Rabanus Derr

Rabanus Derr, Robert C. Williamson

Fair Machine Learning endeavors to prevent unfairness arising in the context of machine learning applications embedded in society. Despite the variety of definitions of fairness and proposed "fair algorithms", there remain unresolved conceptual problems regarding fairness. In this paper, we dissect the role of statistical independence in fairness and randomness notions regularly used in machine learning. Thereby, we are led to a suprising hypothesis: randomness and fairness can be considered equivalent concepts in machine learning. In particular, we obtain a relativized notion of randomness expressed as statistical independence by appealing to Von Mises' century-old foundations for probability. This notion turns out to be "orthogonal" in an abstract sense to the commonly used i.i.d.-randomness. Using standard fairness notions in machine learning, which are defined via statistical independence, we then link the ex ante randomness assumptions about the data to the ex post requirements for fair predictions. This connection proves fruitful: we use it to argue that randomness and fairness are essentially relative and that both concepts should reflect their nature as modeling assumptions in machine learning.

James Bailie, Rabanus Derr

Property elicitation studies which attributes of a probability distribution can be determined by minimising a risk. We investigate a generalisation of property elicitation to imprecise probabilities (IP). This investigation is motivated by multi-distribution learning, which takes the classical machine learning paradigm of minimising a single risk over a (precise) probability and replaces it with $Γ$-maximin risk minimization over an IP. We provide necessary conditions for elicitability of a IP-property. Furthermore, we explain what an elicitable IP-property actually elicits through Bayes pairs -- the elicited IP-property is the corresponding standard property of the maximum Bayes risk distribution.

Rabanus Derr, Jessie Finocchiaro, Robert C. Williamson

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.

Armando J. Cabrera Pacheco, Rabanus Derr, Robert C. Williamson

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.

Rabanus Derr, Robert C. Williamson

Machine learning is about forecasting. When the forecasts come with an evaluation metric the forecasts become useful. What are reasonable evaluation metrics? How do existing evaluation metrics relate? In this work, we provide a general structure which subsumes many currently used evaluation metrics in a two-dimensional hierarchy, e.g., external and swap regret, loss scores, and calibration scores. The framework embeds those evaluation metrics in a large set of single-instance-based comparisons of forecasts and observations which respect a meta-criterion for reasonable forecast evaluations which we term ``fairness''. In particular, this framework sheds light on the relationship on regret-type and calibration-type evaluation metrics showing a theoretical equivalence in their ability to evaluate, but practical incomparability of the obtained scores.