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Finite-sample Analysis of M-estimators using Self-concordance. (arXiv:1810.06838v1 [math.ST])
来源于:arXiv
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
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