We determine the order of magnitude of the nth ℓp-polarization constant of the unit sphere S d−1 for every n, d 1 and p > 0. For p = 2, we prove that extremizers are isotropic vector sets, whereas for p = 1, we show that the polarization problem is equivalent to that of maximizing the norm of signed vector sums. Finally, for d = 2, we discuss the optimality of equally spaced configurations on the unit circle.2010 Mathematics Subject Classification. 52A40(primary), and 31C20(secondary).

We study Stackelberg games where a principal repeatedly interacts with a long-lived, nonmyopic agent, without knowing the agent's payoff function. Although learning in Stackelberg games is well-understood when the agent is myopic, non-myopic agents pose additional complications. In particular, non-myopic agents may strategically select actions that are inferior in the present to mislead the principal's learning algorithm and obtain better outcomes in the future.We provide a general framework that reduces learning in presence of non-myopic agents to robust bandit optimization in the presence of myopic agents. Through the design and analysis of minimally reactive bandit algorithms, our reduction trades off the statistical efficiency of the principal's learning algorithm against its effectiveness in inducing near-best-responses. We apply this framework to Stackelberg security games (SSGs), pricing with unknown demand curve, strategic classification, and general finite Stackelberg games. In each setting, we characterize the type and impact of misspecifications present in near-best-responses and develop a learning algorithm robust to such misspecifications.Along the way, we improve the query complexity of learning in SSGs with 𝑛 targets from the state-of-the-art 𝑂 (𝑛 3 ) to a near-optimal 𝑂 (𝑛) by uncovering a fundamental structural property of such games. This result is of independent interest beyond learning with non-myopic agents.

The Wasserstein distance is a metric on a space of probability measures that has seen a surge of applications in statistics, machine learning, and applied mathematics. However, statistical aspects of Wasserstein distances are bottlenecked by the curse of dimensionality, whereby the number of data points needed to accurately estimate them grows exponentially with dimension. Gaussian smoothing was recently introduced as a means to alleviate the curse of dimensionality, giving rise to a parametric convergence rate in any dimension, while preserving the Wasserstein metric and topological structure. To facilitate valid statistical inference, in this work, we develop a comprehensive limit distribution theory for the empirical smooth Wasserstein distance. The limit distribution results leverage the functional delta method after embedding the domain of the Wasserstein distance into a certain dual Sobolev space, characterizing its Hadamard directional derivative for the dual Sobolev norm, and establishing weak convergence of the smooth empirical process in the dual space. To estimate the distributional limits, we also establish consistency of the nonparametric bootstrap. Finally, we use the limit distribution theory to study applications to generative modeling via minimum distance estimation with the smooth Wasserstein distance, showing asymptotic normality of optimal solutions for the quadratic cost.

Statistical distances, i.e., measures of discrepancy between probability distributions, are ubiquitous in probability theory, statistics and machine learning. To combat the curse of dimensionality when estimating these distances from data, recent work has proposed smoothing out local irregularities in the measured distributions via convolution with a Gaussian kernel. Motivated by the scalability of the smooth framework to high dimensions, we conduct an in-depth study of the structural and statistical behavior of the Gaussian-smoothed p-Wasserstein distance W (σ) p , for arbitrary p ≥ 1. We start by showing that W (σ) p can be expressed as a maximum mean discrepancy (MMD), enabling efficient computation.