10,600 research outputs found
Differentially Private Release and Learning of Threshold Functions
We prove new upper and lower bounds on the sample complexity of differentially private algorithms for releasing approximate answers to
threshold functions. A threshold function over a totally ordered domain
evaluates to if , and evaluates to otherwise. We
give the first nontrivial lower bound for releasing thresholds with
differential privacy, showing that the task is impossible
over an infinite domain , and moreover requires sample complexity , which grows with the size of the domain. Inspired by the
techniques used to prove this lower bound, we give an algorithm for releasing
thresholds with samples. This improves the
previous best upper bound of (Beimel et al., RANDOM
'13).
Our sample complexity upper and lower bounds also apply to the tasks of
learning distributions with respect to Kolmogorov distance and of properly PAC
learning thresholds with differential privacy. The lower bound gives the first
separation between the sample complexity of properly learning a concept class
with differential privacy and learning without privacy. For
properly learning thresholds in dimensions, this lower bound extends to
.
To obtain our results, we give reductions in both directions from releasing
and properly learning thresholds and the simpler interior point problem. Given
a database of elements from , the interior point problem asks for an
element between the smallest and largest elements in . We introduce new
recursive constructions for bounding the sample complexity of the interior
point problem, as well as further reductions and techniques for proving
impossibility results for other basic problems in differential privacy.Comment: 43 page
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Differentially Private Release and Learning of Threshold Functions
We prove new upper and lower bounds on the sample complexity of (ε, δ) differentially private algorithms for releasing approximate answers to threshold functions. A threshold function cx over a totally ordered domain X evaluates to cx(y)=1 if y ≤ x, and evaluates to 0 otherwise. We give the first nontrivial lower bound for releasing thresholds with (ε, δ) differential privacy, showing that the task is impossible over an infinite domain X, and moreover requires sample complexity n ≥ Ω(log∗ |X|), which grows with the size of the domain. Inspired by the techniques used to prove this lower bound, we give an algorithm for releasing thresholds with n ≤ 2(1+o(1)) log∗ |X| samples. This improves the previous best upper bound of 8(1+o(1)) log∗ |X| (Beimel et al., RANDOM ’13). Our sample complexity upper and lower bounds also apply to the tasks of learning distributions with respect to Kolmogorov distance and of properly PAC learning thresholds with differential privacy. The lower bound gives the first separation between the sample complexity of properly learning a concept class with (ε, δ) differential privacy and learning without privacy. For properly learning thresholds in dimensions, this lower bound extends to n ≥ Ω( · log∗ |X|). To obtain our results, we give reductions in both directions from releasing and properly learning thresholds and the simpler interior point problem. Given a database D of elements from X, the interior point problem asks for an element between the smallest and largest elements in D. We introduce new recursive constructions for bounding the sample complexity of the interior point problem, as well as further reductions and techniques for proving impossibility results for other basic problems in differential privacy.Engineering and Applied Science
Order-Revealing Encryption and the Hardness of Private Learning
An order-revealing encryption scheme gives a public procedure by which two
ciphertexts can be compared to reveal the ordering of their underlying
plaintexts. We show how to use order-revealing encryption to separate
computationally efficient PAC learning from efficient -differentially private PAC learning. That is, we construct a concept
class that is efficiently PAC learnable, but for which every efficient learner
fails to be differentially private. This answers a question of Kasiviswanathan
et al. (FOCS '08, SIAM J. Comput. '11).
To prove our result, we give a generic transformation from an order-revealing
encryption scheme into one with strongly correct comparison, which enables the
consistent comparison of ciphertexts that are not obtained as the valid
encryption of any message. We believe this construction may be of independent
interest.Comment: 28 page
Sample Complexity Bounds on Differentially Private Learning via Communication Complexity
In this work we analyze the sample complexity of classification by
differentially private algorithms. Differential privacy is a strong and
well-studied notion of privacy introduced by Dwork et al. (2006) that ensures
that the output of an algorithm leaks little information about the data point
provided by any of the participating individuals. Sample complexity of private
PAC and agnostic learning was studied in a number of prior works starting with
(Kasiviswanathan et al., 2008) but a number of basic questions still remain
open, most notably whether learning with privacy requires more samples than
learning without privacy.
We show that the sample complexity of learning with (pure) differential
privacy can be arbitrarily higher than the sample complexity of learning
without the privacy constraint or the sample complexity of learning with
approximate differential privacy. Our second contribution and the main tool is
an equivalence between the sample complexity of (pure) differentially private
learning of a concept class (or ) and the randomized one-way
communication complexity of the evaluation problem for concepts from . Using
this equivalence we prove the following bounds:
1. , where is the Littlestone's (1987)
dimension characterizing the number of mistakes in the online-mistake-bound
learning model. Known bounds on then imply that can be much
higher than the VC-dimension of .
2. For any , there exists a class such that but .
3. For any , there exists a class such that the sample complexity of
(pure) -differentially private PAC learning is but
the sample complexity of the relaxed -differentially private
PAC learning is . This resolves an open problem of
Beimel et al. (2013b).Comment: Extended abstract appears in Conference on Learning Theory (COLT)
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