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Dimension-Free Bounds for Chasing Convex Functions
Proceedings of Thirty Third Conference on Learning Theory, PMLR 125:219-241, 2020.
Abstract
We consider the problem of chasing convex functions, where functions arrive over time. The player takes actions after seeing the function, and the goal is to achieve a small function cost for these actions, as well as a small cost for moving between actions. While the general problem requires a polynomial dependence on the dimension, we show how to get dimension-independent bounds for well-behaved functions. In particular, we consider the case where the convex functions are κ-well-conditioned, and give an algorithm that achieves an O(√κ)-competitiveness. Moreover, when the functions are supported on k-dimensional affine subspaces—e.g., when the function are the indicators of some affine subspaces—we get O(min-competitive algorithms for request sequences of length T. We also show some lower bounds, that well-conditioned functions require \Omega(\kappa^{1/3})-competitiveness, and k-dimensional functions require \Omega(\sqrt{k})-competitiveness.