Scaling Laws and Representation Learning in Simple Hierarchical Languages: Transformers vs. Convolutional Architectures

Abstract: How do neural LLMs acquire a language's structure when trained for next-token prediction? We address this question by deriving theoretical scaling laws for neural network performance on synthetic datasets generated by the Random Hierarchy Model (RHM) -- an ensemble of probabilistic context-free grammars designed to capture the hierarchical structure of natural language while remaining analytically tractable. Previously, we developed a theory of representation learning based on data correlations that explains how deep learning models capture the hierarchical structure of the data sequentially, one layer at a time. Here, we extend our theoretical framework to account for architectural differences. In particular, we predict and empirically validate that convolutional networks, whose structure aligns with that of the generative process through locality and weight sharing, enjoy a faster scaling of performance compared to transformer models, which rely on global self-attention mechanisms. This finding clarifies the architectural biases underlying neural scaling laws and highlights how representation learning is shaped by the interaction between model architecture and the statistical properties of data.

Scaling Laws and Representation Learning: Analyzing Hierarchical Language Models

The paper presents a comprehensive study on how neural language models acquire the hierarchical structure of language through scalable architectures, specifically focusing on the performance comparison between transformer and convolutional architectures. Using synthetic datasets generated by the Random Hierarchy Model (RHM), the authors explore architectural biases inherent to scaling laws and representation learning. The RHM model uses probabilistic context-free grammars (PCFGs) to simulate hierarchical language structures, making it analytically tractable and allowing detailed exploration of neural network behaviors in language acquisition.

Overview of Key Concepts

1. Random Hierarchy Model (RHM): The model derives synthetic datasets with attributes reminiscent of hierarchical language structures found in natural languages. The constraints and fixed tree topology in the RHM enable closed-form solutions for data statistics, aiding in clear analysis of parameter scaling.

2. Training Dynamics: The paper delineates how deep networks sequentially acquire language structure, revealing the progressive interaction between model architecture and the statistical properties of data. Theoretical scaling laws are derived and validated, predicting that convolutional networks scale performance faster than transformers. This is attributed to the local connectivity and weight sharing inherent in convolutional architectures.

3. Architectural Differences: The study extends existing theoretical frameworks to include architectural variations. Convolutional networks, due to their structure, are able to access stronger correlations in data, thereby outperforming transformers that rely on global self-attention mechanisms.

Empirical and Theoretical Insights

The learning dynamics of deep models trained on RHM data illuminate the role of architectural priors on scaling laws. The authors leverage the fixed structure in RHM to isolate mechanisms in representation learning, showing that CNNs achieve significantly faster scaling due to their inherent local connectivity principles. Transformers, while versatile in capturing long-range dependencies, demonstrated slower performance improvement tracing to their reliance on non-hierarchical $n$-gram statistics.

Transformer models exhibit a stagewise learning curve, transitioning between approximation stages corresponding to different levels of hierarchical structure comprehension. This behavior contrasts with the faster adaptation seen in CNNs, further emphasizing the influence of inductive biases on learning efficiency.

Implications and Future Directions

This study indicates the importance of architectural alignment with data-generating processes, suggesting tailored convolutional configurations when dealing with hierarchical structures. Such insights can inform decisions in practical applications of language modeling, guiding the selection of model architectures based on predicted data modality structures.

Future advancements could extend these findings to variable tree topologies and context-sensitive data, posing potential challenges that might benefit from transformer flexibility. Moreover, probing real-world data with known hierarchical structures through the lens of developed theoretical scaling laws could offer deeper comprehension and improved model training strategies.

In conclusion, the interplay between architecture and hierarchical statistical phenomena offers a framework for enhancing understanding of neural scaling laws, representation learning, and their practical implications in AI systems. This work suggests continued exploration of architectural biases in broader contexts, seeking optimal resource allocation strategies and better comprehension of language model capabilities.