Generalizing the Fano inequality further

2026-01-17

AI Summary

This work addresses the lack of information-theoretic lower bounds for tail-sensitive objectives—such as Conditional Value-at-Risk (CVaR)—in interactive statistical decision-making, where existing bounds primarily target expected risk. The authors propose a generalized Fano framework that replaces the deterministic success event in classical Fano inequalities with a randomized one-bit statistic derived from an arbitrary bounded transformation of the loss function. By integrating Bernoulli f-divergences, the Rockafellar–Uryasev representation of CVaR, and a Pinsker-type inequality, they establish the first information-theoretic lower bound on Bayesian CVaR under bounded losses. This bound is explicitly expressed in terms of mutual information and extends prior interactive Fano results under both KL divergence and mixture reference distributions, thereby providing a theoretical foundation for tail-risk optimization.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Calibration & Uncertainty Quantification
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Security and Privacy: Large-scale security measurements

Abstract

Interactive statistical decision making (ISDM) features algorithm-dependent data generated through interaction. Existing information-theoretic lower bounds in ISDM largely target expected risk, while tail-sensitive objectives are less developed. We generalize the interactive Fano framework of Chen et al. by replacing the hard success event with a randomized one-bit statistic representing an arbitrary bounded transform of the loss. This yields a Bernoulli f-divergence inequality, which we invert to obtain a two-sided interval for the transform, recovering the previous result as a special case. Instantiating the transform with a bounded hinge and using the Rockafellar-Uryasev representation, we derive lower bounds on the prior-predictive (Bayesian) CVaR of bounded losses. For KL divergence with the mixture reference distribution, the bound becomes explicit in terms of mutual information via Pinsker's inequality.

Problem

• interactive statistical decision making
• tail-sensitive objectives
• Bayesian CVaR
• information-theoretic lower bounds
• Fano inequality

Innovation

• interactive Fano inequality
• f-divergence
• Bayesian CVaR
• mutual information
• statistical decision making