分布式仿真与分布式推理
Distributed Simulation and Distributed Inference
摘要：
来自大小为$k$的域上的未知概率分布$\bf p$的独立样本分布在$n$参与者之间，每个参与者持有一个样本。每个参与者可以在通信的同时消息传递模型中将$\ell$比特传送给中央裁判，以帮助裁判推断未知$\bf p$的属性。在$\ell<\log k$的通信匮乏设置中，推理所需的最少参与者数量是多少？我们开始探索一个通用的“模拟和推断”的策略，这样的推理问题，中心模拟所需数量的样本从未知的分布和应用标准的推理算法的配置设置。我们的第一个结果表明，对于$\ell<\log k$，即使是单个样本的完美模拟也是不可能的。尽管如此，我们提出了一个拉斯维加斯算法，模拟一个单一的样本从未知的分布使用$O(k/2^\ell)$样本的期望。作为直接推论，我们得到模拟-推断达到了$Θ(k^2/2^\ellε^2)$的最佳样本复杂度，用于学习未知分布到总变异距离$ε$。对于身份测试的原型测试问题，模拟和推断与$O(k^{3/2}/2^\ellε^2)$样本一起工作，这似乎是所有不使用任何额外资源的通信协议所固有的要求。有趣的是，我们可以使用公共硬币打破这一障碍。具体来说，我们展示了一个公共硬币通信协议，使用$O(k/\sqrt{2^\ell}ε^2)$样本进行身份测试。此外，我们表明，这是最佳的常数因子。我们理论上的样本最优协议很容易在实践中实现。我们的证明下限需要显示收缩$χ^2$距离的产品分布由于通信的限制，可能是独立的利益。
英文摘要：
Independent samples from an unknown probability distribution $\bf p$ on a domain of size $k$ are distributed across $n$ players, with each player holding one sample. Each player can communicate $\ell$ bits to a central referee in a simultaneous message passing model of communication to help the referee infer a property of the unknown $\bf p$. What is the least number of players for inference required in the communication-starved setting of $\ell<\log k$? We begin by exploring a general "simulate-and-infer" strategy for such inference problems where the center simulates the desired number of samples from the unknown distribution and applies standard inference algorithms for the collocated setting. Our first result shows that for $\ell<\log k$ perfect simulation of even a single sample is not possible. Nonetheless, we present a Las Vegas algorithm that simulates a single sample from the unknown distribution using $O(k/2^\ell)$ samples in expectation. As an immediate corollary, we get that simulate-and-infer attains the optimal sample complexity of $Θ(k^2/2^\ellε^2)$ for learning the unknown distribution to total variation distance $ε$. For the prototypical testing problem of identity testing, simulate-and-infer works with $O(k^{3/2}/2^\ellε^2)$ samples, a requirement that seems to be inherent for all communication protocols not using any additional resources. Interestingly, we can break this barrier using public coins. Specifically, we exhibit a public-coin communication protocol that performs identity testing using $O(k/\sqrt{2^\ell}ε^2)$ samples. Furthermore, we show that this is optimal up to constant factors. Our theoretically sample-optimal protocol is easy to implement in practice. Our proof of lower bound entails showing a contraction in $χ^2$ distance of product distributions due to communication constraints and may be of independent interest.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
