A particularly challenging problem in AI safety is providing guarantees on the behavior of high-dimensional autonomous systems. Verification approaches centered around reachability analysis fail to scale, and purely statistical approaches are constrained by the distributional assumptions about the sampling process. Instead, we pose a distributionally robust version of the statistical verification problem for black-box systems, where our performance guarantees hold over a large family of distributions. This paper proposes a novel approach based on uncertainty quantification using concepts from imprecise probabilities. A central piece of our approach is an ensemble technique called Imprecise Neural Networks, which provides the uncertainty quantification. Additionally, we solve the allied problem of exploring the input set using active learning. The active learning uses an exhaustive neural-network verification tool Sherlock to collect samples. An evaluation on multiple physical simulators in the openAI gym Mujoco environments with reinforcement-learned controllers demonstrates that our approach can provide useful and scalable guarantees for high-dimensional systems.

Quadrupedal locomotion via Reinforcement Learning (RL) is commonly addressed using the teacher-student paradigm, where a privileged teacher guides a proprioceptive student policy. However, key challenges such as representation misalignment between privileged teacher and proprioceptive-only student, covariate shift due to behavioral cloning, and lack of deployable adaptation; lead to poor generalization in real-world scenarios. We propose Teacher-Aligned Representations via Contrastive Learning (TAR), a framework that leverages privileged information with self-supervised contrastive learning to bridge this gap. By aligning representations to a privileged teacher in simulation via contrastive objectives, our student policy learns structured latent spaces and exhibits robust generalization to Out-of-Distribution (OOD) scenarios, surpassing the fully privileged “Teacher”. Results showed accelerated training by 2× compared to state-of-the-art baselines to achieve peak performance. OOD scenarios showed better generalization by 40% on average compared to existing methods. Moreover, TAR transitions seamlessly into learning during deployment without requiring privileged states, setting a new benchmark in sample-efficient, adaptive locomotion and enabling continual fine-tuning in real-world scenarios. Open-source code and videos are available at https://amrmousa.com/TARLoco/.

Quantifying differences between probability distributions is fundamental to statistics and machine learning, primarily for comparing statistical uncertainty. In contrast, epistemic uncertainty---due to incomplete knowledge---requires richer representations than those offered by classical probability. Imprecise probability (IP) theory offers such models, capturing ambiguity and partial belief. This has driven growing interest in imprecise probabilistic machine learning (IPML), where inference and decision-making rely on broader uncertainty models---highlighting the need for metrics beyond classical probability. This work introduces the Integral Imprecise Probability Metric (IIPM) framework, a Choquet integral-based generalisation of classical Integral Probability Metric to the setting of capacities---a broad class of IP models encompassing many existing ones, including lower probabilities, probability intervals, belief functions, and more. Theoretically, we establish conditions under which IIPM serves as a valid metric and metrises a form of weak convergence of capacities. Practically, IIPM not only enables comparison across different IP models but also supports the quantification of epistemic uncertainty~(EU) within a single IP model. In particular, by comparing an IP model with its conjugate, IIPM gives rise to a new class of epistemic uncertainty measures---Maximum Mean Imprecision (MMI) ---which satisfy key axiomatic properties proposed in the Uncertainty Quantification literature. We validate MMI through selective classification experiments, demonstrating strong empirical performance against established EU measures, and outperforming them when classical methods struggle to scale to a large number of classes. Our work advances both theory and practice in Imprecise Probabilistic Machine Learning, offering a principled framework for comparing and quantifying epistemic uncertainty under imprecision.

In the past couple of years, various approaches to representing and quantifying different types of predictive uncertainty in machine learning, notably in the setting of classification, have been proposed on the basis of second-order probability distributions, i.e., predictions in the form of distributions on probability distributions. A completely conclusive solution has not yet been found, however, as shown by recent criticisms of commonly used uncertainty measures associated with second-order distributions, identifying undesirable theoretical properties of these measures. In light of these criticisms, we propose a set of formal criteria that meaningful uncertainty measures for predictive uncertainty based on second-order distributions should obey. Moreover, we provide a general framework for developing uncertainty measures to account for these criteria, and offer an instantiation based on the Wasserstein distance, for which we prove that all criteria are satisfied.

Statistical learning theory is the foundation of machine learning, providing theoretical bounds for the risk of models learned from a (single) training set, assumed to issue from an unknown probability distribution. In actual deployment, however, the data distribution may (and often does) vary, causing domain adaptation/generalization issues. In this paper we lay the foundations for a `credal' theory of learning, using convex sets of probabilities (credal sets) to model the variability in the data-generating distribution. Such credal sets, we argue, may be inferred from a finite sample of training sets. Bounds are derived for the case of finite hypotheses spaces (both assuming realizability or not), as well as infinite model spaces, which directly generalize classical results.

Adequate uncertainty representation and quantification have become imperative in various scientific disciplines, especially in machine learning and artificial intelligence. As an alternative to representing uncertainty via one single probability measure, we consider credal sets (convex sets of probability measures). The geometric representation of credal sets as d-dimensional polytopes implies a geometric intuition about (epistemic) uncertainty. In this paper, we show that the volume of the geometric representation of a credal set is a meaningful measure of epistemic uncertainty in the case of binary classification, but less so for multi-class classification. Our theoretical findings highlight the crucial role of specifying and employing uncertainty measures in machine learning in an appropriate way, and for being aware of possible pitfalls.