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

2025-06-02

AI Summary

This work addresses the limitations of gradient descent and conditional gradient methods (Frank–Wolfe) in non-Euclidean optimization. The proposed generalized framework unifies their strengths through three key contributions: (1) the first $(L_0, L_1)$-smoothness theory, which rigorously characterizes the coupled effects of gradients and curvature in non-Euclidean spaces; (2) a generalized gradient norm clipping mechanism that guarantees strict descent under this smoothness condition; and (3) a principled integration of weight decay with the Frank–Wolfe short-step rule, augmented by momentum-based gradient estimation, achieving the optimal $O(n^{-1/4})$ convergence rate for stochastic optimization. Empirically, the method significantly improves training stability and generalization performance on image classification and language modeling benchmarks.

Technology Category

• Search and Optimization: Non-convex Optimization
• Natural Language Processing: Learning & Optimization for NLP
• Machine Learning: Optimization

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

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 and demonstrate their properties on image classification and language modeling.

Problem

• Generalizes gradient norm clipping for hybrid optimization
• Establishes descent under non-Euclidean (L0,L1)-smoothness
• Achieves optimal convergence in stochastic settings

Innovation

• Hybrid non-Euclidean optimization method
• Generalized gradient norm clipping
• Momentum based gradient estimator