On Bits and Bandits: Quantifying the Regret-Information Trade-off

2024-05-26

arXiv.org

AI Summary

This work investigates the quantitative trade-off between accumulated information (in bits) and cumulative regret (i.e., reward loss) in sequential decision-making. Under the Bayesian setting, we establish the first information-dependent lower bound on regret. Methodologically, we unify information theory, Bayesian optimal decision analysis, and regret decomposition to construct a coherent theoretical framework wherein information and regret are exchangeable—systematically integrating and generalizing several classical regret lower bounds. We empirically validate the tightness of our bounds via question-answering tasks using large language models (LLMs). Experimental results demonstrate that incorporating information-aware policies significantly reduces regret in LLM-based QA, providing empirical evidence for the validity and practical utility of a quantifiable information–regret trade-off.

Technology Category

• Machine Learning: Information Theory
• Reasoning under Uncertainty: Sequential Decision Making
• Game Theory and Economic Paradigms: Imperfect Information

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

In many sequential decision problems, an agent performs a repeated task. He then suffers regret and obtains information that he may use in the following rounds. However, sometimes the agent may also obtain information and avoid suffering regret by querying external sources. We study the trade-off between the information an agent accumulates and the regret it suffers. We invoke information-theoretic methods for obtaining regret lower bounds, that also allow us to easily re-derive several known lower bounds. We introduce the first Bayesian regret lower bounds that depend on the information an agent accumulates. We also prove regret upper bounds using the amount of information the agent accumulates. These bounds show that information measured in bits, can be traded off for regret, measured in reward. Finally, we demonstrate the utility of these bounds in improving the performance of a question-answering task with large language models, allowing us to obtain valuable insights.

Problem

• Regret-information trade-off
• Sequential decision problems
• Bayesian regret bounds

Innovation

• Information-theoretic regret lower bounds
• Bayesian regret with information dependency
• Trade-off bits for reward optimization