Yitong Yin

Hongyang Liu, Yitong Yin, Yiyao Zhang

A canonical approach to approximating the partition function of a Gibbs distribution via sampling is simulated annealing. This method has led to efficient reductions from counting to sampling, including: $\bullet$ classic non-adaptive (parallel) algorithms with sub-optimal cost (Dyer-Frieze-Kannan '89; Bezáková-Štefankovič-Vazirani-Vigoda '08); $\bullet$ adaptive (sequential) algorithms with near-optimal cost (Štefankovič-Vempala-Vigoda '09; Huber '15; Kolmogorov '18; Harris-Kolmogorov '24). We present an algorithm that achieves both near-optimal total work and efficient parallelism, providing a reduction from counting to sampling with logarithmic depth and near-optimal work. As consequences, we obtain work-efficient parallel counting algorithms for several important models, including the hardcore and Ising models within the uniqueness regime.

Xiaoyu Chen, Zhe Ju, Tianshun Miao et al.

We prove two results on the mixing times of Markov chains for two-spin systems. First, we show that the Glauber dynamics mixes in polynomial time for the Gibbs distributions of antiferromagnetic two-spin systems at the critical threshold of the uniqueness phase transition of the Gibbs measure on infinite regular trees. This completes the computational phase transition picture for antiferromagnetic two-spin systems, which includes near-linear-time optimal mixing in the uniqueness regime [Chen--Liu--Vigoda, STOC '21; Chen--Feng--Yin--Zhang, FOCS '22], NP-hardness of approximate sampling in the non-uniqueness regime [Sly--Sun, FOCS '12], and polynomial-time mixing at criticality (this work). Second, we prove an optimal $O(\log n)$ mixing time bound as well as an optimal $Ω(1)$ spectral gap for the Swendsen--Wang dynamics for the ferromagnetic Ising model with an external field on bounded-degree graphs. To the best of our knowledge, these are the first sharp bounds on the mixing rate of this classical global Markov chain beyond mean-field or strong spatial mixing (SSM) regimes, and resolve a conjecture of [Feng--Guo--Wang, IANDC '23]. A key ingredient in both proofs is a new family of localization schemes that extends the field dynamics of [Chen--Feng--Yin--Zhang, FOCS '21] by tilting general edge (or hyperedge) weights rather than vertex fields. This framework, which subsumes the classical Swendsen--Wang dynamics as a special case, extends the localization framework of [Chen--Eldan, FOCS '22] beyond stochastic and field localizations, and enables controlled tilting of interaction strengths while preserving external fields.

David G. Harris, Vladimir Kolmogorov, Hongyang Liu et al.

The computational equivalence between approximate counting and sampling is well established for polynomial-time algorithms. The most efficient general reduction from counting to sampling is achieved via simulated annealing, where the counting problem is formulated in terms of estimating the ratio $Q={Z(β_{\max})}/{Z(β_{\min})}$ between partition functions $Z(β)=\sum_{x\in Ω} \exp(βH(x))$ of Gibbs distributions $μ_β$ over $Ω$ with Hamiltonian $H$, given access to a sampling oracle that produces samples from $μ_β$ for $β\in [β_{\min}, β_{\max}]$. The best bound achieved by known annealing algorithms with relative error $\varepsilon$ is $O(q \log h / \varepsilon^2)$, where $q, h$ are parameters which respectively bound $\ln Q$ and $H$. However, all known algorithms attaining this near-optimal complexity are inherently sequential, or *adaptive*: the queried parameters $β$ depend on previous samples. We develop a simple non-adaptive algorithm for approximate counting using $O(q \log^2 h / \varepsilon^2)$ samples, as well as an algorithm that achieves $O(q \log h / \varepsilon^2)$ samples with just two rounds of adaptivity, matching the best sample complexity of sequential algorithms. These algorithms naturally give rise to work-efficient parallel (RNC) counting algorithms. We discuss applications to RNC counting algorithms for several classic models, including the anti-ferromagnetic 2-spin, monomer-dimer and ferromagnetic Ising models.

Weiming Feng, Thomas P. Hayes, Yitong Yin

The Metropolis-Hastings algorithm is a fundamental Markov chain Monte Carlo (MCMC) method for sampling and inference. With the advent of Big Data, distributed and parallel variants of MCMC methods are attracting increased attention. In this paper, we give a distributed algorithm that can correctly simulate sequential single-site Metropolis chains without any bias in a fully asynchronous message-passing model. Furthermore, if a natural Lipschitz condition is satisfied by the Metropolis filters, our algorithm can simulate $N$-step Metropolis chains within $O(N/n+\log n)$ rounds of asynchronous communications, where $n$ is the number of variables. For sequential single-site dynamics, whose mixing requires $Ω(n\log n)$ steps, this achieves an optimal linear speedup. For several well-studied important graphical models, including proper graph coloring, hardcore model, and Ising model, our condition for linear speedup is weaker than the respective uniqueness (mixing) conditions. The novel idea in our algorithm is to resolve updates in advance: the local Metropolis filters can often be executed correctly before the full information about neighboring spins is available. This achieves optimal parallelism without introducing any bias.

Weiming Feng, Nisheeth K. Vishnoi, Yitong Yin

In this paper, we study the problem of sampling from a graphical model when the model itself is changing dynamically with time. This problem derives its interest from a variety of inference, learning, and sampling settings in machine learning, computer vision, statistical physics, and theoretical computer science. While the problem of sampling from a static graphical model has received considerable attention, theoretical works for its dynamic variants have been largely lacking. The main contribution of this paper is an algorithm that can sample dynamically from a broad class of graphical models over discrete random variables. Our algorithm is parallel and Las Vegas: it knows when to stop and it outputs samples from the exact distribution. We also provide sufficient conditions under which this algorithm runs in time proportional to the size of the update, on general graphical models as well as well-studied specific spin systems. In particular we obtain, for the Ising model (ferromagnetic or anti-ferromagnetic) and for the hardcore model the first dynamic sampling algorithms that can handle both edge and vertex updates (addition, deletion, change of functions), both efficient within regimes that are close to the respective uniqueness regimes, beyond which, even for the static and approximate sampling, no local algorithms were known or the problem itself is intractable. Our dynamic sampling algorithm relies on a local resampling algorithm and a new "equilibrium" property that is shown to be satisfied by our algorithm at each step, and enables us to prove its correctness. This equilibrium property is robust enough to guarantee the correctness of our algorithm, helps us improve bounds on fast convergence on specific models, and should be of independent interest.