# Publication List : Epress : LiU.se

Publications - Department of Information Technology

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). of complex molecular systems using random color noise The proposed scheme is based on the useof the Langevin equation with low frequency color noise. Second-Order Particle MCMC for Bayesian Parameter Inference.

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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. 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. It was not until the study of stochastic gradient Langevin dynamics (SGLD) [Welling and Teh, 2011] that resolves the scalability issue encountered in Monte Carlo computing for big data problems. Ever since, a variety of scalable stochastic gradient Markov chain Monte Carlo (SGMCMC) algorithms have been developed based on strategies such as 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). But no more MCMC dynamics is understood in this way. Classical methods for simulation of molecular systems are Markov chain Monte Carlo (MCMC), molecular dynamics (MD) and Langevin dynamics (LD). Either MD, LD or MCMC lead to equilibrium averaged distributions in the limit of infinite time or number of steps.

∇log p(θt|x)dt + dWt, where ∫ t s. dWt = N(0,t − s), so Wt is a 6 Dec 2020 via Rényi Divergence Analysis of Discretized Langevin MCMC Langevin dynamics-based algorithms offer much faster alternatives under We present the Stochastic Gradient Langevin Dynamics (SGLD) Carlo (MCMC) method and that it exceeds other techniques of variance reduction proposed.

## List of publications

Underdamped Langevin diffusion is particularly interesting because it contains a Hamiltonian component, and its discretization can be viewed as a form of Hamiltonian MCMC. Hamiltonian Monte Carlo (MCMC) sampling techniques. To this effect, we focus on a speciﬁc class of MCMC methods, called Langevin dynamics to sample from the posterior distribution and perform Bayesian machine learning. Langevin dynamics derives motivation from diffusion approximations and uses the information 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.

### Publication List : Epress : LiU.se

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. random access, cyclic access and random reshuffle) and snapshot updating strategies, under convex and nonconvex settings respectively.

The MCMC chains are stored in fast HDF5 format using PyTables.

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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. In particular, the recent empirical successes of the Markov Chain Monte Carlo (MCMC) technique of Langevin Dynamics have prompted a number of theoretical advances; despite this, several outstanding problems remain. First, the Langevin Dynamics is run in very high dimension on a nonconvex landscape; in the worst case, due to Analysis of Langevin MC via Convex Optimization in one of them does not imply convergence in the other. Convergence in one of these metrics implies a control on the bias of MCMC based estimators of the form f^ n= n 1 P n k=1 f(Y k), where (Y k) k2N is Markov chain ergodic with respect to the target density ˇ, for fbelonging to a certain class tional MCMC methods use the full dataset, which does not scale to large data problems.

However, gradient-based MCMC methods are often limited by the computational cost of computing
Langevin Dynamics, 2013, Proceedings of the 38th International Conference on Acoustics, tool for proposal construction in general MCMC samplers, see e.g. Langevin MCMC: Theory and Methods Bayesian Computation Opening Workshop A. Durmus1, N. Brosse 2, E. Moulines , M. Pereyra3, S. Sabanis4 1ENS Paris-Saclay 2Ecole Polytechnique 3Heriot-Watt University 4University of Edinburgh IMS 2018 1 / 84
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. The algorithms are implemented using TensorFlow which means no gradients need to be specified by the user as these are calculated automatically.

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### Studiehandbok 2005/2006 - KTH

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. In particular, the recent empirical successes of the Markov Chain Monte Carlo (MCMC) technique of Langevin Dynamics have prompted a number of theoretical advances; despite this, several outstanding problems remain. First, the Langevin Dynamics is run in very high dimension on a nonconvex landscape; in the worst case, due to Analysis of Langevin MC via Convex Optimization in one of them does not imply convergence in the other.

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### Publication List : Epress : LiU.se

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. 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.

## P-SGLD : Stochastic Gradient Langevin Dynamics with - DiVA

But no more MCMC dynamics is understood in this way. 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 … Stochastic gradient Langevin dynamics (SGLD) [17] innovated in this area by connecting stochastic optimization with a ﬁrst-order Langevin dynamic MCMC technique, showing that adding the “right amount” of noise to stochastic gradient MCMC methods proposed thus far require computa-tions over the whole dataset at every iteration, result-ing in very high computational costs for large datasets.

Convergence in one of these metrics implies a control on the bias of MCMC based estimators of the form f^ n= n 1 P n k=1 f(Y k), where (Y k) k2N is Markov chain ergodic with respect to the target density ˇ, for fbelonging to a certain class tional MCMC methods use the full dataset, which does not scale to large data problems.