  We propose a totally corrective boosting algorithm with explicit cardinality
regularization. The resulting combinatorial optimization problems are not known
to be efficiently solvable with existing classical methods, but emerging
quantum optimization technology gives hope for achieving sparser models in
practice. In order to demonstrate the utility of our algorithm, we use a
distributed classical heuristic optimizer as a stand-in for quantum hardware.
Even though this evaluation methodology incurs large time and resource costs on
classical computing machinery, it allows us to gauge the potential gains in
generalization performance and sparsity of the resulting boosted ensembles. Our
experimental results on public data sets commonly used for benchmarking of
boosting algorithms decidedly demonstrate the existence of such advantages. If
actual quantum optimization were to be used with this algorithm in the future,
we would expect equivalent or superior results at much smaller time and energy
costs during training. Moreover, studying cardinality-penalized boosting also
sheds light on why unregularized boosting algorithms with early stopping often
yield better results than their counterparts with explicit convex
regularization: Early stopping performs suboptimal cardinality regularization.
The results that we present here indicate it is beneficial to explicitly solve
the combinatorial problem still left open at early termination.
