We present a new ensemble learning algorithm, DeepBoost, which can use as base classifiers a hypothesis set containing deep decision trees, or members of other rich or complex families, and succeed in achieving high accuracy without over- fitting the data. The key to the success of the al-gorithm is a capacity-conscious criterion for the selection of the hypotheses. We give new data- dependent learning bounds for convex ensembles expressed in terms of the Rademacher complexi-ties of the sub-families composing the base classifier set, and the mixture weight assigned to each sub-family. Our algorithm directly benefits from these guarantees since it seeks to minimize the corresponding learning bound. We give a full description of our algorithm, including the details of its derivation, and report the results of several experiments showing that its performance compares favorably to that of AdaBoost and Logistic Regression and their Li -regularized variants.