Tempered fractional gradient descent: Theory, algorithms, and robust learning applications

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|=(αj) 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 [Formula: see text] , while stochastic variants attain O(1/k^α) 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 demonstrates TFGD's superiority across diverse benchmarks: 98.25 % accuracy on Breast Cancer Wisconsin (vs. 92.11 % SGD), 99.1 % on MNIST (vs. 99.0 % Adam), and 95.83 % on noisy digits (vs. 75.83 % SGD). TFGD converges 2× faster than SGD in medical classification tasks and exhibits smoother optimization trajectories in non-convex settings (MNIST autoencoder). The tempered memory mechanism proves particularly effective where feature correlations benefit from stable gradient averaging, while hyperparameter analysis reveals optimal operating regimes (α=0.6-0.7, λ=0.3-0.5) for noisy data. These results position TFGD as a robust alternative to conventional optimizers in both theoretical and applied machine learning.