Papers › Optimal adaptation for early stopping in statistical inverse problems
Optimal adaptation for early stopping in statistical inverse problems
Gilles Blanchard, Marc Hoffmann, Markus Reiß
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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,…) and halt at a data-dependent iteration τ using some stopping rule, typically based on a discrepancy principle, so that the weak (or prediction) squared-error 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 squared-error E[μ̂^((τ))-μ²]. 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 L²-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.
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