Data-Dependent Regret and Polyak Corrections for Constrained Online Convex Optimization

Abstract: In constrained online convex optimization, the learner must minimize regret against adversarially chosen convex costs while satisfying a convex constraint at every round, a requirement that arises naturally in safety-critical domains such as power systems, autonomous control, and clinical decision-making. A natural and computationally efficient approach augments online gradient descent with a Polyak feasibility step: a closed-form half-space projection requiring only one constraint evaluation and one subgradient per round. This approach is known to achieve $O(\sqrt{T})$ regret with per-round feasibility, yet we prove that its existing analysis is strictly loose by identifying two quantities it unnecessarily discards. Specifically, replacing the worst-case gradient envelope $G_f^2 T$ with the observed accumulation $\mathcal{G}_T = \sum_t \lVert \nabla f_t(x_t) \rVert^2$ yields a data-dependent bound without any algorithmic modification. Furthermore, we introduce the Polyak correction $\mathcal{P}_T \geq 0$, which captures the cumulative squared displacement of the feasibility projection and enters the regret bound with a strictly negative sign, a term that all prior proofs lose entirely through the Pythagorean inequality. The total improvement $\Delta_T = \tfrac{\eta}{2}(G_f^2 T - \mathcal{G}_T) + \tfrac{1}{2\eta}\mathcal{P}_T$ is provably non-negative and decomposes into two independent, complementary sources that vanish only in a degenerate corner case. Building on these analytical insights, we propose AdaOGD-PFS, an adaptive-step-size variant that achieves $O(\sqrt{\mathcal{G}_T})$ regret, potentially much smaller than $O(G_f\sqrt{T})$, while preserving per-round constraint satisfaction. Experiments on ball-constrained and halfspace-constrained instances confirm bound improvements of 38--43%, with both data-dependent gradients and Polyak corrections contributing meaningfully.

Submission Type: Regular submission (no more than 12 pages of main content)

Assigned Action Editor: ~Zhiyu_Zhang1

Submission Number: 8193