  We show that top-down decision tree learning heuristics are amenable to
highly efficient learnability estimation: for monotone target functions, the
error of the decision tree hypothesis constructed by these heuristics can be
estimated with polylogarithmically many labeled examples, exponentially smaller
than the number necessary to run these heuristics, and indeed, exponentially
smaller than information-theoretic minimum required to learn a good decision
tree. This adds to a small but growing list of fundamental learning algorithms
that have been shown to be amenable to learnability estimation.
  En route to this result, we design and analyze sample-efficient minibatch
versions of top-down decision tree learning heuristics and show that they
achieve the same provable guarantees as the full-batch versions. We further
give "active local" versions of these heuristics: given a test point $x^\star$,
we show how the label $T(x^\star)$ of the decision tree hypothesis $T$ can be
computed with polylogarithmically many labeled examples, exponentially smaller
than the number necessary to learn $T$.
