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.

翻译：我们开发了基于哈密顿动力学的光滑凸优化算法，实现了加速收敛率。通过利用平均哈密顿流轨迹的收缩性而非要求轨迹端点的收缩性，我们证明基于哈密顿动力学的优化方法具有确定性和加速收敛保证，拓展了先前仅限于二次目标函数或仅在期望意义下成立的工作。我们分析了一种理想化的连续时间算法，并推导了具有最优一阶复杂度的实用离散时间实现，从而确立了哈密顿动力学作为确定性加速凸优化中一种有用的算法基元。