BAR Conjecture: the Feasibility of Inference Budget-Constrained LLM Services with Authenticity and Reasoning

2025-07-30

AI Summary

This paper addresses the fundamental trade-off among inference budget, factual accuracy, and reasoning capability in large language model (LLM) serving—formalizing this tripartite conflict for the first time and rigorously proving its inevitability via the BAR Theorem, which establishes a theoretical impossibility of simultaneously optimizing all three objectives. Methodologically, we propose a multi-objective constrained optimization framework, incorporating feasibility analysis and formal proofs to characterize Pareto-optimal trade-off curves under varying budget constraints. Our contributions are threefold: (1) deriving the theoretical lower bound for joint optimization of the three attributes; (2) providing principled design guidelines enabling application-aware, rational trade-offs; and (3) empirically validating consistent trade-off patterns across diverse model configurations, thereby laying a foundational theory for trustworthy LLM deployment.

Technology Category

• Machine Learning: Large Multimodal Models (LMMs)
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Natural Language Processing: (Large) Language Models

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

When designing LLM services, practitioners care about three key properties: inference-time budget, factual authenticity, and reasoning capacity. However, our analysis shows that no model can simultaneously optimize for all three. We formally prove this trade-off and propose a principled framework named The BAR Theorem for LLM-application design.

Problem

• Trade-off between inference budget, authenticity, and reasoning in LLMs
• No model optimizes all three key properties simultaneously
• Proposes BAR Theorem framework for principled LLM design

Innovation

• Proves trade-off in LLM properties optimization
• Introduces BAR Theorem framework
• Balances budget, authenticity, reasoning