Generalized Gradient Norm Clipping & Non-Euclidean $(L_0,L_1)$-Smoothness

TL;DR

This paper proposes a hybrid non-Euclidean optimization method that generalizes gradient norm clipping using ($L_0$,$L_1$)-smoothness, combining steepest descent and conditional gradient techniques, with applications to deep learning.

Contribution

It introduces a novel optimization algorithm that unifies gradient clipping and Frank-Wolfe methods under a generalized smoothness framework, with theoretical and practical insights.

Findings

Achieves an $O(n^{-1/4})$ convergence rate in stochastic settings.

Demonstrates effectiveness on image classification and language modeling tasks.

Provides a principled way to incorporate weight decay in non-Euclidean optimization.

Abstract

This work introduces a hybrid non-Euclidean optimization method which generalizes gradient norm clipping by combining steepest descent and conditional gradient approaches. The method achieves the best of both worlds by establishing a descent property under a generalized notion of ($L_0$,$L_1$)-smoothness. Weight decay is incorporated in a principled manner by identifying a connection to the Frank-Wolfe short step. In the stochastic case, we show an order optimal $O(n^{-1/4})$ convergence rate by leveraging a momentum based gradient estimator. We discuss how to instantiate the algorithms for deep learning, which we dub Clipped Scion, and demonstrate their properties on image classification and language modeling. The code is available at https://github.com/LIONS-EPFL/ClippedScion.