In Defense of Softmax Parametrization for Calibrated and Consistent Learning to Defer

Yuzhou Cao, Hussein Mozannar, Lei Feng, Hongxin Wei, Bo An

Keywords: Classification, Learning to Defer, Probability Estimation

TL;DR: We justify the use of softmax in L2D with probability estimation by studying the cause of invalid probability estimators and then propose an asymmetric softmax that can induce both consistent loss and a valid probability estimator for L2D.

Abstract: Enabling machine learning classifiers to defer their decision to a downstream expert when the expert is more accurate will ensure improved safety and performance. This objective can be achieved with the learning-to-defer framework which aims to jointly learn how to classify and how to defer to the expert. In recent studies, it has been theoretically shown that popular estimators for learning to defer parameterized with softmax provide unbounded estimates for the likelihood of deferring which makes them uncalibrated. However, it remains unknown whether this is due to the widely used softmax parameterization and if we can find a softmax-based estimator that is both statistically consistent and possesses a valid probability estimator. In this work, we first show that the cause of the miscalibrated and unbounded estimator in prior literature is due to the symmetric nature of the surrogate losses used and not due to softmax. We then propose a novel statistically consistent asymmetric softmax-based surrogate loss that can produce valid estimates without the issue of unboundedness. We further analyze the non-asymptotic properties of our proposed method and empirically validate its performance and calibration on benchmark datasets.

Submission Number: 8331