Accelerated Convex Optimization via Hamiltonian Dynamics with Deterministic Integration Time
For the optimization community, it provides a deterministic accelerated method for general smooth convex problems, addressing limitations of prior Hamiltonian-based approaches.
The paper develops Hamiltonian dynamics-based algorithms for smooth convex optimization that achieve accelerated convergence rates, extending prior work beyond quadratic objectives or expected guarantees. The discrete-time implementations achieve optimal first-order complexity.
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.