Learning the potentials of interacting particle systems is a fundamental task across various scientific disciplines. A major challenge is that unlabeled data collected at discrete time points lack trajectory information due to limitations in data collection methods or privacy constraints. We address this challenge by introducing a trajectory-free self-test loss function that leverages the weak-form stochastic evolution equation of the empirical distribution. The loss function is quadratic in potentials, supporting parametric and nonparametric regression algorithms for robust estimation that scale to large, high-dimensional systems with big data. Systematic numerical tests show that our method outperforms baseline methods that regress on trajectories recovered via label matching, tolerating large observation time steps. We establish the convergence of parametric estimators as the sample size increases, providing a theoretical foundation for the proposed approach.

Dynamical systems describe how a physical system evolves over time. Physical processes can evolve faster or slower in different environmental conditions. We use time-warping as rescaling the time in a model of a physical system. This thesis proposes a new method of transfer learning for Recurrent Neural Networks (RNNs) based on time-warping. We prove that for a class of linear, first-order differential equations known as time lag models, an LSTM can approximate these systems with any desired accuracy, and the model can be time-warped while maintaining the approximation accuracy. The Time-Warping method of transfer learning is then evaluated in an applied problem on predicting fuel moisture content (FMC), an important concept in wildfire modeling. An RNN with LSTM recurrent layers is pretrained on fuels with a characteristic time scale of 10 hours, where there are large quantities of data available for training. The RNN is then modified with transfer learning to generate predictions for fuels with characteristic time scales of 1 hour, 100 hours, and 1000 hours. The Time-Warping method is evaluated against several known methods of transfer learning. The Time-Warping method produces predictions with an accuracy level comparable to the established methods, despite modifying only a small fraction of the parameters that the other methods modify.

We study privacy-preserving sparse linear regression in the high-dimensional regime, focusing on the LASSO estimator. We analyze two widely used mechanisms for differential privacy: output perturbation, which injects noise into the estimator, and objective perturbation, which adds a random linear term to the loss function. Using approximate message passing (AMP), we characterize the typical behavior of these estimators under random design and privacy noise. To quantify privacy, we adopt typical-case measures, including the on-average KL divergence, which admits a hypothesis-testing interpretation in terms of distinguishability between neighboring datasets. Our analysis reveals that sparsity plays a central role in shaping the privacy-accuracy trade-off: stronger regularization can improve privacy by stabilizing the estimator against single-point data changes. We further show that the two mechanisms exhibit qualitatively different behaviors. In particular, for objective perturbation, increasing the noise level can have non-monotonic effects, and excessive noise may destabilize the estimator, leading to increased sensitivity to data perturbations. Our results demonstrate that AMP provides a powerful framework for analyzing privacy-accuracy trade-offs in high-dimensional sparse models.

Reinforcement learning from human feedback (RLHF) has emerged as a central framework for aligning large language models (LLMs) with human preferences. Despite its practical success, RLHF raises fundamental statistical questions because it relies on noisy, subjective, and often heterogeneous feedback to learn reward models and optimize policies. This survey provides a statistical perspective on RLHF, focusing primarily on the LLM alignment setting. We introduce the main components of RLHF, including supervised fine-tuning, reward modeling, and policy optimization, and relate them to familiar statistical ideas such as Bradley-Terry-Luce (BTL) model, latent utility estimation, active learning, experimental design, and uncertainty quantification. We review methods for learning reward functions from pairwise preference data and for optimizing policies through both two-stage RLHF pipelines and emerging one-stage approaches such as direct preference optimization. We further discuss recent extensions including reinforcement learning from AI feedback, inference-time algorithms, and reinforcement learning from verifiable rewards, as well as benchmark datasets, evaluation protocols, and open-source frameworks that support RLHF research. We conclude by highlighting open challenges in RLHF. An accompanying GitHub demo https://github.com/Pangpang-Liu/RLHF_demo illustrates key components of the RLHF pipeline.

The massive scale of pretrained models has made efficient compression essential for practical deployment. Low-rank decomposition based on the singular value decomposition (SVD) provides a principled approach for model reduction, but its exact computation is expensive for large weight matrices. Randomized alternatives such as randomized SVD (RSVD) improve efficiency, yet they can suffer from poor approximation quality when the singular value spectrum decays slowly, a regime commonly observed in modern pretrained models. In this work, we address this limitation from both theoretical and empirical perspectives. First, we establish a connection between low-rank approximation error and predictive performance by analyzing softmax perturbations, showing that deviations in class probabilities are controlled by the spectral error of the compressed weights. Second, we demonstrate that RSVD is inadequate, and we propose randomized subspace iteration (RSI) as a more effective alternative. By incorporating multiple power iterations, RSI improves spectral separation and provides a controllable mechanism for enhancing approximation quality. We evaluate our approach on both convolutional networks and transformer-based architectures. Our results show that RSI achieves near-optimal approximation quality while outperforming RSVD in predictive accuracy under aggressive compression, enabling efficient model compression.

This paper develops a penalized GMM (PGMM) framework for automatic debiased inference on functionals of nonparametric instrumental variable estimators. We derive convergence rates for the PGMM estimator and provide conditions for root-n consistency and asymptotic normality of debiased functional estimates, covering both linear and nonlinear functionals. Monte Carlo experiments on average derivative show that the PGMM-based debiased estimator performs on par with the analytical debiased estimator that uses the known closed-form Riesz representer, achieving 90-96% coverage while the plug-in estimator falls below 5%. We apply our procedure to estimate mean own-price elasticities in a semiparametric demand model for differentiated products. Simulations confirm near-nominal coverage while the plug-in severely undercovers. Applied to IRI scanner data on carbonated beverages, debiased semiparametric estimates are approximately 20% more elastic compared to the logit benchmark, and debiasing corrections are heterogeneous across products, ranging from negligible to several times the standard error.

Suppose we observe data from a distribution $P$ and we wish to test the composite null hypothesis that $P\in\mathscr P$ against a composite alternative $P\in \mathscr Q\subseteq \mathscr P^c$. Herbert Robbins and coauthors pointed out around 1970 that, while no batch test can have a level $α\in(0,1)$ and power equal to one, sequential tests can be constructed with this fantastic property. Since then, and especially in the last decade, a plethora of sequential tests have been developed for a wide variety of settings. However, the literature has not yet provided a clean and general answer as to when such power-one sequential tests exist. This paper provides a remarkably general sufficient condition (that we also prove is not necessary). Focusing on i.i.d. laws in Polish spaces without any further restriction, we show that there exists a level-$α$ sequential test for any weakly compact $\mathscr P$, that is power-one against $\mathscr P^c$ (or any subset thereof). We show how to aggregate such tests into an $e$-process for $\mathscr P$ that increases to infinity under $\mathscr P^c$. We conclude by building an $e$-process that is asymptotically relatively growth rate optimal against $\mathscr P^c$, an extremely powerful result.

We study high-dimensional convex empirical risk minimization (ERM) under general non-Gaussian data designs. By heuristically extending the Convex Gaussian Min-Max Theorem (CGMT) to non-Gaussian settings, we derive an asymptotic min-max characterization of key statistics, enabling approximation of the mean $μ_{\hatθ}$ and covariance $C_{\hatθ}$ of the ERM estimator $\hatθ$. Specifically, under a concentration assumption on the data matrix and standard regularity conditions on the loss and regularizer, we show that for a test covariate $x$ independent of the training data, the projection $\hatθ^\top x$ approximately follows the convolution of the (generally non-Gaussian) distribution of $μ_{\hatθ}^\top x$ with an independent centered Gaussian variable of variance $\text{Tr}(C_{\hatθ}\mathbb{E}[xx^\top])$. This result clarifies the scope and limits of Gaussian universality for ERMs. Additionally, we prove that any $\mathcal{C}^2$ regularizer is asymptotically equivalent to a quadratic form determined solely by its Hessian at zero and gradient at $μ_{\hatθ}$. Numerical simulations across diverse losses and models are provided to validate our theoretical predictions and qualitative insights.

We study high-dimensional mediation analysis in which exposures, mediators, and outcomes are all multivariate, and both exposures and mediators may be high-dimensional. We formalize this as a many (exposures)-to-many (mediators)-to-many (outcomes) (MMM) mediation analysis problem. Methodologically, MMM mediation analysis simultaneously performs variable selection for high-dimensional exposures and mediators, estimates the indirect effect matrix (i.e., the coefficient matrices linking exposure-to-mediator and mediator-to-outcome pathways), and enables prediction of multivariate outcomes. Theoretically, we show that the estimated indirect effect matrices are consistent and element-wise asymptotically normal, and we derive error bounds for the estimators. To evaluate the efficacy of the MMM mediation framework, we first investigate its finite-sample performance, including convergence properties, the behavior of the asymptotic approximations, and robustness to noise, via simulation studies. We then apply MMM mediation analysis to data from the Alzheimer's Disease Neuroimaging Initiative to study how cortical thickness of 202 brain regions may mediate the effects of 688 genome-wide significant single nucleotide polymorphisms (SNPs) (selected from approximately 1.5 million SNPs) on eleven cognitive-behavioral and diagnostic outcomes. The MMM mediation framework identifies biologically interpretable, many-to-many-to-many genetic-neural-cognitive pathways and improves downstream out-of-sample classification and prediction performance. Taken together, our results demonstrate the potential of MMM mediation analysis and highlight the value of statistical methodology for investigating complex, high-dimensional multi-layer pathways in science. The MMM package is available at https://github.com/THELabTop/MMM-Mediation.

Feature maps associated with positive definite kernels play a central role in kernel methods and learning theory, where regularity properties such as Lipschitz continuity are closely related to robustness and stability guarantees. Despite their importance, explicit characterizations of the Lipschitz constant of kernel feature maps are available only in a limited number of cases. In this paper, we study the Lipschitz regularity of feature maps associated with integral kernels under differentiability assumptions. We first provide sufficient conditions ensuring Lipschitz continuity and derive explicit formulas for the corresponding Lipschitz constants. We then identify a condition under which the feature map fails to be Lipschitz continuous and apply these results to several important classes of kernels. For infinite width two-layer neural network with isotropic Gaussian weight distributions, we show that the Lipschitz constant of the associated kernel can be expressed as the supremum of a two-dimensional integral, leading to an explicit characterization for the Gaussian kernel and the ReLU random neural network kernel. We also study continuous and shift-invariant kernels such as Gaussian, Laplace, and Matérn kernels, which admit an interpretation as neural network with cosine activation function. In this setting, we prove that the feature map is Lipschitz continuous if and only if the weight distribution has a finite second-order moment, and we then derive its Lipschitz constant. Finally, we raise an open question concerning the asymptotic behavior of the convergence of the Lipschitz constant in finite width neural networks. Numerical experiments are provided to support this behavior.