Motivated by recent work on the experts problem in the streaming model, we consider the experts problem in the sliding window model. The sliding window model is a well-studied model that captures applications such as traffic monitoring, epidemic tracking, and automated trading, where recent information is more valuable than older data. Formally, we have $n$ experts, $T$ days, the ability to query the predictions of $q$ experts on each day, a limited amount of memory, and should achieve the (near-)optimal regret $\sqrt{nW}\text{polylog}(nT)$ regret over any window of the last $W$ days. While it is impossible to achieve such regret with $1$ query, we show that with $2$ queries we can achieve such regret and with only $\text{polylog}(nT)$ bits of memory. Not only are our algorithms optimal for sliding windows, but we also show for every interval $\mathcal{I}$ of days that we achieve $\sqrt{n|\mathcal{I}|}\text{polylog}(nT)$ regret with $2$ queries and only $\text{polylog}(nT)$ bits of memory, providing an exponential improvement on the memory of previous interval regret algorithms. Building upon these techniques, we address the bandit problem in data streams, where $q=1$, achieving $n T^{2/3}\text{polylog}(T)$ regret with $\text{polylog}(nT)$ memory, which is the first sublinear regret in the streaming model in the bandit setting with polylogarithmic memory; this can be further improved to the optimal $\mathcal{O}(\sqrt{nT})$ regret if the best expert's losses are in a random order.

翻译：受近期流模型中专家问题研究的启发，我们考虑滑动窗口模型中的专家问题。滑动窗口模型是一种经过充分研究的模型，适用于流量监测、疫情追踪和自动交易等应用场景，其中近期信息比旧数据更具价值。形式化地，我们设有$n$位专家、$T$天时间、每天可查询$q$位专家的预测能力、有限的内存容量，并应在最近$W$天的任意窗口上实现（近似）最优遗憾度$\sqrt{nW}\text{polylog}(nT)$。虽然通过$1$次查询无法实现该遗憾度，但我们证明通过$2$次查询配合仅$\text{polylog}(nT)$比特内存即可达成。我们的算法不仅对滑动窗口达到最优，还证明对于任意天数区间$\mathcal{I}$，通过$2$次查询与仅$\text{polylog}(nT)$比特内存即可实现$\sqrt{n|\mathcal{I}|}\text{polylog}(nT)$遗憾度，这相较于先前区间遗憾算法的内存消耗实现了指数级改进。基于这些技术，我们进一步解决了数据流中的赌博机问题（$q=1$），通过$\text{polylog}(nT)$内存实现$n T^{2/3}\text{polylog}(T)$遗憾度，这是在多对数内存条件下赌博机设置中流模型首次实现次线性遗憾；若最优专家的损失呈随机顺序，该结果可进一步优化至最优遗憾度$\mathcal{O}(\sqrt{nT})$。