What can be sampled locally?

Abstract

The local computation of Linial [FOCS'87] and Naor and Stockmeyer [STOC'93] concerns with the question of whether a locally definable distributed computing problem can be solved locally: more specifically, for a given local CSP whether a CSP solution can be constructed by a distributed algorithm using local information. In this paper, we consider the problem of sampling a uniform CSP solution by distributed algorithms, and ask whether a locally definable joint distribution can be sampled from locally. More broadly, we consider sampling from Gibbs distributions induced by weighted local CSPs, especially the Markov random fields (MRFs), in the LOCAL model.We give two Markov chain based distributed algorithms which we believe to represent two fundamental approaches for sampling from Gibbs distributions via distributed algorithms. The first algorithm generically parallelizes the single-site sequential Markov chain by iteratively updating a random independent set of variables in parallel, and achieves an O(∆ log n) time upper bound in the LOCAL model, where ∆ is the maximum degree, when the Dobrushin's condition for the Gibbs distribution is satisfied. The second algorithm is a novel parallel Markov chain which proposes to update all variables simultaneously yet still guarantees to converge correctly with no bias. It surprisingly parallelizes an intrinsically sequential process: stabilizing to a joint distribution with massive local dependencies, and may achieve an optimal O(log n) time upper bound independent of the maximum degree ∆ under a stronger mixing condition.We also show a strong Ω(diam) lower bound for sampling: in particular for sampling independent set in graphs with maximum degree ∆ ≥ 6. Independent sets are trivial to construct locally and the sampling lower bound holds even when every node is aware of the entire graph. This gives a strong separation between sampling and constructing locally checkable labelings.

Perfect sampling for Gibbs point processes using partial rejection sampling

“…In that case the expected running time complexity of the PRS algorithm is simply of order of the expected number of iterations, which is O log 1 r . See, for example, Feng and Yin (2018); Feng et al (2017) for recent works on distributed sampling. Theorem 1.…”

Dynamic sampling from graphical models

“…Unlike the MCMC sampling, our algorithm is a Las Vegas sampler that knows when it terminates -this is important in simulations. Also, besides being dynamic, our sampling algorithm is parallelizable, and can be implemented as communicationefficient distributed algorithms in a distributed sampling model considered [7,11]. Remark 3.4 (Comparison with algorithms for constructing and sampling LLL solution).…”

“…This bound is asymptotically tight. In the "non-uniqueness regime" where e −2|β| < 1 − 2 ∆ , for the anti-ferromagnetic Ising model, even static and approximate sampling is intractable [13]; and for the ferromagnetic Ising model, by an argument as in [7] there cannot exist such local and parallel sampling algorithms (even for static and approximate sampling) due to the reconstructibility of the ferromagnetic Ising model in the non-uniqueness regime on locally tree-like graphs [3,14].…”

On Local Distributed Sampling and Counting

“…Previous studies in local computation were focused on the complexity of constructing a feasible solution. The studies of sampling problems in local computation were started very recently [2,3]. Several fundamental questions regarding the local complexities of sampling and counting need to be answered.…”

“…Combining with the state of the arts of strong spatial mixing, we obtain efficient sampling algorithms in the LOCAL model for various important sampling problems, including: an O( √ ∆ log 3 n)-round algorithm for exact sampling matchings in graphs with maximum degree ∆, and an O(log 3 n)-round algorithm for sampling according to the hardcore model (weighted independent sets) in the uniqueness regime, which along with the Ω(diam) lower bound in [2] for sampling according to the hardcore model in the non-uniqueness regime, gives the first computational phase transition for distributed sampling.…”