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Non-standard data, which fall outside classical statistical data formats, challenge state-of-the-art analysis. Examples of non-standard data include partial orders and mixed categorical-numeric-spatial data. Most statistical methods required to represent them by classical statistical spaces. However, this representation can distort their inherent structure and thus the results and interpretation. For applicants, this creates a dilemma: using standard statistical methods can risk misrepresenting the data, while preserving their true structure often lead these methods to be inapplicable. To address this dilemma, we introduce the union-free generic depth (ufg-depth) which is a novel framework that respects the true structure of non-standard data while enabling robust statistical analysis. The ufg-depth extends the concept of simplicial depth from normed vector spaces to a much broader range of data types, by combining formal concept analysis and data depth. We provide a systematic analysis of the theoretical properties of the ufg-depth and demonstrate its application to mixed categorical-numerical-spatial data and hierarchical-nominal data. The ufg-depth is a unified approach that bridges the gap between preserving the data structure and applying statistical methods. With this, we provide a new perspective for non-standard data analysis.

We demonstrate that a wide array of machine learning algorithms are specific instances of one single paradigm: reciprocal learning. These instances range from active learning over multi-armed bandits to self-training. We show that all these algorithms do not only learn parameters from data but also vice versa: They iteratively alter training data in a way that depends on the current model fit. We introduce reciprocal learning as a generalization of these algorithms using the language of decision theory. This allows us to study under what conditions they converge. The key is to guarantee that reciprocal learning contracts such that the Banach fixed-point theorem applies. In this way, we find that reciprocal learning algorithms converge at linear rates to an approximately optimal model under relatively mild assumptions on the loss function, if their predictions are probabilistic and the sample adaption is both non-greedy and either randomized or regularized. We interpret these findings and provide corollaries that relate them to specific active learning, self-training, and bandit algorithms., Comment: Accepted at NeurIPS 2024. v2: fixed typos, added future work. v3: changed def. 4 and proof of thm. 4, added illustrations

Given the vast number of classifiers that have been (and continue to be) proposed, reliable methods for comparing them are becoming increasingly important. The desire for reliability is broken down into three main aspects: (1) Comparisons should allow for different quality metrics simultaneously. (2) Comparisons should take into account the statistical uncertainty induced by the choice of benchmark suite. (3) The robustness of the comparisons under small deviations in the underlying assumptions should be verifiable. To address (1), we propose to compare classifiers using a generalized stochastic dominance ordering (GSD) and present the GSD-front as an information-efficient alternative to the classical Pareto-front. For (2), we propose a consistent statistical estimator for the GSD-front and construct a statistical test for whether a (potentially new) classifier lies in the GSD-front of a set of state-of-the-art classifiers. For (3), we relax our proposed test using techniques from robust statistics and imprecise probabilities. We illustrate our concepts on the benchmark suite PMLB and on the platform OpenML., Comment: CJ, GS,JR and HB equally contributed to this work

In this article we introduce a notion of depth functions for data types that are not given in standard statistical data formats. We focus on data that cannot be represented by one specific data structure, such as normed vector spaces. This covers a wide range of different data types, which we refer to as non-standard data. Depth functions have been studied intensively for normed vector spaces. However, a discussion of depth functions for non-standard data is lacking. In this article, we address this gap by using formal concept analysis to obtain a unified data representation. Building on this representation, we then define depth functions for non-standard data. Furthermore, we provide a systematic basis by introducing structural properties using the data representation provided by formal concept analysis. Finally, we embed the generalised Tukey depth into our concept of data depth and analyse it using the introduced structural properties. Thus, this article presents the mathematical formalisation of centrality and outlyingness for non-standard data and increases the number of spaces in which centrality can be discussed. In particular, we provide a basis for defining further depth functions and statistical inference methods for non-standard data.

We propose a framework for descriptively analyzing sets of partial orders based on the concept of depth functions. Despite intensive studies in linear and metric spaces, there is very little discussion on depth functions for non-standard data types such as partial orders. We introduce an adaptation of the well-known simplicial depth to the set of all partial orders, the union-free generic (ufg) depth. Moreover, we utilize our ufg depth for a comparison of machine learning algorithms based on multidimensional performance measures. Concretely, we provide two examples of classifier comparisons on samples of standard benchmark data sets. Our results demonstrate promisingly the wide variety of different analysis approaches based on ufg methods. Furthermore, the examples outline that our approach differs substantially from existing benchmarking approaches, and thus adds a new perspective to the vivid debate on classifier comparison., Comment: arXiv admin note: substantial text overlap with arXiv:2304.09872

Spaces with locally varying scale of measurement, like multidimensional structures with differently scaled dimensions, are pretty common in statistics and machine learning. Nevertheless, it is still understood as an open question how to exploit the entire information encoded in them properly. We address this problem by considering an order based on (sets of) expectations of random variables mapping into such non-standard spaces. This order contains stochastic dominance and expectation order as extreme cases when no, or respectively perfect, cardinal structure is given. We derive a (regularized) statistical test for our proposed generalized stochastic dominance (GSD) order, operationalize it by linear optimization, and robustify it by imprecise probability models. Our findings are illustrated with data from multidimensional poverty measurement, finance, and medicine., Comment: Accepted for the 39th Conference on Uncertainty in Artificial Intelligence (UAI 2023)

We propose a framework for descriptively analyzing sets of partial orders based on the concept of depth functions. Despite intensive studies of depth functions in linear and metric spaces, there is very little discussion on depth functions for non-standard data types such as partial orders. We introduce an adaptation of the well-known simplicial depth to the set of all partial orders, the union-free generic (ufg) depth. Moreover, we utilize our ufg depth for a comparison of machine learning algorithms based on multidimensional performance measures. Concretely, we analyze the distribution of different classifier performances over a sample of standard benchmark data sets. Our results promisingly demonstrate that our approach differs substantially from existing benchmarking approaches and, therefore, adds a new perspective to the vivid debate on the comparison of classifiers., Comment: Accepted to ISIPTA 2023; Forthcoming in: Proceedings of Machine Learning Research

This short note describes and proves a connectedness property which was introduced in Blocher et al. [2023] in the context of data depth functions for partial orders. The connectedness property gives a structural insight into union-free generic sets. These sets, presented in Blocher et al. [2023], are defined by using a closure operator on the set of all partial orders which naturally appears within the theory of formal concept analysis. In the language of formal concept analysis, the property of connectedness can be vividly proven. However, since within Blocher et al. [2023] we did not discuss formal concept analysis, we outsourced the proof to this note.

Self-training is a simple yet effective method within semi-supervised learning. The idea is to iteratively enhance training data by adding pseudo-labeled data. Its generalization performance heavily depends on the selection of these pseudo-labeled data (PLS). In this paper, we aim at rendering PLS more robust towards the involved modeling assumptions. To this end, we propose to select pseudo-labeled data that maximize a multi-objective utility function. The latter is constructed to account for different sources of uncertainty, three of which we discuss in more detail: model selection, accumulation of errors and covariate shift. In the absence of second-order information on such uncertainties, we furthermore consider the generic approach of the generalized Bayesian alpha-cut updating rule for credal sets. As a practical proof of concept, we spotlight the application of three of our robust extensions on simulated and real-world data. Results suggest that in particular robustness w.r.t. model choice can lead to substantial accuracy gains., Comment: 9 pages, 1 figure, under review

The quality of consequences in a decision making problem under (severe) uncertainty must often be compared among different targets (goals, objectives) simultaneously. In addition, the evaluations of a consequence's performance under the various targets often differ in their scale of measurement, classically being either purely ordinal or perfectly cardinal. In this paper, we transfer recent developments from abstract decision theory with incomplete preferential and probabilistic information to this multi-target setting and show how -- by exploiting the (potentially) partial cardinal and partial probabilistic information -- more informative orders for comparing decisions can be given than the Pareto order. We discuss some interesting properties of the proposed orders between decision options and show how they can be concretely computed by linear optimization. We conclude the paper by demonstrating our framework in an artificial (but quite real-world) example in the context of comparing algorithms under different performance measures.

Although being a crucial question for the development of machine learning algorithms, there is still no consensus on how to compare classifiers over multiple data sets with respect to several criteria. Every comparison framework is confronted with (at least) three fundamental challenges: the multiplicity of quality criteria, the multiplicity of data sets and the randomness of the selection of data sets. In this paper, we add a fresh view to the vivid debate by adopting recent developments in decision theory. Based on so-called preference systems, our framework ranks classifiers by a generalized concept of stochastic dominance, which powerfully circumvents the cumbersome, and often even self-contradictory, reliance on aggregates. Moreover, we show that generalized stochastic dominance can be operationalized by solving easy-to-handle linear programs and moreover statistically tested employing an adapted two-sample observation-randomization test. This yields indeed a powerful framework for the statistical comparison of classifiers over multiple data sets with respect to multiple quality criteria simultaneously. We illustrate and investigate our framework in a simulation study and with a set of standard benchmark data sets., Comment: Accepted for publication in: Journal of Machine Learning Research (JMLR)

This paper presents a computationally feasible method to compute rigorous bounds on the interval-generalisation of regression analysis to account for epistemic uncertainty in the output variables. The new iterative method uses machine learning algorithms to fit an imprecise regression model to data that consist of intervals rather than point values. The method is based on a single-layer interval neural network which can be trained to produce an interval prediction. It seeks parameters for the optimal model that minimizes the mean squared error between the actual and predicted interval values of the dependent variable using a first-order gradient-based optimization and interval analysis computations to model the measurement imprecision of the data. An additional extension to a multi-layer neural network is also presented. We consider the explanatory variables to be precise point values, but the measured dependent values are characterized by interval bounds without any probabilistic information. The proposed iterative method estimates the lower and upper bounds of the expectation region, which is an envelope of all possible precise regression lines obtained by ordinary regression analysis based on any configuration of real-valued points from the respective y-intervals and their x-values., Comment: 35 pages, 8 figures

In this paper we propose efficient methods for elicitation of complexly structured preferences and utilize these in problems of decision making under (severe) uncertainty. Based on the general framework introduced in Jansen, Schollmeyer and Augustin (2018, Int. J. Approx. Reason), we now design elicitation procedures and algorithms that enable decision makers to reveal their underlying preference system (i.e. two relations, one encoding the ordinal, the other the cardinal part of the preferences) while having to answer as few as possible simple ranking questions. Here, two different approaches are followed. The first approach directly utilizes the collected ranking data for obtaining the ordinal part of the preferences, while their cardinal part is constructed implicitly by measuring meta data on the decision maker's consideration times. In contrast, the second approach explicitly elicits also the cardinal part of the decision maker's preference system, however, only an approximate version of it. This approximation is obtained by additionally collecting labels of preference strength during the elicitation procedure. For both approaches, we give conditions under which they produce the decision maker's true preference system and investigate how their efficiency can be improved. For the latter purpose, besides data-free approaches, we also discuss ways for effectively guiding the elicitation procedure if data from previous elicitation rounds is available. Finally, we demonstrate how the proposed elicitation methods can be utilized in problems of decision under (severe) uncertainty. Precisely, we show that under certain conditions optimal decisions can be found without fully specifying the preference system.

Searching for new non-perturbatively renormalizable quantum gravity theories, functional renormalization group (RG) flows are studied on a theory space of action functionals depending on the metric and the torsion tensor, the latter parameterized by three irreducible component fields. A detailed comparison with Quantum Einstein-Cartan Gravity (QECG), Quantum Einstein Gravity (QEG), and "tetrad-only" gravity, all based on different theory spaces, is performed. It is demonstrated that, over a generic theory space, the construction of a functional RG equation (FRGE) for the effective average action requires the specification of a metric on the infinite-dimensional field manifold as an additional input. A modified FRGE is obtained if this metric is scale-dependent, as it happens in the metric-torsion system considered., Comment: 73 pages, multiple figures