  This paper establishes a precise high-dimensional asymptotic theory for
boosting on separable data, taking statistical and computational perspectives.
We consider a high-dimensional setting where the number of features (weak
learners) $p$ scales with the sample size $n$, in an overparametrized regime.
Under a class of statistical models, we provide an exact analysis of the
generalization error of boosting when the algorithm interpolates the training
data and maximizes the empirical $\ell_1$-margin. Further, we explicitly pin
down the relation between the boosting test error and the optimal Bayes error,
as well as the proportion of active features at interpolation (with zero
initialization). In turn, these precise characterizations answer certain
questions raised in \cite{breiman1999prediction, schapire1998boosting}
surrounding boosting, under assumed data generating processes. At the heart of
our theory lies an in-depth study of the maximum-$\ell_1$-margin, which can be
accurately described by a new system of non-linear equations; to analyze this
margin, we rely on Gaussian comparison techniques and develop a novel uniform
deviation argument. Our statistical and computational arguments can handle (1)
any finite-rank spiked covariance model for the feature distribution and (2)
variants of boosting corresponding to general $\ell_q$-geometry, $q \in [1,
2]$. As a final component, via the Lindeberg principle, we establish a
universality result showcasing that the scaled $\ell_1$-margin (asymptotically)
remains the same, whether the covariates used for boosting arise from a
non-linear random feature model or an appropriately linearized model with
matching moments.
