$O(\sqrt{T})$ Static Regret and Instance Dependent Constraint Violation for Constrained Online Convex Optimization

TL;DR

This work advances constrained online convex optimization (COCO) by exploiting the geometry of nested constraint sets $S_t$ to control cumulative constraint violation (CCV) without sacrificing regret. It introduces a simple online gradient method, $\mathrm{coco\_alg\_1}$, that interleaves a gradient step with projections onto $S_{t-1}$ and $S_t$, achieving regret $O(\sqrt{T})$ and an instance-dependent CCV bound $O\big(\sqrt{d}\,(1/c^*)^{d}D\big)$, where $c^*$ captures the geometric width of the constraint set intersection. A Switch mechanism combines this with a universal COCO method from Sinha2024 to guarantee CCV no larger than $\min\{\mathcal{V}, O(\sqrt{T}\log T)\}$, providing best-of-both-worlds performance. For favorable geometries (e.g., spheres or axis-aligned $S_t$) and especially in 2D with monotone projection angles, CCV is tightly $O(1)$, and in the OCS special case, CCV improves to $O(1)$. These results substantially sharpen prior universal CCV bounds and highlight the role of constraint geometry in COCO performance.

Abstract

The constrained version of the standard online convex optimization (OCO) framework, called COCO is considered, where on every round, a convex cost function and a convex constraint function are revealed to the learner after it chooses the action for that round. The objective is to simultaneously minimize the static regret and cumulative constraint violation (CCV). An algorithm is proposed that guarantees a static regret of $O(\sqrt{T})$ and a CCV of $\min\{\cV, O(\sqrt{T}\log T) \}$, where $\cV$ depends on the distance between the consecutively revealed constraint sets, the shape of constraint sets, dimension of action space and the diameter of the action space. For special cases of constraint sets, $\cV=O(1)$. Compared to the state of the art results, static regret of $O(\sqrt{T})$ and CCV of $O(\sqrt{T}\log T)$, that were universal, the new result on CCV is instance dependent, which is derived by exploiting the geometric properties of the constraint sets.

Figures (6)

• Figure 1: Figure representing the cone $C_{w_t}(c_t)$ that contains the convex hull of $m_t$ and $S_{t}$ with unit vector $w_t$.
• Figure 2: Definition of $F_t$'s.
• Figure 3: Figure representing the cone $C_{w_t}(c_t)$ that contains the convex hull of $m_t$ and $S_{t}$ with respect to the unit vector $w_t$. $u$ is a unit vector perpendicular to $H_u$ an hyperplane that is a supporting hyperplane $C_t$ at $m_t$ such that $\mathcal{C}_t \cap H_u = \{m_t\}$ and $u^T (x_t-m_t)\ge 0$
• Figure 4: Figure corresponding to Example \ref{['exm:phasedef']}.
• Figure 5: Illustration of definition of $z_t(\kappa)$ for $t\in \mathcal{T}(\kappa)$. In this example, for phase $1$, $t^\star(1)=3$ since the distance of $y_3$ from ${\mathsf{c}}$ is the farthest for phase $1$ that consists of time slots $\mathcal{T}(1) = \{2,3\}$. Hence $z_{t^\star(1)+1}(1)=x_4$. For $t \in \mathcal{T}(1) \backslash t^\star(1)+1$, $z_{t}(1)$ are such $z_{t+1}(1)$ is a projection of $z_{t}(1)$ onto $F_t$.

Theorems & Definitions (38)

• Definition 4
• Theorem 5
• Lemma 6
• Remark 1
• Lemma 7
• Lemma 8
• Lemma 9
• Definition 10
• Remark 2
• Remark 3