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The marginals of a continuous diffusion model can be reproduced by a deterministic ordinary differential equation (ODE), known as the probability flow ODE, which provides a foundation for accelerated sampling and consistency distillation. In this paper we address the question whether an analogous formulation exists for discrete diffusion models and answer it affirmatively by introducing exponential residual flows (ERFs). Our framework augments the discrete state with a continuous residual variable for each token and position. These residuals evolve deterministically according to piecewise-defined ODEs whose vector fields depend on the current discrete state and the forward and backward rate matrices of the prescribed diffusion model. The discrete state changes whenever a residual reaches zero. We prove that this construction preserves the marginals of the original diffusion. We derive a practical discretization of our sampler and specialize it to masked and uniform diffusion. The deterministic trajectories of ERFs support consistency distillation and yield empirical improvements over previous approaches across a variety of benchmarks including code and text generation tasks.

We establish structural properties of the nonparametric maximum likelihood estimator (NPMLE) for $d$-dimensional Gaussian mixtures whose means lie in a hypercube. We show that, with high probability, the NPMLE based on $n$ independent observations self-regularizes by fitting a mixture model with $O((\frac{\log n}{\log\log n})^d)$ components, achieving the optimal model complexity without explicit model selection. Furthermore, the NPMLE exhibits a structural adaptation property: For data drawn from a finite mixture, the number of fitted components is bounded in probability as $n\to\infty$. Our proof applies a multivariate Jensen-type bound to control the typical number of critical points of the KKT function associated with the first-order optimality conditions. The key step is to lower bound its gradient norm uniformly on complex spheres. We establish this lower bound by decomposing the KKT function into an empirical fluctuation term and a relative density estimation error term, combining small-ball probability estimates for the former with polynomial approximation and Hellinger error bounds for the latter.

We study high-dimensional integration on bounded convex domains. Given a non-negative integrand $r$, the goal is to estimate $Z=\int_{D}r(x)\,\mathrm dx$ on $D\subset\mathbb R^d$. Standard Quasi-Monte Carlo does not adapt its evaluation points to the integrand, while effective importance sampling requires a proposal that captures its concentration. Learned diffusion samplers can supply such a proposal, but existing methods are formulated on $\mathbb R^d$ and do not respect the domain constraint $x\in D$. We instead build the proposal from reflected dynamics that stay in $D$ by construction. A reflected diffusion progressively transforms the target distribution toward the uniform distribution on $D$, and a neural score model is trained to reverse this process. A change-of-measure formula for reflected diffusions yields a training objective without additional boundary terms and an unbiased importance-sampling estimator of $Z$. For non-smooth integrands, where reparameterization gradients may be undefined or uninformative, we derive a REINFORCE estimator of the training gradient using Gaussian proposal densities before reflection, without evaluating reflected transition densities or differentiating $r$. We evaluate on numerous synthetic targets and real-world problems, demonstrating strong performance of the proposed method, as well as favorable scaling with the dimension of the problem and the number of points used for the Monte Carlo estimate.

Predictive models trained at well-resourced data-rich institutions often fail when deployed in new populations. Motivated by the goal of building clinical diagnosis and decision-making algorithms for data-scarce rural populations using the general knowledge learned from data-rich hospitals, we formulate the fundamental question of \emph{domain generalization with adversarially chosen domains}, which asks a learner to perform on any unseen target domain nearly as well as the worst hypothesis consistent with the evidence from the observed source domains. Assuming only that the hypothesis class contains a "universal" predictor with small error on all $k$ source domains, we characterize the sample complexity of this objective up to logarithmic factors. Letting $d$ denote the VC dimension of the hypothesis class, we find, somewhat surprisingly, that the sample complexity reduces significantly from $Ω(dk)$ to $O(d)$ under a small relaxation of the learning goal in which, roughly speaking, we allow the learned model to compete with hypotheses that perform well on 99% instead of 100% of the source domains. By applying this insight to the meta-distributional setting, in which source domains are selected according to a distribution on distributions, we obtain a corresponding improvement over known sample complexity upper bounds. We also study the effect of data access to the target domain and show that a small labeled sample from the target domain does not improve the worst-case sample complexity on the source domains.

We introduce a new tool, Express, for converting a non-causal attention approximation into a causal approximation with matching approximation guarantees. When combined with the state-of-the-art Thinformer approximation, Express improves upon the best known causal attention guarantees, delivering $\log^{3/2}(n)/s$ approximation error with only $O(s)$ memory and $O(s^2 \log^2(n))$ compression overhead for a sequence of length $n$. We pair these developments with an efficient I/O-aware Triton implementation, demonstrate substantial speedups over FlashAttention 2, and use Express to overcome four resource bottlenecks in the language modeling pipeline: long-context prefill, KV cache compression, long-form memory-constrained decoding, and long-form compute-constrained decoding.

Model combination offers an alternative to selecting a single forecasting model, particularly when individual models are sensitive to random initialization and hyperparameter choices. We study online aggregation through Follow-the-Leader (FTL) and Hedge-type algorithms, with an application to ensembles of Multi-Frequency Echo State Networks (MFESNs) for macroeconomic forecasting. On the theoretical side, we derive Hoeffding- and Bernstein-type expected-regret bounds for FTL and decreasing Hedge under independent and $\varphi$-mixing expert losses. Under a positive gap between the best expert's expected loss and those of its competitors, the bounds are uniform in the number of forecasting rounds and exhibit logarithmic dependence on the number of experts. Monte Carlo experiments investigate the effects of temporal dependence and the separation between experts' expected losses. Empirically, we construct MFESN ensembles for forecasting U.S. quarterly GDP growth using mixed-frequency macroeconomic and financial indicators. The results show that adaptive aggregation, particularly FTL and AdaHedge, achieves substantial reductions in out-of-sample mean squared forecasting error relative to individual MFESN models and conventional forecast combinations.

We consider the problem of finite-sample identification of an unknown LTI system from a finite hypothesis class. In contrast to existing results that assume the hypothesis class contains the true system, we consider the misspecified setting, where the true system does not belong to the hypothesis class. We first characterize the resulting estimation target and its dependence on the data-generation scheme. We then establish a finite-sample lower bound on the failure probability that any identification algorithm must incur. Finally, we analyze the robustness of an existing exponential weighting-based identification algorithm to the misspecification and derive a finite-sample upper bound on its failure probability. Our results illustrate when using a misspecified finite hypothesis class can provide an advantage over identification without prior knowledge of the candidate systems.

Artificial intelligence offers statisticians a new paradigm for modelling and analysing complex datasets. Agents based on large language models can reason about data and automate key stages of the statistical modelling workflow, from model fitting to validation. In this paper we propose a method that uses such an agent to estimate the conditional density of a response given a vector of covariates from an iid sample. The agent examines the sample, chooses a model for the conditional distribution of the response and the procedure to fit it, and returns the estimator as a short computer program. The program is built from a library of probability distributions. The returned law can therefore be a parametric distribution, a mixture or a multivariate factorization, with distribution parameters given by explicit formulas in the covariates. We call this approach agentic distributional regression (ADR). Unlike in the classical approach, the statistician no longer chooses the estimator, but asks an agent to build one. Treating the returned program as a random conditional density estimator, we derive a finite-sample error bound under assumptions on the number of admissible models and on the accuracy of the agent's fit, and an oracle inequality for the selection of a model among fitted candidates using independent validation data. Finally, assuming that, as the sample size grows, the agent selects the same smooth parametric model with probability tending to one and fits it accurately, we show that the estimated parameters are asymptotically normal. In simulation studies, on twelve real datasets and on a new dataset whose responses postdate the agents' training data, ADR with Claude Opus 5.5 has the best average performance among the existing methods from the statistical literature with which it is compared.

We study high-dimensional nonparametric regression under multisource covariate shift, where $\mathrm J$ sources of training data share a common regression function but differ in their covariate distributions, sample sizes, and noise variances. Our goal is to combine these sources to estimate the regression function accurately under a target covariate distribution. We propose an oracle weighted neural estimator and establish its minimax optimality up to logarithmic factors through matching upper and lower bounds. The resulting rate is $\mathsf n_{\rm eff}^{-2p/(2p+K)}$, where $p$ and $K$ denote the effective smoothness and dimension of the regression function, and the effective sample size is \begin{align} \mathsf n_{\rm eff} =\sum_{j=1}^{\mathrm J} \frac{n_j}{(1+σ_{ρ,j})σ_{ε,j}^{2}}. \nonumber \end{align} Here, $n_j$ is the sample size of source $j$, $σ_{ε,j}^{2}$ is its noise variance, and $σ_{ρ,j}$ controls the deviation between the covariate distribution of source $j$ and the target covariate distribution. We also develop an adaptive procedure that requires no knowledge of source noise variances or covariate-shift parameters. Under suitable conditions, this procedure attains the oracle minimax rate up to logarithmic factors.

This paper studies decision-making and statistical inference for two-sided matching markets via matrix completion. In contrast to the independent sampling assumed in classical matrix completion literature, the observed entries, which arise from past matching data, are constrained by matching capacity. This matching-induced dependence poses new challenges for both estimation and inference in the matrix completion framework. We propose a non-convex algorithm based on Grassmannian gradient descent and establish near-optimal entrywise convergence rates for three canonical mechanisms, i.e., one-to-one matching, one-to-many matching with one-sided random arrival, and two-sided random arrival. To facilitate valid uncertainty quantification and hypothesis testing on matching decisions, we further develop a general debiasing and projection framework for arbitrary linear forms of the reward matrix, deriving asymptotic normality with finite-sample guarantees under matching-induced dependent sampling. Our empirical experiments demonstrate that the proposed approach provides accurate estimation, valid confidence intervals, and efficient evaluation of matching policies.

In this paper, we establish Berry-Esseen-type bounds for federated linear stochastic approximation (LSA). Our results provide the first federated Gaussian approximations for LSA that explicitly capture communication-computation trade-offs and heterogeneity-aware error terms, quantifying the effects of local step size, number of local updates, and heterogeneity on convergence rates. We present results for both (i) constant step size regime and (ii) decreasing step size with an increasing number of local iterations, recovering the recent rates of Bonnerjee et al. [2025] as a special case. As a primary application of our results, we develop an online multiplier bootstrap procedure for inference on the last iterate, which avoids explicit estimation of the asymptotic covariance matrix, and obtain non-asymptotic validity guarantees for this procedure.