Finite-Sample Analysis Of M-Estimators Using Self-Concordance

The classical asymptotic theory for parametric M-estimators guarantees that, in the limit of infinite sample size, the excess risk has a chi-square type distribution, even in the misspecified case. We demonstrate how self-concordance of the loss allows to characterize the critical sample size sufficient to guarantee a chi-square type in-probability bound for the excess risk. Specifically, we consider two classes of losses: (i) self-concordant losses in the classical sense of Nesterov and Nemirovski, i.e., whose third derivative is uniformly bounded with the 3/2 power of the second derivative; (ii) pseudo self-concordant losses, for which the power is removed. These classes contain losses corresponding to several generalized linear models, including the logistic loss and pseudo-Huber losses.Our basic result under minimal assumptions bounds the critical sample size by O(d . d(eff)), where d the parameter dimension and d(eff) the effective dimension that accounts for model misspecification. In contrast to the existing results, we only impose local assumptions that concern the population risk minimizer theta(*). Namely, we assume that the calibrated predictors, i.e., predictors scaled by the square root of the second derivative of the loss, is subgaussian at theta(*). Besides, for type-ii losses we require boundedness of certain measure of curvature of the population risk at theta(*).Our improved result bounds the critical sample size from above asO(max{d(eff), d log d})under slightly stronger assumptions. Namely, the local assumptions must hold in the neighborhood of theta(*) given by the Dikin ellipsoid of the population risk. Interestingly, we find that, for logistic regression with Gaussian design, there is no actual restriction of conditions: the subgaussian parameter and curvature measure remain near-constant over the Dikin ellipsoid. Finally, we extend some of these results to l(1)-penalized estimators in high dimensions.