This work presents a new boosting algorithm that, for any class $\mathcal{F}$ closed under $O(\log \frac{1}{\gamma})$-XOR, strong learns $\mathcal{F}$ using $O(\log \frac{1}{\epsilon})$ calls to a $\gamma$-advantage weak learner and a single batch of $\tilde{O}(\log(\frac{1}{\epsilon
Abstract
Boosting is a fundamental technique for generically improving the accuracy of learning algorithms (Schapire 1989). Existing boosting algorithms construct a strong learner using $O(\log(\frac{1}{\epsilon})/\gamma^2)$ calls to a $\gamma$-advantage weak learner, and this round complexity is known to be optimal for generic boosters that succeed on all concept classes (Freund 1995). We show that this lower bound can be circumvented for concept classes that satisfy a mild closure property. Specifically, we present a new boosting algorithm that, for any class $\mathcal{F}$ closed under $O(\log \frac{1}{\gamma})$-XOR, strong learns $\mathcal{F}$ using $O(\log \frac{1}{\epsilon})$ calls to a $\gamma$-advantage weak learner and a single batch of $\tilde{O}(\log(\frac{1}{\epsilon})/\gamma^2)$ additional samples. Our algorithm arises from a new and simple connection between boosting and list-decodable codes. Viewing the target function as a message, we run the weak learner on its encoding and view the resulting weak hypothesis as a corrupted codeword. Feeding this corrupted codeword to a list decoder, we obtain a small list of candidate hypotheses, at least one of which is a strong hypothesis for the original function. Using additional samples, we identify and output this strong hypothesis.
The upper bound proof follows by combining the known fact that AdaBoost outputs a voting classifier whose voting function has zero empirical $\gamma/2$-margin loss with what is, to the best of the authors' knowledge, a new margin-based generalization bound for voting classifiers.
{AdaBoost.MH} reduces multi-class classification to a collection of binary subproblems and enjoys the classical boosting-type convergence guarantee under a weak learning condition. A more structured variant, Factorized {AdaBoost.MH}, uses base classifiers of the form $\mathbf{h}(x)=\alpha \mathbf{v} \bm{\varphi}(x)$, where a single binary classifier $\bm{\varphi}$ is shared across all classes and the label dependence is carried by a vote vector $\mathbf{v} \in\{\pm1\}^K$. This factorization is algorithmically attractive and achieves better performance in practice, but its convergence depends on whether one can always choose a vote vector with sufficiently large induced binary weight mass. Previous work resolved this question with a lower bound $\max\{1/n,1/\sqrt{2K}\}$, which still leaves a dimension-dependent slowdown relative to the original {AdaBoost.MH} analysis. In this paper, we sharpen this combinatorial step. For the minimax quantity $\mathfrak{W}_{n,K}$ governing the factorized edge, we prove $\mathfrak{W}_{n,K} = C_{\min\{n+1,K\}}$, where $C_q=1$ for $q=1$, $C_q=q/(3q-4)$ for even $q\ge2$, and $C_q=(q+1)/(3q-1)$ for odd $q\ge2$. Since $C_q\downarrow 1/3$, our bounds show that $\mathfrak{W}_{n,K}=\Theta(1)$ uniformly over $n$ and $K$. Consequently, Factorized {AdaBoost.MH} achieves the same boosting-type convergence rate as {AdaBoost.MH} up to a universal constant factor, removing the previously suggested additional dependence on $n$ or $K$ in the number of boosting rounds.
This work improves the upper bound to match the best known lower bound, thus establishing the optimal error guarantee for learning under Tsybakov noise.
Steve Hanneke, Hongao Wang, Mingyue Xu· 0 citations
This paper settles the sample complexity of agnostic PAC learning up to universal constants at every fixed $L^*$, matching the lower bounds of Devroye, Gyorfi, and Lugosi.
Markus Engelund Mathiasen, Jian Qian, Nikita Zhivotovskiy· 0 citations
Worst-case multiclass bounds do not become smaller when the best classifier is already nearly correct: what is missing is an optimistic rate, a guarantee whose fluctuation scales with the oracle risk itself. For a class of Natarajan dimension $d_N$ and Daniely-Shalev-Shwartz dimension $d_{DS}$, the optimal excess risk is known at the two endpoints ($d_{DS}/n$ realizable, $\sqrt{d_N/n}+d_{DS}/n$ agnostic [HMZ24, CEH+26, Pab26]) and open in between. We close the gap: at every fixed oracle risk $L^\star$, the optimal excess risk is $\widetilde{\Theta}(\sqrt{L^\star d_N/n}+d_{DS}/n)$, uniformly in the alphabet size, attained by a learner that knows neither $L^\star$ nor the confidence level. The upper bound composes the cover-menu-compression architecture of [CEH+26], at the realizable rate of [Pab26], with a new comparator-facing relative compression theorem: a size-$k$ compression rule that empirically dominates a comparator $h$ has population risk at most $L(h)+O(\sqrt{L(h)\Gamma}+\Gamma)$ with $\Gamma=(k\log n+\log(1/\delta))/n$, without stability; this transfers the comparison principle of the sharp binary theory [MQZ26] while discarding its Boolean-cube geometry, which does not lift to multiclass labels. The lower bound forces both terms using one class and one distribution at every fixed $L^\star$, by a pair-Assouad scheme calibrated to $L^\star$ and a fiber argument on the pseudo-cubes underlying the Natarajan-versus-DS separation of [BCD+22]. Both theorems extend to list learning: against the best $r$-tuple of hypotheses, the same architecture and the same two engines yield an optimistic rate and a lower bound of the same shape, forcing the fluctuation term that [Pab26] expected to be necessary against list comparators, and removing the factor $r$ from the known realizable list lower bound.
Xiaoyu Li, Andi Han, Jiaojiao Jiang et al.· 1 citation
This analysis reveals a scale-sensitive interaction between the statistical estimation of classification error and its amplification by robustness, sharply explaining the transition in the agnostic rate.
Elad Aigner-Horev, Daniel Rosenberg, Roi Weiss· 1 citation· ⚡1