Georgia Institute of Technology College of Sciences

(Joint work with Timo Seppäläinen) We establish estimates for the coalescence time of semi-infinite directed geodesics in the planar corner growth model with i.i.d. exponential weights. There are four estimates: upper and lower bounds on the probabilities of both fast and slow coalescence on the correct spatial scale with exponent 3/2. Our proofs utilize a geodesic duality introduced by Pimentel and properties of the increment-stationary last-passage percolation process. For fast coalescence our bounds are new and they have matching optimal exponential order of magnitude. For slow coalescence, we reproduce bounds proved earlier with integrable probability inputs, except that our upper bound misses the optimal order by a logarithmic factor.

This talk will introduce a precise high-dimensional asymptotic theory for Boosting (AdaBoost) on separable data, taking both statistical and computational perspectives. We will consider the common modern setting where the number of features p and the sample size n are both large and comparable, and in particular, look at scenarios where the data is asymptotically separable. Under a class of statistical models, we will provide an (asymptotically) exact analysis of the generalization error of AdaBoost, when the algorithm interpolates the training data and maximizes an empirical L1 margin. On the computational front, we will provide a sharp analysis of the stopping time when boosting approximately maximizes the empirical L1 margin. Our theory provides several insights into properties of Boosting; for instance, the larger the dimensionality ratio p/n, the faster the optimization reaches interpolation. At the heart of our theory lies an in-depth study of the maximum L1-margin, which can be accurately described by a new system of non-linear equations; we analyze this margin and the properties of this system, using Gaussian comparison techniques and a novel uniform deviation argument. Time permitting, I will present analogous results for a new class of boosting algorithms that correspond to Lq geometry, for q>1. This is based on joint work with Tengyuan Liang.

Markov chain Monte Carlo (MCMC) methods are state-of-the-art techniques for numerical integration. MCMC methods yield estimators that converge to integrals of interest in the limit of the number of iterations, obtained from Markov chains that converge to stationarity. This iterative asymptotic justification is not ideal. Indeed the literature offers little practical guidance on how many iterations should be performed, despite decades of research on the topic. This talk will describe a computational approach to address some of these issues. The key idea, pioneered by Glynn and Rhee in 2014, is to generate couplings of Markov chains, whereby pairs of chains contract, coalesce or even "meet" after a random number of iterations; we will see that these meeting times, which can be simulated in many practical settings, contain useful information about the finite-time marginal distributions of the chains. This talk will provide an overview of this line of research, joint work with John O'Leary, Yves Atchadé and various collaborators.
The main reference is available here: https://rss.onlinelibrary.wiley.com/doi/abs/10.1111/rssb.12336

Right and left eigenvectors of non-Hermitian matrices form a bi-orthogonal system to which one can associate homogeneous quantities known as overlaps. The matrix of overlaps quantifies the stability of the spectrum and characterizes the joint eigenvalues increments under Dyson-type dynamics. Overlaps first appeared in the physics literature: Chalker and Mehlig calculated their conditional expectation for complex Ginibre matrices (1998). For the same model, we extend their results by deriving the distribution of the overlaps and their correlations (joint work with P. Bourgade). Similar results can be obtained for quaternionic Gaussian matrices, as well as matrices from the spherical and truncated-unitary ensembles.