Posted January 31, 2023
Last modified February 6, 2023
Nadia Drenska, Johns Hopkins University
A PDE Interpretation of Prediction with Expert Advice
Abstract: We study the problem of prediction of binary sequences with expert advice in the online setting, which is a classic example of online machine learning. We interpret the binary sequence as the price history of a stock, and view the predictor as an investor, which converts the problem into a stock prediction problem. In this framework, an investor, who predicts the daily movements of a stock, and an adversarial market, who controls the stock, play against each other over N turns. The investor combines the predictions of n ≥ 2 experts in order to make a decision about how much to invest at each turn, and aims to minimize their regret with respect to the best-performing expert at the end of the game. We consider the problem with history-dependent experts, in which each expert uses the previous d days of history of the market in making their predictions. The prediction problem is played (in part) over a discrete graph called the d-dimensional de Bruijn graph. We focus on an appropriate continuum limit. Using methods from optimal control, graph theory, and partial differential equations, we discuss strategies for the investor and the adversarial market. We prove that the value function for this game, rescaled appropriately, converges as N goes to infinity at a rate of O(N^−1/2) (for C^4 payoff functions) to the viscosity solution of a nonlinear degenerate parabolic PDE. It can be understood as the Hamilton-Jacobi-Issacs equation for the two-person game. As a result, we are able to deduce asymptotically optimal strategies for the investor. This is joint work with Robert Kohn and Jeff Calder.
Posted February 8, 20233:30 pm - 4:20 pm Lockett 232
Xiaoqi Huang, University of Maryland
Spectral Cluster and Weyl Remainder Estimates for the Laplacian and on compact manifolds
Abstract: In the first part of the talk, we shall discuss generalizations of classical versions of the Weyl formula involving Schrödinger operators on compact boundaryless Riemannian manifolds with critically singular potentials V. In particular, we extend the classical results of Avakumović, Levitan and Hörmander by obtaining sharp bounds for the error term in the Weyl formula when we merely assume that V belongs to the Kato class, which is the minimal assumption to ensure that H_V is essentially self-adjoint and bounded from below or has favorable heat kernel bounds. In the second part, we will discuss the problem of trying to obtain improved eigenfunction estimates under geometric assumptions, such as the presence of negative sectional curvatures, which is based on the use of microlocal Kakeya-Nikodym estimates along with a combination of local and global harmonic analysis that further exploit geometric assumptions. We will also discuss some new sharp estimates involving the tori eigenfunctions.
Posted January 19, 2023
Last modified January 20, 2023
Irina Markina, University of Bergen
On rolling of manifolds
In the talk, we will introduce the notion of rolling one manifold over another. The idea of the rolling map originated as a simple mathematical model of rolling a ball over a plate with the constraints of no-slip and no-twist motion in the works of S. Chaplygin (1897), K. Nomizu (1978), R.Bryan and L.Hsu (1993). The geometric features are closely related to the distributions of E.Cartan type (1910). Later this idea was extended to the rolling of Riemannian manifolds of any dimension, as an isometry map preserving the parallelism of vector fields. After a historical overview and necessary definitions, we also mention some applications in the interpolation on nonlinear spaces and construction of stochastic processes on manifolds.