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Double descent refers to the phase transition that is exhibited by thegeneralization error of unregularized learning models when varying the ratiobetween the number of parameters and the number of training samples. The recentsuccess of highly over-parameterized machine learning models such as deepneural networks has motivated a theoretical analysis of the double descentphenomenon in classical models such as linear regression which can alsogeneralize well in the over-parameterized regime. We provide the first exactnon-asymptotic expressions for double descent of the minimum norm linearestimator. Our approach involves constructing a special determinantal pointprocess which we call surrogate random design, to replace the standard i.i.d.design of the training sample. This surrogate design admits exact expressionsfor the mean squared error of the estimator while preserving the key propertiesof the standard design. We also establish an exact implicit regularizationresult for over-parameterized training samples. In particular, we show that,for the surrogate design, the implicit bias of the unregularized minimum normestimator precisely corresponds to solving a ridge-regularized least squaresproblem on the population distribution. In our analysis we introduce a newmathematical tool of independent interest: the class of random matrices forwhich determinant commutes with expectation.

Exact expressions for double descent and implicit regularization via surrogate random design