Bagging Robustly Learns VC Classes with Linear Sample Complexity

We study the problem of \emph{adversarially robust} PAC learning. In this framework, the learner observes independent samples from an unknown distribution over $\mathcal{X} \times \{0,1\}$, as in classical PAC learning. However, given a perturbation map $\mathcal{U} : \mathcal{X} \to 2^{\mathcal{X}}$ known to the learner, the goal is to output, with high probability, a predictor that correctly classifies \emph{every} perturbation $z \in \mathcal{U}(x)$ of most future examples $(x,y)$ drawn from the same underlying distribution. We determine the \emph{optimal} $\mathcal{U}$-independent sample complexity of this problem in both the realizable and agnostic settings. More specifically, for every concept class $\mathcal{H}$ of $\operatorname{VC}$ dimension $d$, we prove upper bounds of $\mathcal{O} \big( d/ε+ \log(1/δ)/ε\big)$ in the realizable setting and $\mathcal{O} \big( d/ε^2 + \log(1/δ)/ε^2 \big)$ in the agnostic setting, together with an optimal first-order refinement of the latter. These bounds match the corresponding lower bounds for classical PAC learning. Consequently, and perhaps surprisingly, adversarial robustness incurs \emph{no additional} distribution-free statistical cost, uniformly over all perturbation maps. Our bounds improve exponentially on those of [Montasser, Hanneke, and Srebro; COLT '19]. On the technical side, we present short and elementary proofs based on a new algorithmic principle that we call \emph{binomial-bagging}. We believe that binomial-bagging and its analysis may be of independent interest.

A monotone adversary observes an i.i.d. labeled sample and appends a finite number of further examples of its choice, every one of them labeled correctly by the target hypothesis. The learner sees a uniform shuffle of the combined sample and is scored on the original distribution. Every example is correctly labeled, but the insertions depend on the clean sample, so the combined sample is not exchangeable. Larsen, Pabbaraju, and Shetty, who introduced this model, showed that empirical risk minimization attains expected error $O((d/n)\log(n/d))$ for classes of VC dimension $d$, and that every known optimal learner can be pushed away from the $Θ(d/n)$ rate, optimal for PAC learning. They asked whether the extra logarithm is an artifact of those particular algorithms or an inherent consequence of the lack of exchangeability. We show that this additional cost is inherent beyond VC dimension one. In the worst case over classes of VC dimension $d$ and over known finite insertion budgets, the minimax expected error is $Θ(1/n)$ at $d=1$ and $Θ((d/n)\log(n/d))$ for $d\geq 2$. The same rates hold with Littlestone dimension $d_{\mathrm L}$ in place of $d$, so the clean online-to-batch rate $O(d_{\mathrm L}/n)$ is unattainable as well. Thus, somewhat counterintuitively, adding correctly labeled examples can make learning harder by a logarithmic factor, even for classes that admit finite mistake bounds in online learning. The dimension-one upper bound is achieved by a simple improper learner whose analysis adapts the leave-one-out argument underlying the one-inclusion graph. All of our lower bounds are elementary and come from a single construction: an explicit class and prior on which two target hypothesis, which differ a point of nonnegligible mass, produce the same sample.

In this paper we show that the generalization error of AdaBoost is $Θ\big(\tfrac{d\ln(nγ^{2}/d)}{nγ^2}+\tfrac{\ln(1/δ)}{n}\big)$, where $γ$ is the advantage guaranteed by the weak learner, $d$ is the VC-dimension of the class containing the weak hypotheses, $n$ is the sample size, and $δ$ is the confidence parameter. The contribution of this paper is the upper bound; the matching lower bound follows from prior work. The upper bound proof follows by combining the known fact that AdaBoost outputs a voting classifier whose voting function has zero empirical $γ/2$-margin loss with what is, to the best of our knowledge, a new margin-based generalization bound for voting classifiers.