AI 中文总结
本文扩展CONES框架至时变损失函数，提出投影近端算法，在凸损失下实现遗憾与移动成本的同时最优权衡，在强凸损失下实现常数遗憾和对数移动成本，并给出相应下界。
AI 中文摘要
嵌套演化可行集下的凸优化（CONES）在文献\cite{CONESVaze}中被提出，其中目标函数\\(f\\)保持固定，但可行区域随时间演化为嵌套序列\\(S_1 \supseteq S_2 \supseteq \cdots \supseteq S_T\\)。在线算法的目标是在始终确保可行性的同时，相对于事后静态最优基准最小化遗憾，并最小化总移动成本\\(M_\cA(T)\\)。CONES是著名的嵌套凸体追踪（NCBC）的面向优化的推广。在本文中，我们将CONES扩展为允许损失函数\\(f_t\\)也随时间变化。当所有损失函数为凸函数时，我们证明投影近端算法在时间范围\\(T\\)内，对于任意\\(\beta \in [0,1)\\)，同时实现\\(O(T^{1-\beta})\\)的遗憾和\\(O(T^\beta)\\)的移动成本。我们还证明，任何具有\\(O(T^\beta)\\)遗憾的弱自适应在线算法，对于任意\\(\beta \in [0,1)\\)，其移动成本为\\(\Omega\left(T^{\frac{1-\beta}{2}}\right)\\)。当所有损失函数为强凸函数时，我们证明投影近端算法同时实现\\(O(1)\\)的遗憾和\\(O(\log T)\\)的移动成本。与此互补，我们证明任何具有次线性随时遗憾的在线算法，其移动成本为\\(\Omega\left(\log T\right)\\)。
英文摘要
Convex Optimization with Nested Evolving Feasible Sets (CONES)} was introduced in \cite{CONESVaze} where the objective function \(f\) remains fixed but the feasible region evolves over time as a nested sequence \(S_1 \supseteq S_2 \supseteq \cdots \supseteq S_T\). The goal of an online algorithm is to simultaneously minimize the regret with respect to hindsight static optimal benchmark and the total movement cost $M_\cA(T)$ while ensuring feasibility at all times. CONES is an optimization-oriented generalization of the well-known \emph{nested convex body chasing} (NCBC). In this paper, we extend CONES to allow for loss functions $f_t'$s to also change over time. When all loss functions are convex, we show that the projected proximal algorithm achieves $O(T^{1-β}), O(T^β)$ simultaneous regret and movement cost, respectively, for any $β\in [0,1)$, over a time horizon of $T$. We also show that any {\it weakly adaptive} online algorithm with $O(T^β)$ regret has a movement cost of $Ω\left(T^{\frac{1-β}{2}}\right)$ for any $β\in [0,1)$. When all loss functions are strongly convex, we show that the projected proximal algorithm simultaneously achieves $O(1)$ regret and a movement cost of $O(\log T)$. To complement this, we show that any online algorithm with sublinear {\it anytime} regret has a movement cost of $Ω\left(\log T\right)$.
