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Non-asymptotic convergence bounds for modified tamed unadjusted langevin algorithm in non-convex setting

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 2025年第1期543卷

作者： Neufeld, Ariel Ng, Matthew (Cheng En) Zhang, Ying Nanyang Technol Univ Div Math Sci 21 Nanyang Link Singapore 637371 Singapore Hong Kong Univ Sci & Technol Financial Technol Thrust Guangzhou Peoples R China

We consider the problem of sampling from a high-dimensional target distribution pi beta on R-d with density proportional to theta bar right arrow e(-beta U(theta)) using explicit numerical schemes based on discretising the langevin stochastic differential equation (SDE). In recent literature, taming has been proposed and studied as a method for ensuring stability of langevin-based numerical schemes in the case of super-linearly growing drift coefficients for the langevin SDE. In particular, the Tamed Unadjusted langevin algorithm (TULA) was proposed in [2] to sample from such target distributions with the gradient of the potential U being super-linearly growing. However, theoretical guarantees in Wasserstein distances for langevin-based algorithms have traditionally been derived assuming strong convexity of the potential U. In this paper, we propose a novel taming factor and derive, under a setting with possibly non-convex potential U and super-linearly growing gradient of U, non-asymptotic theoretical bounds in Wasserstein-1 and Wasserstein-2 distances between the law of our algorithm, which we name the modified Tamed Unadjusted langevin algorithm (mTULA), and the target distribution pi(beta). We obtain respective rates of convergence O(lambda) and O(lambda(1/2)) in Wasserstein-1 and Wasserstein-2 distances for the discretisation error of mTULA in step size lambda. High-dimensional numerical simulations which support our theoretical findings are presented to showcase the applicability of our algorithm. (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

关键词： Modified tamed unadjusted langevin algorithm langevin SDE Super-linearly growing diffusion coefficients High-dimensional sampling Non-asymptotic convergence bounds

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Marvels and Pitfalls of the langevin algorithm in Noisy High-Dimensional Inference

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Physical Review X 2020年第1期10卷 011057-011057页

作者： Stefano Sarao Mannelli Giulio Biroli Chiara Cammarota Florent Krzakala Pierfrancesco Urbani Lenka Zdeborová Université Paris-Saclay CNRS CEA Institut de Physique Théorique 91191 Gif-sur-Yvette France Laboratoire de Physique de l’Ecole Normale Supérieure ENS Université PSL CNRS Sorbonne Université Université Paris-Diderot Sorbonne Paris Cité Paris France Department of Mathematics King’s College London Strand London WC2R 2LS United Kingdom Laboratoire de Physique Statistique CNRS & Université Pierre & Marie Curie & Ecole Normale Supérieure & PSL Université 75005 Paris France

Gradient-descent-based algorithms and their stochastic versions have widespread applications in machine learning and statistical inference. In this work, we carry out an analytic study of the performance of the algorithm most commonly considered in physics, the langevin algorithm, in the context of noisy high-dimensional inference. We employ the langevin algorithm to sample the posterior probability measure for the spiked mixed matrix-tensor model. The typical behavior of this algorithm is described by a system of integrodifferential equations that we call the langevin state evolution, whose solution is compared with the one of the state evolution of approximate message passing (AMP). Our results show that, remarkably, the algorithmic threshold of the langevin algorithm is suboptimal with respect to the one given by AMP. This phenomenon is due to the residual glassiness present in that region of parameters. We also present a simple heuristic expression of the transition line, which appears to be in agreement with the numerical results.

关键词： Phase diagrams Phase transitions Disordered systems Spin glasses Cavity methods langevin algorithm Replica theory

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langevin algorithms for Markovian Neural Networks and Deep Stochastic Control

Langevin Algorithms for Markovian Neural Networks and Deep S...

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International Joint Conference on Neural Networks (IJCNN)

作者： Bras, Pierre Pages, Gilles Sorbonne Univ Lab Probabilites Stat & Modelisat Paris France

ISBN: (纸本)9781665488679

Stochastic Gradient Descent langevin Dynamics (SGLD) algorithms, which add noise to the classic gradient descent, are known to improve the training of neural networks in some cases where the neural network is very deep. In this paper we study the possibilities of training acceleration for the numerical resolution of stochastic control problems through gradient descent, where the control is parametrized by a neural network. If the control is applied at many discretization times then solving the stochastic control problem reduces to minimizing the loss of a very deep neural network. We numerically show that langevin and Layer-langevin algorithms improve the training on various stochastic control problems like hedging and resource management, and for different choices of gradient descent methods.

关键词： langevin algorithm SGLD Markovian neural network Stochastic control Deep neural network Stochastic optimization

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langevin algorithms for Very Deep Neural Networks with Application to Image Classification

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Procedia Computer Science 2023年 222卷 303-310页

作者： Pierre Bras

Training a very deep neural network is a challenging task, as the deeper a neural network is, the more non-linear it is. We compare the performances of various preconditioned langevin algorithms with their non-langevin counterparts for the training of neural networks of increasing depth. For shallow neural networks, langevin algorithms do not lead to any improvement, however the deeper the network is and the greater are the gains provided by langevin algorithms. Adding noise to the gradient descent allows to escape from local traps, which are more frequent for very deep neural networks. Following this heuristic we introduce a new langevin algorithm called Layer langevin, which consists in adding langevin noise only to the weights associated to the deepest layers. We then prove the benefits of langevin and Layer langevin algorithms for the training of popular deep residual architectures for image classification.

关键词： langevin algorithm SGLD Deep neural network Image classification

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Multilevel langevin Pathwise Average for Gibbs Approximation

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MATHEMATICS OF OPERATIONS RESEARCH 2025年第1期50卷 1-781 C2页

作者： Egea, Maxime Panloup, Fabien Univ Angers CNRS LAREMA SFR MATHSTIC F-49000 Angers France

We propose and study a new multilevel method for the numerical approximation of a Gibbs distribution pi on Rd, based on (overdamped) langevin diffusions. This method relies on a multilevel occupation measure, that is, on an appropriate combination of R occupation measures of (constant -step) Euler schemes with respective steps gamma r � gamma 02-r, r � 0,.. .,R. We first state a quantitative result under general assumptions that guarantees an epsilon-approximation (in an L2 -sense) with a cost of the order epsilon-2 or epsilon-2|log epsilon |3 under less contractive assumptions. We then apply it to overdamped langevin diffusions with strongly convex potential U : Rd -> R and obtain an epsilon-complexity of the order O(d epsilon-2log3(d epsilon-2)) or O(d epsilon-2) under additional assumptions on U. More precisely, up to universal constants, an appropriate choice of the parameters leads to a cost controlled by (lambda U proves 1)2 lambda-3U d epsilon-2 (where lambda U and lambda U respectively denote the supremum and the infimum of the largest and lowest eigenvalue of D2U). We finally complete these theoretical results with some numerical illustrations, including comparisons to other algorithms in Bayesian learning and opening to the non-strongly convex setting.

关键词： multilevel Monte Carlo ergodic diffusion langevin algorithm

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(Non)-penalized multilevel methods for non-uniformly log-concave distributions

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ELECTRONIC JOURNAL OF PROBABILITY 2024年第none期29卷 1-43页

作者： Egea, Maxime Univ Angers Angers France

We study and develop multilevel methods for the numerical approximation of a log-concave probability pi on R-d, based on (over-damped) langevin diffusion. In the continuity of [12] concentrated on the uniformly log-concave setting, we here study the procedure in the absence of the uniformity assumption. More precisely, we first adapt an idea of [8] by adding a penalization term to the potential to recover the uniformly convex setting. Such approach leads to an epsilon of the order epsilon(-5)pi(|.|(2))(9/2)d (up to logarithmic terms). Then, in the spirit of \cite{art:gadat2020cost}, we propose to explore the robustness of the method in a weakly convex parametric setting where the lowest eigenvalue of the Hessian of the potential U is controlled by the function U(x)(-r) for r is an element of(0,1). In this intermediary framework between the strongly convex setting (r=0) and the ``Laplace case'' (r=1), we show that with the help of the control of exponential moments of the Euler scheme, we can adapt some fundamental properties for the efficiency of the method. In the ``best'' setting where U is C-3 and U(x)(-r) control the largest eigenvalue of the Hessian, we obtain an epsilon-complexity of the order c(rho,delta)epsilon(-2-rho)d(1+rho/2+(4-rho+delta)r) for any rho>0 (but with a constant c(rho,delta) which increases when rho and delta go to 0).

关键词： multilevel ergodic diffusion langevin algorithm Euler scheme weakly convex.

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Bayes analysis of the generalized gamma AFT models for left truncated and right censored data

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JOURNAL OF STATISTICAL COMPUTATION AND SIMULATION 2023年第12期93卷 2026-2051页

作者： Shukla, Asmita Ranjan, Rakesh Upadhyay, Satyanshu K. Banaras Hindu Univ Fac Sci DST Ctr Interdisciplinary Math Sci Varanasi India Banaras Hindu Univ Fac Sci Dept Stat Varanasi India

This article considers the Bayes analysis of generalized gamma accelerated failure time model and its two components Weibull and gamma when the given observations are left truncated and right censored. In order to perform the analysis, the paper proposes the use of an improved version of the Metropolis-Hastings algorithm, namely, the Metropolis-adjusted langevin algorithm. Besides, the paper also checks the model compatibility and compares the considered models with its components using the Bayes factor computed on the basis of a recent methodology. A numerical illustration is provided based on a simulated as well as a real dataset. The real dataset consists of individuals infected with human immunodeficiency virus who are at the risk of acquired immunodeficiency syndrome and subsequent deaths. The numerical illustration is further extended to check the effect of different therapies including the highly active antiretroviral therapy on the lifetime of individuals.

关键词： Generalized gamma distribution weibull distribution gamma distribution langevin algorithm Metropolis-Hastings algorithm accelerated failure time model Bayes factor highly active antiretroviral therapy

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Exploring noisy Jeffery orbits: A combined Fokker-Planck and langevin analysis in two and three dimensions

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Physical Review E 2024年第4期110卷 044143页

作者： J. Talbot C. Antoine P. Claudin E. Somfai T. Börzsönyi HUN-REN

The behavior of nonspherical particles in a shear flow is of significant practical and theoretical interest. These systems have been the object of numerous investigations since the pioneering work of Jeffery a century ago. His eponymous orbits describe the deterministic motion of an isolated, rodlike particle in a shear flow. Subsequently, the effect of adding noise was investigated. The theory has been applied to colloidal particles, macromolecules, anisometric granular particles, and most recently to microswimmers, for example, bacteria. We study the Jeffery orbits of elongated (uniaxial, prolate) particles subject to noise using langevin simulations and a Fokker-Planck equation. We extend the analytical solution for infinitely thin needles (β=1) obtained by Doi and Edwards to particles with arbitrary shape factor (0≤β≤1) and validate the theory by comparing it with simulations. We examine the rotation of the particle around the vorticity axis and study the orientational order matrix. We use the latter to obtain scalar order parameters s and r describing nematic ordering and biaxiality from the orientational distribution function. The value of s (nematic ordering) increases monotonically with increasing Péclet number, while r (measure of biaxiality) displays a maximum value. From perturbation theory, we obtain simple expressions that provide accurate descriptions at low noise (or large Péclet numbers). We also examine the orientational distribution in the v-grad v plane and in the perpendicular direction. Finally, we present the solution of the Fokker-Planck equation for a strictly two-dimensional (2D) system. For the same noise amplitude, the average rotation speed of the particle in 3D is larger than in 2D.

关键词： Fluctuations & noise Fluid-particle interactions Granular flows Shear flows Stochastic processes Brownian dynamics Fokker–Planck equation langevin algorithm

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Multimodality of Ir187 fission studied by the langevin approach

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Physical Review C 2024年第3期109卷 034609-034609页

作者： Y. G. Huang (黄英格) F. C. Gu (顾富昌) Y. J. Feng (冯玉洁) H. Wang (王慧) E. X. Xiao (肖尔熙) X. Lei (雷昕) L. Zhu (祝龙) J. Su (苏军) Sino-French Institute of Nuclear Engineering and Technology Sun Yat-sen University Zhuhai 519082 China Key Laboratory of Nuclear Data China Institute of Atomic Energy Beijing 102413 China

Background: The fission mechanism of sub-lead nuclides remains unclear, especially the types of fission modes involved and their corresponding shell effects.

关键词： Fission Nuclear reactions 150 ≤ A ≤ 189 langevin algorithm

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Subconductance states in a semimicroscopic model for a tetrameric pore

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Physical Review E 2024年第4期109卷 044402-044402页

作者： L. Ramírez-Piscina J. M. Sancho Departament de Física Aplicada EPSEB Universitat Politécnica de Catalunya Avinguda Doctor Marañón 44 E-08028 Barcelona Spain Universitat de Barcelona Departament de Física de la Matèria Condensada Martí i Franqués 1 E-08028 Barcelona Spain

A physical model for a structured tetrameric pore is studied. The pore is modeled as a device composed of four subunits, each one exhibiting two possible states (open and closed). The pore is located within a membrane that separates two reservoirs with ionic solutions. All variables of the model follow physical dynamical equations accounting for the internal structure of the pore, derived from a single energy functional and supplemented with thermal noises. An extensive study of the resulting ionic intensity is performed for different values of the control parameters, mainly membrane potential and reservoir ion concentrations. Two possible physical devices are studied: voltage-gated (including a voltage sensor in each subunit) and non-voltage-gated pores. The ionic flux through the pore exhibits several distinct dynamical configurations, in particular subconductance states, which indicate very different dynamical internal states of the subunits. Such subconductance states become much easier to observe in sensorless pores. These results are compared with available experimental data on tetrameric K channels and analytical predictions.

关键词： Biomolecular & subcellular processes Brownian motion Ionic transport Cell membrane Brownian dynamics Fokker–Planck equation langevin algorithm langevin equation Stochastic differential equations

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