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Rethinking the Variational Interpretation of Accelerated Optimization Methods
Peiyuan Zhang · Antonio Orvieto · Hadi Daneshmand

Thu Dec 09 12:30 AM -- 02:00 AM (PST) @

The continuous-time model of Nesterov's momentum provides a thought-provoking perspective for understanding the nature of the acceleration phenomenon in convex optimization. One of the main ideas in this line of research comes from the field of classical mechanics and proposes to link Nesterov's trajectory to the solution of a set of Euler-Lagrange equations relative to the so-called Bregman Lagrangian. In the last years, this approach led to the discovery of many new (stochastic) accelerated algorithms and provided a solid theoretical foundation for the design of structure-preserving accelerated methods. In this work, we revisit this idea and provide an in-depth analysis of the action relative to the Bregman Lagrangian from the point of view of calculus of variations. Our main finding is that, while Nesterov's method is a stationary point for the action, it is often not a minimizer but instead a saddle point for this functional in the space of differentiable curves. This finding challenges the main intuition behind the variational interpretation of Nesterov's method and provides additional insights into the intriguing geometry of accelerated paths.

Author Information

Peiyuan Zhang (ETH Zurich)
Antonio Orvieto (ETH Zurich)

PhD Student at ETH Zurich. I’m interested in the design and analysis of optimization algorithms for deep learning. Interned at DeepMind, MILA, and Meta. All publications at http://orvi.altervista.org/ Looking for postdoc positions! :) antonio.orvieto@inf.ethz.ch

Hadi Daneshmand (INRIA PARIS)

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