Hessian-Free High-Resolution Nesterov Acceleration for Sampling

We propose an accelerated-gradient-based MCMC method. It relies on a modification of the Nesterov's accelerated gradient method for strongly convex functions (NAG-SC): We first reformulate NAG-SC as a Hessian-Free High-Resolution ODE, then release the high-resolution coefficient as a free hyperparameter, and finally inject appropriate noise and discretize the diffusion process. Accelerated sampling enabled by this new hyperparameter is not only experimentally demonstrated on several learning tasks, but also theoretically quantified, both at the continuous level and after discretization. For (not-necessarily-strongly-) convex and $L$-smooth potentials, exponential convergence in $\chi^2$ divergence is proved, with a rate analogous to state-of-the-art results of underdamped Langevin dynamics, plus an additional acceleration. At the same time, the method also works for nonconvex potentials, for which we also establish exponential convergence as long as the potential satisfies a Poincar\'e inequality

NySALT: Nystr\"{o}m-type inference-based schemes adaptive to large time-stepping

Large time-stepping is important for efficient long-time simulations of deterministic and stochastic Hamiltonian dynamical systems. Conventional structure-preserving integrators, while being successful for generic systems, have limited tolerance to time step size due to stability and accuracy constraints. We propose to use data to innovate classical integrators so that they can be adaptive to large time-stepping and are tailored to each specific system. In particular, we introduce NySALT, Nystr\"{o}m-type inference-based schemes adaptive to large time-stepping. The NySALT has optimal parameters for each time step learnt from data by minimizing the one-step prediction error. Thus, it is tailored for each time step size and the specific system to achieve optimal performance and tolerate large time-stepping in an adaptive fashion. We prove and numerically verify the convergence of the estimators as data size increases. Furthermore, analysis and numerical tests on the deterministic and stochastic Fermi-Pasta-Ulam (FPU) models show that NySALT enlarges the maximal admissible step size of linear stability, and quadruples the time step size of the St\"{o}rmer--Verlet and the BAOAB when maintaining similar levels of accuracy.Comment: 26 pages, 7 figure

Doubly Robust Augmented Model Accuracy Transfer Inference with High Dimensional Features

Due to label scarcity and covariate shift happening frequently in real-world studies, transfer learning has become an essential technique to train models generalizable to some target populations using existing labeled source data. Most existing transfer learning research has been focused on model estimation, while there is a paucity of literature on transfer inference for model accuracy despite its importance. We propose a novel $\mathbf{D}$oubly $\mathbf{R}$obust $\mathbf{A}$ugmented $\mathbf{M}$odel $\mathbf{A}$ccuracy $\mathbf{T}$ransfer $\mathbf{I}$nferen$\mathbf{C}$e (DRAMATIC) method for point and interval estimation of commonly used classification performance measures in an unlabeled target population using labeled source data. Specifically, DRAMATIC derives and evaluates the risk model for a binary response $Y$ against some low dimensional predictors $\mathbf{A}$ on the target population, leveraging $Y$ from source data only and high dimensional adjustment features $\mathbf{X}$ from both the source and target data. The proposed estimators are doubly robust in the sense that they are $n^{1/2}$ consistent when at least one model is correctly specified and certain model sparsity assumptions hold. Simulation results demonstrate that the point estimation have negligible bias and the confidence intervals derived by DRAMATIC attain satisfactory empirical coverage levels. We further illustrate the utility of our method to transfer the genetic risk prediction model and its accuracy evaluation for type II diabetes across two patient cohorts in Mass General Brigham (MGB) collected using different sampling mechanisms and at different time points

Non-intrusive and structure preserving multiscale integration of stiff ODEs, SDEs and Hamiltonian systems with hidden slow dynamics via flow averaging

We introduce a new class of integrators for stiff ODEs as well as SDEs. These integrators are (i) {\it Multiscale}: they are based on flow averaging and so do not fully resolve the fast variables and have a computational cost determined by slow variables (ii) {\it Versatile}: the method is based on averaging the flows of the given dynamical system (which may have hidden slow and fast processes) instead of averaging the instantaneous drift of assumed separated slow and fast processes. This bypasses the need for identifying explicitly (or numerically) the slow or fast variables (iii) {\it Nonintrusive}: A pre-existing numerical scheme resolving the microscopic time scale can be used as a black box and easily turned into one of the integrators in this paper by turning the large coefficients on over a microscopic timescale and off during a mesoscopic timescale (iv) {\it Convergent over two scales}: strongly over slow processes and in the sense of measures over fast ones. We introduce the related notion of two-scale flow convergence and analyze the convergence of these integrators under the induced topology (v) {\it Structure preserving}: for stiff Hamiltonian systems (possibly on manifolds), they can be made to be symplectic, time-reversible, and symmetry preserving (symmetries are group actions that leave the system invariant) in all variables. They are explicit and applicable to arbitrary stiff potentials (that need not be quadratic). Their application to the Fermi-Pasta-Ulam problems shows accuracy and stability over four orders of magnitude of time scales. For stiff Langevin equations, they are symmetry preserving, time-reversible and Boltzmann-Gibbs reversible, quasi-symplectic on all variables and conformally symplectic with isotropic friction.Comment: 69 pages, 21 figure

Spatial distribution and potential sources of arsenic and water-soluble ions in the snow at Ili River Valley, China

Trace elements and water-soluble ions in snow can be used as indicators to reveal natural and anthropogenic emissions. To understand the chemical composition, characteristics of snow and their potential sources in the Ili River Valley (IRV), snow samples were collected from 17 sites in the IRV from December 2018 to March 2019. Inverse distance weighting, enrichment factor (EF) analysis, and backward trajectory modelling were applied to evaluate the spatial distributions and sources of water-soluble ions and dissolved arsenic (As) in snow. The re-sults indicate that Ca2+ and SO42-were the dominant ions, and the concentrations of As ranged from 0.09 to 0.503 mu g L-1. High concentrations of As were distributed in the northwest and middle of the IRV, and the concentrations of the major ions were high in the west of the IRV. The strong correlation of As with F-, SO42-, and NO2- demonstrates that As mainly originated from coal-burning and agricultural activities. Principal component analysis showed that the ions originated from a combination of anthropogenic and crustal sources. The EFs showed that K+, SO42-, and Mg2+ were mainly influenced by human activities. Backward trajectory cluster analysis suggested that the chemical composition of snow was affected by soil dust transport from the western air mass, the unique terrain, and local anthropogenic activities. These results provide important sci-entific insights for atmospheric environmental management and agricultural production within the IRV