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.