Theoretical Framework for Tempered Fractional Gradient Descent: Application to Breast Cancer Classification

Abstract

This paper introduces Tempered Fractional Gradient Descent (TFGD), a novel optimization framework that synergizes fractional calculus with exponential tempering to enhance gradient-based learning. Traditional gradient descent methods often suffer from oscillatory updates and slow convergence in high-dimensional, noisy landscapes. TFGD addresses these limitations by incorporating a tempered memory mechanism, where historical gradients are weighted by fractional coefficients $|w_j| = \binom{\alpha}{j}$ and exponentially decayed via a tempering parameter $\lambda$. Theoretical analysis establishes TFGD's convergence guarantees: in convex settings, it achieves an $\mathcal{O}(1/K)$ rate with alignment coefficient $d_{\alpha,\lambda} = (1 - e^{-\lambda})^{-\alpha}$, while stochastic variants attain $\mathcal{O}(1/k^\alpha)$ error decay. The algorithm maintains $\mathcal{O}(n)$ time complexity equivalent to SGD, with memory overhead scaling as $\mathcal{O}(d/\lambda)$ for parameter dimension $d$. Empirical validation on the Breast Cancer Wisconsin dataset demonstrates TFGD's superiority, achieving 98.25\% test accuracy (vs. 92.11\% for SGD) and 2$\times$ faster convergence. The tempered memory mechanism proves particularly effective in medical classification tasks, where feature correlations benefit from stable gradient averaging. These results position TFGD as a robust alternative to conventional optimizers in both theoretical and applied machine learning.