Developing simple, sample-efficient learning algorithms for robust classification is a pressing issue in today's tech-dominated world, and current theoretical techniques requiring exponential sample complexity and complicated improper learning rules fall far from answering the need. In this work we study the fundamental paradigm of (robust) empirical risk minimization (RERM), a simple process in which the learner outputs any hypothesis minimizing its training error. RERM famously fails to robustly learn VC classes (Montasser et al., 2019a), a bound we show extends even to `nice' settings such as (bounded) halfspaces. As such, we study a recent relaxation of the robust model called tolerant robust learning (Ashtiani et al., 2022) where the output classifier is compared to the best achievable error over slightly larger perturbation sets. We show that under geometric niceness conditions, a natural tolerant variant of RERM is indeed sufficient for γ-tolerant robust learning VC classes over ℝd, and requires only Õ (VC(H)dlogDγδϵ2) samples for robustness regions of (maximum) diameter D.
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Boosting Simple Learners
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 in order 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/γ2) weak hypotheses with γ-margin are sometimes necessary, our new method requires only Õ(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 identify a combinatorial-geometric parameter which captures the expressivity of base-classes in boosting. As a corollary we provide an affirmative answer to the second question for many well-studied classes, including half-spaces and decision stumps. Along the way, we establish and exploit connections with Discrepancy Theory.
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- Award ID(s):
- 1855464
- PAR ID:
- 10338582
- Date Published:
- Journal Name:
- STOC '21: 53rd Annual ACM SIGACT Symposium on Theory of Computing
- Page Range / eLocation ID:
- 481 to 489
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Vapnik-Chervonenkis (VC) theory has so far been unable to explain the small generalization error of overparametrized neural networks. Indeed, existing applications of VC theory to large networks obtain upper bounds on VC dimension that are proportional to the number of weights, and for a large class of networks, these upper bound are known to be tight. In this work, we focus on a subclass of partially quantized networks that we refer to as hyperplane arrangement neural networks (HANNs). Using a sample compression analysis, we show that HANNs can have VC dimension significantly smaller than the number of weights, while being highly expressive. In particular, empirical risk minimization over HANNs in the overparametrized regime achieves the minimax rate for classification with Lipschitz posterior class probability. We further demonstrate the expressivity of HANNs empirically. On a panel of 121 UCI datasets, overparametrized HANNs match the performance of state-of-the-art full-precision models.more » « less
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/10.1145/3406325.3451030
