Differentially Private Generalized Linear Models Revisited
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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 Õ(‖ w^*‖/√(n) + min{‖ w^* ‖^2/(nϵ)^2/3,√(d)‖ w^*‖^2/nϵ}), 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 ‖ w^∗‖, our bound is essentially tight in all parameters. In particular, we show a lower bound of Ω̃(1/√(n) + min{‖ w^*‖^4/3/(nϵ)^2/3, √(d)‖ w^*‖/nϵ}). 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(‖ w^*‖/√(n) + min{‖ w^*‖/√(nϵ),√(rank)‖ w^*‖/nϵ}), 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 ‖ w^*‖.
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