On the Robustness of CountSketch to Adaptive Inputs

\texttt{CountSketch} is a popular dimensionality reduction technique that maps vectors to a lower-dimension using a randomized set of linear measurements. The sketch has the property that the $\ell_2$-heavy hitters of a vector (entries with $v_i^2 \geq \frac{1}{k}\|\boldsymbol{v}\|^2_2$) can be recovered from its sketch. We study the robustness of the sketch in adaptive settings, such as online optimization, where input vectors may depend on the output from prior inputs. We show that the classic estimator can be attacked with a number of queries of the order of the sketch size and propose a robust estimator (for a slightly modified sketch) that allows for quadratic number of queries. We improve robustness by a factor of $\sqrt{k}$ (for $k$ heavy hitters) over prior approaches.

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