Adaptive Simulated Annealing: A Near-optimal Connection between Sampling and Counting

Abstract

We present a near-optimal reduction from approximately counting the cardinality of a discrete set to approximately sampling elements of the set. An important application of our work is to approximating the partition function Z of a discrete system, such as the Ising model, matchings or colorings of a graph. The typical approach to estimating the partition function Z(β * ) at some desired inverse temperature β * is to define a sequence, which we call a cooling schedule,where Z(0) is trivial to compute and the ratios Z(β i+1 )/Z(β i ) are easy to estimate by sampling from the distribution corresponding to Z(β i ). Previous approaches required a cooling schedule of length O * (ln A) where A = Z(0), thereby ensuring that each ratio Z(β i+1 )/Z(β i ) is bounded. We present a cooling schedule of length ℓ = O * ( √ ln A). For well-studied problems such as estimating the partition function of the Ising model, or approximating the number of colorings or matchings of a graph, our cooling schedule is of length O * ( √ n), which implies an overall savings of O * (n) in the running time of the approximate counting algorithm (since roughly ℓ samples are needed to estimate each ratio).A similar improvement in the length of the cooling schedule was recently obtained by Lovász and Vempala in the context of estimating the volume of convex bodies. While our reduction is inspired by theirs, the discrete analogue of their result turns out to be significantly more difficult. Whereas a fixed schedule suffices in their setting, we prove that in the discrete setting we need an adaptive schedule, i. e., the schedule depends on Z. More precisely, we prove any non-adaptive cooling schedule has length at least O * (ln A), and we present an algorithm to find an adaptive schedule of length O * ( √ ln A).

Random Construction of Interpolating Sets for High-Dimensional Integration

“…So there is an optimal value of α that balances these considerations (see the proof of Lemma 3, below). Dyer and Frieze [3] proved the following lemma, presented here in a form given in [19].…”

“…. , n},Stefankovic et al [19] gave an algorithm (SVV) that yields an (1 + ε, δ)-randomized approximation algorithm for A that used at least As shown in Section 4.1, it is possible to obtain such an approximation using TPA with the average number of samples given in (1). While TPA is O(ln(A) 2 ) and SVV is O(ln(A) ln(ln(A)) 5 ), the respective constants of 2 and 10 10 mean that TPA is the logical choice to use even for very large problems.…”

“…This compares favorably with previous algorithms such as selfreducibility [9], which uses roughly 150 times as many samples to achieve the same result (see Section 2.1). A recent algorithm for approximating normalizing constants of Gibbs distributions byStefankovic et al [19] (here referred to as SVV) uses a number of samples that is nearly linear rather than quadratic in ln(μ(B)/μ(B )). However, the constant hidden by the big O notation is at least 10 10 , making the algorithm impractical for real problems.…”

“…The strength of TPA is the generality of the procedure, but that same generality means that it is possible to do better in restricted circumstances. For instance, when f (x) falls into the class of Gibbs distributions,Stefankovic et al [19] were able to give anÕ(ln(Z)) algorithm for approximating Z, but the high constants involved in their algorithm make it solely of theoretical interest. (Here theÕ notation hides factors of ln(ln(Z)).)…”

Improved inapproximability results for counting independent sets in the hard‐core model

“…The Glauber dynamics is a simple Markov chain that is used to sample from the Gibbs distribution (and hence to approximate the partition function via now standard techniques, see Refs. ). For an input graph and activity , the state space of the chain is .…”