"Understanding Scaling Laws with Statistical and Approximation Theory for Transformer Neural Networks on Intrinsically Low-dimensional Data"

Mathematical proof shows why bigger transformers get smarter: it's all about data's hidden dimensions

Hidden geometry controlling transformer model performance

https://arxiv.org/abs/2411.06646

Original Problem 🤔:

Understanding why transformer models follow power-law scaling behavior lacks rigorous mathematical explanation, especially when input data lies on low-dimensional manifolds.

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Solution in this Paper 🔧:

→ Developed a mathematical framework showing transformers can universally approximate Hölder continuous functions on d-dimensional manifolds using O(log(d)) depth

→ Established statistical estimation theory proving generalization error bounds exponentially dependent on intrinsic data dimension d

→ Created novel computation method for covering number of transformer network classes

→ Validated theoretical predictions through experiments on three datasets: OpenWebText, SQL portion of The Stack, and Tiny Stories

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Key Insights 🎯:

→ Intrinsic data dimension d is crucial for determining model performance and scaling behavior

→ Transformers need only O(log(d)) layers for function approximation, independent of accuracy

→ Scaling laws accurately reflect dataset complexity - simpler datasets show faster convergence

→ The estimated intrinsic dimension d remains stable across model architectures and sizes

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Results 📊:

→ Close agreement (±0.02) between predicted and observed data scaling exponents

→ Achieved O(log(d)) depth efficiency, outperforming traditional ReLU networks

→ Successfully validated on three datasets with varying complexities, showing faster convergence for simpler datasets