Why Neural Network Training Boundaries Are Fractals: The Math of Gradient Descent
S_Conradi · x · 2026-07-24
This post explores an interesting phenomenon in neural network hyperparameter tuning: the boundary between learning rates that converge and those that diverge is literally a fractal.
The author points out that gradient descent is mathematically an iterated map, $\theta \to \theta - \eta\nabla L(\theta)$, exactly like the complex function $z \to z^2 + c$ that draws the Mandelbrot set. When you iterate a nonlinear function and ask if its values remain bounded, you naturally generate a fractal structure. This explains why exploring the parameter space for optimal learning rates feels as complex and self-similar as navigating a coastline.
Related event: Research Shows Neural Network Learning Rate Boundaries are Fractal(4 posts)→
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