Fractional Optimizers Meet Fractal Activation Functions: An Empirical Study of Multi-Scale Optimization in Neural Network
arXiv:2608.14636v1 Announce Type: new Abstract: Fractional optimization methods and fractal activation functions are two independent directions for improving neural network training. Fractional optimizers extend first-order optimization through fractional derivatives and memory effects, whereas fractal activations introduce multi-scale nonlinear representations based on self-similar Weierstrass- and Blancmange-type functions. Here, we investigate their interaction within a unified experimental framework. We evaluate fractional optimizer families on Ackley and Himmelblau benchmark surfaces, in standard form and with additive Weierstrass-type perturbations, and then in feed-forward neural networks with conventional and fractal activations on ten classification datasets. The comparison includes standard methods, regularization-style optimizers, explicit and adaptive memory-based fractional optimizers, and other representative literature methods. Overall, fractional optimization and fractal activations show useful but selective pairings. Regularization-style fractional scaling performs well with selected fractal activations in network training, while Gr\"unwald--Letnikov memory is most relevant on perturbed surfaces. Adaptive memory improves plain memory substitution in several cases, supporting controlled fractional memory as a promising direction rather than a universal replacement.
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[Submitted on 28 Jul 2026]
Title:Fractional Optimizers Meet Fractal Activation Functions: An Empirical Study of Multi-Scale Optimization in Neural Network
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Abstract:Fractional optimization methods and fractal activation functions are two independent directions for improving neural network training. Fractional optimizers extend first-order optimization through fractional derivatives and memory effects, whereas fractal activations introduce multi-scale nonlinear representations based on self-similar Weierstrass- and Blancmange-type functions. Here, we investigate their interaction within a unified experimental framework. We evaluate fractional optimizer families on Ackley and Himmelblau benchmark surfaces, in standard form and with additive Weierstrass-type perturbations, and then in feed-forward neural networks with conventional and fractal activations on ten classification datasets. The comparison includes standard methods, regularization-style optimizers, explicit and adaptive memory-based fractional optimizers, and other representative literature methods. Overall, fractional optimization and fractal activations show useful but selective pairings. Regularization-style fractional scaling performs well with selected fractal activations in network training, while Grünwald--Letnikov memory is most relevant on perturbed surfaces. Adaptive memory improves plain memory substitution in several cases, supporting controlled fractional memory as a promising direction rather than a universal replacement.
Comments: Quite extensive paper, more than 100 pages
Subjects:
Machine Learning (cs.LG); Artificial Intelligence (cs.AI)
Cite as: arXiv:2608.14636 [cs.LG]
(or arXiv:2608.14636v1 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2608.14636
arXiv-issued DOI via DataCite (pending registration)
Submission history
From: Sebastian Raubitzek Dr. techn. MSc. BSc. [view email] [v1] Tue, 28 Jul 2026 08:31:24 UTC (24,213 KB)
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