On the Implicit Geometry of Cross-Entropy Parameterizations for Label-Imbalanced Data

TL;DR

This paper investigates how different cross-entropy parameterizations influence the geometry of classifiers in imbalanced data settings, extending implicit bias theory from linear to non-linear models, and provides formulas for classifier geometry.

Contribution

It characterizes the geometry of classifiers learned by various CE parameterizations in non-linear models and derives formulas linking geometry to class imbalance and hyperparameters.

Findings

Logit-adjusted parameterizations can be tuned to learn symmetric geometries regardless of imbalance.

Derived formulas relate classifier angles and norms to class imbalance and hyperparameters.

Experiments confirm the theoretical predictions on deep networks.

Abstract

Various logit-adjusted parameterizations of the cross-entropy (CE) loss have been proposed as alternatives to weighted CE for training large models on label-imbalanced data far beyond the zero train error regime. The driving force behind those designs has been the theory of implicit bias, which for linear(ized) models, explains why they successfully induce bias on the optimization path towards solutions that favor minorities. Aiming to extend this theory to non-linear models, we investigate the implicit geometry of classifiers and embeddings that are learned by different CE parameterizations. Our main result characterizes the global minimizers of a non-convex cost-sensitive SVM classifier for the unconstrained features model, which serves as an abstraction of deep nets. We derive closed-form formulas for the angles and norms of classifiers and embeddings as a function of the number of classes, the imbalance and the minority ratios, and the loss hyperparameters. Using these, we show that logit-adjusted parameterizations can be appropriately tuned to learn symmetric geometries irrespective of the imbalance ratio. We complement our analysis with experiments and an empirical study of convergence accuracy in deep-nets.