Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Geometry of Sign Gradient Descent

Lukas Balles, Fabian Pedregosa, Nicolas Le Roux

TL;DR

The paper analyzes sign-based optimization methods and their relationship to Adam, arguing that analysis under $\,\ell_\infty$-smoothness is the natural and weaker framework for these methods. It proves that separable smoothness implies Linf-smoothness with $L_\infty=\sum_i l_i$, and that Linf-smoothness suffices for existing sign-based convergence results, thereby unifying prior work. By linking $L_\infty$ to Hessian properties, the authors show sign-based methods benefit when the Hessian is diagonally concentrated and the largest eigenvalue far exceeds the average, a regime common in deep networks. They also experimentally demonstrate that Adam behaves similarly to signSGD with momentum, suggesting the elementwise adaptivity plays a secondary role. Overall, the work offers a geometric understanding of when sign-based methods are advantageous and provides a unified theoretical framework with practical implications for deep learning optimization.

Abstract

Sign-based optimization methods have become popular in machine learning due to their favorable communication cost in distributed optimization and their surprisingly good performance in neural network training. Furthermore, they are closely connected to so-called adaptive gradient methods like Adam. Recent works on signSGD have used a non-standard "separable smoothness" assumption, whereas some older works study sign gradient descent as steepest descent with respect to the $\ell_\infty$-norm. In this work, we unify these existing results by showing a close connection between separable smoothness and $\ell_\infty$-smoothness and argue that the latter is the weaker and more natural assumption. We then proceed to study the smoothness constant with respect to the $\ell_\infty$-norm and thereby isolate geometric properties of the objective function which affect the performance of sign-based methods. In short, we find sign-based methods to be preferable over gradient descent if (i) the Hessian is to some degree concentrated on its diagonal, and (ii) its maximal eigenvalue is much larger than the average eigenvalue. Both properties are common in deep networks.

The Geometry of Sign Gradient Descent

TL;DR

The paper analyzes sign-based optimization methods and their relationship to Adam, arguing that analysis under -smoothness is the natural and weaker framework for these methods. It proves that separable smoothness implies Linf-smoothness with , and that Linf-smoothness suffices for existing sign-based convergence results, thereby unifying prior work. By linking to Hessian properties, the authors show sign-based methods benefit when the Hessian is diagonally concentrated and the largest eigenvalue far exceeds the average, a regime common in deep networks. They also experimentally demonstrate that Adam behaves similarly to signSGD with momentum, suggesting the elementwise adaptivity plays a secondary role. Overall, the work offers a geometric understanding of when sign-based methods are advantageous and provides a unified theoretical framework with practical implications for deep learning optimization.

Abstract

Sign-based optimization methods have become popular in machine learning due to their favorable communication cost in distributed optimization and their surprisingly good performance in neural network training. Furthermore, they are closely connected to so-called adaptive gradient methods like Adam. Recent works on signSGD have used a non-standard "separable smoothness" assumption, whereas some older works study sign gradient descent as steepest descent with respect to the -norm. In this work, we unify these existing results by showing a close connection between separable smoothness and -smoothness and argue that the latter is the weaker and more natural assumption. We then proceed to study the smoothness constant with respect to the -norm and thereby isolate geometric properties of the objective function which affect the performance of sign-based methods. In short, we find sign-based methods to be preferable over gradient descent if (i) the Hessian is to some degree concentrated on its diagonal, and (ii) its maximal eigenvalue is much larger than the average eigenvalue. Both properties are common in deep networks.

Paper Structure

This paper contains 45 sections, 17 theorems, 109 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Analysis of Sign-Based Methods
  3. Contributions
  4. Sign Gradient Descent and Adam
  5. Separable Smoothness and linf-Smoothness
  6. Smoothness and Steepest Descent
  7. Smoothness w.r.t. Arbitrary Norms
  8. Steepest Descent
  9. Convergence of Steepest Descent
  10. Separable Smoothness and linf-Smoothness
  11. Consequences of the Unification
  12. Understanding linf-Smoothness
  13. Smoothness as a Bound on the Hessian
  14. Linf is Sensitive to Axis-Alignment
  15. Comparing Linf and L2
  16. ...and 30 more sections

Key Result

Lemma 1

If $f$ is $L$-smooth w.r.t. $\Vert\cdot\Vert$, then for all $\boldsymbol x, \boldsymbol x^\prime\in\mathbb{R}^d$.

Figures (3)

  • Figure 1: Original Adam compared to two variants based on the decomposition in Eq. \ref{['eq:Adam_decomposition']}. The factors $\boldsymbol{\gamma}_t$ are (i) randomly shuffled or (ii) averaged across coordinates. We add SGD with momentum for comparison. Both variants perform very similarly to original Adam, suggesting that Adam is primarily a sign-based method and the elementwise adaptivity plays a secondary role. Experimental details may be found in Appendix \ref{['apx:experimental_details']}.
  • Figure 2: For $\boldsymbol H\in\mathbb{R}^{2\times 2}$, we plot a contour line of $f(\boldsymbol x)=\frac{1}{2}\boldsymbol x^T\boldsymbol H\boldsymbol x$, which forms an ellipse with principal axes given by the eigenvectors of $\boldsymbol H$ and axis lengths given by the inverse eigenvalues. We fix $\lambda_\text{min}= 1$ and vary $\lambda_\text{max} > 1$ as well as the angle $\theta$ between the principal axes of the ellipse and the coordinate axes. (Mathematical details can be found in Appendix \ref{['apx:example_2d_quadratic']}.) (a) The $\ell_\infty$-smoothness constant $L_\infty=\Vert \boldsymbol H\Vert_{\infty, 1}$ is sensitive to the axis-alignment of the objective. This is in contrast to the Euclidean smoothness constant, which is simply given by $L_2=\lambda_\text{max}$. (b) Separable smoothness bounds $\boldsymbol H$ by a diagonal matrix, $\mathop{\mathrm{diag}}\nolimits(l_1,\dotsc, l_d) \succeq \boldsymbol H$, corresponding to an axis-aligned ellipse that lies fully within the $\boldsymbol H$-ellipse. The "best" bounding ellipse is given by Eq. \ref{['eq:Lsep_def']}. This bound changes with the axis-alignment, becoming both smaller and more circular (i.e., larger and more similar $l_i$) as we rotate further from the coordinate axes. In contrast to that, Euclidean smoothness bounds $\boldsymbol H$ by $\lambda_\text{max} \boldsymbol I\succeq \boldsymbol H$, i.e., a rotation-invariant circle.
  • Figure 3: We consider quadratic objectives varying across two axes: $\lambda_\text{max} / \bar{\lambda}$ as well as a rotation value $\theta$. The left plot depicts the ratio of the two relevant smoothness constants. $L_\infty$ is sensitive to $\theta$ and grows relative to $L_2=\lambda_\text{max}$ as the problem becomes less axis-aligned. The right plot depicts the relative performance of gradient descent and sign gradient descent on these problems. GD drastically (the colormap is clipped) outperforms signGD for mildly-conditioned (small $\lambda_\text{max}/\bar{\lambda}$) and non-axis-aligned (large $\theta$) problems. However, sign gradient descent is preferable for problems with high $\lambda_\text{max}/\bar{\lambda}$, given that they have some degree of axis-alignment (small $\theta$). The dashed line represents equal performance of both algorithms.

Theorems & Definitions (37)

  • Lemma 1
  • proof
  • Lemma 2
  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Proposition 5
  • Proposition 6
  • Proposition 7
  • ...and 27 more