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 tildeOleft(fracVert w^*Vert√n + minleftfracVert w^* Vert2(nε)2/3,frac√dVert w^*Vert2nεrightright), 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^astVert, our bound is essentially tight in all parameters. In particular, we show a lower bound of tildeOmegaleft(frac1√n + minleftfracVert w^*Vert4/3(nε)2/3, frac√dVert w^*Vertnεrightright). 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) Thetaleft(fracVert w^*Vert√n + minleftfracVert w^*Vert√nε,frac√rankVert w^*Vertnεrightright), where 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.