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

TL;DR

This paper defends the use of softmax parametrization in learning-to-defer models, introducing a new asymmetric softmax loss that ensures calibration and consistency, addressing prior issues of unbounded estimates.

Contribution

It demonstrates that unbounded estimates are due to symmetric surrogate losses, not softmax, and proposes a novel asymmetric softmax loss for better calibration and consistency.

Findings

The new loss produces well-calibrated probability estimates.

Empirical results show improved performance on benchmark datasets.

The method is statistically consistent and non-asymptotically reliable.

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 method and empirically validate its performance and calibration on benchmark datasets.