We consider Gibbs distributions, which are families of probability distributions over a discrete space $\Omega$ with probability mass function of the form $\mu^\Omega_\beta(\omega) \propto e^{\beta H(\omega)}$ for $\beta$ in an interval $[\beta_{\min}, \beta_{\max}]$ and $H( \omega ) \in \{0 \} \cup [1, n]$. The partition function is the normalization factor $Z(\beta)=\sum_{\omega \in\Omega}e^{\beta H(\omega)}$. Two important parameters of these distributions are the log partition ratio $q = \log \tfrac{Z(\beta_{\max})}{Z(\beta_{\min})}$ and the counts $c_x = |H^{-1}(x)|$. These are correlated with system parameters in a number of physical applications and sampling algorithms. Our first main result is to estimate the counts $c_x$ using roughly $\tilde O( \frac{q}{\varepsilon^2})$ samples for general Gibbs distributions and $\tilde O( \frac{n^2}{\varepsilon^2} )$ samples for integer-valued distributions (ignoring some second-order terms and parameters), and we show this is optimal up to logarithmic factors. We illustrate with improved algorithms for counting connected subgraphs, independent sets, and perfect matchings. As a key subroutine, we also develop algorithms to compute the partition function $Z$ using $\tilde O(\frac{q}{\varepsilon^2})$ samples for general Gibbs distributions and using $\tilde O(\frac{n^2}{\varepsilon^2})$ samples for integer-valued distributions.

翻译：我们考虑吉布斯分布——离散空间 $\Omega$ 上的一族概率分布，其概率质量函数形式为 $\mu^\Omega_\beta(\omega) \propto e^{\beta H(\omega)}$，其中参数 $\beta$ 位于区间 $[\beta_{\min}, \beta_{\max}]$ 内，$H( \omega ) \in \{0 \} \cup [1, n]$。配分函数为归一化因子 $Z(\beta)=\sum_{\omega \in\Omega}e^{\beta H(\omega)}$。这些分布的两个重要参数是对数配分比 $q = \log \tfrac{Z(\beta_{\max})}{Z(\beta_{\min})}$ 和计数 $c_x = |H^{-1}(x)|$。在诸多物理应用和采样算法中，这些参数与系统参数存在关联。我们的第一个主要结果是：对于一般吉布斯分布，利用约 $\tilde O( \frac{q}{\varepsilon^2})$ 个样本可估计计数 $c_x$；对于整数取值分布，则需约 $\tilde O( \frac{n^2}{\varepsilon^2} )$ 个样本（忽略部分二阶项和参数），且我们证明了该结果在忽略对数因子的意义下是最优的。我们通过改进连通子图计数、独立集计数和完美匹配计数算法进行了验证。作为关键子程序，我们还开发了计算配分函数 $Z$ 的算法：对于一般吉布斯分布需 $\tilde O(\frac{q}{\varepsilon^2})$ 个样本，对于整数取值分布需 $\tilde O(\frac{n^2}{\varepsilon^2})$ 个样本。