Spin glasses are fundamental probability distributions at the core of statistical physics, the theory of average-case computational complexity, and modern high-dimensional statistical inference. In the mean-field setting, we design deterministic quasipolynomial-time algorithms for estimating the partition function to arbitrarily high accuracy for all inverse temperatures in the second moment regime. In particular, for the Sherrington--Kirkpatrick model, our algorithms succeed for the entire replica-symmetric phase. To achieve this, we study the locations of the zeros of the partition function. Notably, our methods are conceptually simple, and apply equally well to the spherical case and the case of Ising spins.

Many natural Markov chains fail to mix to their stationary distribution in polynomially many steps. Often, this slow mixing is inevitable since it is computationally intractable to sample from their stationary measure. Nevertheless, Markov chains can be shown to always converge quickly to measures that are locally stationary, i.e., measures that don't change over a small number of steps. These locally stationary measures are analogous to local minima in continuous optimization, while stationary measures correspond to global minima. While locally stationary measures can be statistically far from stationary measures, do they enjoy provable theoretical guarantees that have algorithmic implications? We study this question in this work and demonstrate three algorithmic applications of locally stationary measures: 1)We show that Glauber dynamics on the hardcore model can be used to find large independent sets in triangle-free graphs of bounded degree. 2)We prove that Glauber dynamics on the Ising model defined by a spiked matrix model finds a vector with constant correlation with the planted spike. 3)We show that for sufficiently large constant signal-to-noise ratio, Glauber dynamics on the Ising model finds a vector that has constant correlation with the hidden community vector. In other words, Glauber dynamics subsumes the spectral method for spiked Wigner and community detection, by weakly recovering the planted spike. The full version of this paper can be found on arXiv(arXiv ID: 2405.20849).

Motivated by the community detection problem in Bayesian inference, as well as the recent explosion of interest in spin glasses from statistical physics, we study the classical Glauber dynamics for sampling from Ising models with sparse random interactions. It is now well-known that when the in- teraction matrix has spectral diameter less than 1, Glauber dynamics mixes in near-linear time. Unfortunately, such criteria fail dramatically for interactions supported on arguably the most well-studied sparse random graph: the Erdos-Renyi random graph. There is a scarcity of positive results in this setting due to the presence of almost linearly many outlier eigenvalues of unbounded magnitude. We prove that for the Viana-Bray spin glass, where the interactions are supported on a random graph and randomly assigned signs, Glauber dynamics mixes in almost-linear time with high probability at sufficiently high temperatures, and we conjecture that our results are tight up to constants. We further extend our results to random graphs drawn according to the 2-community stochastic block model, as well as when the interactions are given by a “centered” version of the adjacency matrix. The latter setting is particularly relevant for the inference problem in community detection. Indeed, we build on this result to demonstrate that Glauber dynamics succeeds at recovering communities in the stochastic block model in a companion paper. The primary technical ingredient in our proof is showing that with high probability, a sparse random graph can be decomposed into two parts - a bulk which behaves like a graph with bounded maximum degree and a well-behaved spectrum, and a near- forest with favorable pseudorandom properties. We then use this decomposition to design a localization procedure that interpolates to simpler Ising models supported only on the near-forest, and then execute a pathwise analysis to establish a modified log- Sobolev inequality. The full version of this paper can be found on arXiv (arXiv ID: 2405.06616).

We optimize pipeline parallelism for deep neural network (DNN) inference by partitioning model graphs into $k$ stages and minimizing the running time of the bottleneck stage, including communication. We give practical and effective algorithms for this NP-hard problem, but our emphasis is on tackling the practitioner’s dilemma of deciding when a solution is good enough. To this end, we design novel mixed integer programming (MIP) relaxations for proving lower bounds. Applying these methods to a diverse testbed of 369 production models, for $k \in \\{2, 4, 8, 16, 32, 64\\}$, we empirically show that these lower bounds are strong enough to be useful in practice. Our lower bounds are substantially stronger than standard combinatorial bounds. For example, evaluated via geometric means across a production testbed with $k = 16$ pipeline stages, our MIP formulations raise the lower bound from 0.4598 to 0.9452, expressed as a fraction of the best partition found. In other words, our improved lower bounds close the optimality gap by a factor of 9.855x.

Strong spatial mixing (SSM) is an important quantitative notion of correlation decay for Gibbs distributions arising in statistical physics, probability theory, and theoretical computer science. A longstanding conjecture is that the uniform distribution on proper q-colorings on a $\Delta$ regular tree exhibits SSM whenever $q \geq \Delta+1$. Moreover, it is widely believed that as long as SSM holds on bounded-degree trees with q colors, one would obtain an efficient sampler for q-colorings on all bounded-degree graphs via simple Markov chain algorithms. It is surprising that such a basic question is still open, even on trees, but then again it also highlights how much we still have to learn about random colorings. In this paper, we show the following: (1)For any $\Delta \geq 3$, SSM holds for random q-colorings on trees of maximum degree $\Delta$ whenever $q \geq \Delta+3$. Thus we almost fully resolve the aforementioned conjecture. Our result substantially improves upon the previously best bound which requires $q \geq 1.59 \Delta+\gamma^{*}$ for an absolute constant $\gamma^{*}\gt0$.(2)For any $\Delta \geq 3$ and $g=\Omega_{\Delta}(1)$, we establish optimal mixing of the Glauber dynamics for q-colorings on graphs of maximum degree $\Delta$ and girth g whenever $q \geq \Delta+3$. Our approach is based on a new general reduction from spectral independence on large-girth graphs to SSM on trees that is of independent interest. Using the same techniques, we also prove near-optimal bounds on weak spatial mixing (WSM), a closely-related notion to SSM, for the antiferromagnetic Potts model on trees.

Abstract. For general antiferromagnetic 2-spin systems, including the hardcore model on weighted independent sets and the antiferromagnetic Ising model, there is an [Formula: see text] for the partition function on graphs of maximum degree [Formula: see text] when the infinite regular tree lies in the uniqueness region by Li, Lu, and Yin [ Correlation Decay up to Uniqueness in Spin Systems, preprint, https://arxiv.org/abs/1111.7064 , 2021]. Moreover, in the tree nonuniqueness region, Sly in [ Computational transition at the uniqueness threshold, in Proceedings of the 51st Annual IEEE Symposium on Foundations of Computer Science, 2010, pp. 287–296] showed that there is no [Formula: see text] to estimate the partition function unless [Formula: see text]. The algorithmic results follow from the correlation decay approach due to Weitz [ Counting independent sets up to the tree threshold, in Proceedings of the 38th Annual ACM Symposium on Theory of Computing, 2006, pp. 140–149] or the polynomial interpolation approach developed by Barvinok [ Combinatorics and Complexity of Partition Functions, Springer, 2016]. However, the running time is only polynomial for constant [Formula: see text]. For the hardcore model, recent work of Anari, Liu, and Oveis Gharan [ Spectral independence in high-dimensional expanders and applications to the hardcore model, in Proceedings of the 61st Annual IEEE Symposium on Foundations of Computer Science, 2020, pp. 1319–1330] establishes rapid mixing of the simple single-site Markov chain, known as the Glauber dynamics, in the tree uniqueness region. Our work simplifies their analysis of the Glauber dynamics by considering the total pairwise influence of a fixed vertex [Formula: see text] on other vertices, as opposed to the total influence of other vertices on [Formula: see text], thereby extending their work to all 2-spin models and improving the mixing time. More important, our proof ties together the three disparate algorithmic approaches: we show that contraction of the so-called tree recursions with a suitable potential function, which is the primary technique for establishing efficiency of Weitz’s correlation decay approach and Barvinok’s polynomial interpolation approach, also establishes rapid mixing of the Glauber dynamics. We emphasize that this connection holds for all 2-spin models (both antiferromagnetic and ferromagnetic), and existing proofs for the correlation decay and polynomial interpolation approaches immediately imply rapid mixing of the Glauber dynamics. Our proof utilizes the fact that the graph partition function is a divisor of the partition function for Weitz’s self-avoiding walk tree. This fact leads to new tools for the analysis of the influence of vertices and may be of independent interest for the study of complex zeros.

We show that the existence of a "good" coupling w.r.t. Hamming distance for any local Markov chain on a discrete product space implies rapid mixing of the Glauber dynamics in a blackbox fashion. More specifically, we only require the expected distance between successive iterates under the coupling to be summable, as opposed to being one-step contractive in the worst case. Combined with recent local-to-global arguments [Chen et al., 2021], we establish asymptotically optimal lower bounds on the standard and modified log-Sobolev constants for the Glauber dynamics for sampling from spin systems on bounded-degree graphs when a curvature condition [Ollivier, 2009] is satisfied. To achieve this, we use Stein’s method for Markov chains [Bresler and Nagaraj, 2019; Reinert and Ross, 2019] to show that a "good" coupling for a local Markov chain yields strong bounds on the spectral independence of the distribution in the sense of [Anari et al., 2020]. Our primary application is to sampling proper list-colorings on bounded-degree graphs. In particular, combining the coupling for the flip dynamics given by [Vigoda, 2000; Chen et al., 2019] with our techniques, we show optimal O(nlog n) mixing for the Glauber dynamics for sampling proper list-colorings on any bounded-degree graph with maximum degree Δ whenever the size of the color lists are at least ({11/6 - ε}) Δ, where ε ≈ 10^{-5} is small constant. While O(n²) mixing was already known before, our approach additionally yields Chernoff-type concentration bounds for Hamming Lipschitz functions in this regime, which was not known before. Our approach is markedly different from prior works establishing spectral independence for spin systems using spatial mixing [Anari et al., 2020; Z. {Chen} et al., 2020; Chen et al., 2021; Feng et al., 2021], which crucially is still open in this regime for proper list-colorings.

We show that the natural Glauber dynamics mixes rapidly and generates a random proper edge-coloring of a graph with maximum degree$\Delta$whenever the number of colors is at least$q\geq(\frac{10}{3}+\epsilon)\Delta$, where$\epsilon > 0$is arbitrary and the maximum degree satisfies$\Delta\geq C$for a constant$C=C(\epsilon)$depending only on$\epsilon$, For edge-colorings, this improves upon prior work [Vig99; Che+19] which show rapid mixing when$q\geq(\frac{11}{3}-\epsilon_{0})\Delta$, where$\epsilon_{0}\approx 10^{-5}$is a small fixed constant. At the heart of our proof, we establish a matrix trickle-down theorem, generalizing Oppenheim's influential result, as a new technique to prove that a high dimensional simplicial complex is a local spectral expander.

This paper formalizes connections between stability of polynomials and convergence rates of Markov Chain Monte Carlo (MCMC) algorithms. We prove that if a (multivariate) partition function is nonzero in a region around a real point$\lambda$then spectral independence holds at$\lambda$. As a consequence, for Holant-type problems (e.g., spin systems) on bounded-degree graphs, we obtain optimal$O(n\ \text{log}\ n)$mixing time bounds for the single-site update Markov chain known as the Glauber dynamics. Our result significantly improves the running time guarantees obtained via the polynomial interpolation method of Barvi-nok (2017), refined by Patel and Regts (2017). There are a variety of applications of our results. In this paper, we focus on Holant-type (i.e., edge-coloring) problems, including weighted edge covers and weighted even subgraphs. For the weighted edge cover problem (and several natural generalizations) we obtain an$O$($n$log n) sampling algorithm on bounded-degree graphs. The even subgraphs problem corresponds to the high-temperature expansion of the ferromagnetic Ising model. We obtain an$O$($n$log n) sampling algorithm for the ferromagnetic Ising model with a nonzero external field on bounded-degree graphs, which improves upon the classical result of Jerrum and Sinclair (1993) for this class of graphs. We obtain further applications to antiferromagnetic two-spin models on line graphs, weighted graph homomorphisms, tensor networks, and more.

We prove tight mixing time bounds for natural random walks on bases of matroids, determinantal distributions, and more generally distributions associated with log-concave polynomials. For a matroid of rank k on a ground set of n elements, or more generally distributions associated with log-concave polynomials of homogeneous degree k on n variables, we show that the down-up random walk, started from an arbitrary point in the support, mixes in time O(klogk). Our bound has no dependence on n or the starting point, unlike the previous analyses of Anari et al. (STOC 2019), Cryan et al. (FOCS 2019), and is tight up to constant factors. The main new ingredient is a property we call approximate exchange, a generalization of well-studied exchange properties for matroids and valuated matroids, which may be of independent interest. In particular, given a distribution µ over size-k subsets of [n], our approximate exchange property implies that a simple local search algorithm gives a kO(k)-approximation of maxS µ(S) when µ is generated by a log-concave polynomial, and that greedy gives the same approximation ratio when µ is strongly Rayleigh. As an application, we show how to leverage down-up random walks to approximately sample random forests or random spanning trees in a graph with n edges in time O(nlog2 n). The best known result for sampling random forest was a FPAUS with high polynomial runtime recently found by Anari et al. (STOC 2019), Cryan et al. (FOCS 2019). For spanning tree, we improve on the almost-linear time algorithm by Schild (STOC 2018). Our analysis works on weighted graphs too, and is the first to achieve nearly-linear running time for these problems. Our algorithms can be naturally extended to support approximately sampling from random forests of size between k1 and k2 in time O(n log2 n), for fixed parameters k1, k2.

We prove an optimal mixing time bound for the single-site update Markov chain known as the Glauber dynamics or Gibbs sampling in a variety of settings. Our work presents an improved version of the spectral independence approach of Anari et al. (2020) and shows O(nlogn) mixing time on any n-vertex graph of bounded degree when the maximum eigenvalue of an associated influence matrix is bounded. As an application of our results, for the hard-core model on independent sets weighted by a fugacity λ, we establish O(nlogn) mixing time for the Glauber dynamics on any n-vertex graph of constant maximum degree Δ when λ<λc(Δ) where λc(Δ) is the critical point for the uniqueness/non-uniqueness phase transition on the Δ-regular tree. More generally, for any antiferromagnetic 2-spin system we prove O(nlogn) mixing time of the Glauber dynamics on any bounded degree graph in the corresponding tree uniqueness region. Our results apply more broadly; for example, we also obtain O(nlogn) mixing for q-colorings of triangle-free graphs of maximum degree Δ when the number of colors satisfies q > α Δ where α ≈ 1.763, and O(mlogn) mixing for generating random matchings of any graph with bounded degree and m edges.

We say a probability distribution $\mu$ is spectrally independent if an associated pairwise influence matrix has a bounded largest eigenvalue for the distribution and all of its conditional distributions. We prove that if $\mu$ is spectrally independent, then the corresponding high-dimensional simplicial complex is a local spectral expander. Using a line of recent works on mixing time of high-dimensional walks on simplicial complexes [T. Kaufman and D. Mass, Proceedings of ITCS, 2017, pp. 4:1–4:27; I. Dinur and T. Kaufman, Proceedings of the IEEE 58th Annual Symposium on Foundations of Computer Science, 2017, pp. 974–985; T. Kaufman and I. Oppenheim, Proceedings of APPROX/RANDOM, 2018, pp. 47:1–47:17; V. L. Alev and L. C. Lau, Proceedings of the 52nd Annual ACM Symposium on Theory of Computing, 2020], this implies that the corresponding Glauber dynamics mixes rapidly and generates (approximate) samples from $\mu$. As an application, we show that natural Glauber dynamics mixes rapidly (in polynomial time) to generate a random independent set from the hardcore model up to the uniqueness threshold. This improves the quasi-polynomial running time of Weitz's deterministic correlation decay algorithm [D. Weitz, Proceedings of the 38th Annual ACM Symposium on Theory of Computing, 2006, pp. 140–149] for estimating the hardcore partition function, also answering a long-standing open problem of mixing time of Glauber dynamics [M. Luby and E. Vigoda, Proceedings of the 29th Annual ACM Symposium on Theory of Computing, 1997, pp. 682–687; M. Luby and E. Vigoda, Random Structures Algorithms, 15 (1999), pp. 229–241; M. Dyer and C. Greenhill, J. Algorithms, 35 (2000), pp. 17–49; E. Vigoda, Electron. J. Combin., 8 (2001); C. Efthymiou et al., Proceedings of FOCS, 2016, pp. 704–713].

For general antiferromagnetic 2-spin systems, including the hardcore model on weighted independent sets and the antiferromagnetic Ising model, there is an FPTAS for the partition function on graphs of maximum degree Δ when the infinite regular tree lies in the uniqueness region by Li et al. (2013). Moreover, in the tree non-uniqueness region, Sly (2010) showed that there is no FPRAS to estimate the partition function unless NP=RP. The algorithmic results follow from the correlation decay approach due to Weitz (2006) or the polynomial interpolation approach developed by Barvinok (2016). However the running time is only polynomial for constant Δ. For the hardcore model, recent work of Anari et al. (2020) establishes rapid mixing of the simple single-site Markov chain known as the Glauber dynamics in the tree uniqueness region. Our work simplifies their analysis of the Glauber dynamics by considering the total pairwise influence of a fixed vertex v on other vertices, as opposed to the total influence of other vertices on v, thereby extending their work to all 2-spin models and improving the mixing time. More importantly our proof ties together the three disparate algorithmic approaches: we show that contraction of the so-called tree recursions with a suitable potential function, which is the primary technique for establishing efficiency of Weitz's correlation decay approach and Barvinok's polynomial interpolation approach, also establishes rapid mixing of the Glauber dynamics. We emphasize that this connection holds for all 2-spin models (both antiferromagnetic and ferromagnetic), and existing proofs for the correlation decay or polynomial interpolation approach immediately imply rapid mixing of the Glauber dynamics. Our proof utilizes that the graph partition function is a divisor of the partition function for Weitz's self-avoiding walk tree. This fact leads to new tools for the analysis of the influence of vertices, and may be of independent interest for the study of complex zeros.

We design an FPRAS to count the number of bases of any matroid given by an independent set oracle, and to estimate the partition function of the random cluster model of any matroid in the regime where 0<q<1. Consequently, we can sample random spanning forests in a graph and estimate the reliability polynomial of any matroid. We also prove the thirty year old conjecture of Mihail and Vazirani that the bases exchange graph of any matroid has edge expansion at least 1.