Fast Rates for Bandit PAC Multiclass Classification

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 (ε, δ)-PAC version of the problem, with sample complexity of O^{((poly(K) + 1/ε2) log(|H |/δ))} for any finite hypothesis class H. In terms of the leading dependence on ε, this improves upon existing bounds for the problem, that are of the form O(K/ε²). We also provide an extension of this result to general classes and establish similar sample complexity bounds in which log |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 Θ(K). We complement this by revealing a stark contrast with the agnostic case, where the price of bandit feedback is only O(1) as ε → 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 H.