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We investigate a generalized framework for estimating latent low-rank tensorsin an online setting, encompassing both linear and generalized linear models.This framework offers a flexible approach for handling continuous orcategorical variables. Additionally, we investigate two specific applications:online tensor completion and online binary tensor learning. To address thesechallenges, we propose the online Riemannian gradient descent algorithm, whichdemonstrates linear convergence and the ability to recover the low-rankcomponent under appropriate conditions in all applications. Furthermore, weestablish a precise entry-wise error bound for online tensor completion.Notably, our work represents the first attempt to incorporate noise in theonline low-rank tensor recovery task. Intriguingly, we observe a surprisingtrade-off between computational and statistical aspects in the presence ofnoise. Increasing the step size accelerates convergence but leads to higherstatistical error, whereas a smaller step size yields a statistically optimalestimator at the expense of slower convergence. Moreover, we conduct regretanalysis for online tensor regression. Under the fixed step size regime, afascinating trilemma concerning the convergence rate, statistical error rate,and regret is observed. With an optimal choice of step size we achieve anoptimal regret of $O(\sqrt{T})$. Furthermore, we extend our analysis to theadaptive setting where the horizon T is unknown. In this case, we demonstratethat by employing different step sizes, we can attain a statistically optimalerror rate along with a regret of $O(\log T)$. To validate our theoreticalclaims, we provide numerical results that corroborate our findings and supportour assertions.

Online Tensor Learning: Computational and Statistical Trade-offs, Adaptivity and Optimal Regret