重命中估计的查询复杂度
Query complexity of heavy hitter estimation
摘要：
我们考虑的问题，确定的元素的子集$\mathcal{S}^γ_{\mathcal{P}}$的支持下的底层分布$\mathcal{P}$的概率值大于一个给定的阈值$γ$，通过主动查询一个oracle，以获得有关序列$X_1, X_2, \ldots$的$i.i.d.$样本从$\mathcal{P}$。我们考虑两种查询模型：$(a)$每个查询是索引$i$并且oracle返回值$X_i$和$(b)$每个查询是一对$(i,j)$并且oracle给出确认是否为$X_i = X_j$的二进制答案。对于这些查询模型中的每一个，我们设计了顺序估计算法，在每一轮，要么决定什么查询发送到Oracle的整个历史的响应，或决定停止并输出估计的$\mathcal{S}^γ_{\mathcal{P}}$，这是需要正确的一些预先指定的大概率。我们提供的查询复杂度的算法的任何分布$\mathcal{P}$的上限，也得到了最佳的查询复杂度的下限下的两个查询模型。我们还考虑了两个查询模型的噪声版本，并提出了鲁棒的估计，可以有效地对抗噪音的预言机响应。
英文摘要：
We consider the problem of identifying the subset $\mathcal{S}^γ_{\mathcal{P}}$ of elements in the support of an underlying distribution $\mathcal{P}$ whose probability value is larger than a given threshold $γ$, by actively querying an oracle to gain information about a sequence $X_1, X_2, \ldots$ of $i.i.d.$ samples drawn from $\mathcal{P}$. We consider two query models: $(a)$ each query is an index $i$ and the oracle return the value $X_i$ and $(b)$ each query is a pair $(i,j)$ and the oracle gives a binary answer confirming if $X_i = X_j$ or not. For each of these query models, we design sequential estimation algorithms which at each round, either decide what query to send to the oracle depending on the entire history of responses or decide to stop and output an estimate of $\mathcal{S}^γ_{\mathcal{P}}$, which is required to be correct with some pre-specified large probability. We provide upper bounds on the query complexity of the algorithms for any distribution $\mathcal{P}$ and also derive lower bounds on the optimal query complexity under the two query models. We also consider noisy versions of the two query models and propose robust estimators which can effectively counter the noise in the oracle responses.
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