发表机构
Massachusetts Institute of Technology; Purdue University（麻省理工学院; 普渡大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对两类在线优化问题，提出嵌套凸体追踪算法，将损失控制与几何运动分离，在降低维度依赖的同时保证了遗憾和约束违反的理论边界，还证明了二维场景下的最优时间范围依赖性。
AI 中文摘要
我们研究两种场景下可行集不断收缩的在线优化问题：嵌套演化可行集的凸优化（CONES）与对抗性约束在线凸优化（COCO）。我们的算法将损失控制与几何运动分离：约束极小化器和累积损失测试保证了遗憾（regret）的理论保障，而确定性可重置嵌套凸体追踪器则限制了运动范围。对于在直径为D的定义域上具有G-Lipschitz、μ-强凸目标函数的CONES问题，我们追踪当前可行集与自适应目标次水平集的交集。利用欧氏追踪比O(√(d log(1+d)))，我们在每个前缀都获得非正遗憾，且运动范围为O(√(d log(1+d) GD log(eT)/μ))。该边界可适应约束最优值的增长。在二维空间中，当所有其他参数固定时，所有终端期望遗憾为O(T^β)（β<1）的随机算法，在某些确定性嵌套序列上会遭受Ω(√log T)的期望运动，证明了最优的时间范围依赖性。在远离约束极小化集的线性增长条件下，Steiner点追踪产生的运动与T无关。对于一般凸COCO问题，带正则化领导者重置的一步延迟追踪可得到遗憾O(G_f D √(d log(1+d) T))和累积约束违反O(G_g D √(d log(1+d) T))。对于强凸损失，当其他参数固定时，两者均为O(d log(1+d) log(eT))。这些改进将先前分析中的O(d^{d/2})投影路径因子替换为欧氏嵌套凸体追踪的多项式维度依赖性。
英文摘要
We study online optimization with nested shrinking feasible regions in two settings: convex optimization with nested evolving feasible sets (CONES) and adversarial constrained online convex optimization (COCO). Our algorithms separate loss control from geometric movement: constrained minimizers and cumulative-loss tests preserve regret guarantees, while a deterministic resettable nested convex-body chaser limits movement. For CONES with a $G$-Lipschitz, $μ$-strongly convex objective on a diameter-$D$ domain, we chase intersections of the current feasible set with adaptive objective sublevel sets. Using the Euclidean chasing ratio $O(\sqrt{d\log(1+d)})$, we obtain nonpositive regret at every prefix and movement $O(\sqrt{d\log(1+d)\,GD\log(eT)/μ})$. The bound adapts to the increase in the constrained optimum value. In dimension two, with all other parameters fixed, every randomized algorithm with terminal expected regret $O(T^β)$, $β<1$, suffers $Ω(\sqrt{\log T})$ expected movement on some deterministic nested sequence, proving optimal horizon dependence. Under linear growth away from the constrained minimizer set, Steiner-point tracking yields movement independent of $T$. For general convex COCO, one-step-delayed chasing with regularized-leader resets gives regret $O(G_fD\sqrt{d\log(1+d)T})$ and cumulative constraint violation $O(G_gD\sqrt{d\log(1+d)T})$. For strongly convex losses, both are $O(d\log(1+d)\log(eT))$ when other parameters are fixed. These reductions replace the $O(d^{d/2})$ projection-path factor in prior analyses by the polynomial dimension dependence of Euclidean nested convex-body chasing.
