Constrained online optimization achieves low regret with minimal violations
$\tilde{O}(\sqrt{T})$ Regret and Polylogarithmic Constraint Violation for COCO
Machine Learning
Summary
The paper looks at a way to make decisions step-by-step when faced with changing rules and costs, aiming to do nearly as well as the best fixed choice, while rarely breaking constraints. The authors improve prior work by showing it’s possible to keep the number of rule violations very low—growing only slowly with time—without sacrificing performance. They do this by cleverly trimming down options when constraints are broken, which quickly reduces problematic choices. Their method balances learning and constraints better than before.
What this means in practice
- •For online ad schedulers: Improve automated decision-making with fewer budget or policy violations and near-optimal cost performance over many ad placement rounds.
- •For network traffic managers: Manage resource allocation in networks while controlling constraint violations to maintain service quality with minimal performance loss.
Authors
Dhruv Sarkar, Abhishek Sinha
Abstract
We study constrained online convex optimization with adversarial convex losses and constraints ($\mathsf{COCO}$). At each round \(t\in[T]\), a learner selects \(x_t\) from a \(d\)-dimensional convex decision set \(\mathcal X\), after which an adaptive adversary reveals a convex cost function \(f_t\) and constraint function \(g_t\). Consequently, the learner incurs cost \(f_t(x_t)\) and constraint violation \(\max\{0,g_t(x_t)\}\), and aims to simultaneously minimize regret and cumulative constraint violation ($\mathsf{CCV}$) over the entire horizon. Existing algorithms achieve \(O(\sqrt{T})\) regret and \(\widetilde O(\sqrt{T})\) $\mathsf{CCV}$. We show that an online policy can achieve \(O(\sqrt{T\log T})\) regret and \(O(\log^2 T)\) $\mathsf{CCV}$, reducing the $\mathsf{CCV}$ from polynomial to polylogarithmic while retaining near-optimal regret. Our approach combines continuous Hedge with elimination on shrinking feasible sets. The key observation is that whenever the mean of the Hedge distribution violates a constraint, Grünbaum's inequality guarantees that a constant fraction of the Hedge probability mass is eliminated. We use an adaptive learning-rate schedule and a potential function coupling the surviving volume with the learning rate to convert this probability-mass reduction into a bound of \(O(\log^2 T)\) on the $\mathsf{CCV}$.