## Upcoming Seminars

Math Seminar: Convergence of Langevin Monte Carlo: The Interplay between Tail Growth and Smoothness
Dr.
Murat Erdgodu,  University of Toronto

April 22nd, 2022
11:00 AM
BSB 117

Abstract: We study sampling from a target distribution $e^{-f}$ using the Langevin Monte Carlo (LMC) algorithm. For any potential function $f$ whose tails behave like $|x|^\alpha$ for $\alpha \in [1,2]$, and has $\beta$-H\”older continuous gradient, we derive the sufficient number of steps to reach the $\eps$-neighborhood of a $d$-dimensional target distribution as a function of $\alpha$ and $\beta$. Our result is the first convergence guarantee for LMC under a functional inequality interpolating between the Poincar\’e and log-Sobolev settings (also covering the edge cases).

## Past Seminars

Math Seminar: Convergence properties of shallow neural networks: implications and applications in scientific computing
Dr.
Grant Rotskoff, Stanford University

April 1st, 2022
11:00 AM
BSB 132

Abstract: The surprising flexibility and undeniable empirical success of machine learning algorithms have inspired many theoretical explanations for the efficacy of neural networks. Here, I will briefly introduce one perspective that provides not only asymptotic guarantees of trainability and accuracy in high-dimensional learning problems but also provides some prescriptions and design principles for learning. Bolstered by the favorable scaling of these algorithms in high dimensional problems, I will turn to the problem of variational high dimensional PDEs. From the perspective of an applied mathematician, these problems often appear hopeless; they are not only high-dimensional but also dominated by rare events. However, with neural networks in the toolkit, at least the dimensionality is somewhat less intimidating. I will describe an algorithm that combines stochastic gradient descent with importance sampling to optimize a function representation of the solution. Finally, I will provide numerical evidence of the power and limitations of this approach.

Math Seminar:  The Manifold Joys of Sampling
Dr.
Santosh Vempala, Frederick G. Storey Chair of Computing and Professor – College of Computing, Georgia Tech

March 4th, 2022
11:00 AM
BSB 132

Abstract: Sampling high-dimensional sets and distributions is a fundamental problem with many applications. The state-of-the-art is that arbitrary logconcave densities can be sampled to arbitrarily small error in time polynomial in the dimension using simple Markov chains based on Euclidean geometry.  In this talk, we describe algorithms that exploit varying local geometry and can be viewed as sampling Riemannian manifolds. This approach will let us derive more efficient algorithms for some cases of interest, as well as analyze affine-invariant versions of Euclidean algorithms, such as the Dikin walk, Hamiltonian Monte-Carlo and Riemannian Langevin.

Math Seminar: Minimax Mixing Time of the Metropolis-Adjusted Langevin Algorithm for Log-concave Sampling
Dr.
Yuansi Chen, Duke University, Department of Statistical Sciences

February 25th, 2022
11:00 AM
BSB 132

Abstract: We study the problem of using the Metropolis-adjusted Langevin algorithm (MALA) to sample from a log-smooth and strongly log-concave distribution in dimension d with condition number $\kappa$. We establish its optimal minimax mixing time under a warm start. First, we demonstrate that MALA with a warm start mixes in $O(d^{1/2} \kappa)$ iterations up to logarithmic factors; this improves upon the previous work on the dependency of either the condition number $\kappa$ or the dimension d. Our proof relies on comparing the leapfrog integrator with the continuous Hamiltonian dynamics, where we establish a new concentration bound for the acceptance rate. Second, we provide an explicit mixing time lower bound for reversible MCMC algorithms on general state spaces. We use this result to show that MALA requires at least $\Omega(d^{1/2} \kappa)$ steps in the worst case, matching our upper bound in terms of both the condition number and the dimension.

Math Seminar: Analysis of two-component Gibbs samplers using the theory of two projections
Dr.
Qian Qin, University of Minnesota

February 18th, 2022
11:00 AM
BSB 132

Abstract: Gibbs samplers are a class of Markov chain Monte Carlo (MCMC) algorithms commonly used in statistics for sampling from intractable probability distributions. In this talk, I will demonstrate how Halmos’s (1969) theory of two projections can be applied to study Gibbs samplers with two components. I will first give an introduction to MCMC algorithms, particularly Gibbs algorithms. Then, I will explain how problems regarding the asymptotic variance and convergence rate of a two-component Gibbs sampler can be translated into simple linear algebraic problems through Halmos’s theory. In particular, a comparison is made between the deterministic-scan and random-scan versions of two-component Gibbs. It is found that in terms of asymptotic variance, the random-scan version is more robust than the deterministic-scan version, provided that the selection probability is appropriately chosen. On the other hand, the deterministic-scan version has a faster convergence rate. These results suggest that one may use the deterministic-scan version in the burn-in stage, and switch to the random-scan version in the estimation stage.

Math Seminar: Unbiased Multilevel Monte Carlo methods for intractable distributions: MLMC meets MCMC
Dr.
Guanyang Wang, Rutgers University, Department of Statistics

February 11th, 2022
11:00 AM
BSB 132

Abstract: Constructing unbiased estimtors from MCMC outputs has recently increased much attention in statistics and machine learning communities.   However, the existing unbiased MCMC framework only works when  the quantity of interest is an expectation of certain probability distribution.  In this work, we prropose unbiased estimators for functions of expectations.  Our ideas is based on the combination of the unbiased MCMC and MLMC methods.  We prove our estimator has a finite variance, a finite computational complexity, and achieves ε-accuracy with O(1/ε²) computational cost under mild conditions.  We also illustrate our estimator on several numerical examples.  This is a joint work with Tianze Wang.

Math Seminar: Quantitative convergence analysis of hypocoercive sampling dynamics
Dr.
Lihan Wang, Carnegie Mellon University

February 4th, 2022
11:00 AM
BSB 132

Abstract: In this talk, we will discuss some advances on quantitative analysis of convergence of hypocoercive sampling dynamics, including underdamped Langevin dynamics, randomized Hamiltonian Monte Carlo, zigzag process and bouncy particle sampler. The analysis is based on a variational framework for hypocoercivity which combines a Poincare-type inequality in time-augmented state space and an L^2 energy estimate. Joint works with Yu Cao (NYU) and Jianfeng Lu (Duke).