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TR12-075 | 12th June 2012 18:09

Limitations of Local Filters of Lipschitz and Monotone Functions

TR12-075
Publication: 12th June 2012 20:08
Keywords:

Abstract:

We study local filters for two properties of functions $f:\B^d\to \mathbb{R}$: the Lipschitz property and monotonicity. A local filter with additive error $a$ is a randomized algorithm that is given black-box access to a function $f$ and a query point $x$ in the domain of $f$. Its output is a value $F(x)$, such that (i) the {\em reconstructed function} $F(x)$ satisfies the property (in our case, is Lipschitz or monotone) and (ii) if the input function $f$ satisfies the property then for every point $x$ in the domain, with high constant probability the reconstructed value $F(x)$ differs from $f(x)$ by at most $a$.
%A filter is {\em distance-respecting} if $F(x)$ and $f(x)$ do not differ on many points.
Local filters were introduced by Saks and Seshadhri (SICOMP 2010). The relaxed definition we study is due to Bhattacharyya \etal (RANDOM 2010), except that we further relax it by allowing additive error. Local filters for Lipschitz and monotone functions have applications to areas such as data privacy.

We show that every local filter for Lipschitz (resp., monotone) functions runs in time exponential in the dimension $d$ in the worst case. This holds even for filters with significant additive error, as well as for both previously studied definitions. Prior lower bounds (for local filters with no additive error, i.e., with $a=0$) applied only to more restrictive notions of filters, e.g., {\em nonadaptive} filters that are required to specify all their lookups in advance, before obtaining values of $f$ on any points. To prove our lower bounds, we construct families of hard functions, and show that lookups of a local filter on these functions are captured by a combinatorial object that we call a $c$-connector. Then we present a lower bound on the maximum outdegree of a $c$-connector, and show that it implies the desired bounds on the running time of local filters. Our lower bounds, in particular, imply the same bound on the running time for a class of privacy mechanisms.

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