基于哈密顿动力学实现确定性加速凸优化
Accelerated Convex Optimization via Hamiltonian Dynamics with Deterministic Integration Time
- 利用平均哈密顿流的收缩特性设计优化算法
- 在确定性条件下达到最优一阶复杂度
- 适用于需要可靠收敛保证的优化场景
我们提出基于哈密顿动力学的平滑凸优化算法，实现加速收敛。通过利用平均哈密顿流轨迹的收缩性，而非仅依赖轨迹终点的收缩，证明了此类方法可在确定性条件下获得加速收敛保证，突破了以往仅适用于二次目标或仅在期望意义下成立的局限。分析理想连续时间算法并推导出最优一阶复杂度的离散实现，确立了哈密顿动力学作为确定性加速凸优化的有效算法基元。
原文摘要 · Abstract (English)
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.
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