用反绝热项加速哈密顿蒙特卡洛，提升多模态采样效率
Counterdiabatic Hamiltonian Monte Carlo
- 在哈密顿量中加入学习到的反绝热项，实现快速演化
- 相比传统方法，采样速度更快，收敛更稳定
- 适合高维多模态分布采样，尤其适用于复杂优化问题
哈密顿蒙特卡洛（HMC）是采样可微密度分布的先进方法，但在处理多模态难题时收敛缓慢。通过使用随时间变化的哈密顿量从初始易处理分布逐步过渡到目标分布，可缓解此问题。结合加权方案消除偏差后，这可视为一种特殊形式的序贯蒙特卡洛（SMC）采样。然而，该方法效率受限于初始与最终分布间变化过慢。受 extit{Sels et al.} (2017) 启发，我们提出反绝热哈密顿蒙特卡洛（CHMC），通过在哈密顿量中引入学习得到的反绝热项，实现更高效的演化路径。该方法可视为具有更优核函数的SMC采样器，并与近期基于学习漂移项加速梯度采样的工作存在联系。我们在简单基准问题上进行了验证。
原文摘要 · Abstract (English)
Hamiltonian Monte Carlo (HMC) is a state of the art method for sampling from distributions with differentiable densities, but can converge slowly when applied to challenging multimodal problems. Running HMC with a time varying Hamiltonian, in order to interpolate from an initial tractable distribution to the target of interest, can address this problem. In conjunction with a weighting scheme to eliminate bias, this can be viewed as a special case of Sequential Monte Carlo (SMC) sampling \cite{doucet2001introduction}. However, this approach can be inefficient, since it requires slow change between the initial and final distribution. Inspired by \cite{sels2017minimizing}, where a learned \emph{counterdiabatic} term added to the Hamiltonian allows for efficient quantum state preparation, we propose \emph{Counterdiabatic Hamiltonian Monte Carlo} (CHMC), which can be viewed as an SMC sampler with a more efficient kernel. We establish its relationship to recent proposals for accelerating gradient-based sampling with learned drift terms, and demonstrate on simple benchmark problems.
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