HOME-3: High-Order Momentum Estimator with Third-Power Gradient for Convex and Smooth Nonconvex Optimization

TL;DR

This paper introduces high-order momentum using third-power gradients to enhance convergence in convex and nonconvex optimization, supported by theoretical analysis and extensive empirical validation.

Contribution

It presents the novel concept of high-order momentum with third-power gradients, demonstrating improved convergence bounds and empirical performance over traditional methods.

Findings

High-order momentum improves convergence bounds.

Empirical results show superior performance across tasks.

Outperforms conventional low-order momentum methods.

Abstract

Momentum-based gradients are essential for optimizing advanced machine learning models, as they not only accelerate convergence but also advance optimizers to escape stationary points. While most state-of-the-art momentum techniques utilize lower-order gradients, such as the squared first-order gradient, there has been limited exploration of higher-order gradients, particularly those raised to powers greater than two. In this work, we introduce the concept of high-order momentum, where momentum is constructed using higher-power gradients, with a focus on the third-power of the first-order gradient as a representative case. Our research offers both theoretical and empirical support for this approach. Theoretically, we demonstrate that incorporating third-power gradients can improve the convergence bounds of gradient-based optimizers for both convex and smooth nonconvex problems. Empirically, we validate these findings through extensive experiments across convex, smooth nonconvex, and nonsmooth nonconvex optimization tasks. Across all cases, high-order momentum consistently outperforms conventional low-order momentum methods, showcasing superior performance in various optimization problems.