Beyond Logit Adjustment: A Residual Decomposition Framework for Long-Tailed Reranking

Long-tailed classification, where a small number of frequent classes dominate many rare ones, remains challenging because models systematically favor frequent classes at inference time. Existing post-hoc methods such as logit adjustment address this by adding a fixed classwise offset to the base-model logits. However, the correction required to restore the relative ranking of two classes need not be constant across inputs, and a fixed offset cannot adapt to such variation. We study this problem through Bayes-optimal reranking on a base-model top-k shortlist. The gap between the optimal score and the base score, the residual correction, decomposes into a classwise component that is constant within each class, and a pairwise component that depends on the input and competing labels. When the residual is purely classwise, a fixed offset suffices to recover the Bayes-optimal ordering. We further show that when the same label pair induces incompatible ordering constraints across contexts, no fixed offset can achieve this recovery. This decomposition leads to testable predictions regarding when pairwise correction can improve performance and when cannot. We develop REPAIR (Reranking via Pairwise residual correction), a lightweight post-hoc reranker that combines a shrinkage-stabilized classwise term with a linear pairwise term driven by competition features on the shortlist. Experiments on five benchmarks spanning image classification, species recognition, scene recognition, and rare disease diagnosis confirm that the decomposition explains where pairwise correction helps and where classwise correction alone suffices.

On the Prime Number Divisibility by Deep Learning

Certain tasks such as determining whether a given integer can be divided by 2, 3, or other prime numbers may be trivial for human beings, but can be less straightforward for computers in the absence of pre-specified algorithms. In this paper, we tested multiple deep learning architectures and feature engineering approaches, and evaluated the scenario of determining divisibility of large finite integers (up to $2^{32}$) by small prime numbers. It turns out that, regardless of the network frameworks or the complexity of the network structures (CNN, RNN, Transformer, etc.), the ability to predict the prime number divisibility critically depends on the feature space fed into the deep learning models. We also evaluated commercially available Automated Machine Learning (AutoML) pipelines from Amazon, Google and Microsoft, and demonstrated that they failed to address this issue unless appropriately engineered features were provided. We further proposed a closed form solution to the problem using the ordinary linear regression on Fourier series basis vectors, and showed its success. Finally, we evaluated prompt-based learning using ChatGPT and demonstrated its success on small primes and apparent failures on larger primes. We conclude that feature engineering remains an important task to improve the performance, increase the interpretability, and reduce the complexity of machine learning/deep learning models, even in the era of AutoML and large-language models (LLMs).