No Escape from the Quadratic Barrier: Adaptive Attacks on Linear $\ell_2$ Sketches

Key points

• At constant accuracy, randomized Euclidean-norm sketches, notably the Johnson-Lindenstrauss transform, support exponentially many nonadaptive queries in sketch dimension $k$.1
• For adaptive black-box queries, known privacy-based defenses guarantee $\widetildeΩ(k^2)$ and it remained open whether stronger guarantees were possible.1
• We show that the quadratic guarantee is optimal up to logarithmic factors, even with a computationally unbounded responder that knows the matrix, attack algorithm, history, and query distributions and may change its randomized rule after every query.1
• For every fixed hidden $A\in\mathbb{R}^{k\times n}$ with $n>40k$ and target accuracy $α\in (0,1]$, our black-box attack uses $O(k+α^2k^2\log(2+α^2k))$ queries in $\widetilde{O}(1+α^2k)$ adaptive rounds to force, with constant probability, a round in which a constant fraction of answers have $Ω(α)$ relative error.1

Sources (1)

• [1]No Escape from the Quadratic Barrier: Adaptive Attacks on Linear $\ell_2$ Sketches
  arXiv (AI, ML, NLP, CV, robotics, multi-agent) · Oct 18, 04:24 AM

  Linear sketches estimate properties of high-dimensional inputs from a fixed low-dimensional projection.

  At constant accuracy, randomized Euclidean-norm sketches, notably the Johnson-Lindenstrauss transform, support exponentially many nonadaptive queries in sketch dimension $k$.

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