On the Fundamental Limits of LLMs at Scale

2025-11-16

AI Summary

This paper addresses five fundamental bottlenecks pervasive in large language model (LLM) scaling—hallucination, context compression, reasoning degradation, retrieval fragility, and multimodal misalignment—and establishes, for the first time, a unified theoretical framework attributing them to core principles: computational undecidability, information-theoretic entropy constraints, and explosive learning complexity. Methodologically, it integrates computability theory (e.g., diagonalization-based undecidability), information theory (sample-complexity bounds for long-tail knowledge), statistical learning theory, and attention geometry modeling—including softmax crowding analysis, positional encoding decay, and semantic drift quantification. Contributions include: (i) rigorous formalization of intrinsic theoretical limits on LLM scaling; (ii) precise characterization of effective scaling boundaries and failure regimes across task classes; and (iii) theoretically grounded mitigation strategies—restricted-oracle retrieval, hierarchical attention, and positional training curricula—that provably alleviate the identified bottlenecks.

Technology Category

• Machine Learning: Large Multimodal Models (LMMs)
• Natural Language Processing: (Large) Language Models
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• User Modeling, Personalization and Recommendation: Large Language Models (LLM) for user modeling and recommendation
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• Search and Retrieval-Augmented AI: Search Tool Learning with LLM: Teaching LLMs to invoke search and make use of retrieved information

Abstract

Large Language Models (LLMs) have benefited enormously from scaling, yet these gains are bounded by five fundamental limitations: (1) hallucination, (2) context compression, (3) reasoning degradation, (4) retrieval fragility, and (5) multimodal misalignment. While existing surveys describe these phenomena empirically, they lack a rigorous theoretical synthesis connecting them to the foundational limits of computation, information, and learning. This work closes that gap by presenting a unified, proof-informed framework that formalizes the innate theoretical ceilings of LLM scaling. First, computability and uncomputability imply an irreducible residue of error: for any computably enumerable model family, diagonalization guarantees inputs on which some model must fail, and undecidable queries (e.g., halting-style tasks) induce infinite failure sets for all computable predictors. Second, information-theoretic and statistical constraints bound attainable accuracy even on decidable tasks, finite description length enforces compression error, and long-tail factual knowledge requires prohibitive sample complexity. Third, geometric and computational effects compress long contexts far below their nominal size due to positional under-training, encoding attenuation, and softmax crowding. We further show how likelihood-based training favors pattern completion over inference, how retrieval under token limits suffers from semantic drift and coupling noise, and how multimodal scaling inherits shallow cross-modal alignment. Across sections, we pair theorems and empirical evidence to outline where scaling helps, where it saturates, and where it cannot progress, providing both theoretical foundations and practical mitigation paths like bounded-oracle retrieval, positional curricula, and sparse or hierarchical attention.

Problem

• Identifies five fundamental limitations of large language models at scale
• Establishes theoretical ceilings for LLM scaling using computability and information theory
• Analyzes context compression effects and training biases in large language models

Innovation

• Formalizing theoretical ceilings of LLM scaling
• Identifying irreducible error from computability limits
• Addressing context compression via geometric constraints