This paper studies adaptive targeting under network interference in a bandit setting, where treatments applied to one individual may affect others through spillover effects. We consider a linear model in a sparse regime, where each individual's outcome can be affected by at most a few others. We first establish a regret lower bound showing that ignoring the network structure and reducing the problem to a standard linear bandit inevitably leads to inefficient learning, particularly in large populations. To understand how structural information can be leveraged, we analyze regimes with varying levels of knowledge of the interference structure: (1) full support knowledge, (2) knowledge of the column support sizes, and (3) no prior knowledge. For each regime, we establish regret lower bounds characterizing the fundamental limits of learning, and develop algorithms that achieve near-optimal regret. Together, our results provide a unified view of how knowledge of the interference structure governs the efficiency of online learning under interference, and offer practical adaptive targeting algorithms in each setting. Numerical experiments on synthetic and real-world data demonstrate the practical benefits of our algorithms.

Generative models, including diffusion models, are increasingly used as foundation models and adapted through sequential fine-tuning, making continual learning an essential problem setting. However, continual learning in such generative models remains poorly understood: after a task change, what aspects of the learned distribution are most easily lost, and what replay samples should be prioritized? We address these questions through the modern Hopfield energy. Recent links between modern Hopfield networks (MHNs) and diffusion models allow analyses in MHNs to be transferred to diffusion models. We introduce intrinsic forgetting as an increase in Hopfield energy after the task change. In tractable settings in an MHN, we prove that high-energy, outlier-like samples undergo a larger energy increase than cluster-like samples, implying that samples located in sharp, isolated basins are more forgettable. We further analyze memory replay and show that replay is particularly effective for high-energy samples, enabling an energy-based selection of replay samples. We validate these predictions in experiments on MHNs and two diffusion models under continual-learning settings: Stable Diffusion and a pixel-space DDPM. In these diffusion models, Hopfield energy tracks reconstruction-based forgetting, and replay experiments reveal energy-dependent mitigation of forgetting that is consistent with the MHN analysis.

Deep neural networks (DNNs) have achieved remarkable empirical success, yet their training dynamics remain understood mainly from optimization rather than statistical principles. Here we develop a statistical framework for DNN training in the over-parameterized regime by showing that the prediction induced by continuous-time neural tangent kernel (NTK) gradient flow is exactly equivalent to that from a classical random-effects model. In this framework, training time acts as a variance component, or equivalently an empirical Bayes covariance hyperparameter, governing the allocation of variation from noise to structured signal. This equivalence reveals an optimization-inference duality: the gradient-flow path is both an optimization trajectory and an empirical Bayes random-effects inference path. Conditional on training time, the network output is the posterior mean of the latent signal, and estimating training time by restricted maximum likelihood (REML) turns early stopping into likelihood-based empirical Bayes inference rather than external tuning. This perspective yields a two-stage inferential procedure. First, a variance-component test determines whether DNN training captures statistically significant structure beyond initialization. Second, conditional on training being warranted, REML provides a likelihood-based early stopping rule. The resulting stopping time admits a spectral interpretation in the NTK eigenbasis, where training proceeds until spectral loss decorrelation is achieved. We further establish that REML-guided early stopping achieves asymptotically optimal prediction error for fixed-design in-sample prediction and, under additional random-design regularity conditions, for out-of-sample prediction. This work reframes DNN training as statistical inference and provides a principled foundation for deciding whether and how long to train deep neural networks.

Many problems in computational science and engineering become one-to-many after coarse graining, partial observation, or inverse reconstruction: a resolved state may not determine a unique subgrid forcing, a structural descriptor may not determine a unique effective response, and a low-resolution observation may correspond to many plausible high-resolution fields. In such settings, deterministic surrogates may learn a well-defined mathematical object while still missing application-relevant uncertainty. This tutorial develops a self-contained module centered on the conditional-mean barrier: the point at which a squared-loss predictor has reached the conditional mean and the remaining error is irreducible aleatoric variance. We give two diagnostics for locating this barrier, residual-feature orthogonality and the coefficient of determination against its explained-variance ceiling, and prove that adding latent randomness to a squared-loss predictor collapses it back to the conditional mean. Crossing the barrier therefore requires a loss that scores distributions rather than point predictions. We briefly organize common distributional objectives, including negative log-likelihood, moment and observable matching, variational objectives, adversarial divergences, and score matching, by the feature of the conditional law each targets. The emphasis is the boundary itself and a finite-data procedure for recognizing it, rather than a survey of methods beyond it. CPU-based demonstrations on a two-branch law and a two-scale Lorenz-96 closure problem show how the diagnostics distinguish deterministic underfitting from residual distributional variability.

We investigate the connection between Gaussian processes and Gaussian random elements in reproducing kernel Banach spaces. We show that the covariance operator of a weak second-order Radon probability measure on such a space is uniquely determined by a positive definite function. In the Gaussian case, we characterize those positive definite functions that arise from covariance operators in terms of $γ$-radonifying operators. Building on these results, we extend the classical Driscoll theorem to the Banach space setting.

We study the problem of learning to bid when the bidder's value is dynamic, i.e., when the current value depends on past outcomes. Specifically, we consider a bidder participating in repeated second-price auctions whose value depends on the time elapsed since their last successful bid, with auctions arriving in continuous time and only aggregated feedback revealed at the end of the horizon. Such a bidder must (1) balance the immediate benefit of winning the current auction against its impact on future values and (2) learn unknown environmental parameters. We derive regret bounds for a class of learning methods that combine plug-in estimators with a differential-equation characterization of the optimal policy, and show that a specific confidence bound algorithm learns the optimal policy with a near optimal regret of $\widetilde{O}(\log N)$ for piecewise linear primitives, and $\widetilde{O}(N^{1/3})$ for general, smooth primitives, achieving these regrets without explicit randomization. These theoretical results are supported by numerical experiments.

Fairness-accuracy trade-offs are a central concern in the deployment of fairness-aware machine learning methods. When sensitive attributes are unavailable at inference time-the so called unawareness setting, principled methods for obtaining accurate predictions under relaxed fairness constraints are largely missing. In this work, we address this gap by formulating regression under a demographic parity penalty as an optimal transport problem. Our framework unifies both the \emph{aware} and \emph{unaware} settings and characterizes optimal prediction functions via optimal transport maps, under both squared Wasserstein-2 and Total Variation penalties. These results reveal that the choice of penalty reflects fundamentally different fairness philosophies: the Wasserstein penalty induces a smooth, population-wide compromise, while Total Variation enforces exact parity for a subset of individuals. Building on these theoretical characterizations, we propose an algorithm that is simple to implement, computationally efficient, and consistently matches or outperforms state-of-the-art baselines on real-world benchmarks.

We consider the problem of learning a counterfactually fair regressor. We adopt a causal uncertainty view in which counterfactual fairness is defined with resampled noise. We focus on obtaining theoretical fairness guarantees for a new post-processing estimator. We begin by showing that counterfactual fairness is equivalent to satisfying demographic parity conditional on the latent variable. This allows us to provide a closed-form expression of the optimal fair regressor via a barycentric quantile map. In order to handle continuous latent variables, we propose a discretized post-processing method. Then, under mild regularity assumptions, we prove high-probability finite-sample fairness guarantees for our estimator, providing an unfairness decay at rate $\tilde O(n^{-1/3})$, and establishing a matching risk bound of order $\tilde O(n^{-1/3})$. We provide a matching lower bound on the excess risk of almost fair predictions. Finally, we extend our results to the setting of relaxed counterfactual fairness. We validate our approach on real-world and synthetic data.

We study bandit learning in matching markets, where players and arms constitute the two market sides, and the players' utilities are linear in the arm contexts. In each round, new arms arrive with observable contexts. Then, the algorithm matches them to players, aiming to minimize each player's regret against a stable matching benchmark. This contextual structure creates significant complexity: subtle context shifts can slightly alter one player's utility while completely reconfiguring the underlying benchmark, causing large regret spikes for others. We address this in two settings: stochastic contexts, drawn from a latent distribution, and adversarial contexts, which may be arbitrary. For the stochastic case, we introduce a novel minimum preference gap to capture learning difficulty and provide a fully adaptive algorithm with an instance-dependent poly-logarithmic regret upper bound. We also establish matching instance-independent regret upper and lower bounds under a mild distributional assumption. For the adversarial setting, we propose a tractable regret notion that remains valid under arbitrary contexts and achieves an instance-independent sublinear regret bound via an adaptive algorithm.

We introduce a continuous-time generative modeling framework, motivated by the Chow-Rashevskii theorem, that builds expressive flows from a small set of fixed vector fields and learned scalar controls. Instead of learning an unconstrained high-dimensional vector field, our framework constructs the velocity by modulating fixed vector fields with learned scalar control functions. When the fixed fields are bracket-generating, their Lie algebra spans the ambient space, providing a mechanism for expressive transport with only a small number of learned control channels and offering a parameter-efficient geometric alternative to standard vector-field parameterizations. This decoupled formulation yields a structured and interpretable generative model in which the number of learned scalar output channels can be chosen independently of the ambient dimension. We formulate an expressivity principle showing that, under suitable controllability and well-posedness assumptions, such controlled flows can transport a source distribution to a target distribution. We train the resulting model using a continuous-normalizing-flow likelihood objective and present proof-of-concept experiments on synthetic distributions.