Long-tailed problems in healthcare emerge from data imbalance due to variability in the prevalence and representation of different medical conditions, warranting the requirement of precise and dependable classification methods. Traditional loss functions such as cross-entropy and binary cross-entropy are often inadequate due to their inability to address the imbalances between the classes with high representation and the classes with low representation found in medical image datasets. We introduce a novel polynomial loss function based on Pade approximation, designed specifically to overcome the challenges associated with long-tailed classification. This approach incorporates asymmetric sampling techniques to better classify under-represented classes. We conducted extensive evaluations on three publicly available medical datasets and a proprietary medical dataset. Our implementation of the proposed loss function is open-sourced in the public repository:https://github.com/ipankhi/ALPA.

Federated learning faces a critical challenge in balancing communication efficiency with rapid convergence, especially for second-order methods. While Newton-type algorithms achieve linear convergence in communication rounds, transmitting full Hessian matrices is often impractical due to quadratic complexity. We introduce Federated Learning with Enhanced Nesterov-Newton Sketch (FLeNS), a novel method that harnesses both the acceleration capabilities of Nesterov's method and the dimensionality reduction benefits of Hessian sketching. FLeNS approximates the centralized Newton's method without relying on the exact Hessian, significantly reducing communication overhead. By combining Nesterov's acceleration with adaptive Hessian sketching, FLeNS preserves crucial second-order information while preserving the rapid convergence characteristics. Our theoretical analysis, grounded in statistical learning, demonstrates that FLeNS achieves super-linear convergence rates in communication rounds - a notable advancement in federated optimization. We provide rigorous convergence guarantees and characterize tradeoffs between acceleration, sketch size, and convergence speed. Extensive empirical evaluation validates our theoretical findings, showcasing FLeNS's state-of-the-art performance with reduced communication requirements, particularly in privacy-sensitive and edge-computing scenarios. The code is available at https://github.com/sunnyinAI/FLeNS