Differentially Private Generalized Linear Models Revisited

Part of Advances in Neural Information Processing Systems 35 (NeurIPS 2022) Main Conference Track

Bibtex Paper Supplemental


Raman Arora, Raef Bassily, Cristóbal Guzmán, Michael Menart, Enayat Ullah


We study the problem of $(\epsilon,\delta)$-differentially private learning of linear predictors with convex losses. We provide results for two subclasses of loss functions. The first case is when the loss is smooth and non-negative but not necessarily Lipschitz (such as the squared loss). For this case, we establish an upper bound on the excess population risk of $\tilde{O}\left(\frac{\Vert w^*\Vert}{\sqrt{n}} + \min\left\{\frac{\Vert w^* \Vert^2}{(n\epsilon)^{2/3}},\frac{\sqrt{d}\Vert w^*\Vert^2}{n\epsilon}\right\}\right)$, where $n$ is the number of samples, $d$ is the dimension of the problem, and $w^*$ is the minimizer of the population risk. Apart from the dependence on $\Vert w^\ast\Vert$, our bound is essentially tight in all parameters. In particular, we show a lower bound of $\tilde{\Omega}\left(\frac{1}{\sqrt{n}} + {\min\left\{\frac{\Vert w^*\Vert^{4/3}}{(n\epsilon)^{2/3}}, \frac{\sqrt{d}\Vert w^*\Vert}{n\epsilon}\right\}}\right)$. We also revisit the previously studied case of Lipschitz losses \cite{SSTT21}. For this case, we close the gap in the existing work and show that the optimal rate is (up to log factors) $\Theta\left(\frac{\Vert w^*\Vert}{\sqrt{n}} + \min\left\{\frac{\Vert w^*\Vert}{\sqrt{n\epsilon}},\frac{\sqrt{\text{rank}}\Vert w^*\Vert}{n\epsilon}\right\}\right)$, where $\text{rank}$ is the rank of the design matrix. This improves over existing work in the high privacy regime. Finally, our algorithms involve a private model selection approach that we develop to enable attaining the stated rates without a-priori knowledge of $\Vert w^*\Vert$.