Tight Bounds for Adversarially Robust Streams and Sliding Windows via Difference Estimators

Abstract

In the adversarially robust streaming model, a stream of elements is presented to an algorithm and is allowed to depend on the output of the algorithm at earlier times during the stream. In the classic insertion-only model of data streams, Ben-Eliezer et al. (PODS 2020, best paper award) show how to convert a non-robust algorithm into a robust one with a roughly <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$1/\varepsilon$</tex> factor overhead. This was subsequently improved to a <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$1/\sqrt{\varepsilon}$</tex> factor overhead by Hassidim et al. (NeurIPS 2020, oral presentation), suppressing logarithmic factors. For general functions the latter is known to be best-possible, by a result of Kaplan et al. (CRYPTO 2021). We show how to bypass this impossibility result by developing data stream algorithms for a large class of streaming problems, with no overhead in the approximation factor. Our class of streaming problems includes the most well-studied problems such as the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$L_{2}$</tex> -heavy hitters problem, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$F_{p}$</tex> -moment estimation, as well as empirical entropy estimation. We substantially improve upon all prior work on these problems, giving the first optimal dependence on the approximation factor. As in previous work, we obtain a general transformation that applies to any non-robust streaming algorithm and depends on the so-called flip number. However, the key technical innovation is that we apply the transformation to what we call a difference estimator for the streaming problem, rather than an estimator for the streaming prob-lem itself. We then develop the first difference estimators for a wide range of problems. Our difference estimator methodology is not only applicable to the adversarially ro-bust model, but to other streaming models where temporal properties of the data play a central role. To demonstrate the generality of our technique, we additionally introduce a general framework for the related sliding window model of data streams and resolve longstanding open questions in that model, obtaining a drastic improvement from the previous <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$1/\varepsilon^{2+p}$</tex> dependence for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$F_{p}$</tex> -moment estimation for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$p\in$</tex> [1], [2] and integer <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$p &gt; 2$</tex> of Braverman and Ostrovsky (FOCS, 2007), to the optimal <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$1/\varepsilon^{2}$</tex> bound. We also improve the prior <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$1/\varepsilon^{3}$</tex> bound for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$p\in[0,1)$</tex> , and the prior <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$1/-\varepsilon^{4}$</tex> bound for empirical entropy, obtaining the first optimal <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$1/\varepsilon^{2}$</tex> dependence for both of these problems as well. Qualitatively, our results show there is no separation between the sliding window model and the standard data stream model in terms of the approximation factor.