How many AdaBoost iterations? In one R package, ada, the main AdaBoost fitting function (also called ada) takes an argument specifying the number of boosting iterations to perform, with the default being 50.
How many iterations should I use? This question has been asked before with no answer. I am also aware of a paper on the subject but I'm not sure how to apply the results (e.g. how do I choose $\epsilon$)?
 A: The answer seems to be that it depends, based on the problem and on how you interpret the AdaBoost algorithm.
Mease and Wyner (2008) argue that AdaBoost should be run for a long time, until it converges, and that 1,000 iterations should be enough. The main point of the paper is that the intuition gained from the "Statistical view of boosting" due to Friedman, Hastie, and Tibshirani (2000) could be incorrect, and that this applies to their recommendations to regularize AdaBoost with the $\nu$ parameter and early stopping times. Their results suggest that long running times, no regularization, and deep trees as base learners make use of AdaBoost's strengths.
The response to that paper by Bennett demonstrates that AdaBoost converges more quickly and is more resistant to overfitting when deeper trees are used. Bennett concludes that 

For slowly converging problems, AdaBoost will frequently be regularized by early stopping

and that

For more rapidly converging problems, AdaBoost will converge and enter an overtraining phase

where "overtraining" does not mean "overfitting" but rather "further improvement of out-of-sample performance after convergence."
Friedman, Hastie, and Tibshirani respond to Mease and Wyner with a demonstration that AdaBoost with decision stumps, shrinkage, and early stopping can perform better than deep trees, no shrinkage and running until convergence, but their argument depended on running the algorithm to convergence anyway. That is, even if 100 or 10 iterations are optimal, it is necessary to run many hundred iterations to find out where the correct stopping point should be. As user Matthew Drury pointed out in the comments, this only requires a single long run of the algorithm.
The paper itself, as well as the rejoinder to the response by Bickel and Ritov, emphasize that in low-dimensional problems AdaBoost does badly overfit and that early stopping is necessary; it seems that ten dimensions is enough for this to no longer be an issue.
The response by Buja and Stuetzle raises the possibility that the first few iterations of AdaBoost reduce bias and that the later iterations reduce variance, and offer that the Mease and Wyner approach of starting with a relatively unbiased but high-variance base learner like an 8-node tree makes sense for that reason. Therefore if low-variance predictions are desired, a longer running time could be useful.
Therefore what I am going to do is cross-validate over the following grid:


*

*$\nu \in \{0.1, 1.0\}$

*$\text{tree depth} \in \{1, 2, 3, 4\}$

*$\text{iterations} \in \{1, \dots, 1000\}$ (but only ever fitting with 1000 iterations)


Basically, I'm looking into both approaches. This is partly because they both seem valid, and partly because many of my predictors are discrete, in which case the "statistical view" apparently applies more directly (although I wish I had a better understanding of how and why).
