Effective Dimension Aware Fractional-Order Stochastic Gradient Descent for Convex Optimization Problems

Fractional-order stochastic gradient descent (FOSGD) leverages a fractional exponent to capture long-memory effects in optimization, yet its practical impact is often constrained by the difficulty of tuning and stabilizing this exponent. In this work, we introduce 2SED Fractional-Order Stochastic Gradient Descent (2SEDFOSGD), a novel method that synergistically combines the Two-Scale Effective Dimension (2SED) algorithm with FOSGD to automatically calibrate the fractional exponent in a data-driven manner. By continuously gauging model sensitivity and effective dimensionality, 2SED dynamically adjusts the exponent to curb erratic oscillations and enhance convergence rates. Theoretically, we demonstrate how this dimension-aware adaptation retains the benefits of fractional memory while averting the sluggish or unstable behaviors frequently observed in naive fractional SGD. Empirical evaluations across multiple benchmarks confirm that our 2SED-driven fractional exponent approach not only converges faster but also achieves more robust final performance, suggesting broad applicability for fractional-order methodologies in large-scale machine learning and related domains.