Soft Guessing Under Logarithmic Loss Allowing Errors and Variable-Length Source Coding

2025-10-10

AI Summary

This paper investigates the soft guessing problem under tolerable error, using logarithmic loss as the distortion measure, and establishes a deep connection with variable-length lossy source coding. We propose, for the first time, an error-tolerant soft guessing framework and introduce novel information-theoretic tools—conditional Rényi entropy and smooth Rényi entropy—integrated with one-shot and asymptotic information-theoretic techniques. Our analysis yields tight upper and lower bounds on the minimal guessing moment, along with its asymptotic expansion. The results are theoretically optimal in both settings—with and without side information—and naturally yield a fundamental rate–distortion trade-off. This unifies soft guessing theory with lossy compression, providing a coherent information-theoretic characterization bridging guessing and coding.

Technology Category

• Machine Learning: Information Theory
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Social Networks and Social Media: Influence propagation, information diffusion, and the prediction on networks

Abstract

This paper considers the problem of soft guessing under a logarithmic loss distortion measure while allowing errors. We find an optimal guessing strategy, and derive single-shot upper and lower bounds for the minimal guessing moments as well as an asymptotic expansion for i.i.d. sources. These results are extended to the case where side information is available to the guesser. Furthermore, a connection between soft guessing allowing errors and variable-length lossy source coding under logarithmic loss is demonstrated. The Rényi entropy, the smooth Rényi entropy, and their conditional versions play an important role.

Problem

• Optimizing soft guessing strategies under logarithmic loss distortion measure
• Establishing connections between soft guessing and variable-length source coding
• Analyzing asymptotic performance using Rényi entropy and its conditional versions

Innovation

• Soft guessing strategy under logarithmic loss distortion
• Single-shot bounds for minimal guessing moments derivation
• Connection between soft guessing and variable-length coding