强对数凹分布Z形抽样算法的复杂性
Complexity of zigzag sampling algorithm for strongly log-concave distributions
摘要：
研究了强对数凹分布下锯齿形抽样算法的计算复杂性。Z字形过程具有不需要时间离散化来实现的优点，并且每个提出的弹跳事件仅需要对电势的偏导数进行一次评估，而其收敛速率是与维度无关的。使用这些属性，我们证明了锯齿形采样算法实现$\varepsilon$错误卡方发散的计算成本相当于$O\bigl(κ^2 d^\frac{1}{2}(\log\frac{1}{\varepsilon})^{\frac{3}{2}}\bigr)$梯度评价制度$κ\ll \frac{d}{\log d}$下的热启动假设，其中$κ$是条件数和$d$是维度。
英文摘要：
We study the computational complexity of zigzag sampling algorithm for strongly log-concave distributions. The zigzag process has the advantage of not requiring time discretization for implementation, and that each proposed bouncing event requires only one evaluation of partial derivative of the potential, while its convergence rate is dimension independent. Using these properties, we prove that the zigzag sampling algorithm achieves $\varepsilon$ error in chi-square divergence with a computational cost equivalent to $O\bigl(κ^2 d^\frac{1}{2}(\log\frac{1}{\varepsilon})^{\frac{3}{2}}\bigr)$ gradient evaluations in the regime $κ\ll \frac{d}{\log d}$ under a warm start assumption, where $κ$ is the condition number and $d$ is the dimension.
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