Golden Ratio Meets Deep Learning: A New Nature-Inspired Method for Neural Network Initialization and Regularization

This article introduces an open-source project named golden-ratio-deep-learning, which incorporates the golden ratio (Φ≈1.618) into the weight initialization and regularization processes of deep neural networks. Through comparative experiments with the classic Xavier method, it explores the impact of natural mathematical constants on the training stability, convergence speed, and learning efficiency of models. The project was created by developer Wesley Melo, based on the PyTorch framework, aiming to bridge the fields of natural mathematics and artificial intelligence.

In deep learning, the weight initialization method has a profound impact on the training process. Improper initialization may lead to gradient vanishing or explosion problems. To address these issues, classic methods such as Xavier initialization (Glorot initialization) and He initialization have been proposed. Xavier initialization adjusts the weight variance based on the number of input and output neurons to maintain a reasonable amplitude of signals during propagation.

This project proposes to use the golden ratio Φ to modify the scaling factor of traditional initialization. In Xavier initialization, the weight variance is 2/(n_in + n_out), while golden ratio initialization introduces Φ as an additional scaling coefficient, making the initial distribution more naturally "harmonious". The intuition is based on the self-similar property of Φ (Φ² = Φ + 1), which may help maintain the consistency of signal amplitude and reduce gradient fluctuations.

The project also explores the application of the golden ratio in regularization. Traditional L2 regularization limits model complexity through a penalty on the sum of squared weights, while golden ratio regularization combines the penalty term with multiples of Φ, controlling complexity while maintaining dynamic balance in training. Its theoretical basis lies in the irrational nature of Φ, which may break the symmetry of weight updates and explore a richer parameter space.

The experiments are based on the PyTorch framework and conducted on standard benchmark datasets. The comparison dimensions include: 1. Training stability (monitoring whether the loss curve decreases smoothly); 2. Convergence speed (number of epochs required to reach a specific accuracy); 3. Learning efficiency (comprehensive evaluation of validation set accuracy and computational resource consumption).

The project code is modular, including a custom initialization module (encapsulating golden ratio calculations as PyTorch initializers) and a regularization module (custom loss function components). The experiment scripts use a configuration-driven approach; researchers can switch strategies by modifying configurations without changing core code, making it easy to integrate into other PyTorch projects.

Introducing natural mathematical laws into ML is not unprecedented; examples include using the Fibonacci sequence for learning rate scheduling and fractal geometry for network architectures. However, it should be noted that constants common in nature do not necessarily dominate in optimization algorithms, and the effect of the golden ratio may depend on network architecture, datasets, and hyperparameters. This project provides a systematic experimental framework to evaluate the hypothesis.

This project provides a novel perspective for the deep learning community, connecting natural mathematics and AI. Regardless of whether the experimental results are better than traditional methods, the spirit of interdisciplinary exploration is worthy of recognition. It reminds researchers that while pursuing large models and data, they can seek inspiration from mathematical foundations. Interested individuals can obtain the code from the GitHub repository to reproduce the experiments.