  Increasingly high-dimensional data sets require that estimation methods do
not only satisfy statistical guarantees but also remain computationally
feasible. In this context, we consider $ L^{2} $-boosting via orthogonal
matching pursuit in a high-dimensional linear model and analyze a data-driven
early stopping time $ \tau $ of the algorithm, which is sequential in the sense
that its computation is based on the first $ \tau $ iterations only. This
approach is much less costly than established model selection criteria, that
require the computation of the full boosting path. We prove that sequential
early stopping preserves statistical optimality in this setting in terms of a
fully general oracle inequality for the empirical risk and recently established
optimal convergence rates for the population risk. Finally, an extensive
simulation study shows that at an immensely reduced computational cost, the
performance of these type of methods is on par with other state of the art
algorithms such as the cross-validated Lasso or model selection via a high
dimensional Akaike criterion based on the full boosting path.
