A Strong Separation for Adversarially Robust ℓ0 Estimation for Linear Sketches

The majority of streaming problems are defined and analyzed in a static setting, where the data stream is any worst-case sequence of insertions and deletions that is fixed in advance. However, many real-world applications require a more flexible model, where an adaptive adversary may select future stream elements after observing the previous outputs of the algorithm. Over the last few years, there has been increased interest in proving lower bounds for natural problems in the adaptive streaming model. In this work, we give the first known adaptive attack against linear sketches for the well-studied ℓ0-estimation problem over turnstile, integer streams. For any linear streaming algorithm A that uses sketching matrix A∈ mathbbZr × n where n is the size of the universe, this attack makes tildeO(r8) queries and succeeds with high constant probability in breaking the sketch. We also give an adaptive attack against linear sketches for the ℓ0-estimation problem over finite fields mathbbFp, which requires a smaller number of tildeO(r3) queries. Finally, we provide an adaptive attack over ℝn against linear sketches A ∈ ℝr × n for ℓ0-estimation, in the setting where A has all nonzero subdeterminants at least frac1textrmpoly(r). Our results provide an exponential improvement over the previous number of queries known to break an ℓ0-estimation sketch.