Related papers: Concentration Inequalities and UQ Bounds for Hypoc…
Non-convex sampling is a key challenge in machine learning, central to non-convex optimization in deep learning as well as to approximate probabilistic inference. Despite its significance, theoretically there remain many important…
Markov chains can be used to generate samples whose distribution approximates a given target distribution. The quality of the samples of such Markov chains can be measured by the discrepancy between the empirical distribution of the samples…
In this paper, we evaluate the performance of the multilevel Monte Carlo method (MLMC) for deterministic and uncertain hyperbolic systems, where randomness is introduced either in the modeling parameters or in the approximation algorithms.…
We present a nonlinear (in the sense of McKean) generalization of Hamiltonian Monte Carlo (HMC) termed nonlinear HMC (nHMC) capable of sampling from nonlinear probability measures of mean-field type. When the underlying confinement…
In this paper, we survey physically related applications of a class of weighted quasi-Monte Carlo methods from a theoretical, deterministic perspective, and establish quantitative universal rapid convergence results via various regularity…
This paper is concerned with the approximation to invariant measures for Langevin dynamics of McKean--Vlasov type. Under dissipativity and Lipschitz conditions, we prove that the empirical measures of both the mean-field and…
Restricted Boltzmann Machines are simple and powerful generative models that can encode any complex dataset. Despite all their advantages, in practice the trainings are often unstable and it is difficult to assess their quality because the…
Explicit, momentum-based dynamics for optimizing functions defined on Lie groups was recently constructed, based on techniques such as variational optimization and left trivialization. We appropriately add tractable noise to the…
In this paper we prove multilevel concentration inequalities for bounded functionals $f = f(X_1, \ldots, X_n)$ of random variables $X_1, \ldots, X_n$ that are either independent or satisfy certain logarithmic Sobolev inequalities. The…
We describe a new MCMC method optimized for the sampling of probability measures on Hilbert space which have a density with respect to a Gaussian; such measures arise in the Bayesian approach to inverse problems, and in conditioned…
We derive conditional a priori error estimates of a wide class of finite volume and Runge-Kutta discontinuous Galerkin methods with abstract limiting for hyperbolic systems of conservation laws in 1D via the verification of weak consistency…
We obtain $q$-Wasserstein convergence rates in the invariance principle for nonuniformly hyperbolic flows, where $q\ge1$ depends on the degree of nonuniformity. Utilizing a martingale-coboundary decomposition for nonuniformly expanding…
We study the underdamped Langevin diffusion when the log of the target distribution is smooth and strongly concave. We present a MCMC algorithm based on its discretization and show that it achieves $\varepsilon$ error (in 2-Wasserstein…
This paper investigates the robust optimal control of sampled-data stochastic systems with multiplicative noise and distributional ambiguity. We consider a class of discrete-time optimal control problems where the controller \emph{jointly}…
We use here a particle system to prove a convergence result as well as a deviation inequality for solutions of granular media equation when the confinement potential and the interaction potential are no more uniformly convex. Proof is…
This paper derives non-asymptotic error bounds for nonlinear stochastic approximation algorithms in the Wasserstein-$p$ distance. To obtain explicit finite-sample guarantees for the last iterate, we develop a coupling argument that compares…
In this article we study examples of systematic biases that can occur in quantum Monte Carlo methods due to the accumulation of non-linear expectation values, and approaches by which these errors can be corrected. We begin with a study of…
Inverse uncertainty quantification (UQ) tasks such as parameter estimation are computationally demanding whenever dealing with physics-based models, and typically require repeated evaluations of complex numerical solvers. When partial…
We establish an Azuma type inequality under a Lipshitz condition for martingales in the framework of noncommutative probability spaces and apply it to deduce a noncommutative Heoffding inequality as well as a noncommutative McDiarmid type…
Numerical models of complex real-world phenomena often necessitate High Performance Computing (HPC). Uncertainties increase problem dimensionality further and pose even greater challenges. We present a parallelization strategy for…