Epistemic Errors of Imperfect Multitask Learners When Distributions Shift

When data are noisy, a statistical learner's goal is to resolve epistemic uncertainty about the data it will encounter at test-time, i.e., to identify the distribution of test (target) data. Many real-world learning settings introduce sources of epistemic uncertainty that can not be resolved on the basis of training (source) data alone: The source data may arise from multiple tasks (multitask learning), the target data may differ systematically from the source data tasks (distribution shift), and/or the learner may not arrive at an accurate characterization of the source data (imperfect learning). We introduce a principled definition of epistemic error, and provide a generic, decompositional epistemic error bound. Our error bound is the first to (i) consider epistemic error specifically, (ii) accommodate all the sources of epistemic uncertainty above, and (iii) separately attribute the error to each of multiple aspects of the learning procedure and environment. As corollaries of the generic result, we provide (i) epistemic error bounds specialized to the settings of Bayesian transfer learning and distribution shift within $\epsilon$-neighborhoods, and (ii) a set of corresponding generalization bounds. Finally, we provide a novel definition of negative transfer, and validate its insights in a synthetic experimental setting.

Integral Imprecise Probability Metrics

Quantifying differences between probability distributions is fundamental to statistics and machine learning, primarily for comparing statistical uncertainty. In contrast, epistemic uncertainty (EU) -- 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 (IPM) 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 within a single IP model. In particular, by comparing an IP model with its conjugate, IIPM gives rise to a new class of EU measures -- Maximum Mean Imprecision -- 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 IPML, offering a principled framework for comparing and quantifying epistemic uncertainty under imprecision.

TAR: Teacher-Aligned Representations via Contrastive Learning for Quadrupedal Locomotion

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 the privileged teacher and the 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 2x compared to state-of-the-art baselines to achieve peak performance. OOD scenarios showed better generalization by 40 percent on average compared to existing methods. Additionally, 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://ammousa.github.io/TARLoco/.

Conformal Prediction Regions are Imprecise Highest Density Regions

Recently, Cella and Martin proved how, under an assumption called consonance, a credal set (i.e. a closed and convex set of probabilities) can be derived from the conformal transducer associated with transductive conformal prediction. We show that the Imprecise Highest Density Region (IHDR) associated with such a credal set corresponds to the classical Conformal Prediction Region. In proving this result, we relate the set of probability density/mass functions (pdf/pmf's) associated with the elements of the credal set to the imprecise probabilistic concept of a cloud. As a result, we establish new relationships between Conformal Prediction and Imprecise Probability (IP) theories. A byproduct of our presentation is the discovery that consonant plausibility functions are monoid homomorphisms, a new algebraic property of an IP tool.

Conformalized Credal Regions for Classification with Ambiguous Ground Truth

An open question in \emph{Imprecise Probabilistic Machine Learning} is how to empirically derive a credal region (i.e., a closed and convex family of probabilities on the output space) from the available data, without any prior knowledge or assumption. In classification problems, credal regions are a tool that is able to provide provable guarantees under realistic assumptions by characterizing the uncertainty about the distribution of the labels. Building on previous work, we show that credal regions can be directly constructed using conformal methods. This allows us to provide a novel extension of classical conformal prediction to problems with ambiguous ground truth, that is, when the exact labels for given inputs are not exactly known. The resulting construction enjoys desirable practical and theoretical properties: (i) conformal coverage guarantees, (ii) smaller prediction sets (compared to classical conformal prediction regions) and (iii) disentanglement of uncertainty sources (epistemic, aleatoric). We empirically verify our findings on both synthetic and real datasets.

Optimal Transport for $ε$-Contaminated Credal Sets

We provide a version for lower probabilities of Monge's and Kantorovich's optimal transport problems. We show that, when the lower probabilities are the lower envelopes of $\epsilon$-contaminated sets, then our version of Monge's, and a restricted version of our Kantorovich's problems, coincide with their respective classical versions. We also give sufficient conditions for the existence of our version of Kantorovich's optimal plan, and for the two problems to be equivalent. As a byproduct, we show that for $\epsilon$-contaminations the lower probability versions of Monge's and Kantorovich's optimal transport problems need not coincide. The applications of our results to Machine Learning and Artificial Intelligence are also discussed.

Imprecise Markov Semigroups and their Ergodicity

We introduce the concept of imprecise Markov semigroup. It allows us to see Markov chains and processes with imprecise transition probabilities as (a collection of diffusion) operators, and thus to unlock techniques from geometry, functional analysis, and (high dimensional) probability to study their ergodic behavior. We show that, if the initial distribution of an imprecise Markov semigroup is known and invariant, under some conditions that also involve the geometry of the state space, eventually the ambiguity around the transition probability fades. We call this property ergodicity of the imprecise Markov semigroup, and we relate it to the classical (Birkhoff's) notion of ergodicity. We prove ergodicity both when the state space is Euclidean or a Riemannian manifold, and when it is an arbitrary measurable space. The importance of our findings for the fields of machine learning and computer vision is also discussed.

Credal Learning Theory

Statistical learning theory is the foundation of machine learning, providing theoretical bounds for the risk of models learnt 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.

Second-Order Uncertainty Quantification: A Distance-Based Approach

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.

IBCL: Zero-shot Model Generation for Task Trade-offs in Continual Learning

Like generic multi-task learning, continual learning has the nature of multi-objective optimization, and therefore faces a trade-off between the performance of different tasks. That is, to optimize for the current task distribution, it may need to compromise performance on some previous tasks. This means that there exist multiple models that are Pareto-optimal at different times, each addressing a distinct task performance trade-off. Researchers have discussed how to train particular models to address specific trade-off preferences. However, existing algorithms require training overheads proportional to the number of preferences -- a large burden when there are multiple, possibly infinitely many, preferences. As a response, we propose Imprecise Bayesian Continual Learning (IBCL). Upon a new task, IBCL (1) updates a knowledge base in the form of a convex hull of model parameter distributions and (2) obtains particular models to address task trade-off preferences with zero-shot. That is, IBCL does not require any additional training overhead to generate preference-addressing models from its knowledge base. We show that models obtained by IBCL have guarantees in identifying the Pareto optimal parameters. Moreover, experiments on standard image classification and NLP tasks support this guarantee. Statistically, IBCL improves average per-task accuracy by at most 23\% and peak per-task accuracy by at most 15\% with respect to the baseline methods, with steadily near-zero or positive backward transfer. Most importantly, IBCL significantly reduces the training overhead from training 1 model per preference to at most 3 models for all preferences.