Université Paris Dauphine-PSL

• Social ranking for feature selection in machine learning models • Comparison with methods based on the Shapley value • Axiomatic and experimental analysis of solutions for feature selection • Approximation of a lexicographic social ranking solution Various methods based on the Shapley value have enjoyed notable success in recent years within the field of Explainable AI (XAI), in particular as feature selection mechanisms and for providing feature attributions for explaining machine learning models. Nevertheless, recent studies have raised concerns regarding the use of the Shapley value in this framework. In this paper, we delve deeper into these limitations through the lens of the axiomatic analysis of the Shapley value and its implications in the realm of machine learning. Leveraging on specific examples of classification models, we compare the effects of axioms for the Shapley value with other axioms for ranking methods based on a coalitional framework, where features are the “players” and the worth of a coalition of features corresponds to their predictive capacity. As an alternative feature selection method we pay particular attention to the lex-cel, a social ranking solution introduced in the recent literature at the intersection between coalitional games and social choice theory. Our analysis suggests that axioms characterizing the lex-cel, under certain circumstances, are more suitable for ranking features in machine learning models, compared to axioms satisfied by the Shapley value. Furthermore, through experiments conducted on public datasets, we show that the lex-cel outperforms some commonly employed feature selection algorithms based on the Shapley value, in particular with respect to the capacity of selecting less redundant features. An approximated version of the lex-cel, showing a satisfactory compromise between scalability of the approach and selection performance, is also presented and discussed.

Abstract We study the propagation of energy in one-dimensional anharmonic chains subject to a periodic, localized forcing. For the purely harmonic case, forcing frequencies outside the linear spectrum produce exponentially localized responses, preventing equi-distribution of energy per degree of freedom. We extend this result to anharmonic perturbations with bounded second derivatives and boundary dissipation, proving that for small perturbations and non-resonant forcing, the dynamics converges to a periodic stationary state with energy exponentially localized uniformly in the system size. The perturbed periodic state is described by a convergent power type expansion in the strength of the anharmonicity. This excludes chaoticity induced by anharmonicity, independently of the size of the system. Our perturbative scheme can also be applied in higher dimensions.

Abstract In this paper, we review recent results on stability and instability in logarithmic Sobolev inequalities, with a particular emphasis on strong norms. We consider several versions of these inequalities on the Euclidean space, for the Lebesgue and the Gaussian measures, and discuss their differences in terms of moments and stability. We give new and direct proofs, as well as examples and discuss the stability of a logarithmic uncertainty principle. Although we do not cover all aspects of the topic, we hope to contribute to establishing the state of the art.