Two pressing topics in the theory of deep learning are the interpretation of feature learning mechanisms and the determination of implicit bias of networks in the rich regime. Current theories of rich feature learning, often appear in the form of high-dimensional non-linear equations, which require computationally intensive numerical solutions. Given the many details that go into defining a deep learning problem, this complexity is a significant and often unavoidable challenge. Here, we propose a powerful heuristic route for predicting the data and width scales at which various patterns of feature learning emerge. This form of scale analysis is considerably simpler than exact theories and reproduces the scaling exponents of various known results. In addition, we make novel predictions on complex toy architectures, such as three-layer non-linear networks and attention heads, thus extending the scope of first-principle theories of deep learning.

Understanding how humans respond to uncertainty is critical for designing safe and effective physical human-robot interaction (pHRI), as physically working with robots introduces multiple sources of uncertainty, including trust, comfort, and perceived safety. Conventional pHRI control frameworks typically build on optimal control theory, which assumes that human actions minimize a cost function; however, human behavior under uncertainty often departs from such optimal patterns. To address this gap, additional understanding of human behavior under uncertainty is needed. This pilot study implemented a physically coupled target-reaching task in which the robot delivered assistance or disturbances with systematically varied probabilities (10\% to 90\%). Analysis of participants' force inputs and decision-making strategies revealed two distinct behavioral clusters: a "trade-off" group that modulated their physical responses according to disturbance likelihood, and an "always-compensate" group characterized by strong risk aversion irrespective of probability. These findings provide empirical evidence that human decision-making in pHRI is highly individualized and that the perception of probability can differ to its true value. Accordingly, the study highlights the need for more interpretable behavioral models, such as cumulative prospect theory (CPT), to more accurately capture these behaviors and inform the design of future adaptive robot controllers.

Humans interpret safety not as a binary signal but as a continuous, context- and spatially-dependent notion of risk. While risk is subjective, humans form rational mental models that guide action selection in dynamic environments. This work proposes a framework for extracting implicit human risk models by introducing a novel, semantically-conditioned and spatially-varying parametrization of risk, supervised directly from safe human demonstration videos and VLM common sense. Notably, we define risk through a Bayesian formulation. The prior is furnished by a pretrained vision-language model. In order to encourage the risk estimate to be more human aligned, a likelihood function modulates the prior to produce a relative metric of risk. Specifically, the likelihood is a learned ViT that maps pretrained features, to pixel-aligned risk values. Our pipeline ingests RGB images and a query object string, producing pixel-dense risk images. These images that can then be used as value-predictors in robot planning tasks or be projected into 3D for use in conventional trajectory optimization to produce human-like motion. This learned mapping enables generalization to novel objects and contexts, and has the potential to scale to much larger training datasets. In particular, the Bayesian framework that is introduced enables fast adaptation of our model to additional observations or common sense rules. We demonstrate that our proposed framework produces contextual risk that aligns with human preferences. Additionally, we illustrate several downstream applications of the model; as a value learner for visuomotor planners or in conjunction with a classical trajectory optimization algorithm. Our results suggest that our framework is a significant step toward enabling autonomous systems to internalize human-like risk. Code and results can be found at https://riskbayesian.github.io/bayesian_risk/.

Learning about the causal structure of the world is a fundamental problem for human cognition. Causal models and especially causal learning have proved to be difficult for large pretrained models using standard techniques of deep learning. In contrast, cognitive scientists have applied advances in our formal understanding of causation in computer science, particularly within the Causal Bayes Net formalism, to understand human causal learning. In the very different tradition of reinforcement learning, researchers have described an intrinsic reward signal called "empowerment" which maximizes mutual information between actions and their outcomes. "Empowerment" may be an important bridge between classical Bayesian causal learning and reinforcement learning and may help to characterize causal learning in humans and enable it in machines. If an agent learns an accurate causal world model, they will necessarily increase their empowerment, and increasing empowerment will lead to a more accurate causal world model. Empowerment may also explain distinctive features of childrens causal learning, as well as providing a more tractable computational account of how that learning is possible. In an empirical study, we systematically test how children and adults use cues to empowerment to infer causal relations, and design effective causal interventions.

Abstract: Reliable optimal control is challenging when the dynamics of a nonlinear system are unknown and only infrequent, noisy output measurements are available. This work addresses this setting of limited sensing by formulating a Bayesian prior over the continuous-time dynamics and latent state trajectory in state-space form and updating it through a targeted marginal Metropolis-Hastings sampler equipped with a numerical ODE integrator. The resulting posterior samples are used to formulate a scenario-based optimal control problem that accounts for both model and measurement uncertainty and is solved using standard nonlinear programming methods. The approach is validated in a numerical case study on glucose regulation using a Type 1 diabetes model. Keywords: Probabilistic and Bayesian methods for system identification, Nonlinear system identification, Time series modeling, Statistical inference, Learning methods for optimal control, Model predictive control, Data-driven control theory 1. INTRODUCTION Accurate dynamical models are fundamental for the predictive and optimal control of nonlinear systems. Although first-principles models may describe the general structure of many systems, important parameters or effects often remain unknown, limiting their direct use for control.

Estimation of the conditional independence graph (CIG) of high-dimensional multivariate Gaussian time series from multi-attribute data is considered. Existing methods for graph estimation for such data are based on single-attribute models where one associates a scalar time series with each node. In multi-attribute graphical models, each node represents a random vector or vector time series. In this paper we provide a unified theoretical analysis of multi-attribute graph learning for dependent time series using a penalized log-likelihood objective function formulated in the frequency domain using the discrete Fourier transform of the time-domain data. We consider both convex (sparse-group lasso) and non-convex (log-sum and SCAD group penalties) penalty/regularization functions. We establish sufficient conditions in a high-dimensional setting for consistency (convergence of the inverse power spectral density to true value in the Frobenius norm), local convexity when using non-convex penalties, and graph recovery. We do not impose any incoherence or irrepresentability condition for our convergence results. We also empirically investigate selection of the tuning parameters based on the Bayesian information criterion, and illustrate our approach using numerical examples utilizing both synthetic and real data.

Statistically correcting measured cross sections for detector effects is an important step across many applications. In particle physics, this inverse problem is known as \textit{unfolding}. In cases with complex instruments, the distortions they introduce are often known only implicitly through simulations of the detector. Modern machine learning has enabled efficient simulation-based approaches for unfolding high-dimensional data. Among these, one of the first methods successfully deployed on experimental data is the \textsc{OmniFold} algorithm, a classifier-based Expectation-Maximization procedure. In practice, however, the forward model is only approximately specified, and the corresponding uncertainty is encoded through nuisance parameters. Building on the well-studied \textsc{OmniFold} algorithm, we show how to extend machine learning-based unfolding to incorporate nuisance parameters. Our new algorithm, called Profile \textsc{OmniFold}, is demonstrated using a Gaussian example as well as a particle physics case study using simulated data from the CMS Experiment at the Large Hadron Collider.

Inverse reinforcement learning aims to infer the reward function that explains expert behavior observed through trajectories of state--action pairs. A long-standing difficulty in classical IRL is the non-uniqueness of the recovered reward: many reward functions can induce the same optimal policy, rendering the inverse problem ill-posed. In this paper, we develop a statistical framework for Inverse Entropy-regularized Reinforcement Learning that resolves this ambiguity by combining entropy regularization with a least-squares reconstruction of the reward from the soft Bellman residual. This combination yields a unique and well-defined so-called least-squares reward consistent with the expert policy. We model the expert demonstrations as a Markov chain with the invariant distribution defined by an unknown expert policy $π^\star$ and estimate the policy by a penalized maximum-likelihood procedure over a class of conditional distributions on the action space. We establish high-probability bounds for the excess Kullback--Leibler divergence between the estimated policy and the expert policy, accounting for statistical complexity through covering numbers of the policy class. These results lead to non-asymptotic minimax optimal convergence rates for the least-squares reward function, revealing the interplay between smoothing (entropy regularization), model complexity, and sample size. Our analysis bridges the gap between behavior cloning, inverse reinforcement learning, and modern statistical learning theory.

Semi-supervised learning (SSL) constructs classifiers from datasets in which only a subset of observations is labelled, a situation that naturally arises because obtaining labels often requires expert judgement or costly manual effort. This motivates methods that integrate labelled and unlabelled data within a learning framework. Most SSL approaches assume that label absence is harmless, typically treated as missing completely at random or ignored, but in practice, the missingness process can be informative, as the chances of an observation being unlabelled may depend on the ambiguity of its feature vector. In such cases, the missingness indicators themselves provide additional information that, if properly modelled, may improve estimation efficiency. The \textbf{SSLfmm} package for R is designed to capture this behaviour by estimating the Bayes' classifier under a finite mixture model in which each component corresponding to a class follows a multivariate normal distribution. It incorporates a mixed-missingness mechanism that combines a missing completely at random (MCAR) component with a (non-ignorable) missing at random (MAR) component, the latter modelling the probability of label missingness as a logistic function of the entropy based on the features. Parameters are estimated via an Expectation--Conditional Maximisation algorithm. In the two-class Gaussian setting with arbitrary covariance matrices, the resulting classifier trained on partially labelled data may, in some cases, achieve a lower misclassification rate than the supervised version in the case where all the labels are known. The package includes a practical tool for modelling and illustrates its performance through simulated examples.

This Inaugural Lecture was given at Royal Holloway University of London in 1996. It covers an introduction to machine learning and describes various theoretical advances and practical projects in the field. The Lecture here is presented in its original format, but a few remarks have been added in 2025 to reflect recent developments, and the list of references has been updated to enhance the convenience and accuracy for readers. When did machine learning start? Maybe a good starting point is 1949, when Claude Shannon proposed a learning algorithm for chess-playing programs. Or maybe we should go back to the 1930s when Ronald Fisher developed discriminant analysis - a type of learning where the problem is to construct a decision rule that separates two types of vectors. Or could it be the 18th century when David Hume discussed the idea of induction? Or the 14th century, when William of Ockham formulated the principle of "simplicity" known as "Ockham's razor" (Ockham, by the way, is a small village not far from Royal Holloway). Or it may be that, like almost everything else in Western civilisation and culture, the origin of these ideas lies in the Mediterranean. After all, it was Aristotle who said that "we learn some things only by doing things". The field of machine learning has been greatly influenced by other disciplines and the subject is in itself not a very homogeneous discipline, but includes separate, overlapping subfields. There are many parallel lines of research in ML: inductive learning, neural networks, clustering, and theories of learning. They are all part of the more general field of machine learning.