Commentary on "Online Optimization with Gradual Variations"

This commentary is about (Chiang et al., 2012b). This paper is the result of a merge between two papers, (Yang et al., 2012) and (Chiang et al., 2012a). Both papers address the same question: is it possible to obtain regret bounds in various online learning settings that depend on some notion of variation in the costs, rather than the number of periods? Both papers give remarkably similar algorithms for this problem, although the analysis techniques are quite different, and obtain very similar results. While (Yang et al., 2012) gives two algorithms obtaining such regret bounds for general online convex optimization (OCO), (Chiang et al., 2012a) gives a unified framework to obtain such regret bounds for three specific cases of OCO: online linear optimization, online learning with experts, and online expconcave optimization.