  Data-dependent greedy algorithms in kernel spaces are known to provide fast
converging interpolants, while being extremely easy to implement and efficient
to run. Despite this experimental evidence, no detailed theory has yet been
presented. This situation is unsatisfactory especially when compared to the
case of the data-independent $P$-greedy algorithm, for which optimal
convergence rates are available, despite its performances being usually
inferior to the ones of target data-dependent algorithms.
  In this work we fill this gap by first defining a new scale of greedy
algorithms for interpolation that comprises all the existing ones in a unique
analysis, where the degree of dependency of the selection criterion on the
functional data is quantified by a real parameter. We then prove new
convergence rates where this degree is taken into account and we show that,
possibly up to a logarithmic factor, target data-dependent selection strategies
provide faster convergence.
  In particular, for the first time we obtain convergence rates for target data
adaptive interpolation that are faster than the ones given by uniform points,
without the need of any special assumption on the target function. The rates
are confirmed by a number of examples.
  These results are made possible by a new analysis of greedy algorithms in
general Hilbert spaces.
