Modes of Sequence Models and Learning Coefficients

2025-04-25

AI Summary

This work bridges the gap between geometric interpretations of patterns and measurability of loss landscapes in sequence modeling. Methodologically, it maps conditional sequence distributions into a Hilbert space, constructs a data-dependent “effective distribution” by truncating low-magnitude modes above an adaptive threshold, and characterizes its geometric structure via the local learning coefficient (LLC). Theoretically, it proves that LLC is inherently robust to noise modes below the truncation threshold—demonstrating that LLC fundamentally reflects the geometry of the effective (truncated) distribution, not the original data distribution. A key conceptual contribution identifies the inverse temperature parameter in stochastic gradient Langevin dynamics (SGLD) as a “resolution controller” for the loss landscape. Finally, the framework is instantiated in Transformer architectures, enabling interpretable and quantifiable geometric characterization of loss landscapes.

Technology Category

• Machine Learning: Learning with Manifolds
• Natural Language Processing: Learning & Optimization for NLP
• Search and Optimization: Learning to Search

Application Category

• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

We develop a geometric account of sequence modelling that links patterns in the data to measurable properties of the loss landscape in transformer networks. First, we cast conditional sequence distributions into a Hilbert-space framework and apply tensor decompositions to identify their principal modes. Truncating the small-amplitude modes yields an effective data distribution that preserves dominant structure while discarding statistical detail. Second, we show theoretically that Local Learning Coefficient (LLC) estimates are insensitive to modes below a data-dependent threshold. Consequently, the LLC calculated in practice characterises the geometry of the effective rather than the true distribution. This insight clarifies why reliable LLC estimates can be obtained even when a network parameter is not a strict minimiser of the population loss, and it highlights how the inverse temperature in SGLD acts as a resolution dial on the landscape structure.

Problem

• Identify principal modes in sequence distributions using tensor decompositions
• Show LLC estimates ignore modes below a data-dependent threshold
• Clarify why reliable LLC estimates work without strict loss minimisation

Innovation

• Hilbert-space framework for sequence distributions
• Tensor decompositions identify principal modes
• Local Learning Coefficients insensitive to minor modes