10,194 research outputs found
Contraction of Locally Differentially Private Mechanisms
We investigate the contraction properties of locally differentially private
mechanisms. More specifically, we derive tight upper bounds on the divergence
between and output distributions of an
-LDP mechanism in terms of a divergence between the
corresponding input distributions and , respectively. Our first main
technical result presents a sharp upper bound on the -divergence
in terms of and
. We also show that the same result holds for a large family of
divergences, including KL-divergence and squared Hellinger distance. The second
main technical result gives an upper bound on
in terms of total variation distance
and . We then utilize these bounds to
establish locally private versions of the van Trees inequality, Le Cam's,
Assouad's, and the mutual information methods, which are powerful tools for
bounding minimax estimation risks. These results are shown to lead to better
privacy analyses than the state-of-the-arts in several statistical problems
such as entropy and discrete distribution estimation, non-parametric density
estimation, and hypothesis testing
The Role of Interactivity in Local Differential Privacy
We study the power of interactivity in local differential privacy. First, we
focus on the difference between fully interactive and sequentially interactive
protocols. Sequentially interactive protocols may query users adaptively in
sequence, but they cannot return to previously queried users. The vast majority
of existing lower bounds for local differential privacy apply only to
sequentially interactive protocols, and before this paper it was not known
whether fully interactive protocols were more powerful. We resolve this
question. First, we classify locally private protocols by their
compositionality, the multiplicative factor by which the sum of a
protocol's single-round privacy parameters exceeds its overall privacy
guarantee. We then show how to efficiently transform any fully interactive
-compositional protocol into an equivalent sequentially interactive protocol
with an blowup in sample complexity. Next, we show that our reduction is
tight by exhibiting a family of problems such that for any , there is a
fully interactive -compositional protocol which solves the problem, while no
sequentially interactive protocol can solve the problem without at least an
factor more examples. We then turn our attention to
hypothesis testing problems. We show that for a large class of compound
hypothesis testing problems --- which include all simple hypothesis testing
problems as a special case --- a simple noninteractive test is optimal among
the class of all (possibly fully interactive) tests
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