Local Lipschitz Filters for Bounded-Range Functions with Applications to Arbitrary Real-Valued Functions

We study local filters for the Lipschitz property of real-valued functions f : V → [0, r], where the Lipschitz property is defined with respect to an arbitrary undirected graph G = (V, E). We give nearly optimal local Lipschitz filters both with respect to ℓ1-distance and ℓ0-distance. Previous work only considered unbounded-range functions over [n] d . Jha and Raskhodnikova (SICOMP '13) gave an algorithm for such functions with lookup complexity exponential in d, which Awasthi et al. (ACM Trans. Comput. Theory) showed was necessary in this setting. We demonstrate that important applications of local Lipschitz filters can be accomplished with filters for functions whose range is bounded in [0, r]. For functions f : [n] d → [0, r], we achieve running time (d r log n) O(log r) for the ℓ1-respecting filter and d O(r) polylog n for the ℓ0-respecting filter, thus circumventing the lower bound. Our local filters provide a novel Lipschitz extension that can be implemented locally. Furthermore, we show that our algorithms are nearly optimal in terms of the dependence on r for the domain 0, 1 d , an important special case of the domain [n] d . In addition, our lower bound resolves an open question of Awasthi et al., removing one of the conditions necessary for their lower bound for general range. We prove our lower bound via a reduction from distribution-free Lipschitz testing and a new technique for proving hardness for adaptive algorithms.

Finally, we provide two applications of our local filters to real-valued functions, with no restrictions on the range. In the first application, we use them in conjunction with the Laplace mechanism for differential privacy and noisy binary search to provide mechanisms for privately releasing outputs of black-box functions, even in the presence of malicious clients. In particular, our differentially private mechanism for arbitrary real-valued functions runs in time 2 polylog min (r,nd) and, for honest clients, has accuracy comparable to the Laplace mechanism for Lipschitz functions, up to a factor of O(log min(r, nd)). In the second application, we use our local filters to obtain the first nontrivial tolerant tester for the Lipschitz property. Our tester works for functions of the form f : 0, 1 d → R, makes 2 O( √ d) queries, and has tolerance ratio 2.01. Our applications demonstrate that local filters for bounded-range functions can be applied to construct efficient algorithms for arbitrary real-valued functions.