Diffusion models are increasingly used as powerful conditional generators, yet real deployments often involve multiple target distributions arising from different tasks, e.g., diverse prompt domains in text-to-image generation, or multiple environments in robotics with diffusion policies. This naturally leads to a multi-objective learning (MOL) problem. A key challenge is that achieving good Pareto trade-offs can require a generalist model class with substantially larger capacity than what suffices for solving any individual task, thereby increasing statistical cost since sample complexity typically scales with the model complexity. To reconcile this, we develop a principled MOL framework for diffusion models with limited data: a semi-supervised regime where paired (labeled) samples are scarce, but (unlabeled) condition data are abundant. We propose a two-stage training procedure that first fits lightweight specialist models from limited paired data, and then distills them into a generalist model by generating pseudo-samples. We establish generalization bounds showing that the required number of paired samples only depends on the complexity of the specialist model classes. We further extend the theory to diffusion policies for sequential decision making to account for distribution shift in on-policy rollouts. Extensive experiments on robotic control and image restoration tasks are conducted to verify our theoretical results.

We investigate the asymptotic properties of the Lévy Adaptive B-spline (LABS) regression model, a Bayesian nonparametric method that incorporates B-spline kernels into the Lévy Adaptive Regression Kernel (LARK) model. LABS applies splines of varying degrees with independently defined knots, yielding a flexible model class capable of adapting to irregular and locally structured features of the true function. Within the nonparametric regression framework with univariate random design and Gaussian errors, we establish that the LABS posterior contracts around the true function in Besov classes at nearly minimax-optimal rates, up to a logarithmic factor, while adapting automatically to unknown smoothness. This study contributes to filling a gap in the literature, where theoretical results on posterior contraction of the LARK model in Besov spaces remain scarce. Simulation experiments on standard test functions in Besov spaces, including Blocks, Bumps, HeaviSine, and Doppler, complement the theoretical results and demonstrate the practical utility of LABS.

Low-rank matrix completion is a widely studied problem with many variants. Inductive matrix completion (IMC) incorporates row and column side information to significantly narrow the search space. Prior work falls into two regimes: methods that exploit this structure to achieve reduced sample complexity but only in noiseless settings, and methods that handle noise but require sample complexity matching the ambient matrix dimension, forfeiting the sample efficiency that side information should provide. In this paper, we close this gap by studying noisy IMC with a nonconvex projected gradient descent algorithm with spectral initialization. Our main technical contribution is establishing a regularity condition for the IMC loss function that holds at the reduced sample complexity determined by the effective problem size, scaling with the side information dimension a rather than the ambient dimension n. This directly yields linear convergence and an estimation error that both depend only on the effective problem size rather than the ambient matrix dimension. We further extend our analysis to the inexact side information setting, demonstrating that the reduced sample complexity is maintained and the estimation error is order-optimal with respect to the inexactness of the side information. Extensive simulations and real-world experiments on the MovieLens dataset validate our theoretical findings.

Federated Learning is a leading framework for training ML and AI models collaboratively across numerous user devices or databases. We study the trade-offs among estimation accuracy, privacy constraints, and communication cost for differentially private (DP) federated M estimation. The two standard methods in the literature are FedAvg, which may suffer from high federation bias, and FedSGD, which can incur high communication cost. Aimed at improving accuracy at a reduced communication cost, we propose FedHybrid, which uses FedSGD starting with an improved initialization by the FedAvg estimator. We propose FedNewton, which averages local Newton iterations to reduce bias in FedAvg, achieving an estimation accuracy comparable to FedSGD with much fewer communication rounds when the number of clients grows sufficiently slowly. We establish finite sample upper bounds on the mean-squared error rates of the DP versions of these estimators as functions of the number of clients, local sample sizes, privacy budget, and number of iterations. We further derive a minimax lower bound on the MSE of any iterative private federated procedure that provides a benchmark to assess the optimality gap of these methods. We numerically evaluate our methods for training a logistic regression and a neural network on the computer vision datasets MNIST and CIFAR-10.

Score-based generative models are trained in high-dimensional ambient spaces, yet many data distributions are supported on low-dimensional nonlinear structures. We prove that, for compact $d$-dimensional smooth manifolds $\mathcal{M} \subset [0,1]^D$ with $d > 2$ and $β$-Hölder densities strictly positive on $\mathcal{M}$, a variance-preserving SGM estimator attains the intrinsic Wasserstein--1 sample exponent $\tilde{\mathcal{O}}(D^{\mathcal{O}_β(d)}n^{-(β+1)/(d+2β)})$, up to logarithmic factors and explicit geometry and density factors. The full nonasymptotic bound explicitly isolates the finite-order geometry envelope, Hölder radius, density lower bound, ambient dependence, and finite-order correction terms. The analysis separates score approximation into a large-noise tangent-cell regime and a small-noise projection-centered, de-Gaussianized Laplace regime. The key technical ingredient is a ReLU implementation of nearest-projection coordinates via finite intrinsic anchors and Gauss--Newton iterations, rather than approximating the manifold projection as a black-box high-dimensional smooth map. Consequently, for families with polynomially controlled geometry and density lower bounds, the constructed score-network parameters have polynomial ambient dependence.

Nearest-neighbor methods are fundamental to classical and modern machine learning, yet their geometric properties are typically analyzed under independent sampling. In this paper, we study the nearest-neighbor radii under dependent sampling. We consider strong mixing dependent observations and ask whether dependence changes the scale of nearest-neighbor neighborhoods. We establish distribution-free almost sure convergence under polynomial mixing and sharp non-asymptotic moment bounds under geometric mixing. The moment bounds depend on the local intrinsic dimension rather than the ambient dimension, making the results applicable to high-dimensional data concentrated near lower-dimensional manifolds. Synthetic experiments and real-world time-series benchmarks support the theory, showing that nearest-neighbor geometry remains informative under dependence sampling.

Length-dependent logit rescaling is widely used to stabilize long-context self-attention, but existing analyses and methods suggest conflicting inverse-temperature laws for the context length $n$, ranging from $(\log n)^{1/2}$ to $\log n$ and $(\log n)^2$. We provide a general theory showing that the desirable scale is determined by the gap-counting function $N_n$ of each attention row. Counting how many competitors lie within each gap from the maximum, we define an upper-tail accumulation scale and prove that it gives the critical inverse-temperature scale for softmax concentration: below this scale, the top competitors remain unseparated, whereas above it, the attention entropy collapses. This framework unifies prior scaling laws as different $N_n$ and yields a direct diagnostic for attention-score families, from idealized theoretical models to more practical transformers.

We study posterior contraction rates for sparse Bayesian Kolmogorov-Arnold networks (KANs) over anisotropic Besov spaces, providing a statistical foundation of KANs from a Bayesian point of view. We show that sparse Bayesian KANs equipped with spike-and-slab-type sparsity priors attain the near-minimax posterior contraction. In particular, the contraction rate depends on the intrinsic anisotropic smoothness of the underlying function. Moreover, by placing a hyperprior on a single model-size parameter, the resulting posterior adapts to unknown anisotropic smoothness and still achieves the corresponding near-minimax rate. A distinctive feature of our results, compared with those for standard sparse MLP-based models, is that the KAN depth can be kept fixed: owing to the flexibility of learnable spline edge functions, the required approximation complexity is controlled through the network width, spline-grid range and size, and parameter sparsity. Our analysis develops theoretical tools tailored to sparse spline-edge architectures, including approximation and complexity bounds for Bayesian KANs. We then extend to compositional Besov spaces and show that the contraction rates depend on layerwise smoothness and effective dimension of the underlying compositional structure, thereby effectively avoiding the curse of dimensionality. Together, the developed tools and findings advance the theoretical understanding of Bayesian neural networks and provide rigorous statistical foundations for KANs.

Large language models are increasingly deployed with test-time strategies: sample $N$ responses, score them with a reward model or verifier, and return the best. This deployment rule exposes a mismatch in post-training: standard objectives optimize the mean reward of a single response, whereas best-of-$N$ performance is governed by the upper tail of the reward distribution. Recent test-time-aware objectives partly address this mismatch, but typically assume that training can use the same per-prompt rollout budget as deployment, which is impractical when post-training must cover many prompts while deployment can allocate much larger per-prompt test-time compute. We study this budget-mismatch regime, where only $m\ll N$ per-prompt rollouts are available during training but the target objective is best-of-$N$ deployment. Under structural assumptions on the reward tails, we show that the policy gradient of the best-of-$N$ objective can be approximated from a much smaller rollout group by extrapolating upper-tail statistics. This yields a family of Tail-Extrapolated estimators for best-of-$N$-oriented post-training: a simple direct estimator, Tail-Extrapolated Advantage (TEA), and a fixed-order debiased Prefix-TEA estimator based on moment cancellation. Experiments on instruction-following tasks show that TEA and Prefix-TEA improve best-of-$N$ performance across different language models, reward models and datasets under various training and test-time budget settings.

We study methods for simultaneous analysis of many noisy and biased estimates, each paired with an even noisier estimate of its own bias. The analyst's goal is to construct short calibrated intervals for each parameter. The standard debiasing approach, which subtracts the bias estimate from each biased estimate, inflates variance and yields long intervals. In this paper, we propose an empirical Bayes rebiasing strategy that starts from the fully debiased estimates and learns from data how much bias to reintroduce by estimating the unknown bias distribution. We provide convergence rates for the coverage of our intervals when the bias distribution is estimated using nonparametric maximum likelihood. Furthermore, we demonstrate substantial precision gains in prediction-powered inference, including pairwise LLM win-rate evaluations, as well as for inference of direct genetic effects in family-based GWAS.