Achieving All with No Parameters: Adaptive NormalHedge
This work provides a foundational advancement in online learning by solving long-standing open problems and improving applications like competing with time-varying competitors and best pruning tree prediction, though it is incremental relative to prior NormalHedge methods.
The paper tackles the online learning problem of predicting with expert advice by proposing AdaNormalHedge, a parameter-free and adaptive algorithm that achieves small regret for competitors with small loss, almost constant regret for stochastic losses, and competes with any convex combination of experts simultaneously, resolving open problems from Chaudhuri et al. (2009) and Chernov and Vovk (2010).
We study the classic online learning problem of predicting with expert advice, and propose a truly parameter-free and adaptive algorithm that achieves several objectives simultaneously without using any prior information. The main component of this work is an improved version of the NormalHedge.DT algorithm (Luo and Schapire, 2014), called AdaNormalHedge. On one hand, this new algorithm ensures small regret when the competitor has small loss and almost constant regret when the losses are stochastic. On the other hand, the algorithm is able to compete with any convex combination of the experts simultaneously, with a regret in terms of the relative entropy of the prior and the competitor. This resolves an open problem proposed by Chaudhuri et al. (2009) and Chernov and Vovk (2010). Moreover, we extend the results to the sleeping expert setting and provide two applications to illustrate the power of AdaNormalHedge: 1) competing with time-varying unknown competitors and 2) predicting almost as well as the best pruning tree. Our results on these applications significantly improve previous work from different aspects, and a special case of the first application resolves another open problem proposed by Warmuth and Koolen (2014) on whether one can simultaneously achieve optimal shifting regret for both adversarial and stochastic losses.