An Adaptive Interacting Wang–Landau Algorithm for Automatic Density Exploration

Abstract

While statisticians are well-accustomed to performing exploratory analysis in the modeling stage of an analysis, the notion of conducting preliminary general-purpose exploratory analysis in the Monte Carlo stage (or more generally, the model-fitting stage) of an analysis is an area which we feel deserves much further attention. Towards this aim, this paper proposes a general-purpose algorithm for automatic density exploration. The proposed exploration algorithm combines and expands upon components from various adaptive Markov chain Monte Carlo methods, with the Wang-Landau algorithm at its heart. Additionally, the algorithm is run on interacting parallel chains -a feature which both decreases computational cost as well as stabilizes the algorithm, improving its ability to explore the density. Performance of this new parallel adaptive Wang-Landau (PAWL) algorithm is studied in several applications. Through a Bayesian variable selection example, the authors demonstrate the convergence gains obtained with interacting chains. The ability of the algorithm's adaptive proposal to induce mode-jumping is illustrated through a Bayesian mixture modeling application. Lastly, through a 2D Ising model, the authors demonstrate the ability of the algorithm to overcome the high correlations encountered in spatial models. The appendices contain the full algorithmic description in pseudo-code, a tri-modal toy example and remarks on the convergence of the proposed algorithm.

Parallel and interacting stochastic approximation annealing algorithms for global optimisation

“…PISAA can be extended to use an adaptive binning strategy for automatically determining the partition of the sampling space similar to (Bornn et al, 2013), or a smoothing method to estimate the frequency of visiting each subregion similar to (Liang, 2009 (i) The function h τ pθq is bounded and continuously differentiable with respect to both θ and τ , and there exists a non-negative, upper bounded, and continuously differentiable function v τ pθq such that for any ∆ ą δ ą 0,…”

“…The parallel and interacting stochastic approximation annealing (PISAA) builds on the main principles of SAA and the ideas of population MC (Song et al, 2014;Bornn et al, 2013). It works on a population of parallel SAA chains that interact each other appropriately in order to facilitate the the search for the global minimum by improving the self-adjusting mechanism and the exploration of the sampling space.…”

“…In this article, we develop the parallel and interacting stochastic approximation annealing (PISAA), a general purpose stochastic optimisation algorithm, that extends SAA ) by using population Monte Carlo ideas (Song et al, 2014;Bornn et al, 2013;Wong, 2000, 2001;Wu and Liang, 2011).…”

“…1 SAMC (Liang et al, , 2010Wu and Liang, 2011;Bornn et al, 2013;Song et al, 2014) is an adaptive MCMC sampler that aims at addressing the local mode trapping problem that standard MCMC samplers encounter. It is a generalisation of the Wang-Landau algorithm (Wang and Landau, 2001) but equipped with a stochastic approximation scheme (Robbins and Monro, 1951) that adjusts the target distribution.…”

Wang-Landau algorithm as stochastic optimization and its acceleration

“…The Wang-Landau algorithm (WL) [1] has been proven useful in solving a wide range of computational problems in statistical physics [2][3][4][5][6] and statistics [7][8][9]. It directly targets the density of states (the number of all possible configurations for an energy level of a system), thus allows us to calculate thermodynamic quantities over an arbitrary range of temperature within a single simulation based on density estimates.…”