Functional Properties of the Focal-Entropy

Cross-entropy (CE) is the default training loss for supervised classification, but its sample efficiency is limited when labels are scarce. Existing remedies primarily act on the data side, via augmentation, synthesis, or transfer from pretrained models; the training objective itself is rarely revisited. We revisit it here. Drawing on the classical observation that generative classifiers reach their asymptotic error with fewer samples than discriminative ones, we propose Generative Cross-Entropy (GenCE), a drop-in replacement for CE that introduces a generative learning principle into a standard discriminative network without altering the architecture or fitting a separate density model. GenCE follows from a Bayesian rewrite of the class-conditional likelihood and, in the mini-batch approximation, reduces to normalizing each sample's softmax score against the model's predictions on the batch, coupling the training signal across examples sharing a class. We extend the proper-scoring-rule framework to such non-local losses and prove that GenCE is strictly proper under a mild completeness condition: its population risk is uniquely minimized at the true posterior. Across three datasets, on two architectures and in both balanced small-data and class-imbalanced regimes, GenCE outperforms CE and other widely used losses, while also producing better-calibrated probabilities and stronger out-of-distribution detection.

Loss functions play a central role in supervised classification. Cross-entropy (CE) is widely used, whereas the mean absolute error (MAE) loss can offer robustness but is difficult to optimize. Interpolating between the CE and MAE losses, generalized cross-entropy (GCE) has recently been introduced to provide a trade-off between optimization difficulty and robustness. Existing formulations of GCE result in a non-convex optimization over classification margins that is prone to underfitting, leading to poor performances with complex datasets. In this paper, we propose a minimax formulation of generalized cross-entropy (MGCE) that results in a convex optimization over classification margins. Moreover, we show that MGCEs can provide an upper bound on the classification error. The proposed bilevel convex optimization can be efficiently implemented using stochastic gradient computed via implicit differentiation. Using benchmark datasets, we show that MGCE achieves strong accuracy, faster convergence, and better calibration, especially in the presence of label noise.

We ask: Can focusing on likely classes of a single, in-domain sample improve accuracy? Prior work argued "no", we answer "yes" on average. Standard entropy minimization yields largest gains, and we further dissect why by looking at its two core mechanisms: increasing likely and decreasing unlikely class predictions. Simply maximizing logits of the two most likely classes explains most of its benefits and, in some cases, even outperforms entropy minimization. Thus, decreasing classes is a secondary concern. Our controlled experiment and theory investigate the most puzzling case, why simply optimizing the logits of the two most likely classes leads to accuracy gains. Our small scale networks suggest that the degree of uncertainty of a prediction, followed by model choice, and to a lesser extent dataset choice moderate the success chances of the optimization. We show that the true class tends to have larger gradients independent of whether the prediction is correct or not. Our theory thoroughly investigates one of multiple possible explanations from the angle of shared features. Our evaluation focuses on 26 large language models and 12 text datasets, and 18 image recognition models trained on ImageNet. It demonstrates gains of 0.6 percent on average for entropy minimization on uncertain samples using no extra data aside from the given test sample using a fixed learning rate. We also suggest gradient-ray ensembling. When traversing input/embedding space by applying the same gradient with differing learning rates and aggregating probabilities for the same sample further improves accuracy significantly to 0.75 percent relying on a coarse range for learning rates, thus improving accuracy and eliminating the need to find a precise learning rate at the expense of computation.