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.

Stochastic gradient descent and other first-order variants, such as Adam and AdaGrad, are commonly used in the field of deep learning due to their computational efficiency and low-storage memory requirements. However, these methods do not exploit curvature information. Consequently, iterates can converge to saddle points or poor local minima. On the other hand, Quasi-Newton methods compute Hessian approximations which exploit this information with a comparable computational budget. Quasi-Newton methods re-use previously computed iterates and gradients to compute a low-rank structured update. The most widely used quasi-Newton update is the L-BFGS, which guarantees a positive semi-definite Hessian approximation, making it suitable in a line search setting. However, the loss functions in DNNs are non-convex, where the Hessian is potentially non-positive definite. In this paper, we propose using a limited-memory symmetric rank-one quasi-Newton approach which allows for indefinite Hessian approximations, enabling directions of negative curvature to be exploited. Furthermore, we use a modified adaptive regularized cubics approach, which generates a sequence of cubic subproblems that have closed-form solutions with suitable regularization choices. We investigate the performance of our proposed method on autoencoders and feed-forward neural network models and compare our approach to state-of-the-art first-order adaptive stochastic methods as well as other quasi-Newton methods.x