| Summary: | Boosting is a celebrated machine learning approach which is based on the idea
of combining weak and moderately inaccurate hypotheses to a strong and accurate
one. We study boosting under the assumption that the weak hypotheses belong to
a class of bounded capacity. This assumption is inspired by the common
convention that weak hypotheses are "rules-of-thumbs" from an "easy-to-learn
class". (Schapire and Freund~'12, Shalev-Shwartz and Ben-David '14.) Formally,
we assume the class of weak hypotheses has a bounded VC dimension. We focus on
two main questions: (i) Oracle Complexity: How many weak hypotheses are needed
to produce an accurate hypothesis? We design a novel boosting algorithm and
demonstrate that it circumvents a classical lower bound by Freund and Schapire
('95, '12). Whereas the lower bound shows that $\Omega({1}/{\gamma^2})$ weak
hypotheses with $\gamma$-margin are sometimes necessary, our new method
requires only $\tilde{O}({1}/{\gamma})$ weak hypothesis, provided that they
belong to a class of bounded VC dimension. Unlike previous boosting algorithms
which aggregate the weak hypotheses by majority votes, the new boosting
algorithm uses more complex ("deeper") aggregation rules. We complement this
result by showing that complex aggregation rules are in fact necessary to
circumvent the aforementioned lower bound. (ii) Expressivity: Which tasks can
be learned by boosting weak hypotheses from a bounded VC class? Can complex
concepts that are "far away" from the class be learned? Towards answering the
first question we {introduce combinatorial-geometric parameters which capture
expressivity in boosting.} As a corollary we provide an affirmative answer to
the second question for well-studied classes, including half-spaces and
decision stumps. Along the way, we establish and exploit connections with
Discrepancy Theory.
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