Lower bounds on individual sequence regret

Abstract. In this work, we lower bound the individual sequence anytime regret of a large family of online algorithms. This bound depends on the quadratic variation of the sequence, QT , and the learning rate. Nevertheless, we show that any learning rate that guarantees a regret upper bound of O( √ QT ) necessarily implies an Ω( √ QT ) anytime regret on any sequence with quadratic variation QT . The algorithms we consider are linear forecasters whose weight vector at time t + 1 is the gradient of a concave potential function of cumulative losses at time t. We show that these algorithms include all linear Regularized Follow the Leader algorithms. We prove our result for the case of potentials with negative definite Hessians, and potentials for the best expert setting satisfying some natural regularity conditions. In the best expert setting, we give our result in terms of the translation-invariant relative quadratic variation. We apply our lower bounds to Randomized Weighted Majori...

• Type: Journal
• Year: 2016
• Where: ML
• Authors: Eyal Gofer, Yishay Mansour
• Keywords: Machine Learning, ML 2016