Max-Information, Differential Privacy, and Post-Selection Hypothesis Testing

TL;DR

This paper explores how approximate differential privacy guarantees can be used to perform valid adaptive hypothesis testing by controlling max-information, extending the understanding of privacy's role in generalization and statistical validity.

Contribution

It establishes a connection between $(\epsilon,\delta)$-differential privacy and bounded max-information for product distributions, and analyzes composition limitations in this context.

Findings

$(\epsilon,\delta)$-DP algorithms have bounded max-information on product distributions.

Differential privacy can be used to correct $p$-values in adaptive hypothesis testing.

Limitations of composition show the connection only holds for inputs from product distributions.

Abstract

In this paper, we initiate a principled study of how the generalization properties of approximate differential privacy can be used to perform adaptive hypothesis testing, while giving statistically valid $p$-value corrections. We do this by observing that the guarantees of algorithms with bounded approximate max-information are sufficient to correct the $p$-values of adaptively chosen hypotheses, and then by proving that algorithms that satisfy $(\epsilon,\delta)$-differential privacy have bounded approximate max information when their inputs are drawn from a product distribution. This substantially extends the known connection between differential privacy and max-information, which previously was only known to hold for (pure) $(\epsilon,0)$-differential privacy. It also extends our understanding of max-information as a partially unifying measure controlling the generalization properties of adaptive data analyses. We also show a lower bound, proving that (despite the strong composition properties of max-information), when data is drawn from a product distribution, $(\epsilon,\delta)$-differentially private algorithms can come first in a composition with other algorithms satisfying max-information bounds, but not necessarily second if the composition is required to itself satisfy a nontrivial max-information bound. This, in particular, implies that the connection between $(\epsilon,\delta)$-differential privacy and max-information holds only for inputs drawn from product distributions, unlike the connection between $(\epsilon,0)$-differential privacy and max-information.