分布式线性可分计算
Distributed Linearly Separable Computation
摘要：
本文阐述了一个分布式计算问题，其中主要求$N$分布式工作者计算线性可分函数。任务函数可以表示为$K$消息的$K_c$线性组合，其中每个消息是一个数据集的函数。我们的目标是找到计算成本（分配给每个工作者的未编码数据集的数量）和通信成本（主机必须下载的符号数量）之间的最佳权衡，使得从$N$工作者中的任何$N_r$的答案中，主机可以以高概率恢复任务函数，其中$K_c$线性组合的系数是一致的i.i.d.在足够大的有限域上该问题可以看作是分布式梯度编码和分布式线性变换等现有问题的推广。在本文中，我们考虑的具体情况下，计算成本是最小的，并提出新的可扩展性计划和逆界的最佳通信成本。可扩展性和逆界符合一些系统参数，当它们不匹配时，我们证明了可实现的分布式计算计划是最优的约束下，广泛使用的“循环分配”计划的数据集。我们的研究结果还表明，当$K = N$，与Tandon等人提出的最佳分布式梯度编码方案，从主恢复一个线性组合的$K$消息相同的通信成本，我们提出的方案可以让主恢复任何额外的$N_r - 1$独立的线性组合的消息以高概率。
英文摘要：
This paper formulates a distributed computation problem, where a master asks $N$ distributed workers to compute a linearly separable function. The task function can be expressed as $K_c$ linear combinations of $K$ messages, where each message is a function of one dataset. Our objective is to find the optimal tradeoff between the computation cost (number of uncoded datasets assigned to each worker) and the communication cost (number of symbols the master must download), such that from the answers of any $N_r$ out of $N$ workers the master can recover the task function with high probability, where the coefficients of the $K_c$ linear combinations are uniformly i.i.d. over some large enough finite field. The formulated problem can be seen as a generalized version of some existing problems, such as distributed gradient coding and distributed linear transform. In this paper, we consider the specific case where the computation cost is minimum, and propose novel achievability schemes and converse bounds for the optimal communication cost. Achievability and converse bounds coincide for some system parameters; when they do not match, we prove that the achievable distributed computing scheme is optimal under the constraint of a widely used `cyclic assignment' scheme on the datasets. Our results also show that when $K = N$, with the same communication cost as the optimal distributed gradient coding scheme proposed by Tandon et al. from which the master recovers one linear combination of $K$ messages, our proposed scheme can let the master recover any additional $N_r - 1$ independent linear combinations of messages with high probability.
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。
