arXiv subjects

Jaroslaw Blasiok

Publications and source records attributed to Jaroslaw Blasiok.

3 recordsLinked to original sources

arXiv2018

Towards Instance-Optimal Private Query Release

We study efficient mechanisms for the query release problem in differential privacy: given a workload of $m$ statistical queries, output approximate answers to the queries while satisfying the constraints of differential privacy. In particular, we are interested in mechanisms that optimally adapt to the given workload. Building on the projection mechanism of Nikolov, Talwar, and Zhang, and using the ideas behind Dudley's chaining inequality, we propose new efficient algorithms for the query release problem, and prove that they achieve optimal sample complexity for the given workload (up to constant factors, in certain parameter regimes) with respect to the class of mechanisms that satisfy concentrated differential privacy. We also give variants of our algorithms that satisfy local differential privacy, and prove that they also achieve optimal sample complexity among all local sequentially interactive private mechanisms.

cs.DS↗

arXiv2017

Streaming Symmetric Norms via Measure Concentration

We characterize the streaming space complexity of every symmetric norm $l$ (a norm on $\mathbb{R}^n$ invariant under sign-flips and coordinate-permutations), by relating this space complexity to the measure-concentration characteristics of $l$. Specifically, we provide nearly matching upper and lower bounds on the space complexity of calculating a $(1\pmε)$-approximation to the norm of the stream, for every $0<ε\leq 1/2$. (The bounds match up to $poly(ε^{-1} \log n)$ factors.) We further extend those bounds to any large approximation ratio $D\geq 1.1$, showing that the decrease in space complexity is proportional to $D^2$, and that this factor the best possible. All of the bounds depend on the median of $l(x)$ when $x$ is drawn uniformly from the $l_2$ unit sphere. The same median governs many phenomena in high-dimensional spaces, such as large-deviation bounds and the critical dimension in Dvoretzky's Theorem. The family of symmetric norms contains several well-studied norms, such as all $l_p$~norms, and indeed we provide a new explanation for the disparity in space complexity between $p\le 2$ and $p>2$. In addition, we apply our general results to easily derive bounds for several norms that were not studied before in the streaming model, including the top-$k$ norm and the $k$-support norm, which was recently employed for machine learning tasks. Overall, these results make progress on two outstanding problems in the area of sublinear algorithms (Problems 5 and 30 in~\url{http://sublinear.info}).

cs.DS↗

arXiv2013

Chain minors are FPT

Given two finite posets P and Q, P is a chain minor of Q if there exists a partial function f from the elements of Q to the elements of P such that for every chain in P there is a chain C_Q in Q with the property that f restricted to C_Q is an isomorphism of chains. We give an algorithm to decide whether a poset P is a chain minor of o poset Q that runs in time O(|Q| log |Q|) for every fixed poset P. This solves an open problem from the monograph by Downey and Fellows [Parameterized Complexity, 1999] who asked whether the problem was fixed parameter tractable.

cs.DS↗