Extended convexity and smoothness and their applications in deep learning

• URL: http://arxiv.org/abs/2410.05807v1
• Date: Tue, 8 Oct 2024 08:40:07 GMT
• Title: Extended convexity and smoothness and their applications in deep learning
• Authors: Binchuan Qi,
• Abstract summary: In this paper, we introduce the $mathcal$H$smoothness algorithm for non-completely understood gradient and strong convexity. The effectiveness of the proposed methodology is validated through experiments.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: The underlying mechanism by which simple gradient-based iterative algorithms can effectively handle the non-convex problem of deep model training remains incompletely understood within the traditional convex and non-convex analysis frameworks, which often require the Lipschitz smoothness of the gradient and strong convexity. In this paper, we introduce $\mathcal{H}(\phi)$-convexity and $\mathcal{H}(\Phi)$-smoothness, which broaden the existing concepts of smoothness and convexity, and delineate their fundamental properties. Building on these concepts, we introduce the high-order gradient descent and high-order stochastic gradient descent methods, which serve as extensions to the traditional gradient descent and stochastic gradient descent methods, respectively. Furthermore, we establish descent lemmas for the $\mathcal{H}(\phi)$-convex and $\mathcal{H}(\Phi)$-smooth objective functions when utilizing these four methods. On the basis of these findings, we develop the gradient structure control algorithm to address non-convex optimization objectives, encompassing both the functions represented by machine learning models and common loss functions in deep learning. The effectiveness of the proposed methodology is empirically validated through experiments.

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