Any-stepsize Gradient Descent for Separable Data under Fenchel-Young Losses

TL;DR

This paper investigates the convergence of gradient descent with arbitrary stepsizes on separable data using Fenchel-Young losses, revealing the importance of separation margin over self-bounding properties for convergence rates.

Contribution

It extends understanding of GD convergence beyond self-bounding losses by establishing arbitrary-stepsize convergence for Fenchel-Young losses, highlighting the role of separation margin.

Findings

Tsallis entropy achieves a convergence rate of Ω(ε^{-1/2})

Rényi entropy achieves a convergence rate of Ω(ε^{-1/3})

Separation margin, not self-bounding property, influences convergence rates

Abstract

The gradient descent (GD) has been one of the most common optimizer in machine learning. In particular, the loss landscape of a neural network is typically sharpened during the initial phase of training, making the training dynamics hover on the edge of stability. This is beyond our standard understanding of GD convergence in the stable regime where arbitrarily chosen stepsize is sufficiently smaller than the edge of stability. Recently, Wu et al. (COLT2024) have showed that GD converges with arbitrary stepsize under linearly separable logistic regression. Although their analysis hinges on the self-bounding property of the logistic loss, which seems to be a cornerstone to establish a modified descent lemma, our pilot study shows that other loss functions without the self-bounding property can make GD converge with arbitrary stepsize. To further understand what property of a loss function matters in GD, we aim to show arbitrary-stepsize GD convergence for a general loss function based on the framework of \emph{Fenchel--Young losses}. We essentially leverage the classical perceptron argument to derive the convergence rate for achieving $\epsilon$-optimal loss, which is possible for a majority of Fenchel--Young losses. Among typical loss functions, the Tsallis entropy achieves the GD convergence rate $T=\Omega(\epsilon^{-1/2})$, and the R{\'e}nyi entropy achieves the far better rate $T=\Omega(\epsilon^{-1/3})$. We argue that these better rate is possible because of \emph{separation margin} of loss functions, instead of the self-bounding property.