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Rethinking LLM Training through Information Geometry and Quantum Metrics

Riccardo Di Sipio

TL;DR

The paper proposes viewing LLM optimization through information geometry and quantum geometry, using the Fisher information and Fubini-Study/QFI to reinterpret curvature, convergence, and scaling. It argues that classical curvature-based methods are limited by computational costs, while quantum geometry embeds curvature intrinsically, suggesting potential for quantum-inspired or hybrid optimization approaches. The work situates these ideas within existing literature and outlines concrete future research directions, emphasizing the speculative yet potentially transformative value of this interdisciplinary lens. If validated, this perspective could motivate novel algorithms and scaling strategies that leverage curvature-rich information manifolds to improve training efficiency and generalization.

Abstract

Optimization in large language models (LLMs) unfolds over high-dimensional parameter spaces with non-Euclidean structure. Information geometry frames this landscape using the Fisher information metric, enabling more principled learning via natural gradient descent. Though often impractical, this geometric lens clarifies phenomena such as sharp minima, generalization, and observed scaling laws. We argue that curvature-based approaches deepen our understanding of LLM training. Finally, we speculate on quantum analogies based on the Fubini-Study metric and Quantum Fisher Information, hinting at efficient optimization in quantum-enhanced systems.

Rethinking LLM Training through Information Geometry and Quantum Metrics

TL;DR

The paper proposes viewing LLM optimization through information geometry and quantum geometry, using the Fisher information and Fubini-Study/QFI to reinterpret curvature, convergence, and scaling. It argues that classical curvature-based methods are limited by computational costs, while quantum geometry embeds curvature intrinsically, suggesting potential for quantum-inspired or hybrid optimization approaches. The work situates these ideas within existing literature and outlines concrete future research directions, emphasizing the speculative yet potentially transformative value of this interdisciplinary lens. If validated, this perspective could motivate novel algorithms and scaling strategies that leverage curvature-rich information manifolds to improve training efficiency and generalization.

Abstract

Optimization in large language models (LLMs) unfolds over high-dimensional parameter spaces with non-Euclidean structure. Information geometry frames this landscape using the Fisher information metric, enabling more principled learning via natural gradient descent. Though often impractical, this geometric lens clarifies phenomena such as sharp minima, generalization, and observed scaling laws. We argue that curvature-based approaches deepen our understanding of LLM training. Finally, we speculate on quantum analogies based on the Fubini-Study metric and Quantum Fisher Information, hinting at efficient optimization in quantum-enhanced systems.

Paper Structure

This paper contains 18 sections, 9 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Theoretical Background
  3. Optimization and Geometry in LLMs
  4. Geometry of Probability Spaces and Fisher Information
  5. Gradient-Based Optimization and Curvature Awareness
  6. A Glimpse into Quantum Geometry
  7. Scaling Laws and the Limits of Classical Geometry
  8. Related Work and Theoretical Context
  9. Geometry and Curvature in Deep Learning
  10. Quantum Information Geometry and the Quantum Fisher Information
  11. Quantum-Inspired and Hybrid Models
  12. Discussion and Future Directions
  13. Reconsidering the Role of Geometry in Optimization
  14. Implications for Scaling Laws
  15. Inspirations for Algorithm Design
  16. ...and 3 more sections

Figures (1)

  • Figure 1: Curve on a manifold illustrating tangent space and information geometry concepts. A curve $X(t)$ on manifold $M$ passes through point $x$. The tangent space at $x$, denoted as $T_xM$, is depicted as a plane. The tangent vector $\mathbf{A}_x$ is shown, expressed in basis $\{\mathbf{e}^1_x, \mathbf{e}^2_x\}$. Tangent vectors derived from the log- likelihood function $\log p(x; \boldsymbol{\theta})$ define the local structure of information geometry.