A Methodology Establishing Linear Convergence of Adaptive Gradient Methods under PL Inequality

• URL: http://arxiv.org/abs/2407.12629v1
• Date: Wed, 17 Jul 2024 14:56:21 GMT
• Title: A Methodology Establishing Linear Convergence of Adaptive Gradient Methods under PL Inequality
• Authors: Kushal Chakrabarti, Mayank Baranwal,
• Abstract summary: We show that AdaGrad and Adam converge linearly when the cost function is smooth and satisfies the Polyak-Lojasiewicz inequality. Our framework can potentially be utilized in analyzing linear convergence of other variants of Adam.
• Score: 5.35599092568615
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Adaptive gradient-descent optimizers are the standard choice for training neural network models. Despite their faster convergence than gradient-descent and remarkable performance in practice, the adaptive optimizers are not as well understood as vanilla gradient-descent. A reason is that the dynamic update of the learning rate that helps in faster convergence of these methods also makes their analysis intricate. Particularly, the simple gradient-descent method converges at a linear rate for a class of optimization problems, whereas the practically faster adaptive gradient methods lack such a theoretical guarantee. The Polyak-{\L}ojasiewicz (PL) inequality is the weakest known class, for which linear convergence of gradient-descent and its momentum variants has been proved. Therefore, in this paper, we prove that AdaGrad and Adam, two well-known adaptive gradient methods, converge linearly when the cost function is smooth and satisfies the PL inequality. Our theoretical framework follows a simple and unified approach, applicable to both batch and stochastic gradients, which can potentially be utilized in analyzing linear convergence of other variants of Adam.

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