Emergent Mind

The boundary of neural network trainability is fractal

Published Feb 9, 2024 in cs.LG , cs.NE , nlin.CD and


Some fractals -- for instance those associated with the Mandelbrot and quadratic Julia sets -- are computed by iterating a function, and identifying the boundary between hyperparameters for which the resulting series diverges or remains bounded. Neural network training similarly involves iterating an update function (e.g. repeated steps of gradient descent), can result in convergent or divergent behavior, and can be extremely sensitive to small changes in hyperparameters. Motivated by these similarities, we experimentally examine the boundary between neural network hyperparameters that lead to stable and divergent training. We find that this boundary is fractal over more than ten decades of scale in all tested configurations.


  • The paper investigates the boundary between stable and unstable neural network training regimes, discovering it exhibits fractal characteristics.

  • Experiments with a single-hidden-layer neural network using different nonlinear functions and training conditions reveal the fractal nature of trainability boundaries.

  • Fractal dimensions were calculated to quantify fractal complexity, offering insights into the effects of hyperparameter adjustments on training stability.

  • The findings highlight the theoretical and practical challenges in neural network training and suggest future research directions for exploring fractal properties in machine learning.


In a pioneering exploration of the neural network hyperparameter space, researchers have drawn a fascinating parallel between the iterative function-based fractal generation observed in mathematical constructs like the Mandelbrot and Julia sets and the process of neural network training. By delving into the iterative nature of gradient descent methods used in machine learning, the study has revealed that the boundary delineating stable from unstable training regimes for neural networks exhibits fractal characteristics across a wide range of hyperparameter configurations.

Empirical Investigations


The crux of the study lies in the meticulous training of a single-hidden-layer neural network, leveraging a spectrum of nonlinear functions (tanh and ReLU) and experiment conditions (e.g., full batch versus minibatch training), to scrutinize the outcomes of neural network training across varied hyperparameter settings. Notably, the research methodologically examines the impacts of learning rate adjustments and initialization parameters on training stability, painting a comprehensive picture of how slight hyperparameter modifications can sway the training trajectory between convergence and divergence.


A series of controlled experiments, as documented in the paper, yielded a set of compelling observations:

  • The fractal nature of the trainability boundary was consistent across different experimental setups, including variances in network nonlinearities and training batch sizes.

  • Fractal dimensions, a quantitative measure of fractal complexity, were meticulously calculated for each condition, revealing variances that may reflect the underlying geometry of network parameter spaces influenced by specific training or initialization approaches.

  • Visually compelling, high-resolution fractal images along with accompanying animations were generated, providing intuitive insights into the intricate dependence of neural network trainability on hyperparameter selections.



The discovery of fractal boundaries in neural network hyperparameter spaces has several profound implications:

  • Theoretical Insights: The resemblance between fractal generation in mathematical systems and neural network training provides a new lens through which the sensitivity and dynamism of the training process can be understood, particularly highlighting the delicate balance required in hyperparameter tuning.

  • Practical Challenges: For practitioners, the fractal nature of trainability boundaries underscores the inherent unpredictability and potential difficulties in identifying optimal training regimes, especially given the high-dimensional hyperparameter spaces of contemporary neural networks.

  • Meta-learning Paradigms: This research also casts light on the challenges facing meta-learning algorithms, which optimize across these fractal landscapes; understanding the fractal structures may guide the development of more robust meta-learning strategies.

Future Directions

The paper opens myriad avenues for future exploration. One such avenue is the investigation of how the properties of training functions influence fractal geometry, potentially offering novel insights into designing more tractable or efficient training algorithms. Additionally, extending fractal analysis to a broader array of hyperparameters—including those governing data augmentation, regularization, and network architecture—promises to deepen our understanding of neural network training dynamics.


By empirically establishing that the boundary delineating stable and unstable neural network training domains is fractal, this study enriches our comprehension of the intricate dynamics at play in machine learning. Beyond its immediate academic intrigue, this work has practical implications for the field, especially in the realms of hyperparameter optimization and meta-learning, and sets the stage for further exploration into the enigmatic yet captivating fractal landscapes of neural network trainability.

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The boundary of neural network trainability is fractal (1 point, 1 comment) in /r/hackernews
The boundary of neural network trainability is fractal (1 point, 1 comment) in /r/neuralnetworks