Conference Agenda

We study the problem of learning with respect to the squared loss over a convex class of functions. It has long been believed that the sample complexity in this setting is governed by localized Rademacher complexities. We show that, assuming access to coarse information on the covariance structure of the model class, the sample complexity is instead controlled by a localized complexity associated with the limiting Gaussian process. In heavy-tailed regimes, this quantity can be significantly smaller than the Rademacher complexity.

We introduce a new dependence order—the conditional convex order—whose minimal and maximal elements characterize independence and perfect dependence. Moreover, it characterizes conditional independence, satisfies information monotonicity, and exhibits several invariance properties. Consequently, it is an ordering for the strength of functional dependence of a random variable Y on a random vector X . As we show, various recently studied dependence measures—including Chatterjee’s rank correlation, optimal transport-based Wasserstein correlations, and rearranged dependence measures—are increasing in this order and inherit their fundamental properties from it. We characterize the conditional convex order by the Schur order and by the concordance order, and we verify it in settings such as additive error models, the multivariate normal distribution, and various copula-based models. Our results offer a unified perspective on the behavior of dependence measures across statistical models.

Graph causal optimal transport is a recent generalisation of both classical and causal optimal transport in which the allowed couplings satisfy causal restrictions given by a directed graph. Inspired by applications to structural causal models, it was originally introduced in Eckstein and Cheridito (2023). We study fundamental properties of graph causal optimal transport, with a particular focus on its induced Wasserstein distance. Our main result is a full characterisation of the directed graphs for which this graph causal Wasserstein distance is indeed a metric, an open problem in the original paper. We study the topology of the new metric and prove continuity of a class of stochastic team problems with respect to it. Furthermore, we fully characterise the gluing properties of graph causal couplings, prove denseness of Monge maps, and provide a dynamic programming principle. Based on joint work with Jan Obloj.

Finding regular transport maps between measures is an important task in generative modelling and a useful tool to transfer functional inequalities. The most well-known result in this field is Caffarelli’s contraction theorem, which shows that the optimal transport map from a Gaussian to a uniformly log-concave measure is globally Lipschitz. Note that for our purposes optimality of the transport map does not play a role. This is why several works investigate other transport maps, such as those derived from diffusion processes, as introduced by Kim and Milman. Here, we establish a lower bound on the log-semiconcavity along the heat flow for a class of what we call asymptotically log-concave measures. We will see that this implies Lipschitz bounds for the heat flow map introduced by Kim and Milman. We will also show that these log-semiconcavity bounds are sufficient for stability of these maps in entropy and Wasserstein distance.

Based on a joint work with Louis-Pierre Chaintron and Giovanni Conforti, and a joint work in progress with Sinho Chewi and Aram-Alexandre Pooladian.