Differentially Private Generalized Linear Models Revisited
We study the problem of $(ε,δ)$-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 $\tilde{O}\left(\frac{\Vert w^*\Vert}{\sqrt{n}} + \min\left\{\frac{\Vert w^* \Vert^2}{(nε)^{2/3}},\frac{\sqrt{d}\Vert w^*\Vert^2}{nε}\right\}\right)$, 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 $\tildeΩ\left(\frac{1}{\sqrt{n}} + {\min\left\{\frac{\Vert w^*\Vert^{4/3}}{(nε)^{2/3}}, \frac{\sqrt{d}\Vert w^*\Vert}{nε}\right\}}\right)$. 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) $Θ\left(\frac{\Vert w^*\Vert}{\sqrt{n}} + \min\left\{\frac{\Vert w^*\Vert}{\sqrt{nε}},\frac{\sqrt{\text{rank}}\Vert w^*\Vert}{nε}\right\}\right)$, 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 $\Vert w^*\Vert$.