Fractional Optimizers Meet Fractal Activation Functions: An Empirical Study of Multi-Scale Optimization in Neural Network
Fractional optimizers extend first-order optimization via fractional derivatives and memory effects, while fractal activations introduce multi-scale nonlinear representations using Weierstrass- and Blancmange-type functions The study evaluates multiple fractional optimizer families on Ackley and Himmelblau benchmark surfaces, both standard and with additive Weierstrass-type perturbations Feed-forward neural networks with conventional and fractal activations were tested across ten classification
Analysis
TL;DR
- Fractional optimizers extend first-order optimization via fractional derivatives and memory effects, while fractal activations introduce multi-scale nonlinear representations using Weierstrass- and Blancmange-type functions
- The study evaluates multiple fractional optimizer families on Ackley and Himmelblau benchmark surfaces, both standard and with additive Weierstrass-type perturbations
- Feed-forward neural networks with conventional and fractal activations were tested across ten classification datasets
- Regularization-style fractional scaling pairs well with selected fractal activations in network training
- Adaptive memory-based fractional optimizers outperform plain memory substitution, supporting controlled fractional memory as a promising but selective direction rather than a universal replacement
Why It Matters
This research bridges two independent optimization improvement directions—fractional calculus-based optimizers and fractal activation functions—providing empirical guidance on when their combination is beneficial. For AI practitioners exploring alternatives to standard Adam/SGD training, the findings highlight that fractional optimization is not a drop-in replacement but requires careful pairing with compatible activation schemes.
Technical Details
- Fractional optimizers evaluated include standard methods, regularization-style optimizers, explicit memory-based fractional optimizers, and adaptive memory-based variants, all extending first-order optimization through fractional derivatives
- Fractal activations are based on self-similar Weierstrass-type and Blancmange-type functions, introducing multi-scale nonlinear representations into neural network layers
- Benchmark evaluation used Ackley and Himmelblau optimization surfaces in both standard form and with additive Weierstrass-type perturbations to test robustness under fractal noise
- Neural network experiments involved feed-forward architectures trained on ten classification datasets, comparing conventional activations against fractal activations across multiple optimizer families
- Key finding: Grünwald-Letnikov memory effects proved most relevant on perturbed surfaces, while regularization-style fractional scaling performed best with selected fractal activations in end-to-end network training
Industry Insight
- Researchers should treat fractional optimization as a specialized tool rather than a universal optimizer upgrade; pairing choices between fractional memory type and activation function significantly impact performance
- Adaptive memory mechanisms in fractional optimizers show measurable improvement over static memory substitution, suggesting future work should focus on dynamic memory control rather than fixed fractional orders
- The selective nature of these pairings implies that hybrid approaches—combining fractional optimization with fractal activations only in specific architectural contexts—may yield better returns than blanket adoption across all model types
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