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| Main Authors: | , , , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2505.11748 |
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| _version_ | 1866913844371128320 |
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| author | Zhang, Wei Zidan, Arif Hassan Jahin, Afrar Bao, Yu Liu, Tianming |
| author_facet | Zhang, Wei Zidan, Arif Hassan Jahin, Afrar Bao, Yu Liu, Tianming |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_11748 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | HOME-3: High-Order Momentum Estimator with Third-Power Gradient for Convex and Smooth Nonconvex Optimization Zhang, Wei Zidan, Arif Hassan Jahin, Afrar Bao, Yu Liu, Tianming Machine Learning 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. |
| title | HOME-3: High-Order Momentum Estimator with Third-Power Gradient for Convex and Smooth Nonconvex Optimization |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2505.11748 |