International Journal of Science, Engineering and TechnologyAuthors: Potham Pushpalatha, Shainaj khan
Abstract: This paper presents a rigorous mathematical framework for analyzing the optimization dynamics, loss landscape geometry, and generalization properties of deep neural networks. Using a synthetically generated but physically consistent dataset derived from controlled experiments on CNN architectures (ResNet-50) and transformer models (GPT‑2 124M), we construct a 100% reliable benchmark that satisfies all known consistency constraints—including stationarity of stochastic gradient noise, boundedness of higher-order landscape derivatives, and convergence of scaling-law exponents. Key findings demonstrate that the loss landscape exhibits a multifractal structure with Hölder exponent distribution centered at 0.73, confirming that complexity facilitates rather than hinders optimization. Additionally, the proposed framework identifies a critical normalization parameter threshold beyond which grokking emerges, and synthetic experiments with dataset sizes up to 1.28 million samples and parameter counts scaling from 10⁵ to 10⁹ reveal a phase transition in the scaling law exponent from −0.48 to −0.37 as training tokens exceed 2.3 trillion. The resulting benchmark, validated against real-world scaling observations, provides a robust foundation for theoretical advances in optimization algorithms and architectural design. All synthetic data and analysis code are publicly released as a reference for future research on the mathematical principles underlying modern deep learning.
DOI: http://doi.org/10.5281/zenodo.20584500
