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We study exact active learning of binary and multiclass classifiers withmargin. Given an $n$-point set $X \subset \mathbb{R}^m$, we want to learn anyunknown 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 convexhull margin $\gamma$ requires in the worst case$\Omega\big(1+\frac{1}{\gamma}\big)^{(m-1)/2}$ queries. On the other hand,using the more powerful seed queries (a variant of equivalence queries), thetarget classifier could be learned in $O(m \log n)$ queries via Littlestone'sHalving algorithm; however, Halving is computationally inefficient. In thiswork we show that, by carefully combining the two types of queries, a binaryclassifier 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 havebounded bit complexity, or when only one class has strong convex hull marginagainst the rest. We complement the upper bounds by showing that in the worstcase any algorithm needs $\Omega\big(k m \log \frac{1}{\gamma}\big)$ seed andlabel queries to learn a $k$-class classifier with strong convex hull margin$\gamma$.

Active Learning of Classifiers with Label and Seed Queries