Dual Adaptivity: A Universal Algorithm for Minimizing the Adaptive Regret of Convex Functions

Lijun Zhang, Guanghui Wang, Wei-Wei Tu, Wei Jiang, Zhi-hua Zhou

Keywords: Online Convex Optimization, Adaptive Regret, Convex Functions, Strongly Convex Functions, Exponentially Concave Functions

Abstract: To deal with changing environments, a new performance measure—adaptive regret, defined as the maximum static regret over any interval, was proposed in online learning. Under the setting of online convex optimization, several algorithms have been successfully developed to minimize the adaptive regret. However, existing algorithms lack universality in the sense that they can only handle one type of convex functions and need apriori knowledge of parameters. By contrast, there exist universal algorithms, such as MetaGrad, that attain optimal static regret for multiple types of convex functions simultaneously. Along this line of research, this paper presents the first universal algorithm for minimizing the adaptive regret of convex functions. Specifically, we borrow the idea of maintaining multiple learning rates in MetaGrad to handle the uncertainty of functions, and utilize the technique of sleeping experts to capture changing environments. In this way, our algorithm automatically adapts to the property of functions (convex, exponentially concave, or strongly convex), as well as the nature of environments (stationary or changing). As a by product, it also allows the type of functions to switch between rounds.

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TL;DR: We develop a universal algorithm that is able to minimize the adaptive regret of general convex, exp-concave and strongly convex functions simultaneously.