Bandit Multiclass List Classification
Liad Erez, Tomer Koren
We study the problem of multiclass list classification with (semi-)bandit feedback, where input examples are mapped into subsets of size $m$ of a collection of $K$ possible labels. In each round of the interaction, the learner observes feedback consisting of the predicted labels which lie in some underlying set of ground truth labels associated with the given example. Our main result is for the $(\varepsilon,\delta)$-PAC variant of the problem for which we design an algorithm that returns an $\varepsilon$-optimal hypothesis with high probability using a sample complexity of $\smash{\widetilde{O}} \big( (\mathrm{poly}(K/m) + sm / \varepsilon^2) \log (|\mathcal H|/\delta) \big)$ where $\mathcal H$ is the underlying (finite) hypothesis class and $s$ is an upper bound on the number of true labels for a given example. This bound improves upon known bounds for combinatorial semi-bandits whenever $s \ll K$. Moreover, in the regime where $s = O(1)$ the leading terms in our bound match the corresponding full-information rates, implying that bandit feedback essentially comes at no cost. Our PAC learning algorithm is also computationally efficient given access to an ERM oracle for $\mathcal H$. In the special case of single-label classification corresponding to $s=m=1$, we prove a sample complexity bound of $O \big((K^7 + 1/\varepsilon^2)\log (|\mathcal H|/\delta)\big)$ which improves upon recent results in this scenario (Erez et al. '24). Additionally, we consider the regret minimization setting where data can be generated adversarially, and establish a regret bound of $\smash{\widetilde O(|\mathcal H| + \sqrt{smT \log |\mathcal H|})}$. Our results generalize and extend prior work in the simpler single-label setting (Erez et al. '24), and apply more generally to contextual combinatorial semi-bandit problems with $s$-sparse rewards.