Optimal Adaptation for Early Stopping in Statistical Inverse Problems

Abstract

For linear inverse problems Y = Aµ + ξ, it is classical to recover the unknown signal µ by iterative regularization methods (µ (m) , m = 0, 1,. . .) and halt at a data-dependent iteration τ using some stopping rule, typically based on a discrepancy principle, so that the weak (or prediction) squarederror A(µ (τ) − µ) 2 is controlled. In the context of statistical estimation with stochastic noise ξ, we study oracle adaptation (that is, compared to the best possible stopping iteration) in strong squarederror E µ (τ) − µ 2. For a residual-based stopping rule oracle adaptation bounds are established for general spectral regularization 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. Adaptive early stopping for the Landweber method is studied in further detail and illustrated numerically.