Accelerated Convex Optimization via Hamiltonian Dynamics with Deterministic Integration Time

Abstract:We develop Hamiltonian dynamics-based algorithms for smooth convex optimization that achieve accelerated rates of convergence. By exploiting contraction of averaged Hamiltonian flow trajectories rather than requiring contraction at trajectory endpoints, we show that Hamiltonian dynamics-based optimization methods admit deterministic and accelerated convergence guarantees, extending prior work that is limited to quadratic objectives or holds only in expectation. We analyze an idealized continuous-time algorithm and derive practical discrete-time implementations with optimal first-order complexity, thereby establishing Hamiltonian dynamics as a useful algorithmic primitive for deterministic accelerated convex optimization.

51 pages, 7 figures. Accepted to the 39th Annual Conference on Learning Theory (COLT 2026)

Optimization and Control (math.OC); Machine Learning (cs.LG); Machine Learning (stat.ML)

arXiv:2606.17260 [math.OC]

(or arXiv:2606.17260v1 [math.OC] for this version)

https://doi.org/10.48550/arXiv.2606.17260 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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