A Variational Perspective on High-Resolution ODEs
This work addresses optimization challenges in machine learning and numerical analysis, offering incremental improvements through a new theoretical framework and algorithm.
The paper tackles the problem of unconstrained minimization of smooth convex functions by proposing a novel variational perspective using forced Euler-Lagrange equations to study high-resolution ODEs, resulting in a faster convergence rate for gradient norm minimization with Nesterov's accelerated gradient method and a new stochastic algorithm for noisy gradients.
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.