Resolving the Mixing Time of the Langevin Algorithm to Its Stationary Distribution for Log-Concave Sampling

The exact transient sampling distribution is found for the infinite neutral alleles model with mutation, thus extending the stationary distribution found by Ewens (1972). The first eigenfunction in the sampling distribution depends only on the homozygosity. In a large sample the expected number of allele types is close to the expected number in a stationary distribution and depends little on the initial frequencies or the time of sampling. The expected number of types in a sample for a non-stationary population is tabulated.The sampling distribution from a population with no mutation is found. The probability that the population is monomorphic given that the sample contains only one type is tabulated, where initially the population has a large number of alleles with equal frequencies.A study is made of the joint distribution of the allele frequencies in two samples taken timetapart from a stationary population. Of particular interest is the number of allele types in common in two such samples. The distribution of the number of types in common in a population viewed at timetapart is also found and tabulated for different mutation rates and time. The expected frequency of the allele types in common is tabulated.The distribution of allele frequencies in two divergent populations of common ancestry is shown to be the same as the distribution of frequencies in a single population at two time points 2tapart. A sufficient statistic for the time of divergence is shown to be the paired frequencies of the allele types in common. The expected waiting time until the populations have no allele types in common is tabulated.

SummaryIn Bayesian inference, predictive distributions are typically in the form of samples generated via Markov chain Monte Carlo or related algorithms. In this paper, we conduct a systematic analysis of how to make and evaluate probabilistic forecasts from such simulation output. Based on proper scoring rules, we develop a notion of consistency that allows to assess the adequacy of methods for estimating the stationary distribution underlying the simulation output. We then provide asymptotic results that account for the salient features of Bayesian posterior simulators and derive conditions under which choices from the literature satisfy our notion of consistency. Importantly, these conditions depend on the scoring rule being used, such that the choices of approximation method and scoring rule are intertwined. While the logarithmic rule requires fairly stringent conditions, the continuous ranked probability score yields consistent approximations under minimal assumptions. These results are illustrated in a simulation study and an economic data example. Overall, mixture‐of‐parameters approximations that exploit the parametric structure of Bayesian models perform particularly well. Under the continuous ranked probability score, the empirical distribution function is a simple and appealing alternative option.

In this paper, we develop a new "robust mixing" framework for reasoning about adversarially modified Markov Chains (AMMC). Let $\mathbb{P}$ be the transition matrix of an irreducible Markov Chain with stationary distribution $\pi$. An adversary announces a sequence of stochastic matrices $\{\mathbb{A}_t\}_{t > 0}$ satisfying $\pi\mathbb{A}_t = \pi$. An AMMC process involves an application of $\mathbb{P}$ followed by $\mathbb{A}_t$ at time $t$. The robust mixing time of an ergodic Markov Chain $\mathbb{P}$ is the supremum over all adversarial strategies of the mixing time of the corresponding AMMC process. Applications include estimating the mixing times for certain non-Markovian processes and for reversible liftings of Markov Chains. Non-Markovian card shuffling processes: The random-to-cyclic transposition process is a non-Markovian card shuffling process, which at time $t$, exchanges the card at position $L_t := t {\pmod n}$ with a random card. Mossel, Peres and Sinclair (2004) showed a lower bound of $(0.0345+o(1))n\log n$ for the mixing time of the random-to-cyclic transposition process. They also considered a generalization of this process where the choice of $L_t$ is adversarial, and proved an upper bound of $C n\log n + O(n)$ (with $C \approx 4\times 10^5$) on the mixing time. We reduce the constant to $1$ by showing that the random-to-top transposition chain (a Markov Chain) has robust mixing time $\leq n\log n + O(n)$ when the adversarial strategies are limited to holomorphic strategies, i.e. those strategies which preserve the symmetry of the underlying Markov Chain. We also show a $O(n\log^2 n)$ bound on the robust mixing time of the lazy random-to-top transposition chain when the adversary is not limited to holomorphic strategies. Reversible liftings: Chen, Lovasz and Pak showed that for a reversible ergodic Markov Chain $\mathbb{P}$, any reversible lifting $\mathbb{Q}$ of $\mathbb{P}$ must satisfy $\mathcal{T}(\mathbb{P}) \leq \mathcal{T}(\mathbb{Q})\log (1/\pi_*)$ where $\pi_*$ is the minimum stationary probability. Looking at a specific adversarial strategy allows us to show that $\mathcal{T}(\mathbb{Q}) \geq r(\mathbb{P})$ where $r(\mathbb{P})$ is the relaxation time of $\mathbb{P}$. This gives an alternate proof of the reversible lifting result and helps identify cases where reversible liftings cannot improve the mixing time by more than a constant factor.

The problem of sampling from the stationary distribution of a Markov chain finds widespread applications in a variety of fields. The time required for a Markov chain to converge to its stationary distribution is known as the classical mixing time. In this article, we deal with analog quantum algorithms for mixing. First, we provide an analog quantum algorithm that, given a Markov chain, allows us to sample from its stationary distribution in a time that scales as the sum of the square root of the classical mixing time and the square root of the classical hitting time. Our algorithm makes use of the framework of interpolated quantum walks and relies on Hamiltonian evolution in conjunction with von Neumann measurements. There also exists a different notion for quantum mixing: the problem of sampling from the limiting distribution of quantum walks, defined in a time-averaged sense. In this scenario, the quantum mixing time is defined as the time required to sample from a distribution that is close to this limiting distribution. Recently, we provided an upper bound on the quantum mixing time for Erd\ifmmode \mbox{\H{o}}\else \H{o}\fi{}s-R\'enyi random graphs [Phys. Rev. Lett. 124, 050501 (2020)]. Here, we also extend and expand upon our findings therein. Namely, we provide an intuitive understanding of the state-of-the-art random matrix theory tools used to derive our results. In particular, for our analysis we require information about macroscopic, mesoscopic, and microscopic statistics of eigenvalues of random matrices which we highlight here. Furthermore, we provide numerical simulations that corroborate our analytical findings and extend this notion of mixing from simple graphs to any ergodic, reversible, Markov chain.

In this paper, we develop a new “robust mixing” framework for reasoning about adversarially modified Markov Chains (AMMC). Let ℙ be the transition matrix of an irreducible Markov Chain with stationary distribution π. An adversary announces a sequence of stochastic matrices $\{{\mathbb{A}}_t\}_{t > 0}$ satisfying $\pi{\mathbb{A}}_t = \pi$. An AMMC process involves an application of ℙ followed by ${\mathbb{A}}_t$ at time t. The robust mixing time of an irreducible Markov Chain ℙ is the supremum over all adversarial strategies of the mixing time of the corresponding AMMC process. Applications include estimating the mixing times for certain non-Markovian processes and for reversible liftings of Markov Chains. Non-Markovian card shuffling processes: The random-to-cyclic transposition process is a non-Markovian card shuffling process, which at time t, exchanges the card at position $t {\pmod n}$ with a random card. Mossel, Peres and Sinclair (2004) showed that the mixing time of this process lies between (0.0345+o(1))nlogn and Cnlogn + O(n) (with C ≈4 ×105). We reduce the constant C to 1 by showing that the random-to-top transposition chain (a Markov Chain) has robust mixing time ≤nlogn + O(n) when the adversarial strategies are limited to those which preserve the symmetry of the underlying Markov Chain. Reversible liftings: Chen, Lovasz and Pak showed that for a reversible ergodic Markov Chain ℙ, any reversible lifting ℚ of ℙ must satisfy ${\mathcal{T}}({\mathbb{P}}) \leq {\mathcal{T}}(\mathbb{Q})\log (1/\pi_*)$ where π* is the minimum stationary probability. Looking at a specific adversarial strategy allows us to show that ${\mathcal{T}}(\mathbb{Q}) \geq r({\mathbb{P}})$ where r(ℙ) is the relaxation time of ℙ. This helps identify cases where reversible liftings cannot improve the mixing time by more than a constant factor.

This paper introduces the Attracting Random Walks model, which describes the dynamics of a system of particles on a graph with $n$ vertices. At each step, a single particle moves to an adjacent vertex (or stays at the current one) with probability proportional to the exponent of the number of other particles at a vertex. From an applied standpoint, the model captures the rich get richer phenomenon. We show that the Markov chain exhibits a phase transition in mixing time, as the parameter governing the attraction is varied. Namely, mixing time is $O(n\log n)$ when the temperature is sufficiently high and $\exp (\Omega (n))$ when temperature is sufficiently low. When $\mathcal {G}$ is the complete graph, the model is a projection of the Potts model, whose mixing properties and the critical temperature have been known previously. However, for any other graph our model is non-reversible and does not seem to admit a simple Gibbsian description of a stationary distribution. Notably, we demonstrate existence of the dynamic phase transition without decomposing the stationary distribution into phases.

The past few decades have seen robust research on questions regarding the existence, form, and properties of stationary distributions of stochastically modeled reaction networks. When a stochastic model admits a stationary distribution an important practical question is: what is the rate of convergence of the distribution of the process to the stationary distribution? With the exception of [1] pertaining to models whose state space is restricted to the non-negative integers, there has been a notable lack of results related to this rate of convergence in the reaction network literature. This paper begins the process of filling that hole in our understanding. In this paper, we characterize this rate of convergence, via the mixing times of the processes, for two classes of stochastically modeled reaction networks. Specifically, by applying a Foster-Lyapunov criteria we establish exponential ergodicity for two classes of reaction networks introduced in [2]. Moreover, we show that for one of the classes the convergence is uniform over the initial state.

We show that a new probabilistic technique, recently introduced by the first author, yields the sharpest bounds obtained to date on mixing times of Markov chains in terms of isoperimetric properties of the state space (also known as conductance bounds or Cheeger inequalities). We prove that the bounds for mixing time in total variation obtained by Lovasz and Kannan, can be refined to apply to the maximum relative deviation |p n (x,y)/π(y)−1| of the distribution at time n from the stationary distribution π. We then extend our results to Markov chains on infinite state spaces and to continuous-time chains. Our approach yields a direct link between isoperimetric inequalities and heat kernel bounds; previously, this link rested on analytic estimates known as Nash inequalities.

With a vast amount of data available on online social networks, how to enable efficient analytics over such data has been an increasingly important research problem. Given the sheer size of such social networks, many existing studies resort to sampling techniques that draw random nodes from an online social network through its restrictive web/API interface. While these studies differ widely in analytics tasks supported and algorithmic design, almost all of them use the exact same underlying technique ofrandom walk- a Markov Chain Monte Carlo based method which iteratively transits from one node to its random neighbor.Random walk fits naturally with this problem because, for most online social networks, the only query we can issue through the interface is to retrieve the neighbors of a given node (i.e., no access to the full graph topology). A problem with random walks, however, is the "burn-in" period which requires a large number of transitions/queries before the sampling distribution converges to a stationary value that enables the drawing of samples in a statistically valid manner.In this paper, we consider a novel problem of speeding up the fundamental design of random walks (i.e., reducing the number of queries it requires)withoutchanging the stationary distribution it achieves - thereby enabling a more efficient "drop-in" replacement for existing sampling-based analytics techniques over online social networks. Technically, our main idea is to leverage the history of random walks to construct a higher-ordered Markov chain. We develop two algorithms,Circulated NeighborsandGroupby NeighborsRandom Walk (CNRW and GNRW) and rigidly prove that, no matter what the social network topology is, CNRW and GNRW offer better efficiency than baseline random walks while achieving the same stationary distribution. We demonstrate through extensive experiments on real-world social networks and synthetic graphs the superiority of our techniques over the existing ones.

A Glivenko–Cantelli theorem is a fundamental result in statistics. It says that an empirical distribution function uniformly approximates the true distribution function for a sufficiently large sample size. We prove general Glivenko–Cantelli theorems for three types of sequences of random variables: delayed regenerative, delayed stationary and delayed exchangeable. In particular, our results hold for irreducible Harris recurrent Markov chains that admit a stationary probability distribution but are not necessarily in the stationary state. We also do not assume any mixing conditions on the Markov chain. This is useful in the application of Markov chain Monte Carlo methods. A key tool used is a generalized version of Polya's theorem on the convergence of distribution functions. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, it is demonstrated how a nonparametric estimator of the stationary workload distribution function of the M/G/1-queue can be obtained by systematic sampling the workload process. Weak convergence results and bootstrap methods for empirical distribution functions for stationary associated sequences are used to derive asymptotic results and bootstrap methods for inference about the workload distribution function. The potential of the method is illustrated by a simulation study of the M/D/1 model.

We present a deterministic logspace algorithm for solving S-T Connectivity on directed graphs if: (i) we are given a stationary distribution of the random walk on the graph in which both of the input vertices s and t have nonnegligible probability mass and (ii) the random walk which starts at the source vertex s has polynomial mixing time. This result generalizes the recent deterministic logspace algorithm for S-T Connectivity on undirected graphs [Reingold, 2008]. It identifies knowledge of the stationary distribution as the gap between the S-T Connectivity problems we know how to solve in logspace ( L ) and those that capture all of randomized logspace ( RL ).

Unified theory for finite Markov chains

We investigate popular trajectory-based algorithms inspired by biology and physics to answer a question of general significance: when is it beneficial to reject improvements? A distinguishing factor of SSWM (strong selection weak mutation), a popular model from population genetics, compared to the Metropolis algorithm (MA), is that the former can reject improvements, while the latter always accepts them. We investigate when one strategy outperforms the other. Since we prove that both algorithms converge to the same stationary distribution, we concentrate on identifying a class of functions inducing large mixing times, where the algorithms will outperform each other over a long period of time. The outcome of the analysis is the definition of a function where SSWM is efficient, while Metropolis requires at least exponential time. The identified function favours algorithms that prefer high quality improvements over smaller ones, revealing similarities in the optimisation strategies of SSWM and Metropolis respectively with best-improvement (BILS) and first-improvement (FILS) local search. We conclude the paper with a comparison of the performance of these algorithms and a (1,lambda ) RLS on the identified function. The algorithm favours the steepest gradient with a probability that increases with the size of its offspring population. The results confirm that BILS excels and that the (1,lambda ) RLS is efficient only for large enough population sizes.

We consider a variant of random walks on finite groups. At each step, we\nchoose an element from a set of generators ("directions") uniformly, and an\ninteger from a power law ("speed") distribution associated with the chosen\ndirection. We show that if the finite group is nilpotent, the mixing time of\nthis walk is of the same order of magnitude as the diameter of a suitable\npseudo-metric, D(S, a), on the group, which depends only on the generators and\nspeeds. Additionally, we give sharp bounds on the $\\ell^2$-distance between the\ndistribution of the position of the walker and the stationary distribution, and\ncompute D(S,a) for some examples.\n