  Transductive learning considers a training set of $m$ labeled samples and a
test set of $u$ unlabeled samples, with the goal of best labeling that
particular test set. Conversely, inductive learning considers a training set of
$m$ labeled samples drawn iid from $P(X,Y)$, with the goal of best labeling any
future samples drawn iid from $P(X)$. This comparison suggests that
transduction is a much easier type of inference than induction, but is this
really the case? This paper provides a negative answer to this question, by
proving the first known minimax lower bounds for transductive, realizable,
binary classification. Our lower bounds show that $m$ should be at least
$\Omega(d/\epsilon + \log(1/\delta)/\epsilon)$ when $\epsilon$-learning a
concept class $\mathcal{H}$ of finite VC-dimension $d<\infty$ with confidence
$1-\delta$, for all $m \leq u$. This result draws three important conclusions.
First, general transduction is as hard as general induction, since both
problems have $\Omega(d/m)$ minimax values. Second, the use of unlabeled data
does not help general transduction, since supervised learning algorithms such
as ERM and (Hanneke, 2015) match our transductive lower bounds while ignoring
the unlabeled test set. Third, our transductive lower bounds imply lower bounds
for semi-supervised learning, which add to the important discussion about the
role of unlabeled data in machine learning.
