@inproceedings{
erez2024fast,
title={Fast Rates for Bandit {PAC} Multiclass Classification},
author={Liad Erez and Alon Cohen and Tomer Koren and Yishay Mansour and Shay Moran},
booktitle={The Thirty-eighth Annual Conference on Neural Information Processing Systems},
year={2024},
url={https://openreview.net/forum?id=6zOKbzjBO4}
}
We study multiclass PAC learning with bandit feedback, where inputs are classified into one of $K$ possible labels and feedback is limited to whether or not the predicted labels are correct. Our main contribution is in designing a novel learning algorithm for the agnostic $(\varepsilon,\delta)$-PAC version of the problem, with sample complexity of $O\big( (\operatorname{poly}(K) + 1 / \varepsilon^2) 
\log (|\mathcal{H}| / \delta) \big)$ for any finite hypothesis class $\mathcal{H}$. In terms of the leading dependence on $\varepsilon$, this improves upon existing bounds for the problem, that are of the form $O(K/\varepsilon^2)$. We also provide an extension of this result to general classes  and establish similar sample complexity bounds in which $\log |\mathcal{H}|$ is replaced by the Natarajan dimension.
This matches the optimal rate in the full-information version of the problem and resolves an open question studied by Daniely, Sabato, Ben-David, and Shalev-Shwartz (2011) who demonstrated that the multiplicative price of bandit feedback in realizable PAC learning is $\Theta(K)$. We complement this by revealing a stark contrast with the agnostic case, where the price of bandit feedback is only $O(1)$ as $\varepsilon \to 0$. Our algorithm utilizes a stochastic optimization technique to minimize a log-barrier potential based on Frank-Wolfe updates for computing a low-variance exploration distribution over the hypotheses, and is made computationally efficient provided access to an ERM oracle over $\mathcal{H}$.
Fast Rates for Bandit PAC Multiclass Classification
Liad Erez Alon Cohen Tomer Koren Yishay Mansour Shay Moran