Dynamic Regret in Online Convex Optimization with Indicator Switching Costs

2026-09-24

AI Summary

This study addresses dynamic regret minimization in online convex optimization with hinted switching costs, demonstrating that existing static approaches fail in dynamic environments. To overcome this limitation, the work proposes a meta-learning-based adaptive framework that integrates randomized lazy Follow-the-Regularized-Leader base learners with a binary time-scale restart mechanism. Efficient decision aggregation is achieved through maximal coupling sampling without requiring prior environmental knowledge. The authors first establish that direct extensions of static methods are fundamentally infeasible. Subsequently, they show that the proposed approach attains near minimax-optimal dynamic regret bounds under both piecewise-constant and high-frequency moving comparator sequences. These results effectively resolve the longstanding bottleneck in dynamic tracking for this problem setting.

Abstract

We study dynamic regret in online convex optimization with an \emph{indicator switching cost}: a fixed penalty incurred whenever two consecutive decisions differ. This captures startup overheads such as server activation, model deployment, and cache updates, and on a bounded domain it recovers norm-based movement costs as a special case. Existing guarantees for indicator costs handle only static comparators. We show that a direct extension of these techniques to dynamic regret provably fails, motivating a different approach. We propose a meta-learning framework: a set of randomized lazy FTRL base learners restarted at dyadic time scales, aggregated by a movement-aware master that mixes their proposal densities and samples actions via maximal coupling of consecutive mixtures. The resulting algorithm satisfies, in expectation, $\mathcal{R}^{\mathbf{1}}_T \le \tilde{\mathcal{O}}(\min\{\sqrt{T(S_T{+}1)},T^{2/3}(P_T+1)^{1/3}\})$, where $\mathcal{R}^{\mathbf{1}}_T$ is the dynamic regret plus the cumulative indicator switching cost, $S_T$ counts comparator switches, and $P_T$ is the comparator path length. The bound holds simultaneously for all sequences and requires no prior knowledge of $S_T$ or $P_T$: it is minimax-optimal (up to logarithmic factors) for tracking piecewise-constant comparators, and also captures frequently moving comparators with small total path length.

Problem

• Online Convex Optimization
• Dynamic Regret
• Indicator Switching Costs
• Meta-Learning

Innovation

• Online Convex Optimization
• Dynamic Regret
• Indicator Switching Costs
• Meta-Learning
• Maximal Coupling