Optimal Competitive Ratios for Chasing Convex Functions with Quadratic Growth

Abstract

Chasing convex function has attracted much attention in recent years, which can be formulated as a game between a learner and an adversary. In each round, the adversary reveals a convex function, and the learner selects a decision and incurs a hitting cost and a movement cost for changing its decision. The performance of the learner is measured by the \emph{competitive ratio}, which compares its cumulative cost with that of an optimal offline decision sequence. For Euclidean movement, existing upper and lower bounds for $\kappa$-well-conditioned functions exhibit a polynomial gap. For squared Euclidean movement, dimension-free competitive guarantees have been established under quadratic growth, but the optimal competitive ratio remains unclear. In this paper, we establish optimal dimension-free competitive ratios for convex function chasing under two types of movement costs. For Euclidean movement, we propose Clipped Proximal Descent (CPD), which combines a balanced proximal step with a clipped correction toward the current minimizer. CPD achieves an $O(\kappa^{1/3})$ competitive ratio for smooth convex functions satisfying quadratic growth, improving the previous $O(\sqrt{\kappa})$ upper bound and matching the existing $\Omega(\kappa^{1/3})$ lower bound. For squared Euclidean movement, we establish the exact optimal dimension-free competitive ratio for convex functions satisfying quadratic growth. Our result is based on a new tight competitive analysis of Regularized Online Balanced Descent (R-OBD) and a matching lower bound against randomized algorithms.