Optimal adaptation for early stopping in statistical inverse problems
Blanchard, Gilles; Hoffmann, Marc; Reiß, Markus (2018), Optimal adaptation for early stopping in statistical inverse problems, SIAM/ASA Journal on Uncertainty Quantification, 6, 3, p. 1043-1075. 10.1137/17M1154096
TypeArticle accepté pour publication ou publié
External document linkhttps://hal.archives-ouvertes.fr/hal-01426253
Journal nameSIAM/ASA Journal on Uncertainty Quantification
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Institute for Mathematics, University of Potsdam
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Institute for Mathematics, Humboldt university
Abstract (EN)For linear inverse problems Y = Aµ + ξ, it is classical to recover the unknown signal µ by iterative regularisation methods (µ (m) , m = 0, 1,. . .) so that the weak (or prediction) error A(µ (τ) − µ) 2 is controlled for some early stopping rule τ based on a discrepancy principle. In the context of statistical estimation with stochastic noise ξ, we study oracle adaptation in strong squared-error E µ (τ) − µ 2. We give precise lower bounds for estimation by early stopping. For a stopping rule based on the residual process oracle adaptation bounds are established for general linear iterative methods. The proofs use bias and variance transfer techniques from weak prediction error to strong L 2-error as well as convexity arguments and concentration bounds for the stochastic part. For Sobolev balls the adaptation bounds are shown to match the lower bounds. Adaptive early stopping for the Landweber and spectral cutoff methods are studied in further detail.
Subjects / KeywordsLinear inverse problems; Early stopping; Discrepancy principle; Adaptive estimation; Oracle inequalities
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