Algorithmic stability for adaptive data analysis

Abstract

Adaptivity is an important feature of data analysis-the choice of questions to ask about a dataset often depends on previous interactions with the same dataset. However, statistical validity is typically studied in a nonadaptive model, where all questions are specified before the dataset is drawn. Recent work by Dwork et al. (STOC, 2015) and Hardt and Ullman (FOCS, 2014) initiated the formal study of this problem, and gave the first upper and lower bounds on the achievable generalization error for adaptive data analysis.Specifically, suppose there is an unknown distribution P and a set of n independent samples x is drawn from P. We seek an algorithm that, given x as input, accurately answers a sequence of adaptively chosen "queries" about the unknown distribution P. How many samples n must we draw from the distribution, as a function of the type of queries, the number of queries, and the desired level of accuracy?In this work we make two new contributions towards resolving this question:1. We give upper bounds on the number of samples n that are needed to answer statistical queries. The bounds improve and simplify the work of Dwork et al. (STOC, 2015), and have been applied in subsequent work by those authors (Science, 2015; NIPS, 2015). 2. We prove the first upper bounds on the number of samples required to answer more general families of queries. These include arbitrary low-sensitivity queries and an important class of optimization queries (alternatively, risk minimization queries).As in Dwork et al., our algorithms are based on a connection with algorithmic stability in the form of differential privacy. We extend their work by giving a quantitatively optimal, more general, and simpler proof of their main theorem that stable algorithms of the kind guaranteed by differential privacy imply low generalization error. We also show that weaker stability guarantees such as bounded KL divergence and total variation distance lead to correspondingly weaker generalization guarantees. * This work unifies and subsumes the two arXiv manuscripts [BSSU15, NS15].

Error Analysis and Variable Selection for Differential Private Learning Algorithm

“…Remark 1 Similar results are proposed in [6] and [7]. However, there the authors limits the function to take value in [ ] 0,1 or { } 0,1 , our result here extends theirs to the function taking value in +  .…”

“…Recently, in [6], the authors find that the empirical average of the output from a differential private algorithm can converge to its expectation. And [7] provides another analysis of this convergence, which motivates our work.…”

Make Up Your Mind: The Price of Online Queries in Differential Privacy

“…However, we show that this is not the case: Theorem 1.4. For every n ∈ N, there is an (ε, δ)-differentially private algorithm that takes a dataset x ∈ [0, 1] n and answers any set of k = 2 Ω(n) adaptivelychosen queries from Q thresh up to error ±1/100.…”

“…Another motivation for studying the relationship between these models is the recent line of work connecting differential privacy to statistical validity for adaptive data analysis [21,9,24,1], which shows that differentially private algorithms for adaptively-chosen queries in fact yield state-of-the-art algorithms for statistical problems unrelated to privacy. This connection further motivates studying the adaptive model and its relationship to the other models in differential privacy.…”

Gentle measurement of quantum states and differential privacy

“…8 (2) If M is ε-DP on product states, and is a product measurement, 9 then M is O (ε √ n)-gentle on product states. 10 Again, here a "measurement" M corresponds to a specification of output probabilities; for M to be α-gentle means that there exists an α-gentle implementation of M .…”