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**Average stability is invariant to data preconditioning. Implications to exp-concave empirical risk minimization.**
*(English)*
Zbl 1473.62257

Summary: We show that the average stability notion introduced by M. Kearns and D. Ron [“Algorithmic stability and sanity-check bounds for leave-one-out cross-validation”, Neural Comput. 11, No. 6, 1427–1453 (1999; doi:10.1162/089976699300016304)], O. Bousquet and A. Elisseeff [J. Mach. Learn. Res. 2, No. 3, 499–526 (2002; Zbl 1007.68083)] is invariant to data preconditioning, for a wide class of generalized linear models that includes most of the known exp-concave losses. In other words, when analyzing the stability rate of a given algorithm, we may assume the optimal preconditioning of the data. This implies that, at least from a statistical perspective, explicit regularization is not required in order to compensate for ill-conditioned data, which stands in contrast to a widely common approach that includes a regularization for analyzing the sample complexity of generalized linear models. Several important implications of our findings include: a) We demonstrate that the excess risk of empirical risk minimization (ERM) is controlled by the preconditioned stability rate. This immediately yields a relatively short and elegant proof for the fast rates attained by ERM in our context. b) We complement the recent bounds of M. Hardt, B. and Y. Singer [“Train faster, generalize better: stability of stochastic gradient descent”, Preprint, arXiv:1509.01240] on the stability rate of the Stochastic Gradient Descent algorithm.

### MSC:

62J12 | Generalized linear models (logistic models) |

68T05 | Learning and adaptive systems in artificial intelligence |

### Keywords:

average stability notion### Citations:

Zbl 1007.68083
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\textit{A. Gonen} and \textit{S. Shalev-Shwartz}, J. Mach. Learn. Res. 18, Paper No. 222, 13 p. (2018; Zbl 1473.62257)

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