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

2026-07-28

AI Summary

This work addresses online convex optimization under convex constraints in safety-critical settings, where feasibility must be maintained at every round. The authors propose a novel algorithm, AdaOGD-PFS, which integrates online gradient descent with Polyak feasibility steps, requiring only a single constraint evaluation and one subgradient query per iteration while guaranteeing perpetual feasibility and computational efficiency. The key contribution lies in establishing a data-dependent regret bound that replaces the worst-case dependence on \( G_f^2 T \) with the observed cumulative gradient norm \( G_T \), and revealing that the non-negative Polyak correction term \( P_T \) can negatively contribute to—thereby improving—the regret bound. An adaptive step-size strategy is further developed. Experiments under ball and half-space constraints demonstrate 38%–43% improvement in regret over baseline methods, confirming the efficacy of the proposed approach.

Abstract

Constrained online convex optimization requires minimizing regret against adversarial convex costs while satisfying a convex constraint at every round, as needed in safety-critical applications. A computationally efficient method combines online gradient descent with a Polyak feasibility step, using one constraint evaluation and one subgradient per round. Although this method achieves O(sqrt(T)) regret with per-round feasibility, we derive a tighter, data-dependent analysis by retaining two quantities omitted by the standard worst-case argument. First, we replace the gradient envelope G_f^2 T with the observed accumulation G_T = sum_t ||grad f_t(x_t)||^2. Second, we identify a nonnegative Polyak correction P_T that measures the cumulative squared displacement caused by feasibility projections and enters the regret bound with a negative sign. The resulting improvement, Delta_T = (eta/2)(G_f^2 T - G_T) + P_T/(2 eta), is always nonnegative. We further propose AdaOGD-PFS, an adaptive-step-size method that achieves O(sqrt(G_T)) regret while preserving per-round feasibility. Experiments on ball- and halfspace-constrained problems improve the regret bound by 38 to 43 percent, with both data-dependent gradients and Polyak corrections contributing substantially.

Problem

• Constrained Online Convex Optimization
• Regret Minimization
• Feasibility Constraints
• Adversarial Costs
• Safety-Critical Applications

Innovation

• data-dependent regret
• Polyak correction
• constrained online convex optimization
• adaptive step size
• feasibility projection