Differentially Private Generalized Linear Models Revisited

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 $\tilde{O}\left(\frac{\Vert w^{*\Vert}{\sqrt{n}}} + \min\left{\frac{\Vert w^* \Vert^{2}{(n\epsilon){2/3}},\frac{\sqrt{d}\Vert} w^{*\Vert2}{n\epsilon}\right}\right)$,} 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 [SSTT20]. 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 $\Vert w^*\Vert$.