**Keywords:**Active learning, Clustering, Multiclass classification

**TL;DR:**We show that, under a generalized notion of margin, combining two types of queries in active multi-class classification yields a polynomial time algorithm with exponential savings in query complexity.

**Abstract:**We study exact active learning of binary and multiclass classifiers with margin. Given an $n$-point set $X \subset \mathbb{R}^m$, we want to learn an unknown classifier on $X$ whose classes have finite strong convex hull margin, a new notion extending the SVM margin. In the standard active learning setting, where only label queries are allowed, learning a classifier with strong convex hull margin $\gamma$ requires in the worst case $\Omega\big(1+\frac{1}{\gamma}\big)^{\frac{m-1}{2}}$ queries. On the other hand, using the more powerful \emph{seed} queries (a variant of equivalence queries), the target classifier could be learned in $O(m \log n)$ queries via Littlestone's Halving algorithm; however, Halving is computationally inefficient. In this work we show that, by carefully combining the two types of queries, a binary classifier can be learned in time $\operatorname{poly}(n+m)$ using only $O(m^2 \log n)$ label queries and $O\big(m \log \frac{m}{\gamma}\big)$ seed queries; the result extends to $k$-class classifiers at the price of a $k!k^2$ multiplicative overhead. Similar results hold when the input points have bounded bit complexity, or when only one class has strong convex hull margin against the rest. We complement the upper bounds by showing that in the worst case any algorithm needs $\Omega\big(k m \log \frac{1}{\gamma}\big)$ seed and label queries to learn a $k$-class classifier with strong convex hull margin $\gamma$.

**Supplementary Material:**pdf

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