对数凹分布的无穷远保证抽样
Sampling from Log-Concave Distributions with Infinity-Distance Guarantees
摘要：
对于约束到凸体$K$的$d$维对数凹分布$π(θ) \propto e^{-f(θ)}$，在差分私有优化中出现了从分布$ν$输出样本的问题，该分布$ν$是与$π$在无限距离$\sup_{θ\in K} |\log \frac{ν(θ)}{π(θ)}|$上接近的分布。虽然在$π$的全变差距离$\varepsilon$内的采样可以通过其运行时间多项式地依赖于$\frac{1}{\varepsilon}$的算法来完成，但是用于在$\varepsilon$无穷远距离中采样的现有算法具有多项式地依赖于$\frac{1}{\varepsilon}$的运行时间界限。我们提出了一种算法，输出一个点$\varepsilon$ -接近$π$在无限远的距离，需要在最多$\mathrm{poly}(\log \frac{1}{\varepsilon}, d)$调用成员资格的预言$K$和评估的预言$f$，当$f$是Lipschitz的，我们弥合了这一差距。我们的方法从以前的工作出发，构造马尔可夫链上的$\frac{1}{\varepsilon^2}$离散$K$，以实现一个样本与$\varepsilon$无限距离误差，并提出了一种方法，直接转换连续样本从$K$与全变差界的样本与无限界。这种方法也使我们能够获得一个改进的维度$d$的运行时间的问题，从一个对数凹分布的多面体$K$无限距离$\varepsilon$，通过插入在TV距离运行时间界的Dikin Walk马尔可夫链。
英文摘要：
For a $d$-dimensional log-concave distribution $π(θ) \propto e^{-f(θ)}$ constrained to a convex body $K$, the problem of outputting samples from a distribution $ν$ which is $\varepsilon$-close in infinity-distance $\sup_{θ\in K} |\log \frac{ν(θ)}{π(θ)}|$ to $π$ arises in differentially private optimization. While sampling within total-variation distance $\varepsilon$ of $π$ can be done by algorithms whose runtime depends polylogarithmically on $\frac{1}{\varepsilon}$, prior algorithms for sampling in $\varepsilon$ infinity distance have runtime bounds that depend polynomially on $\frac{1}{\varepsilon}$. We bridge this gap by presenting an algorithm that outputs a point $\varepsilon$-close to $π$ in infinity distance that requires at most $\mathrm{poly}(\log \frac{1}{\varepsilon}, d)$ calls to a membership oracle for $K$ and evaluation oracle for $f$, when $f$ is Lipschitz. Our approach departs from prior works that construct Markov chains on a $\frac{1}{\varepsilon^2}$-discretization of $K$ to achieve a sample with $\varepsilon$ infinity-distance error, and present a method to directly convert continuous samples from $K$ with total-variation bounds to samples with infinity bounds. This approach also allows us to obtain an improvement on the dimension $d$ in the running time for the problem of sampling from a log-concave distribution on polytopes $K$ with infinity distance $\varepsilon$, by plugging in TV-distance running time bounds for the Dikin Walk Markov chain.
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