What can be sampled locally?

TL;DR

This paper explores distributed algorithms for sampling solutions from Gibbs distributions in local CSPs, introducing two Markov chain methods that achieve efficient parallel sampling under certain conditions.

Contribution

It presents two novel distributed Markov chain algorithms for sampling from Gibbs distributions, with one achieving near-optimal time bounds and the other providing bias-free parallel updates.

Findings

First algorithm achieves $O(\Delta \log n)$ time under Dobrushin's condition.

Second algorithm attains $O(\log n)$ time with stronger mixing conditions.

Proves a lower bound of $ ext{diam}$ for sampling independent sets in certain graphs.

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: 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 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 $O(\Delta\log n)$ time upper bound in the LOCAL model, where $\Delta$ 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 $\Delta$ under a stronger mixing condition. We also show a strong $\Omega(diam)$ lower bound for sampling independent set in graphs with maximum degree $\Delta\ge 6$. This lower bound holds even when every node is aware of the graph. This gives a strong separation between sampling and constructing locally checkable labelings.