A Neural Scaling Law from the Dimension of the Data Manifold (2004.10802v1)

Abstract: When data is plentiful, the loss achieved by well-trained neural networks scales as a power-law $L \propto N^{-\alpha}$ in the number of network parameters $N$. This empirical scaling law holds for a wide variety of data modalities, and may persist over many orders of magnitude. The scaling law can be explained if neural models are effectively just performing regression on a data manifold of intrinsic dimension $d$. This simple theory predicts that the scaling exponents $\alpha \approx 4/d$ for cross-entropy and mean-squared error losses. We confirm the theory by independently measuring the intrinsic dimension and the scaling exponents in a teacher/student framework, where we can study a variety of $d$ and $\alpha$ by dialing the properties of random teacher networks. We also test the theory with CNN image classifiers on several datasets and with GPT-type LLMs.

Neural Scaling Law from Data Manifold Dimension

The paper presents an analysis of neural networks, demonstrating that their performance on large datasets can be predicted accurately using a scaling law derived from the intrinsic dimension of the data manifold. The authors postulate that well-trained neural networks follow an empirical power-law of the form $L \propto N^{-\alpha}$, where $L$ is the loss and $N$ is the number of model parameters. This relationship holds true across different data modalities, suggesting a degree of universality in neural network behavior.

The central proposal is a theoretical framework predicting the scaling exponents based on the intrinsic dimension $d$ of the data manifold. Specifically, for loss functions like cross-entropy and mean-squared error, the scaling exponents should approximate $\alpha \approx 4/d$. This prediction is validated via independent measurements in teacher/student experiments and subsequently tested on various real data scenarios, such as CNNs for image classification and GPT-type LLMs.

Key Findings and Analysis

1. Scaling Law Confirmation: Using a teacher/student framework, the intrinsic dimension $d$ and the scaling exponents $\alpha$ are measured by varying the properties of random teacher networks. This approach confirms the theoretical prediction about scaling exponents, thereby reinforcing the relationship between model size scaling and data manifold dimensionality.
2. Neural Models and Data Manifolds: The paper argues that neural models effectively perform regression on data manifolds, with the intrinsic dimension dictating the scaling behavior. A manifold of dimension $d$ requires model parameter scaling that is inversely proportional to $d$, explaining the universality of observed scaling laws across domains.
3. Practical Implications: The paper extends the theory to practical applications, demonstrating that model architecture does not significantly alter scaling exponents, thereby suggesting efficiency improvements focus on reducing intrinsic dimensionality. In real-world scenarios, this insight can guide the design of models to harness scaling laws better.
4. Empirical Validation: The theory's predictions are robustly tested with multiple frameworks, including teacher/student setups with different intrinsic dimensions $d$, CNNs for CIFAR10 datasets where exponent measurements match the predicted values, and LLMs where intrinsic dimensional measurements match known scaling exponents.
5. Data Manifold Complexity: The experiments with product manifolds, which can be decomposed into sub-manifolds modeled independently by the network, reveal a nuanced understanding of data manifold complexity. This adds a layer of strategic depth to model design concerning sub-component manipulation.

Future Speculations and Extensions

Looking forward, the authors briefly discuss avenues for future research. The foundational understanding of scaling laws may be extended to include diverse neural architectures like transformers, which exhibit unique structural characteristics impacting scaling behaviors. Moreover, fine-tuning pre-trained generative models on downstream tasks may represent these models' ability to iteratively zoom in on specific sections of the data manifold, refining accuracy and task performance.

LLMs, such as GPT-type structures, are highlighted, with suggestions that their scaling differences may arise from complex manifold architectures or data manifold intricacies inherent to natural language processing tasks. Further work could deepen the exploration of attention mechanisms within transformers, probing their role in modulating data manifold dimensionality and convergence behavior.

In conclusion, this paper provides a comprehensive and theory-backed discussion on neural scaling laws stemming from the intrinsic data manifold dimension, offering insightful implications and a solid groundwork for continued exploration within AI and machine learning paradigms.