Adversarially Robust Streaming via Dense--Sparse Trade-offs

Abstract

A streaming algorithm is adversarially robust if it is guaranteed to perform correctly even in the presence of an adaptive adversary. The development and analysis of such algorithms have been a very active topic recently, and several sophisticated frameworks for robustification of classical streaming algorithms have been developed. One of the main open questions in this area is whether efficient adversarially robust algorithms exist for moment estimation problems (e.g., F 2 -estimation) under the turnstile streaming model, where both insertions and deletions are allowed. So far, the best known space complexity for streams of length m, achieved using differential privacy (DP) based techniques, is of order Õ(m 1/2 ) for computing a constant-factor approximation with high constant probability (the Õ notation hides here terms polynomial in log m and log n, where n is the universe size). In this work, we propose a new simple approach to tracking moments by alternating between two different regimes: a sparse regime, in which we can explicitly maintain the current frequency vector and use standard sparse recovery techniques, and a dense regime, in which we make use of existing DP-based robustification frameworks. The results obtained using our technique break the previous m 1/2 barrier for any fixed p. More specifically, our space complexity for F 2 -estimation is Õ(m 2/5 ) and for F 0estimation, i.e., counting the number of distinct elements, it is Õ(m 1/3 ).All existing robustness frameworks have their space complexity depend multiplicatively on a parameter λ called the flip number of the streaming problem, where λ = m in turnstile moment estimation. The best known dependence in these frameworks (for constant factor approximation) is of order Õ(λ 1/2 ), and it is known to be tight for certain problems. Again, our approach breaks this barrier, achieving a dependence of order Õ(λ 1/2−c(p) ) for F p -estimation, where c(p) > 0 depends only on p.

A Framework for Adversarially Robust Streaming Algorithms

“…Recently, [9] used a hybrid approach combining the differential privacy based framework of [31] with classical results in sparse recovery to obtain improved space bounds for this problem; the dependence in m is for example Õ (m 1/3 ) when p = 0 and Õ (m 2/5 ) when p = 2. This large gap in the best known space requirements, despite the fact that no space complexity separations between static and robust algorithms are known, leads to the following natural question (see [32], [9]):…”

Dynamic Algorithms Against an Adaptive Adversary: Generic Constructions and Lower Bounds

“…It is also possible to remove assumption that the answer is at least k: if there is a separated adaptive algorithm that can take care of the problem when the answer is less than k, then we can run both algorithms in parallel. This idea of combining the two algorithms, one for small answers and another for large answers, was explicitly used in [BEEO21] in the streaming setting.…”