Computing an approximately optimal policy with high probability in this case is known as PAC RL with a generative model.

Arelated latent approach, hard attention, fixesthese issues, butisgenerally harder to train and less accurate.

In these frameworks, templates are constructed using an iterative process of template estimation and alignment, which is often computationally very expensive. Due in part to this shortcoming, most methods compute asingle template for the entire population of images, or a few templates for specific sub-groups of the data.

Causal discovery across multiple datasets is often constrained by data privacy regulations and cross-site heterogeneity, limiting the use of conventional methods that require a single, centralized dataset. To address these challenges, we introduce fedCI, a federated conditional independence test that rigorously handles heterogeneous datasets with non-identical sets of variables, site-specific effects, and mixed variable types, including continuous, ordinal, binary, and categorical variables. At its core, fedCI uses a federated Iteratively Reweighted Least Squares (IRLS) procedure to estimate the parameters of generalized linear models underlying likelihood-ratio tests for conditional independence. Building on this, we develop fedCI-IOD, a federated extension of the Integration of Overlapping Datasets (IOD) algorithm, that replaces its meta-analysis strategy and enables, for the fist time, federated causal discovery under latent confounding across distributed and heterogeneous datasets. By aggregating evidence federatively, fedCI-IOD not only preserves privacy but also substantially enhances statistical power, achieving performance comparable to fully pooled analyses and mitigating artifacts from low local sample sizes. Our tools are publicly available as the fedCI Python package, a privacy-preserving R implementation of IOD, and a web application for the fedCI-IOD pipeline, providing versatile, user-friendly solutions for federated conditional independence testing and causal discovery.

Heavy-tailed distributions are ubiquitous in real-world data, where rare but extreme events dominate risk and variability. However, standard Variational Autoencoders (VAEs) employ simple decoder distributions (e.g., Gaussian) that fail to capture heavy-tailed behavior, while existing heavy-tail-aware extensions remain restricted to predefined parametric families whose tail behavior is fixed a priori. We propose the Phase-Type Variational Autoencoder (PH-VAE), whose decoder distribution is a latent-conditioned Phase-Type (PH) distribution defined as the absorption time of a continuous-time Markov chain (CTMC). This formulation composes multiple exponential time scales, yielding a flexible and analytically tractable decoder that adapts its tail behavior directly from the observed data. Experiments on synthetic and real-world benchmarks demonstrate that PH-VAE accurately recovers diverse heavy-tailed distributions, significantly outperforming Gaussian, Student-t, and extreme-value-based VAE decoders in modeling tail behavior and extreme quantiles. In multivariate settings, PH-VAE captures realistic cross-dimensional tail dependence through its shared latent representation. To our knowledge, this is the first work to integrate Phase-Type distributions into deep generative modeling, bridging applied probability and representation learning.

Understanding minimal assumptions that enable learning and generalization is perhaps the central question of learning theory. Several celebrated results in statistical learning theory, such as the VC theorem and Littlestone's characterization of online learnability, establish conditions on the hypothesis class that allow for learning under independent data and adversarial data, respectively. Building upon recent work bridging these extremes, we study sequential decision making under distributional adversaries that can adaptively choose data-generating distributions from a fixed family $U$ and ask when such problems are learnable with sample complexity that behaves like the favorable independent case. We provide a near complete characterization of families $U$ that admit learnability in terms of a notion known as generalized smoothness i.e. a distribution family admits VC-dimension-dependent regret bounds for every finite-VC hypothesis class if and only if it is generalized smooth. Further, we give universal algorithms that achieve low regret under any generalized smooth adversary without explicit knowledge of $U$. Finally, when $U$ is known, we provide refined bounds in terms of a combinatorial parameter, the fragmentation number, that captures how many disjoint regions can carry nontrivial mass under $U$. These results provide a nearly complete understanding of learnability under distributional adversaries. In addition, building upon the surprising connection between online learning and differential privacy, we show that the generalized smoothness also characterizes private learnability under distributional constraints.

The proliferation of Large Language Models (LLMs) necessitates efficient mechanisms to distinguish machine-generated content from human text. While statistical watermarking has emerged as a promising solution, existing methods suffer from two critical limitations: the lack of a principled approach for selecting sampling distributions and the reliance on fixed-horizon hypothesis testing, which precludes valid early stopping. In this paper, we bridge this gap by developing the first e-value-based watermarking framework, Anchored E-Watermarking, that unifies optimal sampling with anytime-valid inference. Unlike traditional approaches where optional stopping invalidates Type-I error guarantees, our framework enables valid, anytime-inference by constructing a test supermartingale for the detection process. By leveraging an anchor distribution to approximate the target model, we characterize the optimal e-value with respect to the worst-case log-growth rate and derive the optimal expected stopping time. Our theoretical claims are substantiated by simulations and evaluations on established benchmarks, showing that our framework can significantly enhance sample efficiency, reducing the average token budget required for detection by 13-15% relative to state-of-the-art baselines.

We present a novel algorithm to estimate the barycenter of arbitrary probability distributions with respect to the Sinkhorn divergence. Based on a Frank-Wolfe optimization strategy, our approach proceeds by populating the support of the barycenter incrementally, without requiring any pre-allocation.

Humans reason with concepts and metaconcepts: we recognizered and green from visual input; we also understand that theydescribe the same property of objects(i.e.,thecolor).