We demonstrate how self-concordance of the loss can be exploited to obtain
asymptotically optimal rates for M-estimators in finite-sample regimes. We
consider two classes of losses: (i) canonically self-concordant losses in the
sense of Nesterov and Nemirovski (1994), i.e., with the third derivative
bounded with the $3/2$ power of the second; (ii) pseudo self-concordant losses,
for which the power is removed, as introduced by Bach (2010)...
These classes
contain some losses arising in generalized linear models, including logistic
regression; in addition, the second class includes some common pseudo-Huber
losses. Our results consist in establishing the critical sample size sufficient
to reach the asymptotically optimal excess risk for both classes of losses. Denoting $d$ the parameter dimension, and $d_{\text{eff}}$ the effective
dimension which takes into account possible model misspecification, we find the
critical sample size to be $O(d_{\text{eff}} \cdot d)$ for canonically
self-concordant losses, and $O(\rho \cdot d_{\text{eff}} \cdot d)$ for pseudo
self-concordant losses, where $\rho$ is the problem-dependent local curvature
parameter. In contrast to the existing results, we only impose local
assumptions on the data distribution, assuming that the calibrated design,
i.e., the design scaled with the square root of the second derivative of the
loss, is subgaussian at the best predictor $\theta_*$. Moreover, we obtain the
improved bounds on the critical sample size, scaling near-linearly in
$\max(d_{\text{eff}},d)$, under the extra assumption that the calibrated design
is subgaussian in the Dikin ellipsoid of $\theta_*$. Motivated by these
findings, we construct canonically self-concordant analogues of the Huber and
logistic losses with improved statistical properties. Finally, we extend some
of these results to $\ell_1$-regularized M-estimators in high dimensions.
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Abstract