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

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 |wj| = binomαj and exponentially decayed via a tempering parameter λ. Theoretical analysis establishes TFGD's convergence guarantees: in convex settings, it achieves an O(1/K) rate with alignment coefficient dα,λ = (1 - e-λ)-α, while stochastic variants attain O(1/kα) error decay. The algorithm maintains O(n) time complexity equivalent to SGD, with memory overhead scaling as O(d/λ) 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× 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.