Online Structured Prediction with Fenchel--Young Losses and Improved Surrogate Regret for Online Multiclass Classification with Logistic Loss
<|reference_start|>Online Structured Prediction with Fenchel--Young Losses and Improved Surrogate Regret for Online Multiclass Classification with Logistic Loss: This paper studies online structured prediction with full-information feedback. For online multiclass classification, Van der Hoeven (2020) established \emph{finite} surrogate regret bounds, which are independent of the time horizon, by introducing an elegant \emph{exploit-the-surrogate-gap} framework. However, this framework has been limited to multiclass classification primarily because it relies on a classification-specific procedure for converting estimated scores to outputs. We extend the exploit-the-surrogate-gap framework to online structured prediction with \emph{Fenchel--Young losses}, a large family of surrogate losses that includes the logistic loss for multiclass classification as a special case, obtaining finite surrogate regret bounds in various structured prediction problems. To this end, we propose and analyze \emph{randomized decoding}, which converts estimated scores to general structured outputs. Moreover, by applying our decoding to online multiclass classification with the logistic loss, we obtain a surrogate regret bound of $O(\| \mathbf{U} \|_\mathrm{F}^2)$, where $\mathbf{U}$ is the best offline linear estimator and $\| \cdot \|_\mathrm{F}$ denotes the Frobenius norm. This bound is tight up to logarithmic factors and improves the previous bound of $O(d\| \mathbf{U} \|_\mathrm{F}^2)$ due to Van der Hoeven (2020) by a factor of $d$, the number of classes.<|reference_end|>
@article{sakaue2024online,
    title={Online Structured Prediction with Fenchel--Young Losses and Improved
  Surrogate Regret for Online Multiclass Classification with Logistic Loss},
    author={Shinsaku Sakaue, Han Bao, Taira Tsuchiya, Taihei Oki},
    journal={arXiv preprint arXiv:2402.08180},
    year={2024},
    archivePrefix={arXiv},
    eprint={2402.08180},
    primaryClass={cs.LG}
}