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Article overview
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Landscape Complexity for the Empirical Risk of Generalized Linear Models | | Antoine Maillard
; Gérard Ben Arous
; Giulio Biroli
; | | Date: |
4 Dec 2019 | | Abstract: | We present a method to obtain the average and the typical value of the number
of critical points of the empirical risk landscape for generalized linear
estimation problems and variants. This represents a substantial extension of
previous applications of the Kac-Rice method since it allows to analyze the
critical points of high dimensional non-Gaussian random functions. We obtain a
rigorous explicit variational formula for the annealed complexity, which is the
logarithm of the average number of critical points at fixed value of the
empirical risk. This result is simplified, and extended, using the non-rigorous
Kac-Rice replicated method from theoretical physics. In this way we find an
explicit variational formula for the quenched complexity, which is generally
different from its annealed counterpart, and allows to obtain the number of
critical points for typical instances up to exponential accuracy. | | Source: | arXiv, 1912.2143 | | Services: | Forum | Review | PDF | Favorites | |
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