Towards Bregman Loss with Aitchison-Inspired Potential for Label Distribution Learning

Label Distribution Learning (LDL) represents label ambiguity using simplex-valued vectors over the label space, referred to as label distributions. In most applications, the observed training label distribution is aggregated from a finite number of annotations and can, under an unbiased sampling mechanism, be viewed as a finite-sample estimate of an underlying population label distribution rather than an error-free target. Under this assumption, the conditional arithmetic mean of label distributions equals the underlying population label distribution. Consequently, an LDL point prediction should approximate the conditional arithmetic mean, rather than an individual realization. This motivates strict arithmetic-mean properness of the training loss, requiring the conditional arithmetic mean to be its unique conditional Bayes prediction. Although the widely used Kullback-Leibler divergence and squared Euclidean loss satisfy this property, under restricted model capacity or data noise, small errors in probability space can still induce substantial distortions in relative label ratios, particularly near the simplex boundary. Such distortions can be detrimental when relative label proportions are critical to downstream decisions. We therefore investigate how to reconcile mean-properness with ratio-sensitivity in LDL. Since the squared Aitchison loss directly captures relative variation through log-ratios but is generally not arithmetic-mean proper, we seek a mean-proper loss with Aitchison local geometry as its reference. Specifically, we propose a Bregman loss with Aitchison-inspired potential, which is derived by matching the coordinatewise curvature scaling of its potential function to local Aitchison geometry. We establish its mean-properness, quantitatively compare its local geometry with Aitchison, and show its merits over conventional LDL losses. Finally, we conduct extensive experiments to show the effectiveness of our method.