用分数阶积分改进优化器，让模型更好识别少数类数据。
Beyond the Markovian Assumption: Robust Optimization via Fractional Weyl Integrals in Imbalanced Data
- 用加权分数阶Weyl积分替代传统梯度，引入历史信息记忆
- 在金融反欺诈任务中PR-AUC提升约40%，医疗诊断也更抗过拟合
- 适合处理极端不均衡数据，如欺诈检测、罕见病诊断
标准梯度下降及其现代变体假设权重更新具有局部马尔可夫性，对噪声和过拟合敏感。这一缺陷在金融欺诈检测等极端不均衡数据集中尤为严重，主导类的梯度会系统性掩盖少数类的微弱信号。本文提出一种基于分数阶微积分的新优化算法，通过分离广义分数阶导数的核心记忆机制——加权分数阶Weyl积分，将瞬时梯度替换为动态加权的历史序列。该分数阶记忆算子天然具备正则化作用。实验表明，该方法能有效防止过拟合，在医疗诊断任务中表现更稳健，并在金融欺诈检测中使PR-AUC相比经典优化器提升约40%，建立起纯分数拓扑与应用机器学习之间的稳健桥梁。
原文摘要 · Abstract (English)
Standard Gradient Descent and its modern variants assume local, Markovian weight updates, making them highly susceptible to noise and overfitting. This limitation becomes critically severe in extremely imbalanced datasets such as financial fraud detection where dominant class gradients systematically overwrite the subtle signals of the minority class. In this paper, we introduce a novel optimization algorithm grounded in Fractional Calculus. By isolating the core memory engine of the generalized fractional derivative, the Weighted Fractional Weyl Integral, we replace the instantaneous gradient with a dynamically weighted historical sequence. This fractional memory operator acts as a natural regularizer. Empirical evaluations demonstrate that our method prevents overfitting in medical diagnostics and achieves an approximately 40 percent improvement in PR-AUC over classical optimizers in financial fraud detection, establishing a robust bridge between pure fractional topology and applied Machine Learning.
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