AI 中文总结
针对对抗性到达的标量存储系统，提出累积策略重参数化与衰减加权投影次梯度更新，实现 $O(\sqrt T)$ 遗憾及匹配下界，刻画了时间范围和保留时间的联合依赖。
AI 中文摘要
我们研究标量存储系统的在线控制问题，其中到达量为对抗性非负值，保留系数已知，成本函数为凸函数且依赖于状态和动作。每个动作必须遵守当前资源可用性，并在当前到达量和成本函数揭示之前选择。对于现有的单纯形扰动-动作策略类，我们通过累积分配分数和衰减加权投影次梯度更新给出了精确的重新参数化。由此产生的遗憾界与策略记忆长度无关。对于固定的保留系数和成本常数，控制器相对于该类别中最佳固定无限记忆策略实现了 $O(\sqrt T)$ 的遗憾，每轮使用 $O(\log T)$ 的记忆和算术运算以及一次成本次梯度查询。一种针对存储的块构造给出了针对每个因果可行控制器（包括随机控制器）的匹配下界。记 $\tau=(1-\alpha)^{-1}$，对于每个有限记忆单纯形策略类及其无限记忆扩展，当 $\alpha\in[1/2,1)$，$T\ge4$ 且正成本常数固定时，极小极大期望遗憾为 $\Theta(\sqrt T\min\{T,\tau\}^{3/2})$。这确定了这些策略基准的联合时间范围和保留时间依赖性。
英文摘要
We study online control of a scalar storage system with adversarial nonnegative arrivals, known retention coefficient, and convex costs depending on both state and action. Each action must respect current resource availability and is chosen before the current arrival and cost function are revealed. For the existing simplex disturbance-action policy class, we give an exact reparameterization by cumulative allocation fractions and a decay-weighted projected subgradient update. The resulting regret bound is independent of policy memory length. For fixed retention coefficient and cost constants, the controller achieves $O(\sqrt T)$ regret against the best fixed infinite-memory policy in this class, using $O(\log T)$ memory and arithmetic operations per round and one cost-subgradient query. A storage-specific block construction gives a matching lower bound against every causal feasible controller, including randomized controllers. Writing $τ=(1-α)^{-1}$, the minimax expected regret is $Θ(\sqrt T\min\{T,τ\}^{3/2})$ for every finite-memory simplex policy class and its infinite-memory extension, when $α\in[1/2,1)$, $T\ge4$, and the positive cost constants are fixed. This identifies the joint horizon and retention-time dependence for these policy benchmarks.
