  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.
