[论文解读] Consensus Needs Broadcast in Noiseless Models but can be Exponentially Easier in the Presence of Noise

Andrea Clementi, Luciano Gualà+4 more|arXiv (Cornell University)|Jul 15, 2018

Distributed systems and fault tolerance参考文献 52被引用 2

一句话总结

该论文证明，在无噪声的分布式模型中，共识问题至少与广播问题一样困难，且在统一的GOSSIP模型中，其轮数下界为紧致的对数级。相比之下，在如噪声统一PULL这样的噪声模型中，共识问题可比广播问题指数级地更简单：广播需要Θ(ε⁻²n log n)轮，而二元共识仅需Θ(ε⁻² log n)轮，从而因噪声引起的信道信息聚合而展现出指数级的复杂度差距。

ABSTRACT

Consensus and Broadcast are two fundamental problems in distributed computing, whose solutions have several applications. Intuitively, Consensus should be no harder than Broadcast, and this can be rigorously established in several models. Can Consensus be easier than Broadcast? In models that allow noiseless communication, we prove a reduction of (a suitable variant of) Broadcast to binary Consensus, that preserves the communication model and all complexity parameters such as randomness, number of rounds, communication per round, etc., while there is a loss in the success probability of the protocol. Using this reduction, we get, among other applications, the first logarithmic lower bound on the number of rounds needed to achieve Consensus in the uniform GOSSIP model on the complete graph. The lower bound is tight and, in this model, Consensus and Broadcast are equivalent. We then turn to distributed models with noisy communication channels that have been studied in the context of some bio-inspired systems. In such models, only one noisy bit is exchanged when a communication channel is established between two nodes, and so one cannot easily simulate a noiseless protocol by using error-correcting codes. An Ω(ε^{-2} n) lower bound is proved by Boczkowski et al. [PLOS Comp. Bio. 2018] on the convergence time of binary Broadcast in one such model (noisy uniform PULL), where ε is a parameter that measures the amount of noise). We prove an O(ε^{-2} log n) upper bound on the convergence time of binary Consensus in such model, thus establishing an exponential complexity gap between Consensus versus Broadcast. We also prove our upper bound above is tight and this implies, for binary Consensus, a further strong complexity gap between noisy uniform PULL and noisy uniform PUSH. Finally, we show a Θ(ε^{-2} n log n) bound for Broadcast in the noisy uniform PULL.

研究动机与目标

• 研究在分布式系统中共识与广播问题的相对复杂度。
• 确定在某些通信模型中，共识是否可以严格比广播更容易。
• 分析通信噪声对两种问题轮数复杂度的影响。
• 在噪声和无噪声模型中，为广播和共识问题建立紧致的下界与上界。

提出的方法

• 在无噪声模型中，提出一种从广播变体到二元共识的归约，保持复杂度参数不变，证明共识至少与广播一样困难。
• 使用切诺夫不等式和联合界分析噪声模型中k-多数动态的收敛性，证明在存在偏置时可实现快速共识。
• 设计一种两阶段协议NoisyBroadcast，用于噪声统一PULL模型：第一阶段聚合噪声消息，第二阶段运行多数共识。
• 应用Berry-Esseen定理，将消息分布近似为正态分布，以进行概率分析。
• 利用集中不等式，界定NoisyBroadcast第一阶段中错误意见选择的概率。
• 通过概率分析和正态分布的尾部界限，推导出紧致的渐近界。

实验结果

研究问题

• RQ1 在任何分布式模型中，共识是否严格比广播更容易？
• RQ2 通信噪声如何影响共识与广播的轮数复杂度？
• RQ3 能否利用噪声实现共识协议的指数级性能提升？
• RQ4 在噪声统一PULL模型中，广播与共识问题的最紧致轮数复杂度界是什么？

主要发现

• 在无噪声的统一GOSSIP模型中，共识需要Ω(log n)轮，与广播的已知下界一致，证明两者渐近等价。
• 在噪声统一PULL模型中，广播需要Θ(ε⁻²n log n)轮，与已知下界一致，确立了紧致性。
• 在相同噪声模型中，二元共识可在Θ(ε⁻² log n)轮内解决，与广播相比展现出指数级差距。
• 该指数级差距源于噪声促进了信息的快速聚合：节点可利用噪声比特交换，在信噪比较低的情况下仍能迅速收敛。
• 协议NoisyBroadcast以高概率在O(ε⁻²n log n)轮内完成广播，与下界匹配，证明了最优性。
• 分析表明，即使在噪声通信下，经过线性数量的噪声拉取后，仍有一半以上的节点可正确推断出源节点的值，从而实现快速共识。