A Variational Perspective on High-Resolution ODEs

• URL: http://arxiv.org/abs/2311.02002v1
• Date: Fri, 3 Nov 2023 16:00:40 GMT
• Title: A Variational Perspective on High-Resolution ODEs
• Authors: Hoomaan Maskan, Konstantinos C. Zygalakis, Alp Yurtsever
• Abstract summary: We consider unconstrained minimization of convex functions using forced Euler-Lagrange equation. We obtain a faster convergence rate for gradient norm minimization using Nesterov's accelerated gradient method.
• Score: 10.036727981085223
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We consider unconstrained minimization of smooth convex functions. We propose a novel variational perspective using forced Euler-Lagrange equation that allows for studying high-resolution ODEs. Through this, we obtain a faster convergence rate for gradient norm minimization using Nesterov's accelerated gradient method. Additionally, we show that Nesterov's method can be interpreted as a rate-matching discretization of an appropriately chosen high-resolution ODE. Finally, using the results from the new variational perspective, we propose a stochastic method for noisy gradients. Several numerical experiments compare and illustrate our stochastic algorithm with state of the art methods.

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