Convex Games in Banach Spaces

We study the regret of an online learner playing a multi-round game in a Banach space B against an adversary that plays a convex function at each round. We characterize the minimax regret when the adversary plays linear functions in terms of the Martingale type of the dual of B. The cases when the adversary plays bounded and uniformly convex functions respectively are also considered. Our results connect online convex learning to the study of the geometry of Banach spaces. We also show that appropriate modifications of the Mirror Descent algorithm from convex optimization can be used to achieve our regret upper bounds. Finally, we provide a version of Mirror Descent that adapts to the changing exponent of uniform convexity of the adversary's functions. This adaptive mirror descent strategy provides new algorithms even for the more familiar Hilbert space case where the loss functions on each round have varying exponents of uniform convexity (curvature).

• Type: Journal
• Year: 2010
• Where: COLT
• Authors: Karthik Sridharan, Ambuj Tewari
• Keywords: Banach Spaces, COLT 2010, Convex Functions, Machine Learning, Mirror Descent