Sampling procedure for simulating the free energy. To produce simulations of the partition function for any given optimization problem, we use a Metropolis-Hastings procedure to sample solutions at a given temperature 1/β, coupled with an importance sampling cooling schedule scheme to efficiently sample solutions at low temperature levels. Below, we provide a brief review of importance sampling. N Importance sampling. Let us assume that samples from a distribution Q over a random variable X are given. Then, the expectation EPφ(X) of a function φ(X) under a distribution P can be estimated by sampling X under Q with N N N i Q(Xi) Q N E^ = 1 Σ φ(X ) P(Xi) , since E E^ i=1 = 1 Σ E i=1 φ(X ) P(Xi) = E φ(X). (116) N Q Q(Xi) P This method is called the importance sampling since each sample is “reweighted” using the target distribution. where Z(β) = c∈C e−βR(c) is the partition function. Let us assume the parti- We adapt it for the computation of Xxxxx distribution partition functions. Suppose we have Xxxxx distribution at temperature 1/β over a space C defined by a cost functiΣon R : C → R. The probability of c ∈ C is P(c|β) = e−βR(c)/Z(β) tion function Z(β) is given and we can sample from the Xxxxx distribution at temperature 1/βj. Then N N N Z∗ (β, βj) = 1 Σ Z(β)e−(βt−β)R(ci) (117) i=1 ·| is an unbiased estimator of Z(βj) when sampled under P( β). Its precision is controlled by the relative variance: Varrel Z∗ (β, βj) = VarP(·|β)ZN∗ (β, βj) = 1 . Z(2βj − β)Z(β) − 1Σ . (118) E P(·|β) N P(·|β) ZN∗ (β, βj) N Z(βj)2 Observe that when β differs significantly from βj, the variance may be large, leading to poor simulations results. Furthermore, when β is close to βj, the variance is small, thus simulations are more accurate. · · · Our goal is to estimate the partition function for a wide range of β. The difficulties arise mostly for large values of β, since the partition function is then very concentrated. To overcome this, we apply our importance sampling philosophy and simulate first the partition function for small values of β (this makes the partition function more uniform and easier to estimate). Once we have computed the partition function for small β, we use equation (117) to evaluate it for the targeted value βj. But that is not the end of the story since we need to proceed in small steps using a cooling schedule β0 = 0 < β1 < < βk in order to reach the regions of the solution space contributing the most to the partition function. This is called a cooling schedule (Xxxxx, 2012). In practice, we use a Metropoli...
