$\tilde{O}(\sqrt{T})$ Regret and Polylogarithmic Constraint Violation for COCO

arXiv:2610.03983v1 Announce Type: new 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, Gr\"unbaum'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 $\mathsf{CCV}$.

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 $\mathsf{CCV}$.

Machine Learning (cs.LG)

arXiv:2610.03983 [cs.LG]

(or arXiv:2610.03983v1 [cs.LG] for this version)

https://doi.org/10.48550/arXiv.2610.03983 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.

Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.

Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.

analiza aiqbench

$\tilde{O}(\sqrt{T})$ Regret and Polylogarithmic Constraint Violation for COCO

Subiecte conexe