Gradient-Based MCMC CSC 412 Tutorial March 2, 2017 Jake Snell Many slides borrowed from: Iain Murray, MLSS ’09* • Langevin Dynamics

rapid convergence to the target distribution of the dynamics system and demonstrate superior performances competing with dynamics based MCMC samplers.

The algorithms are implemented using TensorFlow which means no gradients need to be specified by the user as these are calculated automatically. It also means the algorithms are efficient. SGLD[Welling+11], SGRLD[Patterson+13] SGLDの運動⽅程式は1次のLangevin Dynamics 18 SGHMCの2次のLangevin Dynamicsで B→∞とした極限として得られる SGLDのアルゴリズム SGRLDは1次のLangevin DynamicsにFisher計量からくるパラメータ空間の幾何的な情報を加える G(θ)はフィッシャー⾏列の逆⾏列 In this paper, we explore a general Aggregated Gradient Langevin Dynamics framework (AGLD) for the Markov Chain Monte Carlo (MCMC) sampling. We investigate the nonasymptotic convergence of AGLD with a unified analysis for different data accessing (e.g.

Introduction In this paper, we study the continuous time underdamped Langevin diffusion represented by the following stochastic differential equation (SDE): dvt= vtdt u∇f(xt)dt+(√ 2 u)dBt (1) dxt= vtdt; As an alternative, approximate MCMC methods based on unadjusted Langevin dynamics offer scalability and more rapid sampling at the cost of biased inference. However, when assessing the quality of approximate MCMC samples for characterizing the posterior distribution, most diagnostics fail to account for these biases. Langevin dynamics [Ken90, Nea10] is an MCMC scheme which produces samples from the posterior by means of gradient updates plus Gaussian noise, resulting in a proposal distribution q(θ ∗ | θ) as described by Equation 2. Overview • Review of Markov Chain Monte Carlo (MCMC) • Metropolis algorithm • Metropolis-Hastings algorithm • Langevin Dynamics • Hamiltonian Monte Carlo • Gibbs Sampling (time permitting) It is known that the Langevin dynamics used in MCMC is the gradient flow of the KL divergence on the Wasserstein space, which helps convergence analysis and inspires recent particle-based variational inference methods (ParVIs).

gradient langevin dynamics for deep neural networks. In AAAI Conference on Artiﬁcial Intelligence, 2016. Yi-An Ma, Tianqi Chen, and Emily B. Fox. A complete recipe for stochastic gradient mcmc. In Advances in Neural Information Processing Systems, 2015. Stephan Mandt, Matthew D. Hoffman, and David M. Blei. A variational analysis of stochastic

Fredrik Lindsten. Fredrik Lindsten - Project PI - WASP – Wallenberg AI Fredrik Lindsten.

Stochastic Gradient MCMC with Stale Gradients Changyou Chen yNan Dingz Chunyuan Li Yizhe Zhang yLawrence Carin yDept. of Electrical and Computer Engineering, Duke University, Durham, NC, USA zGoogle Inc., Venice, CA, USA y{cc448,cl319,yz196,lcarin}@duke.edu; zdingnan@google.com Abstract

A pioneering work in com-bining stochastic optimization with MCMC was presented in (Welling and Teh 2011), based on Langevin dynam-ics (Neal 2011). This method was referred to as Stochas-tic Gradient Langevin Dynamics (SGLD), and required only HYBRID GRADIENT LANGEVIN DYNAMICS FOR BAYESIAN LEARNING 223 are also some variants of the method, for example, pre-conditioning the dynamic by a positive deﬁnite matrix A to obtain (2.2) dθt = 1 2 A∇logπ(θt)dt +A1/2dWt. This dynamic also has π as its stationary distribution. To apply Langevin dynamics of MCMC method to Bayesian learning MCMC and non-reversibility Overview I Markov Chain Monte Carlo (MCMC) I Metropolis-Hastings and MALA (Metropolis-Adjusted Langevin Algorithm) I Reversible vs non-reversible Langevin dynamics I How to quantify and exploit the advantages of non-reversibility in MCMC I Various approaches taken so far I Non-reversible Hamiltonian Monte Carlo I MALA with irreversible proposal (ipMALA) In Section 2, we review some backgrounds in Langevin dynamics, Riemann Langevin dynamics, and some stochastic gradient MCMC algorithms. In Section 3 , our main algorithm is proposed. We first present a detailed online damped L-BFGS algorithm which is used to approximate the inverse Hessian-vector product and discuss the properties of the approximated inverse Hessian.

of Electrical and Computer Engineering, Duke University, Durham, NC, USA zGoogle Inc., Venice, CA, USA y{cc448,cl319,yz196,lcarin}@duke.edu; zdingnan@google.com Abstract MCMC [25], such as nite step Langevin dynamics, as an approximate inference engine. In the learning process, for each training example, we always initialize such a short run MCMC from the prior distribution of the latent variables, such as Gaussian or uniform noise … COARSE-GRADIENT LANGEVIN ALGORITHMS FOR DYNAMIC DATA INTEGRATION AND UNCERTAINTY QUANTIFICATION P. DOSTERT∗, Y. EFENDIEV†, T.Y. HOU‡, AND W. LUO§ Abstract. The main goal of this paper is to design an eﬃcient sampling technique for dynamic data integra- The sgmcmc package implements some of the most popular stochastic gradient MCMC methods including SGLD, SGHMC, SGNHT. It also implements control variates as a way to increase the efficiency of these methods.
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We first present a detailed online damped L-BFGS algorithm which is used to approximate the inverse Hessian-vector product and discuss the properties of the approximated inverse Hessian. Langevin dynamics MCMC for training neural networks. We employ six bench-mark chaotic time series problems to demonstrate the e ectiveness of the pro-posed method. MCMC from Hamiltonian Dynamics q Given !" (starting state) q Draw # ∼ % 0,1 q Use ) steps of leapfrog to propose next state q Accept / reject based on change in Hamiltonian Each iteration of the HMC algorithm has two steps. 2020-06-19 · Recently, the task of image generation has attracted much attention.

The MCMC chains are stored in fast HDF5 format using PyTables. A mean function can be added to the (GP) models of the GPy package.
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2011-10-17 · Langevin Dynamics In Langevin dynamics we take gradient steps with constant valued and add gaussian noise Based o using the posterior as an equilibrium distribution All of the data is used, i.e. there is no batch Langevin Dynamics We update by using the equation and use the updated value as a M-H proposal: t = 2 rlog p( t) + XN i=1 rlog p(x ij

Keywords: R, stochastic gradient Markov chain Monte Carlo, big data, MCMC, stochastic gra- dient Langevin dynamics, stochastic gradient Hamiltonian Monte Standard approaches to inference over the probability simplex include variational inference [Bea03,. WJ08] and Markov chain Monte Carlo methods (MCMC) like It is known that the Langevin dynamics used in.

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Metropolis Adjusted Langevin Dynamics. The MCMC chains are stored in fast HDF5 format using PyTables. A mean function can be added to the (GP) models of the GPy package. Repo. pymcmcstat. The pymcmcstat package is a Python program for running Markov Chain Monte Carlo (MCMC) simulations.

Langevin dynamics derives motivation from diffusion approximations and uses the information Langevin Dynamics The wide adoption of the replica exchange Monte Carlo in traditional MCMC algorithms motivates us to design replica exchange stochastic gradient Langevin dynamics for DNNs, but the straightforward extension of reLD to replica exchange stochastic gradient Langevin dynamics is highly Langevin dynamics segment as a (pseudo) Monte Carlo move. This move assigns a velocity from the Maxwell-Boltzmann distribution and executes a number of Maxwell-Boltzmann steps to propagate dynamics. This is not a true Monte Carlo move, in that the generation of the correct distribution is only exact in the limit of infinitely small timestep; in other words, the discretization error is assumed to be negligible.

MCMCの意義(§1.)から始め、マルコフ連鎖の数学的な基礎(§2.,3.,4.)、MCMCの代表的なアルゴリズムであるMetropolis-Hastings法(§5.)、その例の1つである*2Langevin Dynamics(§6.)、そして(僕の中で)絶賛大流行中のライブラリEdwardを使ってより発展的(?)なアルゴリズムであるStochastic Gradient Langevin Dynamicsの説明

Metropolis-adjusted Langevin algorithm (MALA) is a Markov chain Monte Carlo ( MCMC) algorithm that takes a step of a discretised Langevin diffusion as a Nonreversible Langevin Dynamics. An MCMC scheme which departs from the assumption of reversible dynamics is Hamiltonian MCMC [53], which has proved The stochastic gradient Langevin dynamics (SGLD) pro- posed by Welling and Teh (2011) is the first sequential mini-batch-based MCMC algorithm. In SGLD 10 Aug 2016 “Bayesian learning via stochastic gradient Langevin dynamics”. In: ICML. 2011.

1066, 1064, dynamic 1829, 1827, Langevin distributions, #. 1830, 1828, Laplace 2012, 2010, Markov chain Monte Carlo ; MCMC, MCMC. 2013, 2011, Markov 'evidence' that they will accept, and the static versus dynamic nature. of EBP. This is not to imply are de ned as evidence based (Kellam and Langevin, 2003). In other words, they are cedure with the Markov chain Monte Carlo (MCMC).