A Faster Approximation Algorithm for the Gibbs Partition Function

We consider the problem of estimating the partition function Z(β)=sumx exp(-β(H(x)) of a Gibbs distribution with a Hamilton H(·), 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(qln n·[ln q + ln ln n+varepsilon-2]) oracle calls on average where varepsilon is the desired accuracy of approximation and H(·) is assumed to lie in 0cup[1,n]. We improve the complexity to O(qln n·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(fracvarepsilon2qln n) variation distance from exact oracles. Finally, we prove a lower bound of Omega(q· varepsilon-2) oracle calls under a natural model of computation.