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).

In the last decade, Reinforcement Learning (RL) has achieved remarkable success in the control and decision-making of complex dynamical systems. However, most RL algorithms rely on the Markov Decision Process assumption, which is violated in practical cyber-physical systems affected by sensing delays, actuation latencies, and communication constraints. Such time delays introduce memory effects that can significantly degrade performance and compromise stability, particularly in networked and multi-agent environments. This paper presents a comprehensive survey of RL methods designed to address time delays in control systems. We first formalize the main classes of delays and analyze their impact on the Markov property. We then systematically categorize existing approaches into five major families: state augmentation and history-based representations, recurrent policies with learned memory, predictor-based and model-aware methods, robust and domain-randomized training strategies, and safe RL frameworks with explicit constraint handling. For each family, we discuss underlying principles, practical advantages, and inherent limitations. A comparative analysis highlights key trade-offs among these approaches and provides practical guidelines for selecting suitable methods under different delay characteristics and safety requirements. Finally, we identify open challenges and promising research directions, including stability certification, large-delay learning, multi-agent communication co-design, and standardized benchmarking. This survey aims to serve as a unified reference for researchers and practitioners developing reliable RL-based controllers in delay-affected cyber-physical systems.

Modern large language models (LLMs) such as GPT, Claude, and Gemini have transformed the way we learn, work, and communicate. Y et, their ability to produce highly human-like text raises serious concerns about misinformation and academic integrity, making it an urgent need for reliable algorithms to detect LLMgenerated content. In this paper, we start by presenting a geometric approach to demystify rewrite-based detection algorithms, revealing their underlying rationale and demonstrating their generalization ability. Building on this insight, we introduce a novel rewrite-based detection algorithm that adaptively learns the distance between the original and rewritten text. Theoretically, we demonstrate that employing an adaptively learned distance function is more effective for detection than using a fixed distance. Empirically, we conduct extensive experiments with over 100 settings, and find that our approach demonstrates superior performance over baseline algorithms in the majority of scenarios. In particular, it achieves relative improvements from 57.8% to 80.6% over the strongest baseline across different target LLMs (e.g., GPT, Claude, and Gemini). The past few years have witnessed the emergence and rapid development of large language models (LLMs) such as GPT (Hurst et al., 2024), DeepSeek (Liu et al., 2024), Claude (Anthropic, 2024), Gemini (Comanici et al., 2025), Grok (xAI, 2025) and Qwen (Y ang et al., 2025). Their impact is everywhere, from education, academia and software development to healthcare and everyday life (Arora & Arora, 2023; Chan & Hu, 2023; Hou et al., 2024). On one side of the coin, LLMs can support users with conversational question answering, help students learn more effectively, draft emails, write computer code, prepare presentation slides and more. On the other side, their ability to closely mimic human-written text also raises serious concerns, including the generation of biased or harmful content, the spread of misinformation in the news ecosystem, and the challenges related to authorship attribution and intellectual property (Dave et al., 2023; Fang et al., 2024; Messeri & Crockett, 2024; Mahajan et al., 2025; Laurito et al., 2025). Addressing these concerns requires effective algorithms to distinguish between human-written and LLM-generated text, which has become an active and popular research direction in recent literature (see Crothers et al., 2023; Wu et al., 2025, for reviews).

We study partial-feedback online learning, where each instance admits a set of correct labels, but the learner only observes one correct label per round; any prediction within the correct set is counted as correct. This model captures settings such as language generation, where multiple responses may be valid but data provide only a single reference. We give a near-complete characterization of minimax regret for both deterministic and randomized learners in the set-realizable regime, i.e., in the regime where sublinear regret is generally attainable. For deterministic learners, we introduce the Partial-Feedback Littlestone dimension (PFLdim) and show it precisely governs learnability and minimax regret; technically, PFLdim cannot be defined via the standard version space, requiring a new collection version space viewpoint and an auxiliary dimension used only in the proof. We further develop the Partial-Feedback Measure Shattering dimension (PMSdim) to obtain tight bounds for randomized learners. We identify broad conditions ensuring inseparability between deterministic and randomized learnability (e.g., finite Helly number or nested-inclusion label structure), and extend the argument to set-valued online learning, resolving an open question of Raman et al. [2024b]. Finally, we show a sharp separation from weaker realistic and agnostic variants: outside set realizability, the problem can become information-theoretically intractable, with linear regret possible even for $|H|=2$. This highlights the need for fundamentally new, noise-sensitive complexity measures to meaningfully characterize learnability beyond set realizability.

Popularized by their strong image generation performance, diffusion and related methods for generative modeling have found widespread success in visual media applications. In particular, diffusion methods have enabled new approaches to data compression, where realistic reconstructions can be generated at extremely low bit-rates. This article provides a unifying review of recent diffusion-based methods for generative lossy compression, with a focus on image compression. These methods generally encode the source into an embedding and employ a diffusion model to iteratively refine it in the decoding procedure, such that the final reconstruction approximately follows the ground truth data distribution. The embedding can take various forms and is typically transmitted via an auxiliary entropy model, and recent methods also explore the use of diffusion models themselves for information transmission via channel simulation. We review representative approaches through the lens of rate-distortion-perception theory, highlighting the role of common randomness and connections to inverse problems, and identify open challenges.

This paper investigates two fundamental descriptors of data, i.e., density distribution versus mass distribution, in the context of clustering. Density distribution has been the de facto descriptor of data distribution since the introduction of statistics. We show that density distribution has its fundamental limitation -- high-density bias, irrespective of the algorithms used to perform clustering. Existing density-based clustering algorithms have employed different algorithmic means to counter the effect of the high-density bias with some success, but the fundamental limitation of using density distribution remains an obstacle to discovering clusters of arbitrary shapes, sizes and densities. Using the mass distribution as a better foundation, we propose a new algorithm which maximizes the total mass of all clusters, called mass-maximization clustering (MMC). The algorithm can be easily changed to maximize the total density of all clusters in order to examine the fundamental limitation of using density distribution versus mass distribution. The key advantage of the MMC over the density-maximization clustering is that the maximization is conducted without a bias towards dense clusters.

Reinforcement learning (RL) has achieved remarkable success in real-world decision-making across diverse domains, including gaming, robotics, online advertising, public health, and natural language processing. Despite these advances, a substantial gap remains between RL research and its deployment in many practical settings. Two recurring challenges often underlie this gap. First, many settings offer limited opportunity for the agent to interact extensively with the target environment due to practical constraints. Second, many target environments often undergo substantial changes, requiring redesign and redeployment of RL systems (e.g., advancements in science and technology that change the landscape of healthcare delivery). Addressing these challenges and bridging the gap between basic research and application requires theory and methodology that directly inform the design, implementation, and continual improvement of RL systems in real-world settings. In this paper, we frame the application of RL in practice as a three-component process: (i) online learning and optimization during deployment, (ii) post- or between-deployment offline analyses, and (iii) repeated cycles of deployment and redeployment to continually improve the RL system. We provide a narrative review of recent advances in statistical RL that address these components, including methods for maximizing data utility for between-deployment inference, enhancing sample efficiency for online learning within-deployment, and designing sequences of deployments for continual improvement. We also outline future research directions in statistical RL that are use-inspired -- aiming for impactful application of RL in practice.

Ensuring fairness in matching algorithms is a key challenge in allocating scarce resources and positions. Focusing on Optimal Transport (OT), we introduce a novel notion of group fairness requiring that the probability of matching two individuals from any two given groups in the OT plan satisfies a predefined target. We first propose \texttt{FairSinkhorn}, a modified Sinkhorn algorithm to compute perfectly fair transport plans efficiently. Since exact fairness can significantly degrade matching quality in practice, we then develop two relaxation strategies. The first one involves solving a penalised OT problem, for which we derive novel finite-sample complexity guarantees. This result is of independent interest as it can be generalized to arbitrary convex penalties. Our second strategy leverages bilevel optimization to learn a ground cost that induces a fair OT solution, and we establish a bound guaranteeing that the learned cost yields fair matchings on unseen data. Finally, we present empirical results that illustrate the trade-offs between fairness and performance.

We provide an overview of high dimensional dynamical systems driven by random matrices, focusing on applications to simple models of learning and generalization in machine learning theory. Using both cavity method arguments and path integrals, we review how the behavior of a coupled infinite dimensional system can be characterized as a stochastic process for each single site of the system. We provide a pedagogical treatment of dynamical mean field theory (DMFT), a framework that can be flexibly applied to these settings. The DMFT single site stochastic process is fully characterized by a set of (two-time) correlation and response functions. For linear time-invariant systems, we illustrate connections between random matrix resolvents and the DMFT response. We demonstrate applications of these ideas to machine learning models such as gradient flow, stochastic gradient descent on random feature models and deep linear networks in the feature learning regime trained on random data. We demonstrate how bias and variance decompositions (analysis of ensembling/bagging etc) can be computed by averaging over subsets of the DMFT noise variables. From our formalism we also investigate how linear systems driven with random non-Hermitian matrices (such as random feature models) can exhibit non-monotonic loss curves with training time, while Hermitian matrices with the matching spectra do not, highlighting a different mechanism for non-monotonicity than small eigenvalues causing instability to label noise. Lastly, we provide asymptotic descriptions of the training and test loss dynamics for randomly initialized deep linear neural networks trained in the feature learning regime with high-dimensional random data. In this case, the time translation invariance structure is lost and the hidden layer weights are characterized as spiked random matrices.