We propose and analyze a continuous-time robust reinforcement learning framework for optimal stopping problems under ambiguity. In this framework, an agent chooses a stopping rule motivated by two objectives: robust decision-making under ambiguity and learning about the unknown environment. Here, ambiguity refers to considering multiple probability measures dominated by a reference measure, reflecting the agent's awareness that the reference measure representing her learned belief about the environment would be erroneous. Using the $g$-expectation framework, we reformulate an optimal stopping problem under ambiguity as an entropy-regularized optimal control problem under ambiguity, with Bernoulli distributed controls to incorporate exploration into the stopping rules. We then derive the optimal Bernoulli distributed control characterized by backward stochastic differential equations. Moreover, we establish a policy iteration theorem and implement it as a reinforcement learning algorithm. Numerical experiments demonstrate the convergence and robustness of the proposed algorithm across different levels of ambiguity and exploration.

The training of deep neural networks predominantly relies on a combination of gradient-based optimisation and back-propagation for the computation of the gradient. While incredibly successful, this approach faces challenges such as vanishing or exploding gradients, difficulties with non-smooth activations, and an inherently sequential structure that limits parallelisation. Lifted training methods offer an alternative by reformulating the nested optimisation problem into a higher-dimensional, constrained optimisation problem where the constraints are no longer enforced directly but penalised with penalty terms. This chapter introduces a unified framework that encapsulates various lifted training strategies--including the Method of Auxiliary Coordinates (MAC), Fenchel Lifted Networks, and Lifted Bregman Training--and demonstrates how diverse architectures, such as Multi-Layer Perceptrons (MLPs), Residual Neural Networks (ResNets), and Proximal Neural Networks (PNNs), fit within this structure. By leveraging tools from convex optimisation, particularly Bregman distances, the framework facilitates distributed optimisation, accommodates non-differentiable proximal activations, and can improve the conditioning of the training landscape. We discuss the implementation of these methods using block-coordinate descent (BCD) strategies--including deterministic implementations enhanced by accelerated (e.g., Nesterov, Heavyball) and adaptive (e.g., Adam) optimisation techniques--as well as implicit stochastic gradient methods (ISGM). Furthermore, we explore the application of this framework to inverse problems, detailing methodologies for both the training of specialised networks (e.g., unrolled architectures) and the stable inversion of pre-trained networks. Numerical results on standard imaging tasks validate the effectiveness and stability of the lifted Bregman approach compared to conventional training, particularly for architectures employing proximal activations.

Time series forecasting (TSF) remains a challenging and largely unsolved problem in machine learning, despite significant recent efforts leveraging Large Language Models (LLMs), which predominantly rely on Transformer architectures. Empirical evidence consistently shows that even powerful Transformers often fail to outperform much simpler models, e.g., linear models, on TSF tasks; however, a rigorous theoretical understanding of this phenomenon remains limited. In this paper, we provide a theoretical analysis of Transformers' limitations for TSF through the lens of In-Context Learning (ICL) theory. Specifically, under AR($p$) data, we establish that: (1) Linear Self-Attention (LSA) models $\textit{cannot}$ achieve lower expected MSE than classical linear models for in-context forecasting; (2) as the context length approaches to infinity, LSA asymptotically recovers the optimal linear predictor; and (3) under Chain-of-Thought (CoT) style inference, predictions collapse to the mean exponentially. We empirically validate these findings through carefully designed experiments. Our theory not only sheds light on several previously underexplored phenomena but also offers practical insights for designing more effective forecasting architectures. We hope our work encourages the broader research community to revisit the fundamental theoretical limitations of TSF and to critically evaluate the direct application of increasingly sophisticated architectures without deeper scrutiny.

Disclosing private information via publication of a machine learning model is often a concern. Intuitively, publishing a learned model should be less risky than publishing a dataset. But how much risk is there? In this paper, we present a principled disclosure metric called \emph{gradient uniqueness} that is derived from an upper bound on the amount of information disclosure from publishing a learned model. Gradient uniqueness provides an intuitive way to perform privacy auditing. The mathematical derivation of gradient uniqueness is general, and does not make any assumption on the model architecture, dataset type, or the strategy of an attacker. We examine a simple defense based on monitoring gradient uniqueness, and find that it achieves privacy comparable to classical methods such as DP-SGD, while being substantially better in terms of (utility) testing accuracy.

Modeling clinical time-series data is hampered by the challenge of capturing latent, time-varying dependencies among features. State-of-the-art approaches often rely on black-box mechanisms or simple aggregation, failing to explicitly model how the influence of one clinical variable propagates through others over time. We propose $\textbf{Chain-of-Influence (CoI)}$, an interpretable deep learning framework that constructs an explicit, time-unfolded graph of feature interactions. CoI leverages a multi-level attention architecture: first, a temporal attention layer identifies critical time points in a patient's record; second, a cross-feature attention layer models the directed influence from features at these time points to subsequent features. This design enables the tracing of influence pathways, providing a granular audit trail that shows how any feature at any time contributes to the final prediction, both directly and through its influence on other variables. We evaluate CoI on mortality and disease progression tasks using the MIMIC-IV dataset and a private chronic kidney disease cohort. Our framework significantly outperforms existing methods in predictive accuracy. More importantly, through case studies, we show that CoI can uncover clinically meaningful, patient-specific patterns of disease progression that are opaque to other models, offering unprecedented transparency into the temporal and cross-feature dependencies that inform clinical decision-making.

Recent advances in reasoning domains with neural networks have primarily been enabled by a training recipe that optimizes Large Language Models, previously trained to predict the next-token in a sequence, with reinforcement learning algorithms. We introduce a framework to study the success of this paradigm, and we theoretically expose the optimization mechanisms by which reinforcement learning improves over next-token prediction in this setting. We study learning from mixture distributions of short and long ``chain-of-thought'' sequences encoding a single task. In particular, when the task consists of predicting the parity of $d$ bits and long sequences are rare, we show how reinforcement learning after next-token prediction enables autoregressive transformers to generalize, whereas mere next-token prediction requires extreme statistical or computational resources to do so. We further explain how reinforcement learning leverages increased test-time computation, manifested in longer responses, to facilitate this learning process. In a simplified setting, we theoretically prove that autoregressive linear models following this training recipe can efficiently learn to predict the parity of $d$ bits as long as the proportion of long demonstrations in the data mix is not exponentially small in the input dimension $d$. Finally, we demonstrate these same phenomena in other settings, including the post-training of Llama-series models on mixture variations of common mathematical reasoning benchmarks.

Sequential data - ranging from financial time series to natural language - has driven the growing adoption of autoregressive models. However, these algorithms rely on the presence of underlying patterns in the data, and their identification often depends heavily on human expertise. Misinterpreting these patterns can lead to model misspecification, resulting in increased generalization error and degraded performance. The recently proposed evolving pattern (EvoRate) metric addresses this by using the mutual information between the next data point and its past to guide regression order estimation and feature selection. Building on this idea, we introduce a general framework based on predictive information, defined as the mutual information between the past and the future, $I(X_{past}; X_{future})$. This quantity naturally defines an information-theoretic learning curve, which quantifies the amount of predictive information available as the observation window grows. Using this formalism, we show that the presence or absence of temporal patterns fundamentally constrains the learnability of sequential models: even an optimal predictor cannot outperform the intrinsic information limit imposed by the data. We validate our framework through experiments on synthetic data, demonstrating its ability to assess model adequacy, quantify the inherent complexity of a dataset, and reveal interpretable structure in sequential data.

Generative models frequently suffer miscalibration, wherein class probabilities and other statistics of the sampling distribution deviate from desired values. We frame calibration as a constrained optimization problem and seek the closest model in Kullback-Leibler divergence satisfying calibration constraints. To address the intractability of imposing these constraints exactly, we introduce two surrogate objectives for fine-tuning: (1) the relax loss, which replaces the constraint with a miscalibration penalty, and (2) the reward loss, which converts calibration into a reward fine-tuning problem. We demonstrate that these approaches substantially reduce calibration error across hundreds of simultaneous constraints and models with up to one billion parameters, spanning applications in protein design, image generation, and language modeling.

Diffusion models have demonstrated remarkable performance in generating high-dimensional samples across domains such as vision, language, and the sciences. Although continuous-state diffusion models have been extensively studied both empirically and theoretically, discrete-state diffusion models, essential for applications involving text, sequences, and combinatorial structures, remain significantly less understood from a theoretical standpoint. In particular, all existing analyses of discrete-state models assume score estimation error bounds without studying sample complexity results. In this work, we present a principled theoretical framework for discrete-state diffusion, providing the first sample complexity bound of $\widetilde{\mathcal{O}}(ε^{-2})$. Our structured decomposition of the score estimation error into statistical, approximation, optimization, and clipping components offers critical insights into how discrete-state models can be trained efficiently. This analysis addresses a fundamental gap in the literature and establishes the theoretical tractability and practical relevance of discrete-state diffusion models.

Autonomy, from the Greek autos (self) and nomos (law), refers to the capacity to operate according to internal rules without external control. Autonomous vehicles (AuVs) are therefore understood as systems that perceive their environment and execute pre-programmed tasks independently of external input, consistent with the SAE levels of automated driving. Yet recent research and real-world deployments have begun to showcase vehicles that exhibit behaviors outside the scope of this definition. These include natural language interaction with humans, goal adaptation, contextual reasoning, external tool use, and the handling of unforeseen ethical dilemmas, enabled in part by multimodal large language models (LLMs). These developments highlight not only a gap between technical autonomy and the broader cognitive and social capacities required for human-centered mobility, but also the emergence of a form of vehicle intelligence that currently lacks a clear designation. To address this gap, the paper introduces the concept of agentic vehicles (AgVs): vehicles that integrate agentic AI systems to reason, adapt, and interact within complex environments. It synthesizes recent advances in agentic systems and suggests how AgVs can complement and even reshape conventional autonomy to ensure mobility services are aligned with user and societal needs. The paper concludes by outlining key challenges in the development and governance of AgVs and their potential role in shaping future agentic transportation systems.