Benjamin Roussillon

Patrick Loiseau, Benjamin Roussillon

We consider the problem of finding optimal classifiers in an adversarial setting where the class-1 data is generated by an attacker whose objective is not known to the defender -- an aspect that is key to realistic applications but has so far been overlooked in the literature. To model this situation, we propose a Bayesian game framework where the defender chooses a classifier with no a priori restriction on the set of possible classifiers. The key difficulty in the proposed framework is that the set of possible classifiers is exponential in the set of possible data, which is itself exponential in the number of features used for classification. To counter this, we first show that Bayesian Nash equilibria can be characterized completely via functional threshold classifiers with a small number of parameters. We then show that this low-dimensional characterization enables to develop a training method to compute provably approximately optimal classifiers in a scalable manner; and to develop a learning algorithm for the online setting with low regret (both independent of the dimension of the set of possible data). We illustrate our results through simulations.

Nicolas Gast, Stratis Ioannidis, Patrick Loiseau et al.

Linear regression is a fundamental building block of statistical data analysis. It amounts to estimating the parameters of a linear model that maps input features to corresponding outputs. In the classical setting where the precision of each data point is fixed, the famous Aitken/Gauss-Markov theorem in statistics states that generalized least squares (GLS) is a so-called "Best Linear Unbiased Estimator" (BLUE). In modern data science, however, one often faces strategic data sources, namely, individuals who incur a cost for providing high-precision data. In this paper, we study a setting in which features are public but individuals choose the precision of the outputs they reveal to an analyst. We assume that the analyst performs linear regression on this dataset, and individuals benefit from the outcome of this estimation. We model this scenario as a game where individuals minimize a cost comprising two components: (a) an (agent-specific) disclosure cost for providing high-precision data; and (b) a (global) estimation cost representing the inaccuracy in the linear model estimate. In this game, the linear model estimate is a public good that benefits all individuals. We establish that this game has a unique non-trivial Nash equilibrium. We study the efficiency of this equilibrium and we prove tight bounds on the price of stability for a large class of disclosure and estimation costs. Finally, we study the estimator accuracy achieved at equilibrium. We show that, in general, Aitken's theorem does not hold under strategic data sources, though it does hold if individuals have identical disclosure costs (up to a multiplicative factor). When individuals have non-identical costs, we derive a bound on the improvement of the equilibrium estimation cost that can be achieved by deviating from GLS, under mild assumptions on the disclosure cost functions.