Strongly adaptive online learning over partial intervals

Abstract

To cope with changing environments, strongly adaptive algorithms that almost enjoy the optimal performance on every time interval have been proposed for online learning. However, the best regret bound of existing algorithms on each time interval with length τ is $O\left( {\sqrt {\tau \log \,T} } \right)$, and their complexities are increasing with a factor of O(log T), where T is the time horizon. In real-world applications, T could go to infinity, which means that even these logarithmic factors are unacceptable. In this paper, we propose to remove the logarithmic factors of existing algorithms by utilizing prior information of environments. Specifically, we assume a lower bound τ₁ and an upper bound τ₂ on how long the environment changes are given, and only focus on the performance over time intervals with length in [τ₁, τ₂]. Then, we propose a new algorithm with a refined set of intervals that can reduce the complexity and a simple weighting method that can cooperate with our interval set. Theoretical analysis reveals that the regret bound of our algorithm on any focused interval is optimal up to a constant factor. Both the regret bound and the computational cost per iteration are independent of T. Experimental results show that our algorithm outperforms the state-of-the-art algorithm.