Boosting Simple Learners
Noga Alon, Alon Gonen, Elad Hazan, Shay Moran
TL;DR
This work studies boosting when weak hypotheses come from a fixed base-class B with bounded VC dimension, focusing on the impact of base simplicity on oracle complexity and expressivity. It introduces Graph Separation Boosting, achieving an oracle complexity of $\tilde{O}(1/\gamma)$ under γ-realizable conditions, and develops combinatorial dimensions (the $γ$-VC and $γ$-interpolation$\,$ID_γ) to quantify expressivity. The results show universal expressivity for natural base-classes such as halfspaces and decision stumps as $\gamma\to 0$, with connections to discrepancy theory guiding the bounds. A key takeaway is that, with simple base-classes, one can go beyond the classic $Ω(1/\gamma^2)$ barrier for oracle complexity, while carefully designed aggregation rules and discrepancy-based analyses control generalization. The findings offer a principled framework for choosing base-classes and aggregation schemes to balance expressivity, efficiency, and generalization in boosting systems.
Abstract
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 $Ω({1}/{γ^2})$ weak hypotheses with $γ$-margin are sometimes necessary, our new method requires only $\tilde{O}({1}/γ)$ 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.