Learning to Defer to Multiple Experts: Consistent Surrogate Losses, Confidence Calibration, and Conformal Ensembles

Abstract

We study the statistical properties of learning to defer (L2D) to multiple experts. In particular, we address the open problems of deriving a consistent surrogate loss, confidence calibration, and principled ensembling of experts. Firstly, we derive two consistent surrogates—one based on a softmax parameterization, the other on a one-vs-all (OvA) parameterization—that are analogous to the single expert losses proposed by Mozannar and Sontag (2020) and Verma and Nalisnick (2022), respectively. We then study the frameworks’ ability to estimate $P( m_j = y | x )$, the probability that the $j$th expert will correctly predict the label for $x$. Theory shows the softmax-based loss causes mis-calibration to propagate between the estimates while the OvA-based loss does not (though in practice, we find there are trade offs). Lastly, we propose a conformal inference technique that chooses a subset of experts to query when the system defers. We perform empirical validation on tasks for galaxy, skin lesion, and hate speech classification.