Overview
This work presents novel change of measure inequalities developed through a unified framework based on the data processing inequality. These inequalities aim to translate divergences between probability measures into explicit bounds on event probabilities.
Research Context
Change of measure inequalities are integral to deriving probabilistic guarantees across multiple domains, including learning theory, information theory, and statistics. Their primary function involves quantifying the relationship between differences in probability distributions and the likelihood of specific events.
Approach
A unified framework was developed for constructing new change of measure inequalities. This framework leverages the data processing inequality and is described as elementary yet capable of yielding novel and tighter inequalities. The resulting inequalities are applicable to a broad family of information measures. Specific examples include $f$-divergences, which encompass Kullback-Leibler divergence and $\chi^2$-divergence as special cases. Rényl divergence is also covered, as is $\alpha$-mutual information, where maximal leakage is a specific instance.
Findings
- Novel change of measure inequalities were generated through a unified framework built upon the data processing inequality.
- These new inequalities provide tighter probabilistic bounds.
- The framework accommodates a wide range of information measures, including:
- $f$-divergences (e.g., Kullback-Leibler divergence, $\chi^2$-divergence)
- Rényl divergence
- $\alpha$-mutual information (e.g., maximal leakage)
- Application of these results to generalization error analysis, PAC-Bayesian theory, differential privacy, and data memorization yielded stronger guarantees.
- The analysis also recovered best-known results, simplifying the derivation process.
Why This Matters
The development of tighter change of measure inequalities offers stronger guarantees in areas such as learning theory, information theory, and statistics. By providing more precise probabilistic bounds across various information measures and applications, this research contributes to a more robust foundation for theoretical analyses in these fields.