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For linear inverse problems $Y=\mathsf{A}\mu+\xi$, it is classical to recover the unknown signal $\mu$ by iterative regularisation methods $(\widehat \mu^{(m)}, m=0,1,\ldots)$ and halt at a data-dependent iteration $\tau$ using some stopping rule, typically based on a discrepancy principle, so that the weak (or prediction) squared-error $\|\mathsf{A}(\widehat \mu^{(\tau)}-\mu)\|^2$ is controlled. In the context of statistical estimation with stochastic noise $\xi$, we study oracle adaptation (that is, compared to the best possible stopping iteration) in strong squared-error $E[\|\hat \mu^{(\tau)}-\mu\|^2]$. For a residual-based stopping rule oracle adaptation bounds are established for general spectral regularisation 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.

Optimal Adaptation for Early Stopping in Statistical Inverse Problems