Online learning of smooth functions

Abstract

In this paper, we study the online learning of real-valued functions where the hidden function is known to have certain smoothness properties. Specifically, for q≥1, let Fq be the class of absolutely continuous functions f:[0,1]→R such that ‖f′‖q≤1. For q≥1 and d∈Z+, let Fq,d be the class of functions f:[0,1]d→R such that any function g:[0,1]→R formed by fixing all but one parameter of f is in Fq. For any class of real-valued functions F and p>0, let optp(F) be the best upper bound on the sum of pth powers of absolute prediction errors that a learner can guarantee in the worst case. In the single-variable setup, we find new bounds for optp(Fq) that are sharp up to a constant factor. We show for all ε∈(0,1) that opt1+ε(F∞)=Θ(ε−12) and opt1+ε(Fq)=Θ(ε−12) for all q≥2. We also show for ε∈(0,1) that opt2(F1+ε)=Θ(ε−1). In addition, we obtain new exact results by proving that optp(Fq)=1 for q∈(1,2) and p≥2+1q−1. In the multi-variable setup, we establish inequalities relating optp(Fq,d) to optp(Fq) and show that optp(F∞,d) is infinite when p<d and finite when p>d. We also obtain sharp bounds on learning F∞,d for p<d when the number of trials is bounded.