Characterising the area under the curve loss function landscape

One of the most common metrics to evaluate neural network classifiers is the area under the receiver operating characteristic curve (AUC). However, optimisation of the AUC as the loss function during network training is not a standard procedure. Here we compare minimising the cross-entropy (CE) loss and optimising the AUC directly. In particular, we analyse the loss function landscape (LFL) of approximate AUC (appAUC) loss functions to discover the organisation of this solution space. We discuss various surrogates for AUC approximation and show their differences. We find that the characteristics of the appAUC landscape are significantly different from the CE landscape. The approximate AUC loss function improves testing AUC, and the appAUC landscape has substantially more minima, but these minima are less robust, with larger average Hessian eigenvalues. We provide a theoretical foundation to explain these results. To generalise our results, we lastly provide an overview of how the LFL can help to guide loss function analysis and selection.