We study the problem of learning a low-degree spherical polynomial of degree $\ell_0 = Θ(1) \ge 1$ defined on the unit sphere in $\RR^d$ by training an over-parameterized two-layer neural network (NN) with channel attention in this paper. Our main result is the significantly improved sample complexity for learning such low-degree polynomials. We show that, for any regression risk $\eps \in (0,1)$, a carefully designed two-layer NN with channel attention and finite width of $m \ge Θ({n^4 \log (2n/δ)}/{d^{2\ell_0}})$ trained by the vanilla gradient descent (GD) requires the lowest sample complexity of $n \asymp Θ(d^{\ell_0}/\eps)$ with probability $1-δ$ for every $δ\in (0,1)$, in contrast with the representative sample complexity $Θ\pth{d^{\ell_0} \max\set{\eps^{-2},\log d}}$, where $n$ is the training daata size. Moreover, such sample complexity is not improvable since the trained network renders a sharp rate of the nonparametric regression risk of the order $Θ(d^{\ell_0}/{n})$ with probability at least $1-δ$. On the other hand, the minimax optimal rate for the regression risk with a kernel of rank $Θ(d^{\ell_0})$ is $Θ(d^{\ell_0}/{n})$, so that the rate of the nonparametric regression risk of the network trained by GD is minimax optimal. The training of the two-layer NN with channel attention consists of two stages. In Stage 1, a provable learnable channel selection algorithm identifies the ground-truth channel number $\ell_0$ from the initial $L \ge \ell_0$ channels in the first-layer activation, with high probability. This learnable selection is achieved by an efficient one-step GD update on both layers, enabling feature learning for low-degree polynomial targets. In Stage 2, the second layer is trained by standard GD using the activation function with the selected channels.

This study proposes Riesz representer estimation methods based on score matching. The Riesz representer is a key component in debiased machine learning for constructing $\sqrt{n}$-consistent and efficient estimators in causal inference and structural parameter estimation. To estimate the Riesz representer, direct approaches have garnered attention, such as Riesz regression and the covariate balancing propensity score. These approaches can also be interpreted as variants of direct density ratio estimation (DRE) in several applications such as average treatment effect estimation. In DRE, it is well known that flexible models can easily overfit the observed data due to the estimand and the form of the loss function. To address this issue, recent work has proposed modeling the density ratio as a product of multiple intermediate density ratios and estimating it using score-matching techniques, which are often used in the diffusion model literature. We extend score-matching-based DRE methods to Riesz representer estimation. Our proposed method not only mitigates overfitting but also provides insights for causal inference by bridging marginal effects and average policy effects through time score functions.

Statistical inference in contextual bandits is complicated by the adaptive, non-i.i.d. nature of the data. A growing body of work has shown that classical least-squares inference may fail under adaptive sampling, and that constructing valid confidence intervals for linear functionals of the model parameter typically requires paying an unavoidable inflation of order $\sqrt{d \log T}$. This phenomenon -- often referred to as the price of adaptivity -- highlights the inherent difficulty of reliable inference under general contextual bandit policies. A key structural property that circumvents this limitation is the \emph{stability} condition of Lai and Wei, which requires the empirical feature covariance to concentrate around a deterministic limit. When stability holds, the ordinary least-squares estimator satisfies a central limit theorem, and classical Wald-type confidence intervals -- designed for i.i.d. data -- become asymptotically valid even under adaptation, \emph{without} incurring the $\sqrt{d \log T}$ price of adaptivity. In this paper, we propose and analyze a penalized EXP4 algorithm for linear contextual bandits. Our first main result shows that this procedure satisfies the Lai--Wei stability condition and therefore admits valid Wald-type confidence intervals for linear functionals. Our second result establishes that the same algorithm achieves regret guarantees that are minimax optimal up to logarithmic factors, demonstrating that stability and statistical efficiency can coexist within a single contextual bandit method. Finally, we complement our theory with simulations illustrating the empirical normality of the resulting estimators and the sharpness of the corresponding confidence intervals.

Federated learning (FL) supports privacy-preserving, decentralized machine learning (ML) model training by keeping data on client devices. However, non-independent and identically distributed (non-IID) data across clients biases updates and degrades performance. To alleviate these issues, we propose Clust-PSI-PFL, a clustering-based personalized FL framework that uses the Population Stability Index (PSI) to quantify the level of non-IID data. We compute a weighted PSI metric, $WPSI^L$, which we show to be more informative than common non-IID metrics (Hellinger, Jensen-Shannon, and Earth Mover's distance). Using PSI features, we form distributionally homogeneous groups of clients via K-means++; the number of optimal clusters is chosen by a systematic silhouette-based procedure, typically yielding few clusters with modest overhead. Across six datasets (tabular, image, and text modalities), two partition protocols (Dirichlet with parameter $α$ and Similarity with parameter S), and multiple client sizes, Clust-PSI-PFL delivers up to 18% higher global accuracy than state-of-the-art baselines and markedly improves client fairness by a relative improvement of 37% under severe non-IID data. These results establish PSI-guided clustering as a principled, lightweight mechanism for robust PFL under label skew.

Survival analysis relies fundamentally on the semi-parametric Cox Proportional Hazards (CoxPH) model and the parametric Accelerated Failure Time (AFT) model. CoxPH assumes constant hazard ratios, often failing to capture real-world dynamics, while traditional AFT models are limited by rigid distributional assumptions. Although deep learning models like DeepAFT address these constraints by improving predictive accuracy and handling censoring, they inherit the significant challenge of black-box interpretability. The recent introduction of CoxKAN demonstrated the successful integration of Kolmogorov-Arnold Networks (KANs), a novel architecture that yields highly accurate and interpretable symbolic representations, within the CoxPH framework. Motivated by the interpretability gains of CoxKAN, we introduce KAN-AFT (Kolmogorov Arnold Network-based AFT), the first framework to apply KANs to the AFT model. Our primary contributions include: (i) a principled AFT-KAN formulation, (ii) robust optimization strategies for right-censored observations (e.g., Buckley-James and IPCW), and (iii) an interpretability pipeline that converts the learned spline functions into closed-form symbolic equations for survival time. Empirical results on multiple datasets confirm that KAN-AFT achieves performance comparable to or better than DeepAFT, while uniquely providing transparent, symbolic models of the survival process.

We introduce two nonlinear sufficient dimension reduction methods for regressions with tensor-valued predictors. Our goal is two-fold: the first is to preserve the tensor structure when performing dimension reduction, particularly the meaning of the tensor modes, for improved interpretation; the second is to substantially reduce the number of parameters in dimension reduction, thereby achieving model parsimony and enhancing estimation accuracy. Our two tensor dimension reduction methods echo the two commonly used tensor decomposition mechanisms: one is the Tucker decomposition, which reduces a larger tensor to a smaller one; the other is the CP-decomposition, which represents an arbitrary tensor as a sequence of rank-one tensors. We developed the Fisher consistency of our methods at the population level and established their consistency and convergence rates. Both methods are easy to implement numerically: the Tucker-form can be implemented through a sequence of least-squares steps, and the CP-form can be implemented through a sequence of singular value decompositions. We investigated the finite-sample performance of our methods and showed substantial improvement in accuracy over existing methods in simulations and two data applications.

\noindent Hyper-parameter selection is a central practical problem in modern machine learning, governing regularization strength, model capacity, and robustness choices. Cross-validation is often computationally prohibitive at scale, while fully Bayesian hyper-parameter learning can be difficult due to the cost of posterior sampling. We develop a generative perspective on hyper-parameter tuning that combines two ideas: (i) optimization-based approximations to Bayesian posteriors via randomized, weighted objectives (weighted Bayesian bootstrap), and (ii) amortization of repeated optimization across many hyper-parameter settings by learning a transport map from hyper-parameters (including random weights) to the corresponding optimizer. This yields a ``generator look-up table'' for estimators, enabling rapid evaluation over grids or continuous ranges of hyper-parameters and supporting both predictive tuning objectives and approximate Bayesian uncertainty quantification. We connect this viewpoint to weighted $M$-estimation, envelope/auxiliary-variable representations that reduce non-quadratic losses to weighted least squares, and recent generative samplers for weighted $M$-estimators.

Goodness-of-fit (GoF) tests are fundamental for assessing model adequacy. Score-based tests are appealing because they require fitting the model only once under the null. However, extending them to powerful nonparametric alternatives is difficult due to the lack of suitable score functions. Through a class of exponentially tilted models, we show that the resulting score-based GoF tests are equivalent to the tests based on integral probability metrics (IPMs) indexed by a function class. When the class is rich, the test is universally consistent. This simple yet insightful perspective enables reinterpretation of classical distance-based testing procedures-including those based on Kolmogorov-Smirnov distance, Wasserstein-1 distance, and maximum mean discrepancy-as arising from score-based constructions. Building on this insight, we propose a new nonparametric score-based GoF test through a special class of IPM induced by kernelized Stein's function class, called semiparametric kernelized Stein discrepancy (SKSD) test. Compared with other nonparametric score-based tests, the SKSD test is computationally efficient and accommodates general nuisance-parameter estimators, supported by a generic parametric bootstrap procedure. The SKSD test is universally consistent and attains Pitman efficiency. Moreover, SKSD test provides simple GoF tests for models with intractable likelihoods but tractable scores with the help of Stein's identity and we use two popular models, kernel exponential family and conditional Gaussian models, to illustrate the power of our method. Our method achieves power comparable to task-specific normality tests such as Anderson-Darling and Lilliefors, despite being designed for general nonparametric alternatives.

Unmeasured confounding can render identification strategies based on adjustment functionals invalid. We study the "Napkin graph", a causal structure that encapsulates patterns of M-bias, instrumental variables, and the classical back-door and front-door models within a single graphical framework, yet requires a nonstandard identification strategy: the average treatment effect is expressed as a ratio of two g-formulas. We develop novel estimators for this functional, including doubly robust one-step and targeted minimum loss-based estimators that remain asymptotically linear when nuisance functions are estimated at slower-than-parametric rates using machine learning. We also show how a generalized independence restriction encoded by the Napkin graph, known as a Verma constraint, can be exploited to improve efficiency, illustrating more generally how such constraints in hidden variable DAGs can inform semiparametric inference. The proposed methods are validated through simulations and applied to the Finnish Life Course study to estimate the effect of educational attainment on income. An accompanying R package, napkincausal, implements all proposed procedures.

Empirical Bayes (EB) improves the accuracy of simultaneous inference "by learning from the experience of others" (Efron, 2012). Classical EB theory focuses on latent variables that are iid draws from a fitted prior (Efron, 2019). Modern applications, however, feature complex structure, like arrays, spatial processes, or covariates. How can we apply EB ideas to these settings? We propose a generalized approach to empirical Bayes based on the notion of probabilistic symmetry. Our method pairs a simultaneous inference problem-with an unknown prior-to a symmetry assumption on the joint distribution of the latent variables. Each symmetry implies an ergodic decomposition, which we use to derive a corresponding empirical Bayes method. We call this methodBayesian empirical Bayes (BEB). We show how BEB recovers the classical methods of empirical Bayes, which implicitly assume exchangeability. We then use it to extend EB to other probabilistic symmetries: (i) EB matrix recovery for arrays and graphs; (ii) covariate-assisted EB for conditional data; (iii) EB spatial regression under shift invariance. We develop scalable algorithms based on variational inference and neural networks. In simulations, BEB outperforms existing approaches to denoising arrays and spatial data. On real data, we demonstrate BEB by denoising a cancer gene-expression matrix and analyzing spatial air-quality data from New York City.