Stochastic Online Linear Regression: the Forward Algorithm to Replace Ridge

Reda Ouhamma, Odalric-Ambrym Maillard, Vianney Perchet

Keywords: Online linear regression, Multi-armed bandit

TL;DR: We propose a new analysis for the celebrated forward algorithm in the setting of stochastic online linear regression, and show the benefits of it replacing ridge regression whenever possible.

Abstract: We consider the problem of online linear regression in the stochastic setting. We derive high probability regret bounds for online $\textit{ridge}$ regression and the $\textit{forward}$ algorithm. This enables us to compare online regression algorithms more accurately and eliminate assumptions of bounded observations and predictions. Our study advocates for the use of the forward algorithm in lieu of ridge due to its enhanced bounds and robustness to the regularization parameter. Moreover, we explain how to integrate it in algorithms involving linear function approximation to remove a boundedness assumption without deteriorating theoretical bounds. We showcase this modification in linear bandit settings where it yields improved regret bounds. Last, we provide numerical experiments to illustrate our results and endorse our intuitions.

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