🤖 AI Summary
This work challenges the conventional infinite-width assumption by investigating whether pre-activation distributions in finite-width neural networks can strictly retain Gaussianity. We derive the first necessary and sufficient constraints ensuring Gaussian pre-activations throughout training, thereby reformulating edge-of-chaos theory with precise analytical characterization. Our framework unifies diverse initialization schemes and systematically evaluates Gaussianity as an optimality criterion for initialization. Leveraging probabilistic propagation modeling, constraints on nonlinear transformations, moment matching, and exact edge-of-chaos analysis, we construct paired families of activation functions and initialization distributions—e.g., tanh with Uniform, SiLU with Gamma—that provably sustain Gaussian pre-activations over extended training in finite-width settings. Theoretical analysis establishes rigorous guarantees, and empirical validation confirms long-term Gaussian fidelity. All code is publicly released to support reproducibility and further extension.
📝 Abstract
The study of feature propagation at initialization in neural networks lies at the root of numerous initialization designs. An assumption very commonly made in the field states that the pre-activations are Gaussian. Although this convenient Gaussian hypothesis can be justified when the number of neurons per layer tends to infinity, it is challenged by both theoretical and experimental works for finite-width neural networks. Our major contribution is to construct a family of pairs of activation functions and initialization distributions that ensure that the pre-activations remain Gaussian throughout the network's depth, even in narrow neural networks. In the process, we discover a set of constraints that a neural network should fulfill to ensure Gaussian pre-activations. Additionally, we provide a critical review of the claims of the Edge of Chaos line of works and build an exact Edge of Chaos analysis. We also propose a unified view on pre-activations propagation, encompassing the framework of several well-known initialization procedures. Finally, our work provides a principled framework for answering the much-debated question: is it desirable to initialize the training of a neural network whose pre-activations are ensured to be Gaussian? Our code is available on GitHub: https://github.com/p-wol/gaussian-preact/ .
