We investigate a generalized framework for estimating latent low-rank tensors in an online setting, encompassing both linear and generalized linear models. This framework offers a flexible approach for handling continuous or categorical variables. Additionally, we investigate two specific applications: online tensor completion and online binary tensor learning. To address these challenges, we propose the online Riemannian gradient descent algorithm, which demonstrates linear convergence and the ability to recover the low-rank component under appropriate conditions in all applications. Furthermore, we establish a precise entry-wise error bound for online tensor completion. Notably, our work represents the first attempt to incorporate noise in the online low-rank tensor recovery task. Intriguingly, we observe a surprising trade-off between computational and statistical aspects in the presence of noise. Increasing the step size accelerates convergence but leads to higher statistical error, whereas a smaller step size yields a statistically optimal estimator at the expense of slower convergence. Moreover, we conduct regret analysis for online tensor regression. Under the fixed step size regime, a fascinating trilemma concerning the convergence rate, statistical error rate, and regret is observed. With an optimal choice of step size we achieve an optimal regret of $O(\sqrt{T})$. Furthermore, we extend our analysis to the adaptive setting where the horizon T is unknown. In this case, we demonstrate that by employing different step sizes, we can attain a statistically optimal error rate along with a regret of $O(\log T)$. To validate our theoretical claims, we provide numerical results that corroborate our findings and support our assertions.

本文提出了一种在线低秩张量恢复的概括性框架，包括线性和广义线性模型，特别地，在线张量补全和在线二进制张量学习的应用中，通过在线黎曼梯度下降算法实现了线性收敛并在所有应用程序中恢复了低秩分量，还在在线张量回归方面进行了悔恨分析，通过数值结果验证了理论结论。

在线张量学习：计算和统计权衡，适应性和最优遗憾