Search for: All records

We give new upper and lower bounds on the power of several restricted classes of arbitrary-order read-once branching programs (ROBPs) and standard-order ROBPs (SOBPs) that have received significant attention in the literature on pseudorandomness for space-bounded computation. - Regular SOBPs of length n and width ⌊w(n+1)/2⌋ can exactly simulate general SOBPs of length n and width w, and moreover an n/2-o(n) blow-up in width is necessary for such a simulation. Our result extends and simplifies prior average-case simulations (Reingold, Trevisan, and Vadhan (STOC 2006), Bogdanov, Hoza, Prakriya, and Pyne (CCC 2022)), in particular implying that weighted pseudorandom generators (Braverman, Cohen, and Garg (SICOMP 2020)) for regular SOBPs of width poly(n) or larger automatically extend to general SOBPs. Furthermore, our simulation also extends to general (even read-many) oblivious branching programs. - There exist natural functions computable by regular SOBPs of constant width that are average-case hard for permutation SOBPs of exponential width. Indeed, we show that Inner-Product mod 2 is average-case hard for arbitrary-order permutation ROBPs of exponential width. - There exist functions computable by constant-width arbitrary-order permutation ROBPs that are worst-case hard for exponential-width SOBPs. - Read-twice permutation branching programs of subexponential width can simulate polynomial-width arbitrary-order ROBPs. 

f-DP has recently been proposed as a generalization of differential privacy allowing a lossless analysis of composition, post-processing, and privacy amplification via subsampling. In the setting of f-DP, we propose the concept of a canonical noise distribution (CND), the first mechanism designed for an arbitrary f-DP guarantee. The notion of CND captures whether an additive privacy mechanism perfectly matches the privacy guarantee of a given f. We prove that a CND always exists, and give a construction that produces a CND for any f. We show that private hypothesis tests are intimately related to CNDs, allowing for the release of private p-values at no additional privacy cost as well as the construction of uniformly most powerful (UMP) tests for binary data, within the general f-DP framework. We apply our techniques to the problem of difference of proportions testing, and construct a UMP unbiased (UMPU) "semi-private" test which upper bounds the performance of any f-DP test. Using this as a benchmark we propose a private test, based on the inversion of characteristic functions, which allows for optimal inference for the two population parameters and is nearly as powerful as the semi-private UMPU. When specialized to the case of (ϵ,0)-DP, we show empirically that our proposed test is more powerful than any (ϵ/sqrt(2))-DP test and has more accurate type I errors than the classic normal approximation test. 

Abstract Economics and social science research often require analyzing datasets of sensitive personal information at fine granularity, with models fit to small subsets of the data. Unfortunately, such fine-grained analysis can easily reveal sensitive individual information. We study regression algorithms that satisfy differential privacy , a constraint which guarantees that an algorithm’s output reveals little about any individual input data record, even to an attacker with side information about the dataset. Motivated by the Opportunity Atlas , a high-profile, small-area analysis tool in economics research, we perform a thorough experimental evaluation of differentially private algorithms for simple linear regression on small datasets with tens to hundreds of records—a particularly challenging regime for differential privacy. In contrast, prior work on differentially private linear regression focused on multivariate linear regression on large datasets or asymptotic analysis. Through a range of experiments, we identify key factors that affect the relative performance of the algorithms. We find that algorithms based on robust estimators—in particular, the median-based estimator of Theil and Sen—perform best on small datasets (e.g., hundreds of datapoints), while algorithms based on Ordinary Least Squares or Gradient Descent perform better for large datasets. However, we also discuss regimes in which this general finding does not hold. Notably, the differentially private analogues of Theil–Sen (one of which was suggested in a theoretical work of Dwork and Lei) have not been studied in any prior experimental work on differentially private linear regression.