Private Isotonic Regression

• URL: http://arxiv.org/abs/2210.15175v1
• Date: Thu, 27 Oct 2022 05:08:07 GMT
• Title: Private Isotonic Regression
• Authors: Badih Ghazi and Pritish Kamath and Ravi Kumar and Pasin Manurangsi
• Abstract summary: We consider the problem of isotonic regression over a partially ordered set (poset) $mathcalX$ and for any Lipschitz loss function. We obtain a pure-DP algorithm that has an expected excess empirical risk of roughly $mathrmwidth(mathcalX) cdot log|mathcalX| / n$, where $mathrmwidth(mathcalX)$ is the width of the poset. We show that the bounds above are essentially the best that can be
• Score: 54.32252900997422
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: In this paper, we consider the problem of differentially private (DP) algorithms for isotonic regression. For the most general problem of isotonic regression over a partially ordered set (poset) $\mathcal{X}$ and for any Lipschitz loss function, we obtain a pure-DP algorithm that, given $n$ input points, has an expected excess empirical risk of roughly $\mathrm{width}(\mathcal{X}) \cdot \log|\mathcal{X}| / n$, where $\mathrm{width}(\mathcal{X})$ is the width of the poset. In contrast, we also obtain a near-matching lower bound of roughly $(\mathrm{width}(\mathcal{X}) + \log |\mathcal{X}|) / n$, that holds even for approximate-DP algorithms. Moreover, we show that the above bounds are essentially the best that can be obtained without utilizing any further structure of the poset. In the special case of a totally ordered set and for $\ell_1$ and $\ell_2^2$ losses, our algorithm can be implemented in near-linear running time; we also provide extensions of this algorithm to the problem of private isotonic regression with additional structural constraints on the output function.

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