Fractal Trainability in Neural Networks

• Fractal trainability is characterized by self-similar, complex boundaries between convergent and divergent regimes in hyperparameter and parameter space.
• Empirical studies reveal that fractal dimensions, often near 1.98 in certain architectures, quantify the intricate, scale-invariant structure of training landscapes.
• The fractal geometry of optimization challenges reproducibility and hyperparameter tuning, necessitating adaptive methods to manage chaotic boundary effects.

Neural network trainability exhibits fractal properties in parameter and hyperparameter space, with boundaries between convergent and divergent regimes displaying self-similar structure over many decades of scale. This fractal nature arises from the high-dimensional, iterated dynamics of gradient-based optimization, inherent non-convexity of loss landscapes, finite-size effects, and the presence of riddled basins of attraction. Fractal geometry thus sets fundamental limits on predictability, tuning, and reproducibility in deep learning.

1. Dynamical Systems Perspective: Fractals in Training Iteration

Neural network training can be rigorously modeled as the iteration of a high-dimensional discrete map, analogous to how fractal sets such as the Mandelbrot set are generated by recursively iterating low-dimensional maps. In full-batch gradient descent on parameters $W \in \mathbb{R}^d$, the update step is

$W_{t+1} = W_t - \eta \nabla_W \ell(W_t),$

with $\eta$ a hyperparameter analogous to the complex constant in fractal literature. The critical question—“for which $\eta$ does $\{W_t\}$ converge or diverge?”—is structurally identical to “for which $c$ does $z_{t+1} = z_t^2 + c$ remain bounded?”. The bifurcation boundary delineating trainable and untrainable regimes in learning-rate space is a high-dimensional fractal, exhibiting intricate, self-similar structure across scales (Sohl-dickstein, 2024).

2. Experimental Confirmation: Measurement and Properties of Fractal Trainability Boundaries

Systematic grid sweeps over pairs of hyperparameters such as learning rates for different network layers ($\eta_0, \eta_1$) or between learning rate and initialization scale, reveal sharp, convoluted boundaries that separate successful from failed training. These boundaries were resolved experimentally at up to $4096 \times 4096$ resolution, with up to 50 dyadic zooms, confirming persistent fractality down to floating-point discretization (Sohl-dickstein, 2024). More complex networks, including transformers, display a two-dimensional trainability landscape whose boundary has a box-counting dimension $D \approx 1.97$, exhibiting statistically indistinguishable distributions of convergence metrics across magnifications, and a core region of robust convergence enveloped by a fractal, chaotic border (Torkamandi, 8 Jan 2025).

The table below summarizes measured fractal dimensions across several architectures and conditions.

Condition	Fractal dimension $W_{t+1} = W_t - \eta \nabla_W \ell(W_t),$0
Deep linear, full batch	1.17
ReLU, full batch	1.20
tanh, dataset size = 1	1.41
tanh, minibatch size 16	1.55
tanh, full batch	1.66
Param-init vs LR sweep	1.98
Decoder-only transformer (zoomed)	1.98

In each case, the border between convergence and divergence exhibits new “pockets” and “peninsulas” at every scale, with quantitative correlation between box-counting dimension and the complexity of the boundary (Sohl-dickstein, 2024, Torkamandi, 8 Jan 2025).

3. Mathematical Mechanisms: Origin of Fractality in Optimization Landscapes

Fractal trainability boundaries can be induced by minimal non-convexity. For example, gradient descent on a quadratic loss perturbed by an additive or multiplicative cosine produces scalar “roughness” parameters ($W_{t+1} = W_t - \eta \nabla_W \ell(W_t),$1 or $W_{t+1} = W_t - \eta \nabla_W \ell(W_t),$2), controlling the onset of fractality (Liu, 2024). There is a critical roughness threshold ($W_{t+1} = W_t - \eta \nabla_W \ell(W_t),$3) at which the loss transitions from convex to non-convex. Below this, the trainability boundary is smooth (dimension zero); above, it becomes fractal, with box-counting dimension increasing monotonically with roughness towards one (additive) or ranging $W_{t+1} = W_t - \eta \nabla_W \ell(W_t),$4–$W_{t+1} = W_t - \eta \nabla_W \ell(W_t),$5 (multiplicative). High-dimensional cases preserve this behavior and show weak dependence of fractal dimension on ambient dimension.

The presence of riddled basins of attraction introduces “fat fractals” in parameter space, such that every neighborhood of an initialization leading to one final solution contains infinitely many initializations leading to different solutions. The fractal boundary’s Hausdorff (or box-counting) dimension approaches the ambient space dimension, and the uncertainty exponent $W_{t+1} = W_t - \eta \nabla_W \ell(W_t),$6 controlling the scaling $W_{t+1} = W_t - \eta \nabla_W \ell(W_t),$7 of outcome unpredictability satisfies $W_{t+1} = W_t - \eta \nabla_W \ell(W_t),$8, so even vast increases in initialization precision yield negligible gains in final state predictability (Ly et al., 7 Oct 2025).

4. Impact of Finite-Size Effects and Network Architectures

Mean-field theory predicts smooth, sharp boundaries between ordered and chaotic regimes in the infinite-width ($W_{t+1} = W_t - \eta \nabla_W \ell(W_t),$9) limit (D'Inverno et al., 5 Aug 2025). For finite width and depth, stochastic fluctuations ($\eta$0) in input-output correlations and random drift in information propagation destroy this idealized transition, causing fractalization of the critical “frontier” between input-separability and chaos. Numerical box-counting yields fractal dimensions in the range $\eta$1–$\eta$2 for the edge-of-chaos boundary in finite-width multilayer perceptrons and convolutional architectures. Fourier-based structured transforms preserve this fractal structure in initialization space (D'Inverno et al., 5 Aug 2025).

Empirically, such fractal frontiers are robust to architectural details: convolutional networks, randomly-connected layers, and structured Fourier-diagonalizable layers all present similar fractal dimensions near order-chaos transitions.

5. Quantitative Diagnostics: Box-Counting, State Space Complexity, and Generalization

A standard metric for fractal complexity is the box-counting dimension $\eta$3, where $\eta$4 is the number of $\eta$5-sized boxes covering the relevant boundary or attractor. In echo-state networks, the mapping from input sequence to hidden state yields a self-similar cloud with a box-counting dimension $\eta$6 (where $\eta$7 is reservoir width), sharply increasing trainability and separability when the dimension approaches $\eta$8 (Mayer et al., 2022).

For stochastic optimizers such as SGD, the stationary distribution of parameters under constant step size is supported on a fractal attractor, whose Hausdorff dimension controls generalization error bounds. Specifically, generalization error scales as

$\eta$9

where $\eta$0 is the fractal dimension of the invariant measure, $\eta$1 and $\eta$2 are model-dependent constants, and $\eta$3 the sample size (Camuto et al., 2021). The fractal dimension itself depends on hyperparameters such as stepsize, batch size, and problem Hessian, decreasing as contraction strengthens.

6. Practical Implications: Hyperparameter Sensitivity, Meta-Learning, and Robust Training

The fractal structure of trainability boundaries has direct implications for hyperparameter search and algorithmic robustness. Multiscale sensitivity implies that tiny changes in step size or learning rate can flip a model from stable to divergent, and that minute “islands” of trainability exist arbitrarily deep within divergent regions. As a result, grid search, Bayesian optimization, or meta-learning methods may struggle or yield chaotic, high-variance results (Sohl-dickstein, 2024).

Mitigation approaches include:

• Constraining training far from the fractal edge, using adaptive optimizers, large batch sizes, or strong regularization to smooth effective dynamics (Sohl-dickstein, 2024, Liu, 2024).
• Layerwise, scale-aware initialization—maximizing the fractal dimension of hidden state clouds for recurrent networks enhances separability and expressiveness (Mayer et al., 2022).
• Iterative, zoom-in search in hyperparameter space is necessary to localize robust optima within fractal boundaries (Torkamandi, 8 Jan 2025).

For reproducibility, the riddled, fat-fractal geometry fundamentally limits predictability: even ideal hardware and fixed pseudo-random seeds cannot guarantee convergence to a specific global or local minimum, as neighborhoods of parameter space contain basins for all possible solutions (Ly et al., 7 Oct 2025).

7. Theoretical Unification and Future Directions

The convergence of fractal geometry, dynamical systems, and deep learning optimization reveals a unifying principle: the iterative, non-linear, high-dimensional nature of neural network training generically produces complex, fractal structures in the landscapes organizing trainability, generalization, and information propagation. This connection extends classical dynamical-systems fractals into native high-dimensional spaces, setting intrinsic barriers on the efficacy of precision, grid search, and theoretical predictability in training large models (Sohl-dickstein, 2024, Ly et al., 7 Oct 2025).

Further research is warranted into:

• The geometry of fractal boundaries in higher-dimensional hyperparameter spaces.
• Approaches to quantify, estimate, and control fractal dimension in large-scale networks.
• Integration of fractal diagnostics into training and architecture selection pipelines.
• The impact of non-convex roughness and the proliferation of riddled basins on safe deployment and AI alignment (Ly et al., 7 Oct 2025, Liu, 2024).

The fractal nature of neural network trainability thus provides a quantitative and conceptual framework for understanding sensitivity, unpredictability, and the limits of classical optimization heuristics in modern learning systems.