[Paper Review] A necessary and sufficient stability notion for adaptive generalization

Katrina Ligett, Moshe Shenfeld|arXiv (Cornell University)|Jun 3, 2019

Privacy-Preserving Technologies in Data23 references4 citations

TL;DR

This paper introduces Local Statistical Stability (LSS), a new stability notion that is both necessary and sufficient for adaptive generalization in bounded-sensitivity linear queries. It quantifies how much a computation's output affects the posterior distribution over data elements, ensuring robust generalization under adaptivity, post-processing, and composition, with differential privacy and other known stability notions shown to imply LSS.

ABSTRACT

We introduce a new notion of the stability of computations, which holds under post-processing and adaptive composition. We show that the notion is both necessary and sufficient to ensure generalization in the face of adaptivity, for any computations that respond to bounded-sensitivity linear queries while providing accuracy with respect to the data sample set. The stability notion is based on quantifying the effect of observing a computation's outputs on the posterior over the data sample elements. We show a separation between this stability notion and previously studied notion and observe that all differentially private algorithms also satisfy this notion.

Motivation & Objective

• Address the open question of how relaxed stability notions can still guarantee generalization under adaptive data analysis.
• Formalize a stability notion that is robust to post-processing and adaptive composition, unlike prior notions.
• Establish a precise equivalence between stability and generalization for bounded-sensitivity linear queries, assuming sample accuracy.
• Provide a new theoretical framework based on posterior distribution shifts to quantify harmful computation outputs.
• Demonstrate that existing mechanisms like differentially private algorithms satisfy the new stability condition, enabling new generalization guarantees.

Proposed method

• Define a prior distribution over data elements and compute the posterior distribution conditioned on a computation’s output.
• Introduce Local Statistical Stability (LSS) as a measure of the statistical distance between prior and posterior distributions over individual data elements.
• Formalize LSS as a probabilistic bound on the likelihood of producing outputs that significantly alter the posterior over any data element.
• Prove that LSS is preserved under post-processing and adaptive composition using information-theoretic and probabilistic arguments.
• Establish implications from known stability notions (e.g., differential privacy, Max Information, compression schemes) to LSS via direct proofs.
• Use separation theorems to show that LSS is strictly weaker than Max Information and Local Max Information, demonstrating its optimality.

Experimental results

Research questions

• RQ1 What stability notion is both necessary and sufficient to ensure generalization in adaptive data analysis for bounded-sensitivity linear queries?
• RQ2 How does the proposed Local Statistical Stability (LSS) notion relate to existing stability concepts like differential privacy and Max Information?
• RQ3 Can LSS be preserved under post-processing and adaptive composition, and how does this affect generalization guarantees?
• RQ4 Are there mechanisms that satisfy LSS but not stronger stability notions like Max Information, indicating a strict hierarchy?
• RQ5 Can LSS be used to derive tighter generalization bounds by leveraging natural data sampling noise?

Key findings

• Local Statistical Stability (LSS) is both necessary and sufficient for adaptive generalization in bounded-sensitivity linear queries, assuming sample accuracy.
• The paper establishes a separation between LSS and previously studied notions: LSS is strictly weaker than Local Max Information (LMI), which in turn is strictly weaker than Max Information (MI).
• Differential privacy, Max Information, Typical Stability, and compression schemes all imply LSS, providing a unifying framework for their generalization properties.
• A mechanism that uniformly samples and outputs one data element is (11√(ln(2n/δ)/n), δ)-LSS but not (1, 1/(2n))-LMI, demonstrating that LSS is strictly weaker than LMI.
• A mechanism outputting the parity of the sample set is (ε,0)-LMI but not (1,1/5)-MI, proving that MI is strictly stronger than LMI.
• The upper bound on LSS requires one more query than the lower bound, indicating near-tightness of the stability-generalization equivalence.