Uniform regret bounds over $R^d$ for the sequential linear regression problem with the square loss
- INRIA（法国国家信息与自动化研究所）
- ENS（巴黎高等师范学院）
- PSL Research University（巴黎文理研究大学）
- Institut de mathématiques de Toulouse（图卢兹数学研究所）
- Université Paul Sabatier（保罗·萨巴蒂埃大学）
- Laboratoire de mathématiques d’Orsay（奥赛数学实验室）
- Université Paris-Sud（巴黎第十一大学）
- CNRS（法国国家科学研究中心）
- Université Paris-Saclay（巴黎萨克雷大学）
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英文摘要：
We consider the setting of online linear regression for arbitrary deterministic sequences, with the square loss. We are interested in the aim set by Bartlett et al. (2015): obtain regret bounds that hold uniformly over all competitor vectors. When the feature sequence is known at the beginning of the game, they provided closed-form regret bounds of $2d B^2 \ln T + \mathcal{O}_T(1)$, where $T$ is the number of rounds and $B$ is a bound on the observations. Instead, we derive bounds with an optimal constant of $1$ in front of the $d B^2 \ln T$ term. In the case of sequentially revealed features, we also derive an asymptotic regret bound of $d B^2 \ln T$ for any individual sequence of features and bounded observations. All our algorithms are variants of the online non-linear ridge regression forecaster, either with a data-dependent regularization or with almost no regularization.
