  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 $O(d \cdot
d_{\text{eff}}),$ where $d$ the parameter dimension and $d_{\text{eff}}$ 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
design, i.e., design 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 a certain measure of curvature of the population risk at
$\theta_*$.Our improved result bounds the critical sample size from above as
$O(\max\{d_{\text{eff}}, d \log d\})$ 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 $\ell_1$-penalized estimators in high dimensions.
