When deploying a single predictor across multiple subpopulations, we propose a fundamentally different approach: interpreting group fairness as a bargaining problem among subpopulations. This game-theoretic perspective reveals that existing robust optimization methods such as minimizing worst-group loss or regret correspond to classical bargaining solutions and embody different fairness principles. We propose relative improvement, the ratio of actual risk reduction to potential reduction from a baseline predictor, which recovers the Kalai-Smorodinsky solution. Unlike absolute-scale methods that may not be comparable when groups have different potential predictability, relative improvement provides axiomatic justification including scale invariance and individual monotonicity. We establish finite-sample convergence guarantees under mild conditions.

We study sequential change-point detection for spatio-temporal point processes, where actionable detection requires not only identifying when a distributional change occurs but also localizing where it manifests in space. While classical quickest change detection methods provide strong guarantees on detection delay and false-alarm rates, existing approaches for point-process data predominantly focus on temporal changes and do not explicitly infer affected spatial regions. We propose a likelihood-free, score-based detection framework that jointly estimates the change time and the change region in continuous space-time without assuming parametric knowledge of the pre- or post-change dynamics. The method leverages a localized and conditionally weighted Hyvärinen score to quantify event-level deviations from nominal behavior and aggregates these scores using a spatio-temporal CUSUM-type statistic over a prescribed class of spatial regions. Operating sequentially, the procedure outputs both a stopping time and an estimated change region, enabling real-time detection with spatial interpretability. We establish theoretical guarantees on false-alarm control, detection delay, and spatial localization accuracy, and demonstrate the effectiveness of the proposed approach through simulations and real-world spatio-temporal event data.

A complete understanding of heterogeneous treatment effects involves characterizing the full conditional distribution of potential outcomes. To this end, we propose the Conditional Counterfactual Mean Embeddings (CCME), a framework that embeds conditional distributions of counterfactual outcomes into a reproducing kernel Hilbert space (RKHS). Under this framework, we develop a two-stage meta-estimator for CCME that accommodates any RKHS-valued regression in each stage. Based on this meta-estimator, we develop three practical CCME estimators: (1) Ridge Regression estimator, (2) Deep Feature estimator that parameterizes the feature map by a neural network, and (3) Neural-Kernel estimator that performs RKHS-valued regression, with the coefficients parameterized by a neural network. We provide finite-sample convergence rates for all estimators, establishing that they possess the double robustness property. Our experiments demonstrate that our estimators accurately recover distributional features including multimodal structure of conditional counterfactual distributions.

Deep learning has achieved remarkable success across a wide range of domains, significantly expanding the frontiers of what is achievable in artificial intelligence. Yet, despite these advances, critical challenges remain -- most notably, ensuring robustness to small input perturbations and generalization to out-of-distribution data. These critical challenges underscore the need to understand the underlying fundamental principles that govern robustness and generalization. Among the theoretical tools available, Lipschitz continuity plays a pivotal role in governing the fundamental properties of neural networks related to robustness and generalization. It quantifies the worst-case sensitivity of network's outputs to small input perturbations. While its importance is widely acknowledged, prior research has predominantly focused on empirical regularization approaches based on Lipschitz constraints, leaving the underlying principles less explored. This thesis seeks to advance a principled understanding of the principles of Lipschitz continuity in neural networks within the paradigm of machine learning, examined from two complementary perspectives: an internal perspective -- focusing on the temporal evolution of Lipschitz continuity in neural networks during training (i.e., training dynamics); and an external perspective -- investigating how Lipschitz continuity modulates the behavior of neural networks with respect to features in the input data, particularly its role in governing frequency signal propagation (i.e., modulation of frequency signal propagation).

Prediction sets provide a means of quantifying the uncertainty in predictive tasks. Using held out calibration data, conformal prediction and risk control can produce prediction sets that exhibit statistically valid error control in a computationally efficient manner. However, in the standard formulations, the error is only controlled on average over many possible calibration datasets of fixed size. In this paper, we extend the control to remain valid with high probability over a cumulatively growing calibration dataset at any time point. We derive such guarantees using quantile-based arguments and illustrate the applicability of the proposed framework to settings involving distribution shift. We further establish a matching lower bound and show that our guarantees are asymptotically tight. Finally, we demonstrate the practical performance of our methods through both simulations and real-world numerical examples.

Identifying and making statistical inferences on differential treatment effects (commonly known as subgroup analysis in clinical research) is central to precision health. Subgroup analysis allows practitioners to pinpoint populations for whom a treatment is especially beneficial or protective, thereby advancing targeted interventions. Tree based recursive partitioning methods are widely used for subgroup analysis due to their interpretability. Nevertheless, these approaches encounter significant limitations, including suboptimal partitions induced by greedy heuristics and overfitting from locally estimated splits, especially under limited sample sizes. To address these limitations, we propose a fused optimal causal tree method that leverages mixed integer optimization (MIO) to facilitate precise subgroup identification. Our approach ensures globally optimal partitions and introduces a parameter fusion constraint to facilitate information sharing across related subgroups. This design substantially improves subgroup discovery accuracy and enhances statistical efficiency. We provide theoretical guarantees by rigorously establishing out of sample risk bounds and comparing them with those of classical tree based methods. Empirically, our method consistently outperforms popular baselines in simulations. Finally, we demonstrate its practical utility through a case study on the Health and Aging Brain Study Health Disparities (HABS-HD) dataset, where our approach yields clinically meaningful insights.

Many fields collect large-scale temporal data through repeated measurements (trials), where each trial is labeled with a set of metadata variables spanning several categories. For example, a trial in a neuroscience study may be linked to a value from category (a): task difficulty, and category (b): animal choice. A critical challenge in time-series analysis is to understand how these labels are encoded within the multi-trial observations, and disentangle the distinct effect of each label entry across categories. Here, we present MILCCI, a novel data-driven method that i) identifies the interpretable components underlying the data, ii) captures cross-trial variability, and iii) integrates label information to understand each category's representation within the data. MILCCI extends a sparse per-trial decomposition that leverages label similarities within each category to enable subtle, label-driven cross-trial adjustments in component compositions and to distinguish the contribution of each category. MILCCI also learns each component's corresponding temporal trace, which evolves over time within each trial and varies flexibly across trials. We demonstrate MILCCI's performance through both synthetic and real-world examples, including voting patterns, online page view trends, and neuronal recordings.

In sensitive applications involving relational datasets, protecting information about individual links from adversarial queries is of paramount importance. In many such settings, the available data are summarized solely through the degrees of the nodes in the network. We adopt the $β$ model, which is the prototypical statistical model adopted for this form of aggregated relational information, and study the problem of minimax-optimal parameter estimation under both local and central differential privacy constraints. We establish finite sample minimax lower bounds that characterize the precise dependence of the estimation risk on the network size and the privacy parameters, and we propose simple estimators that achieve these bounds up to constants and logarithmic factors under both local and central differential privacy frameworks. Our results provide the first comprehensive finite sample characterization of privacy utility trade offs for parameter estimation in $β$ models, addressing the classical graph case and extending the analysis to higher order hypergraph models. We further demonstrate the effectiveness of our methods through experiments on synthetic data and a real world communication network.

Recent progress has rapidly advanced our understanding of the mechanisms underlying in-context learning in modern attention-based neural networks. However, existing results focus exclusively on unimodal data; in contrast, the theoretical underpinnings of in-context learning for multi-modal data remain poorly understood. We introduce a mathematically tractable framework for studying multi-modal learning and explore when transformer-like architectures can recover Bayes-optimal performance in-context. To model multi-modal problems, we assume the observed data arises from a latent factor model. Our first result comprises a negative take on expressibility: we prove that single-layer, linear self-attention fails to recover the Bayes-optimal predictor uniformly over the task distribution. To address this limitation, we introduce a novel, linearized cross-attention mechanism, which we study in the regime where both the number of cross-attention layers and the context length are large. We show that this cross-attention mechanism is provably Bayes optimal when optimized using gradient flow. Our results underscore the benefits of depth for in-context learning and establish the provable utility of cross-attention for multi-modal distributions.

Maximum likelihood estimators (MLE) and control variate estimators (CVE) have been used in conjunction with known information across sketching algorithms and applications in machine learning. We prove that under certain conditions in an exponential family, an optimal CVE will achieve the same asymptotic variance as the MLE, giving an Expectation-Maximization (EM) algorithm for the MLE. Experiments show the EM algorithm is faster and numerically stable compared to other root finding algorithms for the MLE for the bivariate Normal distribution, and we expect this to hold across distributions satisfying these conditions. We show how the EM algorithm leads to reproducibility for algorithms using MLE / CVE, and demonstrate how the EM algorithm leads to finding the MLE when the CV weights are known.