Optimistic Bounds for Multi-output Learning
Abstract
We investigate the challenge of multi-output learning, where the goal is to learn a vector-valued function based on a supervised data set. This includes a range of important problems in Machine Learning including multi-target regression, multi-class classification and multi-label classification. We begin our analysis by introducing the self-bounding Lipschitz condition for multi-output loss functions, which interpolates continuously between a classical Lipschitz condition and a multi-dimensional analogue of a smoothness condition. We then show that the self-bounding Lipschitz condition gives rise to optimistic bounds for multi-output learning, which attain the minimax optimal rate up to logarithmic factors. The proof exploits local Rademacher complexity combined with a powerful minoration inequality due to Srebro, Sridharan and Tewari. As an application we derive a state-of-the-art generalisation bound for multi-class gradient boosting.
BibTeX
@InProceedings{pmlr-v119-reeve20a,
title = {Optimistic Bounds for Multi-output Learning},
author = {Reeve, Henry and Kaban, Ata},
booktitle = {Proceedings of the 37th International Conference on Machine Learning},
pages = {8030--8040},
year = {2020},
editor = {III, Hal Daumé and Singh, Aarti},
volume = {119},
series = {Proceedings of Machine Learning Research},
month = {13--18 Jul},
publisher = {PMLR},
pdf = {http://proceedings.mlr.press/v119/reeve20a/reeve20a.pdf},
url = {https://proceedings.mlr.press/v119/reeve20a.html},
abstract = {We investigate the challenge of multi-output learning, where the goal is to learn a vector-valued function based on a supervised data set. This includes a range of important problems in Machine Learning including multi-target regression, multi-class classification and multi-label classification. We begin our analysis by introducing the self-bounding Lipschitz condition for multi-output loss functions, which interpolates continuously between a classical Lipschitz condition and a multi-dimensional analogue of a smoothness condition. We then show that the self-bounding Lipschitz condition gives rise to optimistic bounds for multi-output learning, which attain the minimax optimal rate up to logarithmic factors. The proof exploits local Rademacher complexity combined with a powerful minoration inequality due to Srebro, Sridharan and Tewari. As an application we derive a state-of-the-art generalisation bound for multi-class gradient boosting.}
}