Interactive Inference Under Information Constraints

Abstract

We study the role of interactivity in distributed statistical inference under information constraints, e.g., communication constraints and local differential privacy. We focus on the tasks of goodness-of-fit testing and estimation of discrete distributions. From prior work, these tasks are well understood under noninteractive protocols. Extending these approaches directly for interactive protocols is difficult due to correlations that can build due to interactivity; in fact, gaps can be found in prior claims of tight bounds of distribution estimation using interactive protocols. We propose a new approach to handle this correlation and establish a unified method to establish lower bounds for both tasks. As an application, we obtain optimal bounds for both estimation and testing under local differential privacy and communication constraints. We also provide an example of a natural testing problem where interactivity helps.

Uniformity Testing in the Shuffle Model: Simpler, Better, Faster

“…Thus, having n ≥ 40 • k 3/4 µ 1/2 α suffices to ensure Pr[Z ≤ τ ] ≤ 1/3 in this case. Taking C = max(40, 4 3/2) = 40 concludes the proof.…”

“…The second term of the expression is the non-private lower bound, and thus holds regardless of the privacy parameter. Since the first term only dominates when k ≥ (ε/α) 4 , we can assume we are in this parameter regime. Suppose by contradiction there exists a private-coin (ε, 0, 1/3)robustly shuffle private protocol uniformity tester with sample complexity o k 3/4 /(αε) + k 1/2 /α 2 .…”

“…The question was then revisited from the privacy perspective by [18], after which [10,12] established the tight sample complexity bound Θ( √ k/α 2 + √ k/(α √ ε) + k 1/3 /(α 4/3 ε 2/3 ) + 1/(αε)) under (central) differential privacy. In the more stringent locally private setting, the question was later raised in [37], and fully answered in a sequence of works which show that the tight sample complexity is Θ(k 3/2 /(α 2 ε 2 )) for non-interactive private-coin protocols and Θ(k/(α 2 ε 2 )) for non-interactive public-coin protocols (where the users have access to a common random seed) [2,6,1], and that allowing interactive protocols does not improve the sample complexity beyond Θ(k/(α 2 ε 2 )) [13,16,4].…”

“…Namely, we employ the "privacy amplification by shuffling" paradigm [28] to convert a locally private uniformity testing algorithm to a robust shuffle private testing algorithm. In order to do so, we leverage the recent result of [29], which provides an amplification-by-shuffling theorem optimal in all privacy regimes -quite fortunately, as an optimal dependence on ε even in the low-privacy regime is exactly what we need to achieve (4).…”

“…The first states that adding properly calibrated Poisson noise to an integer-valued function ensures privacy of the output. 4 Lemma 1 (Poisson Mechanism [31,Theorem 11]). Let f :…”