Evaluating rare-event forecasts is challenging because standard metrics collapse as event prevalence declines. Measures such as F1-score, AUPRC, MCC, and accuracy induce degenerate thresholds -- converging to zero or one -- and their values become dominated by class imbalance rather than tail discrimination. We develop a family of rare-event-stable (RES) metrics whose optimal thresholds remain strictly interior as the event probability approaches zero, ensuring coherent decision rules under extreme rarity. Simulations spanning event probabilities from 0.01 down to one in a million show that RES metrics maintain stable thresholds, consistent model rankings, and near-complete prevalence invariance, whereas traditional metrics exhibit statistically significant threshold drift and structural collapse. A credit-default application confirms these results: RES metrics yield interpretable probability-of-default cutoffs (4-9%) and remain robust under subsampling, while classical metrics fail operationally. The RES framework provides a principled, prevalence-invariant basis for evaluating extreme-risk forecasts.

Differential privacy is increasingly formalized through the lens of hypothesis testing via the robust and interpretable $f$-DP framework, where privacy guarantees are encoded by a baseline Blackwell trade-off function $f_{\infty} = T(P_{\infty}, Q_{\infty})$ involving a pair of distributions $(P_{\infty}, Q_{\infty})$. The problem of choosing the right privacy metric in practice leads to a central question: what is a statistically appropriate baseline $f_{\infty}$ given some prior modeling assumptions? The special case of Gaussian differential privacy (GDP) showed that, under compositions of nearly perfect mechanisms, these trade-off functions exhibit a central limit behavior with a Gaussian limit experiment. Inspired by Le Cam's theory of limits of statistical experiments, we answer this question in full generality in an infinitely divisible setting. We show that suitable composition experiments $(P_n^{\otimes n}, Q_n^{\otimes n})$ converge to a binary limit experiment $(P_{\infty}, Q_{\infty})$ whose log-likelihood ratio $L = \log(dQ_{\infty} / dP_{\infty})$ is infinitely divisible under $P_{\infty}$. Thus any limiting trade-off function $f_{\infty}$ is determined by an infinitely divisible law $P_{\infty}$, characterized by its Levy--Khintchine triplet, and its Esscher tilt defined by $dQ_{\infty}(x) = e^{x} dP_{\infty}(x)$. This characterizes all limiting baseline trade-off functions $f_{\infty}$ arising from compositions of nearly perfect differentially private mechanisms. Our framework recovers GDP as the purely Gaussian case and yields explicit non-Gaussian limits, including Poisson examples. It also positively resolves the empirical $s^2 = 2k$ phenomenon observed in the GDP paper and provides an optimal mechanism for count statistics achieving asymmetric Poisson differential privacy.

Synthetic data generation is an important tool for privacy-preserving data sharing. While diffusion models have set recent benchmarks, flow matching (FM) offers a promising alternative. This paper presents different ways to implement flow matching for tabular data synthesis. We provide a comprehensive empirical study that compares flow matching (FM and variational FM) with a state-of-the-art diffusion method (TabDDPM and TabSyn) in tabular data synthesis. We evaluate both the standard Optimal Transport (OT) and the Variance Preserving (VP) probability paths, and also compare deterministic and stochastic samplers -- something possible when learning to generate using \textit{variational} flow matching -- characterising the empirical relationship between data utility and privacy risk. Our key findings reveal that flow matching, particularly TabbyFlow, outperforms diffusion baselines. Flow matching methods also achieves better performance with remarkably low function evaluations ($\leq$ 100 steps), offering a substantial computational advantage. The choice of probability path is also crucial, as using the OT path demonstrates superior performance, while VP has potential for producing synthetic data with lower disclosure risk. Lastly, our results show that making flows stochastic not only preserves marginal distributions but, in some instances, enables the generation of high utility synthetic data with reduced disclosure risk.

Replicability is a fundamental challenge in reinforcement learning (RL), as RL algorithms are empirically observed to be unstable and sensitive to variations in training conditions. To formally address this issue, we study \emph{list replicability} in the Probably Approximately Correct (PAC) RL framework, where an algorithm must return a near-optimal policy that lies in a \emph{small list} of policies across different runs, with high probability. The size of this list defines the \emph{list complexity}. We introduce both weak and strong forms of list replicability: the weak form ensures that the final learned policy belongs to a small list, while the strong form further requires that the entire sequence of executed policies remains constrained. These objectives are challenging, as existing RL algorithms exhibit exponential list complexity due to their instability. Our main theoretical contribution is a provably efficient tabular RL algorithm that guarantees list replicability by ensuring the list complexity remains polynomial in the number of states, actions, and the horizon length. We further extend our techniques to achieve strong list replicability, bounding the number of possible policy execution traces polynomially with high probability. Our theoretical result is made possible by key innovations including (i) a novel planning strategy that selects actions based on lexicographic order among near-optimal choices within a randomly chosen tolerance threshold, and (ii) a mechanism for testing state reachability in stochastic environments while preserving replicability. Finally, we demonstrate that our theoretical investigation sheds light on resolving the \emph{instability} issue of RL algorithms used in practice. In particular, we show that empirically, our new planning strategy can be incorporated into practical RL frameworks to enhance their stability.

The Influence Maximization (IM) problem aims to select a set of seed nodes within a given budget to maximize the spread of influence in a social network. However, real-world social networks have several structural inequalities, such as dominant majority groups and underrepresented minority groups. If these inequalities are not considered while designing IM algorithms, the outcomes might be biased, disproportionately benefiting majority groups while marginalizing minorities. In this work, we address this gap by designing a fairness-aware IM method using Reinforcement Learning (RL) that ensures equitable influence outreach across all communities, regardless of protected attributes. Fairness is incorporated using a maximin fairness objective, which prioritizes improving the outreach of the least-influenced group, pushing the solution toward an equitable influence distribution. We propose a novel fairness-aware deep RL method, called DQ4FairIM, that maximizes the expected number of influenced nodes by learning an RL policy. The learnt policy ensures that minority groups formulate the IM problem as a Markov Decision Process (MDP) and use deep Q-learning, combined with the Structure2Vec network embedding, earning together with Structure2Vec network embedding to solve the MDP. We perform extensive experiments on synthetic benchmarks and real-world networks to compare our method with fairness-agnostic and fairness-aware baselines. The results show that our method achieves a higher level of fairness while maintaining a better fairness-performance trade-off than baselines. Additionally, our approach learns effective seeding policies that generalize across problem instances without retraining, such as varying the network size or the number of seed nodes.

Hierarchical clustering is a fundamental machine-learning technique for grouping data points into dendrograms. However, existing hierarchical clustering methods encounter two primary challenges: 1) Most methods specify dendrograms without a global objective. 2) Graph-based methods often neglect the significance of graph structure, optimizing objectives on complete or static predefined graphs. In this work, we propose Hyperbolic Continuous Structural Entropy neural networks, namely HypCSE, for structure-enhanced continuous hierarchical clustering. Our key idea is to map data points in the hyperbolic space and minimize the relaxed continuous structural entropy (SE) on structure-enhanced graphs. Specifically, we encode graph vertices in hyperbolic space using hyperbolic graph neural networks and minimize approximate SE defined on graph embeddings. To make the SE objective differentiable for optimization, we reformulate it into a function using the lowest common ancestor (LCA) on trees and then relax it into continuous SE (CSE) by the analogy of hyperbolic graph embeddings and partitioning trees. To ensure a graph structure that effectively captures the hierarchy of data points for CSE calculation, we employ a graph structure learning (GSL) strategy that updates the graph structure during training. Extensive experiments on seven datasets demonstrate the superior performance of HypCSE.

Data privacy is a critical challenge in modern medical workflows as the adoption of electronic patient records has grown rapidly. Stringent data protection regulations limit access to clinical records for training and integrating machine learning models that have shown promise in improving diagnostic accuracy and personalized care outcomes. Synthetic data offers a promising alternative; however, current generative models either struggle with time-series data or lack formal privacy guaranties. In this paper, we enhance a state-of-the-art time-series generative model to better handle longitudinal clinical data while incorporating quantifiable privacy safeguards. Using real data from chronic kidney disease and ICU patients, we evaluate our method through statistical tests, a Train-on-Synthetic-Test-on-Real (TSTR) setup, and expert clinical review. Our non-private model (Augmented TimeGAN) outperforms transformer- and flow-based models on statistical metrics in several datasets, while our private model (DP-TimeGAN) maintains a mean authenticity of 0.778 on the CKD dataset, outperforming existing state-of-the-art models on the privacy-utility frontier. Both models achieve performance comparable to real data in clinician evaluations, providing robust input data necessary for developing models for complex chronic conditions without compromising data privacy.

Random Forests and Gradient Boosting are among the most effective algorithms for supervised learning on tabular data. Both belong to the class of tree-based ensemble methods, where predictions are obtained by aggregating many randomized regression trees. In this paper, we develop a theoretical framework for analyzing such methods through Reproducing Kernel Hilbert Spaces (RKHSs) constructed on tree ensembles--more precisely, on the random partitions generated by randomized regression trees. We establish fundamental analytical properties of the resulting Random Forest kernel, including boundedness, continuity, and universality, and show that a Random Forest predictor can be characterized as the unique minimizer of a penalized empirical risk functional in this RKHS, providing a variational interpretation of ensemble learning. We further extend this perspective to the continuous-time formulation of Gradient Boosting introduced by Dombry and Duchamps (2024a,b), and demonstrate that it corresponds to a gradient flow on a Hilbert manifold induced by the Random Forest RKHS. A key feature of this framework is that both the kernel and the RKHS geometry are data-dependent, offering a theoretical explanation for the strong empirical performance of tree-based ensembles. Finally, we illustrate the practical potential of this approach by introducing a kernel principal component analysis built on the Random Forest kernel, which enhances the interpretability of ensemble models, as well as GVI, a new geometric variable importance criterion.

Many emerging applications - such as adversarial training, AI alignment, and robust optimization - can be framed as zero-sum games between neural nets, with von Neumann-Nash equilibria (NE) capturing the desirable system behavior. While such games often involve non-convex non-concave objectives, empirical evidence shows that simple gradient methods frequently converge, suggesting a hidden geometric structure. In this paper, we provide a theoretical framework that explains this phenomenon through the lens of hidden convexity and overparameterization. We identify sufficient conditions - spanning initialization, training dynamics, and network width - that guarantee global convergence to a NE in a broad class of non-convex min-max games. To our knowledge, this is the first such result for games that involve two-layer neural networks. Technically, our approach is twofold: (a) we derive a novel path-length bound for the alternating gradient descent-ascent scheme in min-max games; and (b) we show that the reduction from a hidden convex-concave geometry to two-sided Polyak-Łojasiewicz (PŁ) min-max condition hold with high probability under overparameterization, using tools from random matrix theory.

We take the novel perspective of incorporating offline RL algorithms as subroutines of tabula rasa online RL. This is feasible because an online learning agent can repurpose its historical interactions as offline dataset. We formalize this idea into a framework that accommodates several variants of offline RL incorporation such as final policy recommendation and online fine-tuning. We further introduce convenient techniques to improve its effectiveness in enhancing online learning efficiency. Our extensive and systematic empirical analyses show that 1) the effectiveness of the proposed framework depends strongly on the nature of the task, 2) our proposed techniques greatly enhance its effectiveness, and 3) existing online fine-tuning methods are overall ineffective, calling for more research therein.