Variance Reduction Using Nonreversible Langevin Samplers

Sampling Efficiency of Transverse Forces in Dense Liquids

Sampling the Boltzmann distribution using forces that violate detailed balance can be faster than with the equilibrium evolution, but the acceleration depends on the nature of the nonequilibrium drive and the physical situation. Here, we study the efficiency of forces transverse to energy gradients in dense liquids through a combination of techniques: Brownian dynamics simulations, exact infinite-dimensional calculation, and a mode-coupling approximation. We find that the sampling speedup varies nonmonotonically with temperature, and decreases as the system becomes more glassy. We characterize the interplay between the distance to equilibrium and the efficiency of transverse forces by means of odd transport coefficients.

Untangling dissipative and Hamiltonian effects in bulk and boundary-driven systems

Using the theory of large deviations, macroscopic fluctuation theory provides a framework to understand the behavior of nonequilibrium dynamics and steady states in diffusive systems. We extend this framework to a minimal model of a nonequilibrium nondiffusive system, specifically an open linear network on a finite graph. We explicitly calculate the dissipative bulk and boundary forces that drive the system towards the steady state, and the nondissipative bulk and boundary forces that drive the system in orbits around the steady state. Using the fact that these forces are orthogonal in a certain sense, we provide a decomposition of the large-deviation cost into dissipative and nondissipative terms. We establish that the purely nondissipative force turns the dynamics into a Hamiltonian system. These theoretical findings are illustrated by numerical examples.

Thermodynamic Bounds on Correlation Times

We derive a variational expression for the correlation time of physical observables in steady-state diffusive systems. As a consequence of this variational expression, we obtain lower bounds on the correlation time, which provide speed limits on the self-averaging of observables. In equilibrium, the bound takes the form of a trade-off relation between the long- and short-time fluctuations of an observable. Out of equilibrium, the trade-off can be violated, leading to an acceleration of self-averaging. We relate this violation to the steady-state entropy production rate, as well as the geometric structure of the irreversible currents, giving rise to two complementary speed limits. One of these can be formulated as a lower estimate on the entropy production from the measurement of time-symmetric observables. Using an illustrating example, we show the intricate behavior of the correlation time out of equilibrium for different classes of observables and how this can be used to partially infer dissipation even if no time-reversal symmetry breaking can be observed in the trajectories of the observable.

Accelerating, to some extent, the p-spin dynamics

We consider a detailed-balance-violating dynamics whose stationary state is a prescribed Boltzmann distribution. Such dynamics have been shown to be faster than any equilibrium counterpart. We quantify the gain in convergence speed for a system whose energy landscape displays one and then an infinite number of energy barriers. In the latter case, we work with the mean-field disordered p spin and show that the convergence to equilibrium or to the nonergodic phase is accelerated during both the β- and α-relaxation stages. An interpretation in terms of trajectories in phase space and of an accidental fluctuation-dissipation theorem is provided.

Accelerated Diffusion-Based Sampling by the Non-Reversible Dynamics with Skew-Symmetric Matrices

Langevin dynamics (LD) has been extensively studied theoretically and practically as a basic sampling technique. Recently, the incorporation of non-reversible dynamics into LD is attracting attention because it accelerates the mixing speed of LD. Popular choices for non-reversible dynamics include underdamped Langevin dynamics (ULD), which uses second-order dynamics and perturbations with skew-symmetric matrices. Although ULD has been widely used in practice, the application of skew acceleration is limited although it is expected to show superior performance theoretically. Current work lacks a theoretical understanding of issues that are important to practitioners, including the selection criteria for skew-symmetric matrices, quantitative evaluations of acceleration, and the large memory cost of storing skew matrices. In this study, we theoretically and numerically clarify these problems by analyzing acceleration focusing on how the skew-symmetric matrix perturbs the Hessian matrix of potential functions. We also present a practical algorithm that accelerates the standard LD and ULD, which uses novel memory-efficient skew-symmetric matrices under parallel-chain Monte Carlo settings.

Role of current fluctuations in nonreversible samplers

It is known that the distribution of nonreversible Markov processes breaking the detailed balance condition converges faster to the stationary distribution compared to reversible processes having the same stationary distribution. This is used in practice to accelerate Markov chain Monte Carlo algorithms that sample the Gibbs distribution by adding nonreversible transitions or nongradient drift terms. The breaking of detailed balance also accelerates the convergence of empirical estimators to their ergodic expectation in the long-time limit. Here, we give a physical interpretation of this second form of acceleration in terms of currents associated with the fluctuations of empirical estimators using the level 2.5 of large deviations, which characterizes the likelihood of density and current fluctuations in Markov processes. Focusing on diffusion processes, we show that there is accelerated convergence because estimator fluctuations arise in general with current fluctuations, leading to an added large deviation cost compared to the reversible case, which shows no current. We study the current fluctuation most likely to arise in conjunction with a given estimator fluctuation and provide bounds on the acceleration, based on approximations of this current. We illustrate these results for the Ornstein-Uhlenbeck process in two dimensions and the Brownian motion on the circle.

A discrete bouncy particle sampler

Most Markov chain Monte Carlo methods operate in discrete time and are reversible with respect to the target probability. Nevertheless, it is now understood that the use of nonreversible Markov chains can be beneficial in many contexts. In particular, the recently proposed bouncy particle sampler leverages a continuous-time and nonreversible Markov process, and empirically shows state-of-the-art performance when used to explore certain probability densities; however, its implementation typically requires the computation of local upper bounds on the gradient of the log target density. We present the discrete bouncy particle sampler, a general algorithm based on a guided random walk, a partial refreshment of direction and a delayed-rejection step. We show that the bouncy particle sampler can be understood as a scaling limit of a special case of our algorithm. In contrast to the bouncy particle sampler, implementing the discrete bouncy particle sampler only requires pointwise evaluation of the target density and its gradient. We propose extensions of the basic algorithm for situations when the exact gradient of the target density is not available. In a Gaussian setting, we establish a scaling limit for the radial process as the dimension increases to infinity. We leverage this result to obtain the theoretical efficiency of the discrete bouncy particle sampler as a function of the partial-refreshment parameter, which leads to a simple and robust tuning criterion. A further analysis in a more general setting suggests that this tuning criterion applies more generally. Theoretical and empirical efficiency curves are then compared for different targets and algorithm variations.

Response theory and phase transitions for the thermodynamic limit of interacting identical systems

We study the response to perturbations in the thermodynamic limit of a network of coupled identical agents undergoing a stochastic evolution which, in general, describes non-equilibrium conditions. All systems are nudged towards the common centre of mass. We derive Kramers–Kronig relations and sum rules for the linear susceptibilities obtained through mean field Fokker–Planck equations and then propose corrections relevant for the macroscopic case, which incorporates in a self-consistent way the effect of the mutual interaction between the systems. Such an interaction creates a memory effect. We are able to derive conditions determining the occurrence of phase transitions specifically due to system-to-system interactions. Such phase transitions exist in the thermodynamic limit and are associated with the divergence of the linear response but are not accompanied by the divergence in the integrated autocorrelation time for a suitably defined observable. We clarify that such endogenous phase transitions are fundamentally different from other pathologies in the linear response that can be framed in the context of critical transitions. Finally, we show how our results can elucidate the properties of the Desai–Zwanzig model and of the Bonilla–Casado–Morillo model, which feature paradigmatic equilibrium and non-equilibrium phase transitions, respectively.

Asymptotic variance for random walk Metropolis chains in high dimensions: logarithmic growth via the Poisson equation

There are two ways of speeding up Markov chain Monte Carlo algorithms: (a) construct more complex samplers that use gradient and higher-order information about the target and (b) design a control variate to reduce the asymptotic variance. While the efficiency of (a) as a function of dimension has been studied extensively, this paper provides the first results linking the efficiency of (b) with dimension. Specifically, we construct a control variate for a d-dimensional random walk Metropolis chain with an independent, identically distributed target using the solution of the Poisson equation for the scaling limit in [30]. We prove that the asymptotic variance of the corresponding estimator is bounded above by a multiple of $\log(d)/d$ over the spectral gap of the chain. The proof hinges on large deviations theory, optimal Young’s inequality and Berry–Esseen-type bounds. Extensions of the result to non-product targets are discussed.

Feature Input Symmetry Algorithm of Multi-Modal Natural Language Library Based on BP Neural Network

When using traditional knowledge retrieval algorithms to analyze whether the feature input of words in multi-modal natural language library is symmetrical, the symmetry of words cannot be analyzed, resulting in inaccurate analysis results. A feature input symmetric algorithm of multi-modal natural language library based on BP (back propagation) neural network is proposed in this paper. A Chinese abstract generation method based on multi-modal neural network is used to extract Chinese abstracts from images in multi-modal natural language library. The Word Sense Disambiguation (WSD) Model is constructed by the BP neural network. After the word or text disambiguation is performed on the Chinese abstract in the multi-modal natural language library, the feature input symmetry problem in the multi-modal natural language library is analyzed according to the sentence similarity. The experimental results show that the proposed algorithm can effectively analyze the eigenvalue symmetry problem of the multi-modal natural language library. The maximum error rate of the analysis algorithm is 7%, the growth rate of the analysis speed is up to 50%, and the average analysis time is 540.56 s. It has the advantages of small error and high efficiency.

Dynamics of the Desai-Zwanzig model in multiwell and random energy landscapes

We analyze a variant of the Desai-Zwanzig model [J. Stat. Phys. 19, 1 (1978)JSTPBS0022-471510.1007/BF01020331]. In particular, we study stationary states of the mean field limit for a system of weakly interacting diffusions moving in a multiwell potential energy landscape, coupled via a Curie-Weiss type (quadratic) interaction potential. The location and depth of the local minima of the potential are either deterministic or random. We characterize the structure and nature of bifurcations and phase transitions for this system, by means of extensive numerical simulations and of analytical calculations for an explicitly solvable model. Our numerical experiments are based on Monte Carlo simulations, the numerical solution of the time-dependent nonlinear Fokker-Planck (McKean-Vlasov) equation, the minimization of the free-energy functional, and a continuation algorithm for the stationary solutions.

Stochastic modelling of urban structure

The building of mathematical and computer models of cities has a long history. The core elements are models of flows (spatial interaction) and the dynamics of structural evolution. In this article, we develop a stochastic model of urban structure to formally account for uncertainty arising from less predictable events. Standard practice has been to calibrate the spatial interaction models independently and to explore the dynamics through simulation. We present two significant results that will be transformative for both elements. First, we represent the structural variables through a single potential function and develop stochastic differential equations to model the evolution. Second, we show that the parameters of the spatial interaction model can be estimated from the structure alone, independently of flow data, using the Bayesian inferential framework. The posterior distribution is doubly intractable and poses significant computational challenges that we overcome using Markov chain Monte Carlo methods. We demonstrate our methodology with a case study on the London, UK, retail system.

Mean Field Limits for Interacting Diffusions in a Two-Scale Potential

In this paper, we study the combined mean field and homogenization limits for a system of weakly interacting diffusions moving in a two-scale, locally periodic confining potential, of the form considered in Duncan et al. (Brownian motion in an N-scale periodic potential, arXiv:1605.05854, 2016b). We show that, although the mean field and homogenization limits commute for finite times, they do not, in general, commute in the long time limit. In particular, the bifurcation diagrams for the stationary states can be different depending on the order with which we take the two limits. Furthermore, we construct the bifurcation diagram for the stationary McKean-Vlasov equation in a two-scale potential, before passing to the homogenization limit, and we analyze the effect of the multiple local minima in the confining potential on the number and the stability of stationary solutions.

Using Perturbed Underdamped Langevin Dynamics to Efficiently Sample from Probability Distributions

In this paper we introduce and analyse Langevin samplers that consist of perturbations of the standard underdamped Langevin dynamics. The perturbed dynamics is such that its invariant measure is the same as that of the unperturbed dynamics. We show that appropriate choices of the perturbations can lead to samplers that have improved properties, at least in terms of reducing the asymptotic variance. We present a detailed analysis of the new Langevin sampler for Gaussian target distributions. Our theoretical results are supported by numerical experiments with non-Gaussian target measures.

Limit theorems for the zig-zag process

Markov chain Monte Carlo (MCMC) methods provide an essential tool in statistics for sampling from complex probability distributions. While the standard approach to MCMC involves constructing discrete-time reversible Markov chains whose transition kernel is obtained via the Metropolis–Hastings algorithm, there has been recent interest in alternative schemes based on piecewise deterministic Markov processes (PDMPs). One such approach is based on the zig-zag process, introduced in Bierkens and Roberts (2016), which proved to provide a highly scalable sampling scheme for sampling in the big data regime; see Bierkenset al.(2016). In this paper we study the performance of the zig-zag sampler, focusing on the one-dimensional case. In particular, we identify conditions under which a central limit theorem holds and characterise the asymptotic variance. Moreover, we study the influence of the switching rate on the diffusivity of the zig-zag process by identifying a diffusion limit as the switching rate tends to ∞. Based on our results we compare the performance of the zig-zag sampler to existing Monte Carlo methods, both analytically and through simulations.

Acceleration of Convergence to Equilibrium in Markov Chains by Breaking Detailed Balance

We analyse and interpret the effects of breaking detailed balance on the convergence to equilibrium of conservative interacting particle systems and their hydrodynamic scaling limits. For finite systems of interacting particles, we review existing results showing that irreversible processes converge faster to their steady state than reversible ones. We show how this behaviour appears in the hydrodynamic limit of such processes, as described by macroscopic fluctuation theory, and we provide a quantitative expression for the acceleration of convergence in this setting. We give a geometrical interpretation of this acceleration, in terms of currents that are antisymmetric under time-reversal and orthogonal to the free energy gradient, which act to drive the system away from states where (reversible) gradient-descent dynamics result in slow convergence to equilibrium.

Noise-induced transitions in rugged energy landscapes

We consider the problem of an overdamped Brownian particle moving in multiscale potential with N+1 characteristic length scales: the macroscale and N separated microscales. We show that the coarse-grained dynamics is given by an overdamped Langevin equation with respect to the free energy and with a space-dependent diffusion tensor, the calculation of which requires the solution of N fully coupled Poisson equations. We study in detail the structure of the bifurcation diagram for one-dimensional problems, and we show that the multiscale structure in the potential leads to hysteresis effects and to noise-induced transitions. Furthermore, we obtain an explicit formula for the effective diffusion coefficient for a self-similar separable potential, and we investigate the limit of infinitely many small scales.