The batched multi-armed bandit (MAB) problem, where rewards are collected in batches, is pivotal in applications like clinical trials. While prior work assumes light-tailed reward distributions, real-world scenarios often exhibit heavy-tailed outcomes. This paper addresses this gap by introducing robust batched bandit algorithms for heavy-tailed rewards in both multi-arm and Lipschitz settings. We uncover somewhat surprising phenomena for such problems-heavier tails require fewer batches to achieve near-optimal regret in the instance-independent setting, as well as the Lipschitz setting. In sharp contrast, in the instance-dependent setting, the number of batches required to achieve near-optimal regret does not depend on the tail heaviness.
We introduce a Bayesian framework for geodesic regression on Grassmann manifolds. Since a geodesic on the manifold is fully determined by its initial position and velocity, we reformulate the regression from function estimation to inferring these two initial conditions. We perform this inference via a Grassmann-specific Lagrangian Hamiltonian Monte Carlo method that exploits the manifold's Riemannian geometry. Solving the associated Lagrangian dynamics reduces to following the geodesic flow, yielding an efficient inference procedure. We analyze the asymptotic properties of the resulting posterior, providing theoretical guarantees, and confirm the computational efficiency and accuracy of the approach experimentally on complex Bayesian modelling tasks.
Network data are ubiquitous in the real world, capturing pairwise and high-order relations among objects. We introduce a class of measure-theoretic network objects, termed partitioned measure networks, that generalize many existing representations of network data, including graphs, hypergraphs and augmented graphs (i.e. graphs whose nodes are associated with categorical classes). We then propose a metric, the partitioned network distance, between partitioned measure networks, which extends both the Gromov-Wasserstein distance for graphs and the co-optimal transport distance for hypergraphs. By characterizing the geometry of the space of partitioned measure networks, we provide a unified theoretical treatment of generalized network structures encompassing both pairwise and higher-order relations. In particular, we show that this metric endows the space with an Alexandrov geometry of non-negative curvature and leverages this structure to define gradients for certain functionals commonly encountered in geometric data analysis. We further extend our framework to incorporate node-level label information and derive efficient computational schemes for utilizing the partitioned network distance in practice. Equipped with these theoretical and computational advances, we demonstrate the utility of our framework across diverse applications, including hypergraph alignment, clustering and dictionary learning from ensemble data, multi-omics alignment, as well as multiscale network alignment.
We introduce a mathematically new approach for quantization of vectorial signals. Namely, we are aimed at quantizing vectorial signals on input to a computational device that calculates some given function of the input. As opposed to the classical approach, which optimizes the quality of the quantized signal, without taking into account any further transformation of the latter by means of the computational device, here we optimize the quality of quantization on the output of this device. Moreover, we quantize components of the signal separately. This leads to a quantization problem qualitatively different from the classical one. We study existence of optimal quantizers (which is not at all obvious in this setting) and estimate the optimal cost for several classes of functions.
We study property testing for graphical models in a setting where an algorithm may adaptively query individual entries of the covariance matrix Sigma, and cost is measured by the number of queried entries. Under natural (strong) faithfulness assumptions, we design divide-and-conquer tests guided by balanced separators that operate on small submatrices and avoid global matrix inversion. Our first result is a tester for whether the underlying graph is a tree; with high probability it decides correctly using a subquadratic number of correlation queries, and for bounded maximum degree Delta the complexity is near-linear up to logarithmic factors. Our second result concerns graphs with a small separation number (hence small treewidth). We present two complementary procedures: a conditional descent test that never breaks when sn(G) <= k and, upon termination, returns an O(k log(n/k)) certificate; and a marginal descent test that, under a 'good run' condition, either certifies sn(G) <= 2k or proves sn(G) > k. The approach extends beyond the Gaussian case whenever reliable conditional-independence queries are available (e.g., non-paranormal models), yielding tests that access only a vanishing fraction of the entries of Sigma.
