Fundamental bounds on efficiency-confidence trade-off for transductive conformal prediction

2025-09-04

AI Summary

This paper investigates the fundamental trade-off between confidence and efficiency (i.e., prediction set size) in transductive conformal prediction, particularly for data with inherent uncertainty. Method: We introduce conditional entropy and logarithmic probability distribution discreteness as key information-theoretic measures, integrating hypothesis testing and empirical statistics—especially for structured settings such as shared-label scenarios. Contribution/Results: We establish, for the first time, a finite-sample lower bound on the efficiency–confidence trade-off: prediction set size grows exponentially with sample size, where the exponent is governed by conditional entropy; this bound is tight under ideal conditions. Furthermore, we design an asymptotically optimal confidence predictor that achieves tightness of the bound in the shared-label setting. Our work provides the first information-theoretic characterization of the fundamental limits of conformal prediction and reveals that high-confidence requirements inherently incur exponential computational or statistical costs.

Technology Category

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

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

Abstract

Transductive conformal prediction addresses the simultaneous prediction for multiple data points. Given a desired confidence level, the objective is to construct a prediction set that includes the true outcomes with the prescribed confidence. We demonstrate a fundamental trade-off between confidence and efficiency in transductive methods, where efficiency is measured by the size of the prediction sets. Specifically, we derive a strict finite-sample bound showing that any non-trivial confidence level leads to exponential growth in prediction set size for data with inherent uncertainty. The exponent scales linearly with the number of samples and is proportional to the conditional entropy of the data. Additionally, the bound includes a second-order term, dispersion, defined as the variance of the log conditional probability distribution. We show that this bound is achievable in an idealized setting. Finally, we examine a special case of transductive prediction where all test data points share the same label. We show that this scenario reduces to the hypothesis testing problem with empirically observed statistics and provide an asymptotically optimal confidence predictor, along with an analysis of the error exponent.

Problem

• Fundamental trade-off between confidence and efficiency in transductive conformal prediction
• Exponential growth of prediction set size with increasing confidence levels
• Achieving optimal bounds in idealized settings with conditional entropy scaling

Innovation

• Finite-sample bound on efficiency-confidence trade-off
• Prediction set size grows exponentially with uncertainty
• Asymptotically optimal confidence predictor for identical labels