Xiaocheng Shang

Benedict Leimkuhler, Xiaocheng Shang

We study numerical methods for sampling probability measures in high dimension where the underlying model is only approximately identified with a gradient system. Extended stochastic dynamical methods are discussed which have application to multiscale models, nonequilibrium molecular dynamics, and Bayesian sampling techniques arising in emerging machine learning applications. In addition to providing a more comprehensive discussion of the foundations of these methods, we propose a new numerical method for the adaptive Langevin/stochastic gradient Nosé--Hoover thermostat that achieves a dramatic improvement in numerical efficiency over the most popular stochastic gradient methods reported in the literature. We also demonstrate that the newly established method inherits a superconvergence property (fourth order convergence to the invariant measure for configurational quantities) recently demonstrated in the setting of Langevin dynamics. Our findings are verified by numerical experiments.

Noe Blassel, Louis Carillo, Shiva Darshan et al.

We review various numerical approaches to compute transport coefficients in molecular dynamics. These approaches can be broadly classified into three groups: (i) nonequilibrium methods based on applying an external driving field to the system, measuring the average response in the system, and evaluating the related linear response coefficient; (ii) approaches reformulating the transport coefficient of interest through a time correlation function for the equilibrium dynamics (the most popular instances being Green--Kubo and Einstein formulas); (iii) transient techniques, where the transport coefficient can be computed by monitoring the return to the steady state of a dynamics perturbed off its stationary distribution. For all three classes of methods, we provide elements of numerical analysis, allowing to estimate or at least quantify the level of numerical errors in the estimator of the transport coefficient; and also briefly present recent attempts to more efficiently compute transport coefficients with variance reduction approaches such as control variates, importance sampling and coupling methods. The computation of transport coefficients remains nonetheless challenging and will continue requiring research efforts in the foreseeable future.

Jiani Wei, Xiaocheng Shang

Stochastic gradient Langevin dynamics and its variants approximate the likelihood of an entire dataset, via random (and typically much smaller) subsets, in the setting of Bayesian sampling. Due to the (often substantial) improvement of the computational efficiency, they have been widely used in large-scale machine learning applications. It has been demonstrated that the so-called covariance-controlled adaptive Langevin (CCAdL) thermostat, which incorporates an additional term involving the covariance matrix of the noisy force, outperforms popular alternative methods. A moving average is used in CCAdL to estimate the covariance matrix of the noisy force, in which case the covariance matrix will converge to a constant matrix in long-time limit. Moreover, it appears in our numerical experiments that the use of a moving average could reduce the stability of the numerical integrators, thereby limiting the largest usable stepsize. In this article, we propose a modified CCAdL (i.e., mCCAdL) thermostat that uses the scaling part of the scaling and squaring method together with a truncated Taylor series approximation to the exponential to numerically approximate the exact solution to the subsystem involving the additional term proposed in CCAdL. We also propose a symmetric splitting method for mCCAdL, instead of an Euler-type discretisation used in the original CCAdL thermostat. We demonstrate in our numerical experiments that the newly proposed mCCAdL thermostat achieves a substantial improvement in the numerical stability over the original CCAdL thermostat, while significantly outperforming popular alternative stochastic gradient methods in terms of the numerical accuracy for large-scale machine learning applications.

Xiaocheng Shang, Zhanxing Zhu, Benedict Leimkuhler et al.

Monte Carlo sampling for Bayesian posterior inference is a common approach used in machine learning. The Markov Chain Monte Carlo procedures that are used are often discrete-time analogues of associated stochastic differential equations (SDEs). These SDEs are guaranteed to leave invariant the required posterior distribution. An area of current research addresses the computational benefits of stochastic gradient methods in this setting. Existing techniques rely on estimating the variance or covariance of the subsampling error, and typically assume constant variance. In this article, we propose a covariance-controlled adaptive Langevin thermostat that can effectively dissipate parameter-dependent noise while maintaining a desired target distribution. The proposed method achieves a substantial speedup over popular alternative schemes for large-scale machine learning applications.