约束在线凸优化（COCO）的OGD+投影算法的累积约束违反的下界
Lower Bound on the Cumulative Constrained Violation for the OGD+Projection algorithm for Constrained Online Convex Optimization (COCO)
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中文总结 AI 辅助
研究约束在线凸优化问题，针对当前最好的OGD+投影算法，给出其累积约束违反的下界为\(\Omega (T^{\frac{d - 1}{2d}})\)，此为该问题的首个下界结果。
中文摘要 AI 辅助
考虑约束在线凸优化问题，在每一轮中，学习者选定动作\(x_t \in \mathcal{X} \subset \mathbb{R}^d\)后，会揭示一个凸损失函数\(f_t\)和一个驱动约束\(g_t(x)\le 0\)的凸约束函数\(g_t\)。目标是与提前知晓所有\(t\)的损失函数和约束函数\(f_t\)和\(g_t\)并选择对所有\(g_t(x)\le 0\)可行的静态最优动作的基准相比，同时最小化静态遗憾和累积约束违反（CCV）。目前已知最好的算法是[Vaze和Sinha，2025]的OGD+投影算法，对于\(d = 2\)，其同时遗憾为\(O(\sqrt{T})\)，CCV为\(O(T^{1/3})\) ，对于任意\(d\)，同时遗憾为\(O(\sqrt{T})\)，CCV为\(O(\sqrt{T})\)。本文表明OGD+投影算法的CCV为\(\Omega (T^{\frac{d - 1}{2d}})\)，这是首个此类下界结果。
英文摘要
The problem of constrained online convex optimization is considered, where at each round, once a learner commits to an action $x_t \in \mathcal{X} \subset \mathbb{R}^d$, a convex loss function $f_t$ and a convex constraint function $g_t$ that drives the constraint $g_t(x)\le 0$ are revealed. The objective is to simultaneously minimize the static regret and cumulative constraint violation (CCV) compared to the benchmark that knows the loss functions and constraint functions $f_t$ and $g_t$ for all $t$ ahead of time, and chooses a static optimal action that is feasible with respect to all $g_t(x)\le 0$. Currently, the best known algorithm is OGD+Projection algorithm of [Vaze and Sinha, 2025] that has simultaneous regret of $O(\sqrt{T})$ and CCV of $O(T^{1/3})$ for $d=2$ [Balasundaram et al., 2026], and simultaneous regret of $O(\sqrt{T})$ and CCV of $O(\sqrt{T})$ for any $d$ [Sarkar and Sinha, 2026]. In this paper, we show that the CCV of the OGD+Projection algorithm is $Ω(T^{\frac{d-1}{2d}})$. This is the first such lower bound result.
