A stochastic first-order method with multi-extrapolated momentum for highly smooth unconstrained optimization

• URL: http://arxiv.org/abs/2412.14488v2
• Date: Fri, 10 Jan 2025 13:52:14 GMT
• Title: A stochastic first-order method with multi-extrapolated momentum for highly smooth unconstrained optimization
• Authors: Chuan He,
• Abstract summary: We show that the proposed SFOM can accelerate optimization by exploiting the high-order smoothness of the objective function $f$. To the best of our knowledge, this is the first SFOM to leverage arbitrary-order smoothness of the objective function for acceleration.
• Score: 3.8919212824749296
• License:
• Abstract: In this paper, we consider an unconstrained stochastic optimization problem where the objective function exhibits high-order smoothness. Specifically, we propose a new stochastic first-order method (SFOM) with multi-extrapolated momentum, in which multiple extrapolations are performed in each iteration, followed by a momentum update based on these extrapolations. We demonstrate that the proposed SFOM can accelerate optimization by exploiting the high-order smoothness of the objective function $f$. Assuming that the $p$th-order derivative of $f$ is Lipschitz continuous for some $p\ge2$, and under additional mild assumptions, we establish that our method achieves a sample complexity of $\widetilde{\mathcal{O}}(\epsilon^{-(3p+1)/p})$ for finding a point $x$ such that $\mathbb{E}[\|\nabla f(x)\|]\le\epsilon$. To the best of our knowledge, this is the first SFOM to leverage arbitrary-order smoothness of the objective function for acceleration, resulting in a sample complexity that improves upon the best-known results without assuming the mean-squared smoothness condition. Preliminary numerical experiments validate the practical performance of our method and support our theoretical findings.

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