Four accuracy bounds and one estimator for frequency estimation under local differential privacy
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We present four lower bounds on the mean squared error of both frequency and distribution estimation under Local Differential Privacy (LDP). Each of these bounds is stated in a different mathematical `language', making them all interesting in their own right. Two are particularly important: First, for a given LDP protocol, we give formulas for the optimal frequency and distribution estimators for the aggregator. This theoretically `solves' the issue of postprocessing, although it may be computationally infeasible in practice. We present methods to approximate the optimal estimator, yielding an estimator that outperforms state of the art methods. Second, we give a lower bound on MSE in terms of the LDP parameter ε, giving a quantitative statement of the privacy-utility tradeoff in LDP. In the high privacy domain, this improves on existing results by giving tighter constants even where earlier results consider a worst-case error metric, or giving constants where up to now, no constants were known.
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