BAPE: Learning an Explicit Bayes Classifier for Long-tailed Visual Recognition

Bayesian decision theory advocates the Bayes classifier as the optimal approach for minimizing the risk in machine learning problems. Current deep learning algorithms usually solve for the optimal classifier by implicitly estimating the posterior probabilities, e.g., by minimizing the Softmax cross-entropy loss. This simple methodology has been proven effective for meticulously balanced academic benchmark datasets. However, it is not applicable to the long-tailed data distributions in the real world, where it leads to the gradient imbalance issue and fails to ensure the Bayes optimal decision rule. To address these challenges, this paper presents a novel approach (BAPE) that provides a more precise theoretical estimation of the data distributions by explicitly modeling the parameters of the posterior probabilities and solving them with point estimation. Consequently, our method directly learns the Bayes classifier without gradient descent based on Bayes' theorem, simultaneously alleviating the gradient imbalance and ensuring the Bayes optimal decision rule. Furthermore, we propose a straightforward yet effective distribution adjustment technique. This method enables the Bayes classifier trained from the long-tailed training set to effectively adapt to the test data distribution with an arbitrary imbalance factor, thereby enhancing performance without incurring additional computational costs. In addition, we demonstrate the gains of our method are orthogonal to existing learning approaches for long-tailed scenarios, as they are mostly designed under the principle of implicitly estimating the posterior probabilities. Extensive empirical evaluations on CIFAR-10-LT, CIFAR-100-LT, ImageNet-LT, and iNaturalist demonstrate that our method significantly improves the generalization performance of popular deep networks, despite its simplicity.