619 research outputs found
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
, where the Lipschitz property is defined with respect to an
arbitrary undirected graph . We give nearly optimal local Lipschitz
filters both with respect to -distance and -distance. Previous
work only considered unbounded-range functions over . Jha and
Raskhodnikova (SICOMP `13) gave an algorithm for such functions with lookup
complexity exponential in , 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 with bounded-range. For functions , we circumvent
the lower bound and achieve running time for the
-respecting filter and for the
-respecting filter. Our local filters provide a novel Lipschitz
extension that can be implemented locally. Furthermore, we show that our
algorithms have nearly optimal dependence on for the domain . 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 {\em adaptive} algorithms. We
provide two applications of our local filters to arbitrary real-valued
functions. 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 the second application, we use our local
filters to obtain the first nontrivial tolerant tester for the Lipschitz
property
Certified Computation from Unreliable Datasets
A wide range of learning tasks require human input in labeling massive data.
The collected data though are usually low quality and contain inaccuracies and
errors. As a result, modern science and business face the problem of learning
from unreliable data sets.
In this work, we provide a generic approach that is based on
\textit{verification} of only few records of the data set to guarantee high
quality learning outcomes for various optimization objectives. Our method,
identifies small sets of critical records and verifies their validity. We show
that many problems only need verifications, to
ensure that the output of the computation is at most a factor of away from the truth. For any given instance, we provide an
\textit{instance optimal} solution that verifies the minimum possible number of
records to approximately certify correctness. Then using this instance optimal
formulation of the problem we prove our main result: "every function that
satisfies some Lipschitz continuity condition can be certified with a small
number of verifications". We show that the required Lipschitz continuity
condition is satisfied even by some NP-complete problems, which illustrates the
generality and importance of this theorem.
In case this certification step fails, an invalid record will be identified.
Removing these records and repeating until success, guarantees that the result
will be accurate and will depend only on the verified records. Surprisingly, as
we show, for several computation tasks more efficient methods are possible.
These methods always guarantee that the produced result is not affected by the
invalid records, since any invalid record that affects the output will be
detected and verified
- …