  We study the cost of parallelizing weak-to-strong boosting algorithms for
learning, following the recent work of Karbasi and Larsen. Our main results are
two-fold:
  - First, we prove a tight lower bound, showing that even "slight"
parallelization of boosting requires an exponential blow-up in the complexity
of training.
  Specifically, let $\gamma$ be the weak learner's advantage over random
guessing. The famous \textsc{AdaBoost} algorithm produces an accurate
hypothesis by interacting with the weak learner for $\tilde{O}(1 / \gamma^2)$
rounds where each round runs in polynomial time.
  Karbasi and Larsen showed that "significant" parallelization must incur
exponential blow-up: Any boosting algorithm either interacts with the weak
learner for $\Omega(1 / \gamma)$ rounds or incurs an $\exp(d / \gamma)$ blow-up
in the complexity of training, where $d$ is the VC dimension of the hypothesis
class. We close the gap by showing that any boosting algorithm either has
$\Omega(1 / \gamma^2)$ rounds of interaction or incurs a smaller exponential
blow-up of $\exp(d)$.
  -Complementing our lower bound, we show that there exists a boosting
algorithm using $\tilde{O}(1/(t \gamma^2))$ rounds, and only suffer a blow-up
of $\exp(d \cdot t^2)$.
  Plugging in $t = \omega(1)$, this shows that the smaller blow-up in our lower
bound is tight. More interestingly, this provides the first trade-off between
the parallelism and the total work required for boosting.
