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1. Generative Model-based Collective Variable Learning with Metastable State Identification in Molecular Dynamics

   Lyu, Liyao; She, Zhiyuan; Lei, Huan (May 2025, arXivorg)

   We propose a generative model-based framework for learning collective variables (CVs) that faithfully capture the individual metastable states of the fulldimensional molecular dynamics (MD) systems. Unlike most existing approaches based on various feature extraction strategies, the new framework transfers the exhausting efforts of feature selection into a generative task of reconstructing the full-dimensional probability density function (PDF) from a set of CVs under a prior distribution with pre-assigned local maxima. By pairing the CVs with a set of auxiliary Gaussian random variables, we seek an invertible mapping that recovers the full-dimensional PDF and meanwhile, preserves the correspondence between the metastable states of the MD space and individual local maxima of the prior distribution. Through identifying the metastable states within MD space that are generally unknown and imposing the correspondence between the two spaces, the constructed CVs retain clear physical interpretations and provide kinetic insight for the molecular systems on the collective scale. We demonstrate the effectiveness of the proposed method with the alanine dipeptide in the aqueous environment. The constructed CVs faithfully capture the essential metastable states of the full MD systems, which show good agreement with kinetic properties such as the transition from the ballistic to the plateau regime for the mean square displacement. 

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2. On Approximate Thompson Sampling with Langevin Algorithms

   Mazumdar, Eric; Pacchiano, Aldo; Ma, Yian; Jordan, Michael; Bartlett, Peter (January 2020, Proceedings of the 37th International Conference on Machine Learning)

   Daumé III, Hal; Singh, Aarti (Ed.)

   Thompson sampling for multi-armed bandit problems is known to enjoy favorable performance in both theory and practice. However, its wider deployment is restricted due to a significant computational limitation: the need for samples from posterior distributions at every iteration. In practice, this limitation is alleviated by making use of approximate sampling methods, yet provably incorporating approximate samples into Thompson Sampling algorithms remains an open problem. In this work we address this by proposing two efficient Langevin MCMC algorithms tailored to Thompson sampling. The resulting approximate Thompson Sampling algorithms are efficiently implementable and provably achieve optimal instance-dependent regret for the Multi-Armed Bandit (MAB) problem. To prove these results we derive novel posterior concentration bounds and MCMC convergence rates for log-concave distributions which may be of independent interest. 

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3. Likelihood level adapted estimation of marginal likelihood for Bayesian model selection

   https://doi.org/10.1016/j.cma.2025.118141  

   De, Subhayan; Farzad, Reza; Brewick, Patrick T; Johnson, Erik A; Wojtkiewicz, Steven F (October 2025, Computer Methods in Applied Mechanics and Engineering)

   In computational mechanics, multiple models are often present to describe a physical system. While Bayesian model selection is a helpful tool to compare these models using measurement data, it requires the computationally expensive estimation of a multidimensional integral — known as the marginal likelihood or as the model evidence (i.e., the probability of observing the measured data given the model) — over the multidimensional parameter domain. This study presents efficient approaches for estimating this marginal likelihood by transforming it into a one-dimensional integral that is subsequently evaluated using a quadrature rule at multiple adaptively-chosen iso-likelihood contour levels. Three different algorithms are proposed to estimate the probability mass at each adapted likelihood level using samples from importance sampling, stratified sampling, and Markov chain Monte Carlo (MCMC) sampling, respectively. The proposed approach is illustrated — with comparisons to Monte Carlo, nested, and MultiNest sampling — through four numerical examples. The first, an elementary example, shows the accuracies of the three proposed algorithms when the exact value of the marginal likelihood is known. The second example uses an 11-story building subjected to an earthquake excitation with an uncertain hysteretic base isolation layer with two models to describe the isolation layer behavior. The third example considers flow past a cylinder when the inlet velocity is uncertain. Based on the these examples, the method with stratified sampling is by far the most accurate and efficient method for complex model behavior in low dimension, particularly considering that this method can be implemented to exploit parallel computation. In the fourth example, the proposed approach is applied to heat conduction in an inhomogeneous plate with uncertain thermal conductivity modeled through a 100 degree-of-freedom Karhunen–Loève expansion. The results indicate that MultiNest cannot efficiently handle the high-dimensional parameter space, whereas the proposed MCMC-based method more accurately and efficiently explores the parameter space. The marginal likelihood results for the last three examples — when compared with the results obtained from standard Monte Carlo sampling, nested sampling, and MultiNest algorithm — show good agreement. 

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4. Learning cycle-consistent cooperative networks via alternating MCMC teaching for unsupervised cross-domain translation

   Xie, J; Zheng, Z; Fang, X; Zhu, S C; Wu, Y N (January 2021, Proceedings of the AAAI Conference on Artificial Intelligence)

   This paper studies the unsupervised cross-domain translation problem by proposing a generative framework, in which the probability distribution of each domain is represented by a generative cooperative network that consists of an energy based model and a latent variable model. The use of generative cooperative network enables maximum likelihood learning of the domain model by MCMC teaching, where the energy-based model seeks to fit the data distribution of domain and distills its knowledge to the latent variable model via MCMC. Specifically, in the MCMC teaching process, the latent variable model parameterized by an encoder-decoder maps examples from the source domain to the target domain, while the energy-based model further refines the mapped results by Langevin revision such that the revised results match to the examples in the target domain in terms of the statistical properties, which are defined by the learned energy function. For the purpose of building up a correspondence between two unpaired domains, the proposed framework simultaneously learns a pair of cooperative networks with cycle consistency, accounting for a two-way translation between two domains, by alternating MCMC teaching. Experiments show that the proposed framework is useful for unsupervised image-to-image translation and unpaired image sequence translation. 

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5. Projected Stochastic Gradient Langevin Algorithms for Constrained Sampling and Non-Convex Learning

   Lamperski, A. (August 2021, Proceedings of Thirty Fourth Conference on Learning Theory)

   Belkin, M.; Kpotufe, S. (Ed.)

   Langevin algorithms are gradient descent methods with additive noise. They have been used for decades in Markov Chain Monte Carlo (MCMC) sampling, optimization, and learning. Their convergence properties for unconstrained non-convex optimization and learning problems have been studied widely in the last few years. Other work has examined projected Langevin algorithms for sampling from log-concave distributions restricted to convex compact sets. For learning and optimization, log-concave distributions correspond to convex losses. In this paper, we analyze the case of non-convex losses with compact convex constraint sets and IID external data variables. We term the resulting method the projected stochastic gradient Langevin algorithm (PSGLA). We show the algorithm achieves a deviation of 𝑂(𝑇−1/4(𝑙𝑜𝑔𝑇)1/2) from its target distribution in 1-Wasserstein distance. For optimization and learning, we show that the algorithm achieves 𝜖-suboptimal solutions, on average, provided that it is run for a time that is polynomial in 𝜖 and slightly super-exponential in the problem dimension. 

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