Accelerated Convex Optimization via Hamiltonian Dynamics with Deterministic Integration Time
Xiuyuan Wang, Vishwak Srinivasan, Qiang Fu, Siddharth Mitra, Ashia Wilson, Andre Wibisono
math.OC, cs.LG, stat.ML
Submitted: 2026-06-15
Comments: 51 pages, 7 figures. Accepted to the 39th Annual Conference on Learning Theory (COLT 2026)
Code: https://github.com/vishwakftw/dHFA
License: http://creativecommons.org/licenses/by/4.0/
The gist: We develop Hamiltonian dynamics-based algorithms for smooth convex optimization that achieve accelerated rates of convergence.
Terminology
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.
Sources
- On Symplectic Optimization
- Tail-Sensitive KL and R'enyi Convergence of Unadjusted Hamiltonian Monte Carlo via One-Shot Couplings
- A geometric alternative to Nesterov's accelerated gradient descent
- Improving Energy Conserving Descent for Machine Learning: Theory and Practice
- Hamiltonian Descent Methods
- Frictionless Hamiltonian Descent and Coordinate Hamiltonian Descent for Strongly Convex Quadratic Problems
- Gradient methods for convex minimization: better rates under weaker conditions
Related papers
- Lions and Muons: Optimization via Stochastic Frank-Wolfe under Heavy-Tailed Noise
- Adam-HNAG: A Convergent Reformulation of Adam with Accelerated Rate
- Incremental Learning in Mirror Flows
- Beyond Aggregate Drift: Optimal Client-Specific Windows in Federated Learning
- Lower Bounds for Randomized Linear-Oracle Optimization over Strongly Convex Sets
- Second-Order Optimization with Asynchronous Curvature