Optimal Competitive Ratios for Chasing Convex Functions with Quadratic Growth

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 competitive ratio, which compares its cumulative cost with that of an optimal offline decision sequence. For Euclidean movement, existing upper and lower bounds for $κ$-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 $O(κ^{1/3})$ competitive ratio for smooth convex functions satisfying quadratic growth, improving the previous $O(\sqrtκ)$ upper bound and matching the existing $Ω(κ^{1/3})$ 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.