论文标题
部分可观测时空混沌系统的无模型预测
On the amortized complexity of approximate counting
论文作者
论文摘要
天真地存储柜台至值$ n $将需要$ω（\ log n）$位置的内存。 Nelson和Yu [NY22]在[Morris78]的工作之后表明，如果查询答案仅需$（1 +ε）$ - 近似值至少$ 1-δ$，则$ o（\ log \ log \ log \ log log \ log \ log \ log \ log \ log \ log \ log \ log \ log \ log \ log \ log \ log \ log \ log（1/Δ）但是，莫里斯（Morris）研究此问题的原始动机以及现代应用程序不仅需要维护一个计数器，而且还需要$ k $柜台，以$ k $大。这激发了以下问题：对于$ k $大，可以同时使用渐近的内存来同时维护$ k $的计数器，而不是$ k $ a $ k $乘以单个计数器的成本？也就是说，这个问题是否受益于改进的{\ it摊销}空间复杂性绑定？ 我们以负面的方式回答这个问题。具体而言，我们证明了几乎全部参数范围的下限，这表明，就存储器使用而言，在存储多个计数器时，没有通过摊销可以通过渐近益处。我们的主要证明利用了Braverman，Garg和Woodruff在2020年最近引入的某种“信息成本”概念，以证明流媒体算法的下限。
Naively storing a counter up to value $n$ would require $Ω(\log n)$ bits of memory. Nelson and Yu [NY22], following work of [Morris78], showed that if the query answers need only be $(1+ε)$-approximate with probability at least $1 - δ$, then $O(\log\log n + \log\log(1/δ) + \log(1/ε))$ bits suffice, and in fact this bound is tight. Morris' original motivation for studying this problem though, as well as modern applications, require not only maintaining one counter, but rather $k$ counters for $k$ large. This motivates the following question: for $k$ large, can $k$ counters be simultaneously maintained using asymptotically less memory than $k$ times the cost of an individual counter? That is to say, does this problem benefit from an improved {\it amortized} space complexity bound? We answer this question in the negative. Specifically, we prove a lower bound for nearly the full range of parameters showing that, in terms of memory usage, there is no asymptotic benefit possible via amortization when storing multiple counters. Our main proof utilizes a certain notion of "information cost" recently introduced by Braverman, Garg and Woodruff in FOCS 2020 to prove lower bounds for streaming algorithms.
