infinitesimal step size) effectively enforces the differences between squared norms across different layers to remain invariant without any explicit regularization. This result implies that if the weights are initially small, gradient flow automatically balances the magnitudes of all layers. Using a discretization argument, we analyze gradient descent with positive step size for the non-convex low-rank asymmetric matrix factorization problem without any regularization. Inspired by our findings for gradient flow, we prove that gradient descent with step sizes $\eta_t = O\left(t{-\left( \frac12+\delta\right)} \right)$ ($0<\delta\le\frac12$) automatically balances two low-rank factors and converges to a bounded global optimum. Furthermore, for rank-$1$ asymmetric matrix factorization we give a finer analysis showing gradient descent with constant step size converges to the global minimum at a globally linear rate. We believe that the idea of examining the invariance imposed by first order algorithms in learning homogeneous models could serve as a fundamental building block for studying optimization for learning deep models.

Citations (218)

Summary

  • The paper demonstrates that gradient flow inherently enforces an invariance in the squared norms of weight matrices across layers, leading to automatic layer balancing.
  • The study provides rigorous theoretical analysis and empirical validation on both fully connected and convolutional architectures, proving convergence to global optima even with finite step sizes.
  • By examining the implicit regularization of gradient descent, the research offers actionable insights for simplifying network initialization and regularization in deep learning models.

Essay on Algorithmic Regularization in Learning Deep Homogeneous Models

The paper "Algorithmic Regularization in Learning Deep Homogeneous Models: Layers are Automatically Balanced" by Simon S. Du, Wei Hu, and Jason D. Lee investigates the implicit regularization effects of gradient descent methods on multilayer homogeneous networks. Specifically, the paper examines how gradient flow, which is an idealized form of gradient descent with infinitesimal step size, contributes to layer magnitude balancing in deep neural networks.

Overview

The authors focus on a class of homogeneous functions critical in modern machine learning, notably those with linear, ReLU, or Leaky ReLU activations within fully connected and convolutional network architectures. They identify and mathematically prove that gradient flow inherently enforces an invariance in the squared norms of weight matrices across these layers. This invariance implies that if weight matrices begin with balanced norms, they remain balanced over the course of training, even in the absence of explicit regularization.

Main Contributions

  1. Theoretical Analysis: The paper provides rigorous analysis showing that during gradient flow training, the differences between squared norms of weights at consecutive layers remain unchanged. This invariance results in the automatic balancing of initial small weight norms across the layers.
  2. Empirical Validation: Through experiments, the authors demonstrate that gradient descent without regularization successfully converges to a globally optimal solution in practice, asserting the theoretical findings in empirical settings.
  3. Low-rank Matrix Factorization: The authors expand their findings to the asymmetric matrix factorization problem, showing that a similar invariance holds even when finite step size gradient descent is applied. They leverage the implicit balancing properties to prove convergence to bounded global optima with specific step sizes.
  4. Extension to Convolutional Architectures: The invariance property is generalized beyond fully connected networks to those with arbitrary sparsity patterns and weight-sharing structures, including CNNs, underlining the broad applicability of their findings.

Implications

The implications of these findings are both practical and theoretical. Practically, the automatic balancing of layers implies potential simplifications in network initialization and regularization requirements during training, as well-balanced weights can improve convergence properties of optimization algorithms. Theoretically, the paper contributes to the understanding of implicit biases imposed by algorithmic choices like gradient descent, offering a foundational insight for future exploration in deep learning optimization.

Future Directions

The investigation opens several avenues for future research. One immediate direction is to explore how the invariance properties extend to gradient-based methods beyond gradient descent, such as those with adaptive learning rates or momentum. Another interesting route is developing theoretical tools that seamlessly transition from analyzing gradient flow to finite step-size gradient descent dynamics. Further, the interaction of these implicit balancing properties with other forms of regularization or constraint imposition can be explored, to better harness and control these inherent biases for improved model training.

This paper deepens the understanding of the internal mechanics of gradient descent in the context of deep learning and solidifies the foundational link between algorithm design and model training outcomes. Such insights could pave the way for more robust and efficient training regimens in various neural network architectures.

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