arXiv subjects

Robert C. Williamson

Publications and source records attributed to Robert C. Williamson.

At least 19 recordsLinked to original sources

arXiv2026

Comparing Corrupted Constrained Learning Problems

A key result in statistics is the data processing inequality, originally proved by Blackwell (1951) and later refined by DeGroot (1962) in terms of statistical uncertainty. The latter statement claims that the Bayes risk achieved on raw data always undercuts the Bayes risk on a processed version of the same data. This seemingly contradicts empirical findings in machine learning: pre-training, representation learning, feature learning, data augmentation are techniques used to improve performance of machine learning models. We reconcile both worlds by simply accepting that machine learning problems are constrained learning problems: the model class used does not include all measurable functions. We present counterexamples showing that the classical data processing inequality fails to hold in such a setting. Hence, we formulate a generalized data processing inequality, requiring the constrained Bayes risk of a joint distribution (with respect to a loss function and a constrained model class) to lower bound the constrained Bayes risk on the stochastically modified data distribution, regardless of the choice of distribution. We show this inequality to be equivalent to a set containment condition on a specific function set induced by the loss and model class, called the superprediction set. Finally, we exploit our characterization, derive sufficient conditions for this containment and quantify the inequality-gap.

cs.LG↗

arXiv2026

Law of Large Numbers: Accuracy as Statistical Measure for AI Compliance and Competition

The machine learning community progresses (in part) by improving the "accuracy" of its systems. The EU AI Act explicitly refers to "accuracy" as part of its compliance measures for high-risk AI systems. Are we talking about the same thing? This work presents "accuracy" as a case-study for differing requirements of social worlds, the technological machine learning community and the legal community. While competition on accuracy contributes to technological development, machine learning scholars simultaneously recognize accuracy's shortcomings regarding the usefulness and effectiveness of machine learning systems. The legal counterpart embraces the vagueness of "accuracy," leaving interpretative flexibility for technological and societal changes. At the same time, accuracy is a core element of compliance within the EU AI Act. We elaborate on five main tensions, (a) nature of accuracy, (b) notion of performance, (c) scope of validity, (d) ends, and (e) statisticalness, to show that the two communities project disparate, and sometimes contradictory, expectations on accuracy. Both legal and technical communities lack precise understanding of "accuracy" beyond the contextual boundaries of their community. The resulting frictions, \eg, based on the empirical or normative understanding of accuracy, are symptoms of an unresolved (and unresolvable) debate on what accuracy is. We constructively use the frictions to recommend baselines and interventional studies in standardization, and demand for tools to extend the validity of accuracy measurements.

cs.CY↗

arXiv2026

Evaluation Metrics as Averaged Outcomes of Fair Gambles

In the current practices of machine learning, the evaluation of forecasts has become a cornerstone of scientific progress. A multitude of evaluation metrics have been suggested and used to qualify "good" forecasts. What do those metrics share? How are they related? In this work, we use a protocol borrowed from game-theoretic probability to show that a large part of evaluation metrics can be viewed as averaged outcomes of fair gambles. Intuitively, a fair gambler is one which a forecaster would expect to fail. Hence, the gambler's ability to gain disproves the quality of the forecast. Standard evaluation metrics are then variants of choices of such fair gambles. In particular, this choice is structured along two dimensions, one of which separates calibration-type and regret-type metrics. In particular, this framework sheds light on the relationship of calibration and regret showing a theoretical equivalence in their ability to evaluate when being scaled appropriately, but the incomparability of obtained scores.

cs.LG↗

arXiv2026

Corruptions of Supervised Learning Problems: Typology and Mitigations

Corruption is notoriously widespread in data collection. Despite extensive research, the existing literature predominantly focuses on specific settings and learning scenarios, lacking a unified view of corruption modelization and mitigation. In this work, we develop a general theory of corruption, which incorporates all modifications to a supervised learning problem, including changes in model class and loss. Focusing on changes to the underlying probability distributions via Markov kernels, our approach leads to three novel opportunities. First, it enables the construction of a novel, provably exhaustive corruption framework, distinguishing among different corruption types. This serves to unify existing models and establish a consistent nomenclature. Second, it facilitates a systematic analysis of corruption's consequences on learning tasks, by comparing Bayes risks in the clean and corrupted scenarios. Notably, while label corruptions affect only the loss function, attribute corruptions additionally influence the hypothesis class. Third, building upon these results, we investigate mitigations for various corruption types. We expand existing loss-correction methods for label corruption to handle dependent corruption types. Our findings highlight the necessity to generalize this classical corruption-corrected learning framework to a new paradigm with weaker requirements to encompass more corruption types. We provide such a paradigm as well as loss correction formulas in the attribute and joint corruption cases.

cs.LG↗

arXiv2026

The Costs of Pretending That There Are Data-Generating Probability Distributions in the Social World

Machine Learning research, including work promoting fair or equitable algorithms, often relies on the concept of a data-generating probability distribution. The standard presumption is that since data points are 'sampled from' such a distribution, one can learn from observed data about this distribution and, thus, predict future data points which are also drawn from it. We argue, however, that such true probability distributions do not exist and that the rhetoric around them is harmful in social settings. We show that alternative frameworks focusing directly on relevant populations rather than abstract distributions are available and leave classical learning theory almost unchanged. Furthermore, we argue that the assumption of true probabilities or data-generating distributions can be misleading and obscure both the choices made and the goals pursued in machine learning practice. Based on these considerations, we suggest avoiding the assumption of data-generating probability distributions in the social world.

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arXiv2026

The Rhetoric of Machine Learning

I examine the technology of machine learning from the perspective of rhetoric, which is simply the art of persuasion. Rather than being a neutral and "objective" way to build "world models" from data, machine learning is (I argue) inherently rhetorical. I explore some of its rhetorical features, and examine one pervasive business model where machine learning is widely used, "manipulation as a service."

cs.LG↗

arXiv2026

Limits to Predicting Online Speech Using Large Language Models

Our paper studies the predictability of online speech -- that is, how well language models learn to model the distribution of user generated content on X (previously Twitter). We define predictability as a measure of the model's uncertainty, i.e. its negative log-likelihood. As the basis of our study, we collect 10M tweets for ``tweet-tuning'' base models and a further 6.25M posts from more than five thousand X (previously Twitter) users and their peers. In our study involving more than 5000 subjects, we find that predicting posts of individual users remains surprisingly hard. Moreover, it matters greatly what context is used: models using the users' own history significantly outperform models using posts from their social circle. We validate these results across four large language models ranging in size from 1.5 billion to 70 billion parameters. Moreover, our results replicate if instead of prompting the model with additional context, we finetune on it. We follow up with a detailed investigation on what is learned in-context and a demographic analysis. Up to 20\% of what is learned in-context is the use of @-mentions and hashtags. Our main results hold across the demographic groups we studied.

cs.CL↗

arXiv2025

Sparse Robust Classification via the Kernel Mean

Many leading classification algorithms output a classifier that is a weighted average of kernel evaluations. Optimizing these weights is a nontrivial problem that still attracts much research effort. Furthermore, explaining these methods to the uninitiated is a difficult task. Letting all the weights be equal leads to a conceptually simpler classification rule, one that requires little effort to motivate or explain, the mean. Here we explore the consistency, robustness and sparsification of this simple classification rule.

stat.ML↗

arXiv2025

Geometry and Stability of Supervised Learning Problems

We introduce a notion of distance between supervised learning problems, which we call the Risk distance. This distance, inspired by optimal transport, facilitates stability results; one can quantify how seriously issues like sampling bias, noise, limited data, and approximations might change a given problem by bounding how much these modifications can move the problem under the Risk distance. With the distance established, we explore the geometry of the resulting space of supervised learning problems, providing explicit geodesics and proving that the set of classification problems is dense in a larger class of problems. We also provide two variants of the Risk distance: one that incorporates specified weights on a problem's predictors, and one that is more sensitive to the contours of a problem's risk landscape.

cs.LG↗

arXiv2025

Formalising causal inference as prediction on a target population

The standard approach to causal modelling especially in social and health sciences is the potential outcomes framework due to Neyman and Rubin. In this framework, observations are thought to be drawn from a distribution over variables of interest, and the goal is to identify parameters of this distribution. Even though the stated goal is often to inform decision making on some target population, there is no straightforward way to include these target populations in the framework. Instead of modelling the relationship between the observed sample and the target population, the inductive assumptions in this framework take the form of abstract sampling and independence assumptions. In this paper, we develop a version of this framework that construes causal inference as treatment-wise predictions for finite populations where all assumptions are testable in retrospect; this means that one can not only test predictions themselves (without any fundamental problem) but also investigate sources of error when they fail. Due to close connections to the original framework, established methods can still be be analysed under the new framework.

stat.ME↗

arXiv2025

Three Types of Calibration with Properties and their Semantic and Formal Relationships

Fueled by discussions around "trustworthiness" and algorithmic fairness, calibration of predictive systems has regained scholars attention. The vanilla definition and understanding of calibration is, simply put, on all days on which the rain probability has been predicted to be p, the actual frequency of rain days was p. However, the increased attention has led to an immense variety of new notions of "calibration." Some of the notions are incomparable, serve different purposes, or imply each other. In this work, we provide two accounts which motivate calibration: self-realization of forecasted properties and precise estimation of incurred losses of the decision makers relying on forecasts. We substantiate the former via the reflection principle and the latter by actuarial fairness. For both accounts we formulate prototypical definitions via properties $Γ$ of outcome distributions, e.g., the mean or median. The prototypical definition for self-realization, which we call $Γ$-calibration, is equivalent to a certain type of swap regret under certain conditions. These implications are strongly connected to the omniprediction learning paradigm. The prototypical definition for precise loss estimation is a modification of decision calibration adopted from Zhao et al. [73]. For binary outcome sets both prototypical definitions coincide under appropriate choices of reference properties. For higher-dimensional outcome sets, both prototypical definitions can be subsumed by a natural extension of the binary definition, called distribution calibration with respect to a property. We conclude by commenting on the role of groupings in both accounts of calibration often used to obtain multicalibration. In sum, this work provides a semantic map of calibration in order to navigate a fragmented terrain of notions and definitions.

cs.LG↗

arXiv2025

Data Models With Two Manifestations of Imprecision

Motivated by recently emerging problems in machine learning and statistics, we propose data models which relax the familiar i.i.d. assumption. In essence, we seek to understand what it means for data to come from a set of probability measures. We show that our frequentist data models, parameterized by such sets, manifest two aspects of imprecision. We characterize the intricate interplay of these manifestations, aggregate (ir)regularity and local (ir)regularity, where a much richer set of behaviours compared to an i.i.d. model is possible. In doing so we shed new light on the relationship between non-stationary, locally precise and stationary, locally imprecise data models. We discuss possible applications of these data models in machine learning and how the set of probabilities can be estimated. For the estimation of aggregate irregularity, we provide a negative result but argue that it does not warrant pessimism. Understanding these frequentist aspects of imprecise probabilities paves the way for deriving generalization of proper scoring rules and calibration to the imprecise case, which can then contribute to tackling practical problems.

math.ST↗

arXiv2024

Scoring Rules and Calibration for Imprecise Probabilities

What does it mean to say that, for example, the probability for rain tomorrow is between 20% and 30%? The theory for the evaluation of precise probabilistic forecasts is well-developed and is grounded in the key concepts of proper scoring rules and calibration. For the case of imprecise probabilistic forecasts (sets of probabilities), such theory is still lacking. In this work, we therefore generalize proper scoring rules and calibration to the imprecise case. We develop these concepts as relative to data models and decision problems. As a consequence, the imprecision is embedded in a clear context. We establish a close link to the paradigm of (group) distributional robustness and in doing so provide new insights for it. We argue that proper scoring rules and calibration serve two distinct goals, which are aligned in the precise case, but intriguingly are not necessarily aligned in the imprecise case. The concept of decision-theoretic entropy plays a key role for both goals. Finally, we demonstrate the theoretical insights in machine learning practice, in particular we illustrate subtle pitfalls relating to the choice of loss function in distributional robustness.

cs.LG↗

arXiv2024

An Axiomatic Approach to Loss Aggregation and an Adapted Aggregating Algorithm

Supervised learning has gone beyond the expected risk minimization framework. Central to most of these developments is the introduction of more general aggregation functions for losses incurred by the learner. In this paper, we turn towards online learning under expert advice. Via easily justified assumptions we characterize a set of reasonable loss aggregation functions as quasi-sums. Based upon this insight, we suggest a variant of the Aggregating Algorithm tailored to these more general aggregation functions. This variant inherits most of the nice theoretical properties of the AA, such as recovery of Bayes' updating and a time-independent bound on quasi-sum regret. Finally, we argue that generalized aggregations express the attitude of the learner towards losses.

cs.LG↗

arXiv2024

Risk Measures and Upper Probabilities: Coherence and Stratification

Machine learning typically presupposes classical probability theory which implies that aggregation is built upon expectation. There are now multiple reasons to motivate looking at richer alternatives to classical probability theory as a mathematical foundation for machine learning. We systematically examine a powerful and rich class of alternative aggregation functionals, known variously as spectral risk measures, Choquet integrals or Lorentz norms. We present a range of characterization results, and demonstrate what makes this spectral family so special. In doing so we arrive at a natural stratification of all coherent risk measures in terms of the upper probabilities that they induce by exploiting results from the theory of rearrangement invariant Banach spaces. We empirically demonstrate how this new approach to uncertainty helps tackling practical machine learning problems.

cs.LG↗

arXiv2024

Insights From Insurance for Fair Machine Learning

We argue that insurance can act as an analogon for the social situatedness of machine learning systems, hence allowing machine learning scholars to take insights from the rich and interdisciplinary insurance literature. Tracing the interaction of uncertainty, fairness and responsibility in insurance provides a fresh perspective on fairness in machine learning. We link insurance fairness conceptions to their machine learning relatives, and use this bridge to problematize fairness as calibration. In this process, we bring to the forefront two themes that have been largely overlooked in the machine learning literature: responsibility and aggregate-individual tensions.

cs.LG↗

arXiv2023

Information Processing Equalities and the Information-Risk Bridge

We introduce two new classes of measures of information for statistical experiments which generalise and subsume $ϕ$-divergences, integral probability metrics, $\mathfrak{N}$-distances (MMD), and $(f,Γ)$ divergences between two or more distributions. This enables us to derive a simple geometrical relationship between measures of information and the Bayes risk of a statistical decision problem, thus extending the variational $ϕ$-divergence representation to multiple distributions in an entirely symmetric manner. The new families of divergence are closed under the action of Markov operators which yields an information processing equality which is a refinement and generalisation of the classical data processing inequality. This equality gives insight into the significance of the choice of the hypothesis class in classical risk minimization.

cs.LG↗

arXiv2023

The Geometry and Calculus of Losses

Statistical decision problems lie at the heart of statistical machine learning. The simplest problems are binary and multiclass classification and class probability estimation. Central to their definition is the choice of loss function, which is the means by which the quality of a solution is evaluated. In this paper we systematically develop the theory of loss functions for such problems from a novel perspective whose basic ingredients are convex sets with a particular structure. The loss function is defined as the subgradient of the support function of the convex set. It is consequently automatically proper (calibrated for probability estimation). This perspective provides three novel opportunities. It enables the development of a fundamental relationship between losses and (anti)-norms that appears to have not been noticed before. Second, it enables the development of a calculus of losses induced by the calculus of convex sets which allows the interpolation between different losses, and thus is a potential useful design tool for tailoring losses to particular problems. In doing this we build upon, and considerably extend existing results on $M$-sums of convex sets. Third, the perspective leads to a natural theory of ``polar'' loss functions, which are derived from the polar dual of the convex set defining the loss, and which form a natural universal substitution function for Vovk's aggregating algorithm.

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