A Faster Approximation Algorithm for the Gibbs Partition Function

We consider the problem of estimating the partition function $Z(β)=\sum_x \exp(-β(H(x))$ of a Gibbs distribution with a Hamilton $H(\cdot)$, or more precisely the logarithm of the ratio $q=\ln Z(0)/Z(β)$. It has been recently shown how to approximate $q$ with high probability assuming the existence of an oracle that produces samples from the Gibbs distribution for a given parameter value in $[0,β]$. The current best known approach due to Huber [9] uses $O(q\ln n\cdot[\ln q + \ln \ln n+\varepsilon^{-2}])$ oracle calls on average where $\varepsilon$ is the desired accuracy of approximation and $H(\cdot)$ is assumed to lie in $\{0\}\cup[1,n]$. We improve the complexity to $O(q\ln n\cdot\varepsilon^{-2})$ oracle calls. We also show that the same complexity can be achieved if exact oracles are replaced with approximate sampling oracles that are within $O(\frac{\varepsilon^2}{q\ln n})$ variation distance from exact oracles. Finally, we prove a lower bound of $Ω(q\cdot \varepsilon^{-2})$ oracle calls under a natural model of computation.