Accelerated Convex Optimization via Hamiltonian Dynamics with Deterministic Integration Time

2026-06-15

AI Summary

Existing deterministic acceleration methods are typically restricted to quadratic objectives or hold only in expectation. This work proposes a general framework grounded in Hamiltonian dynamics, introducing Hamiltonian flows as a novel algorithmic primitive for deterministic accelerated convex optimization—thereby eliminating reliance on quadratic structure or randomness. By analyzing the contraction properties of trajectory-averaged flows rather than endpoint contraction, and by synergistically integrating continuous-time dynamics with discrete-time implementation, the authors construct a first-order optimization algorithm that achieves deterministic accelerated convergence for smooth convex optimization. The method attains the optimal first-order oracle complexity and comes with rigorous theoretical guarantees.

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.

Problem

• convex optimization
• Hamiltonian dynamics
• accelerated convergence
• deterministic algorithms
• first-order methods

Innovation

• Hamiltonian dynamics
• accelerated optimization
• deterministic convergence
• convex optimization
• first-order methods