Sparse Semantic Dimension as a Generalization Certificate for LLMs

TL;DR

This paper introduces the Sparse Semantic Dimension (SSD), a new complexity measure based on sparse internal representations of LLMs, providing insights into their generalization, compressibility, and safety monitoring.

Contribution

The paper formalizes SSD as a measure of low-dimensional, sparse semantic features in LLMs, linking representation geometry to generalization and safety.

Findings

SSD provides non-vacuous generalization certificates at realistic sample sizes.

Larger models like Gemma-2B learn more compressible, distinct semantic structures.

Out-of-distribution inputs cause a spike in active features, signaling uncertainty.

Abstract

Standard statistical learning theory predicts that Large Language Models (LLMs) should overfit because their parameter counts vastly exceed the number of training tokens. Yet, in practice, they generalize robustly. We propose that the effective capacity controlling generalization lies in the geometry of the model's internal representations: while the parameter space is high-dimensional, the activation states lie on a low-dimensional, sparse manifold. To formalize this, we introduce the Sparse Semantic Dimension (SSD), a complexity measure derived from the active feature vocabulary of a Sparse Autoencoder (SAE) trained on the model's layers. Treating the LLM and SAE as frozen oracles, we utilize this framework to attribute the model's generalization capabilities to the sparsity of the dictionary rather than the total parameter count. Empirically, we validate this framework on GPT-2 Small and Gemma-2B, demonstrating that our bound provides non-vacuous certificates at realistic sample sizes. Crucially, we uncover a counter-intuitive "feature sharpness" scaling law: despite being an order of magnitude larger, Gemma-2B requires significantly fewer calibration samples to identify its active manifold compared to GPT-2, suggesting that larger models learn more compressible, distinct semantic structures. Finally, we show that this framework functions as a reliable safety monitor: out-of-distribution inputs trigger a measurable "feature explosion" (a sharp spike in active features), effectively signaling epistemic uncertainty through learned feature violation. Code is available at: https://github.com/newcodevelop/sparse-semantic-dimension.