[论文解读] Stochastic ep load balancing and moment problems via the L-function method

Marco Molinaro|arXiv (Cornell University)|Jan 6, 2019

Advanced Queuing Theory Analysis被引用 6

一句话总结

本文将L-函数方法首次应用于具有p-范数目标的随机负载均衡问题，对随机ep负载均衡问题（StochLoadBalp）实现了常数因子近似，显著优于先前的O(p/ln p)-近似。该方法实现了对随机变量p-次矩的可分、基于边缘的建模，使得即使在具有指数级约束数量的情况下，也能实现高效的凸松弛与舍入。

ABSTRACT

This paper considers stochastic optimization problems whose objective functions involve powers of random variables. For a concrete example, consider the classic Stochasticep Load Balancing Problem (StochLoadBalp): There are m machines and n jobs, and we are given independent random variables Yij describing the distribution of the load incurred on machine i if we assign job j to it. The goal is to assign each job to the machines in order to minimize the expected ep-norm of the total load incurred over the machines. That is, letting Ji denote the jobs assigned to machine i, we want to minimize E(∑i(∑j ∈ JiYij)p)1 / p. While convex relaxations represent one of the most powerful algorithmic tools, in problems such as StochLoadBalp the main difficulty is to capture such objective function in a way that only depends on each random variable separately.In this paper, show how to capture p-power-type objectives in such separable way by using the L-function method. This method was precisely introduced by Latala to capture in a sharp way the moment of sums of random variables through the individual marginals. We first show how this quickly leads to a constant-factor approximation for very general subset selection problem with p-moment objective.Moreover, we give a constant-factor approximation for StochLoadBalp, improving on the recent O(p/ln p)-approximation of [Gupta et al., SODA 18]. Here the application of the method is much more involved. In particular, we need to prove structural results connecting the expected ep-norm of a random vector with the p-moments of its coordinate-marginals (machine loads) in a sharp way, taking into account simultaneously the different scales of the loads that are incurred in the different machines by an unknown assignment. Moreover, our starting convex (indeed linear) relaxation has exponentially many constraints that are not conducive to integral rounding; we need to use the solution of this LP to obtain a reduced LP which can then be used to obtain the desired assignment.

研究动机与目标

• 解决在随机作业分配下最小化机器负载的期望p-范数的挑战。
• 开发一种方法，仅使用单个随机变量的边缘分布，以可分方式捕捉p-次幂类型的目标。
• 通过实现常数因子近似，超越先前对StochLoadBalp的O(p/ln p)-近似。
• 设计一种凸松弛框架，通过结构化线性规划降维，处理指数级数量的约束。
• 建立随机向量的期望p-范数与各坐标边缘分布的p-矩之间在不同负载尺度下的紧密联系。

提出的方法

• 利用Latala最初提出的L-函数方法，仅通过边缘分布精确估计随机变量和的矩。
• 将该方法应用于以可分形式建模p-次幂目标，实现按机器和任务分配的分解。
• 推导出总负载期望p-范数与各台机器负载边缘分布p-矩之间具有结构的不等式。
• 构建一个初始线性规划，包含指数级数量的约束，以建模分配问题。
• 通过求解原问题，将初始线性规划缩减为更小规模、可处理的线性规划，从而实现有效的舍入与分配。
• 利用简化后的线性规划，推导出StochLoadBalp的常数因子近似算法。

实验结果

研究问题

• RQ1 L-函数方法能否被适配，以实现对具有p-范数目标的随机负载均衡的常数因子近似？
• RQ2 如何以精确且尺度不变的方式，将机器负载的p-矩与总负载向量的期望p-范数关联起来？
• RQ3 负载分布的何种结构性质，使得即使约束数量呈指数级增长，仍能通过简化线性规划保持近似保证？
• RQ4 是否可能实现对StochLoadBalp的常数因子近似，从而超越先前工作的O(p/ln p)界？
• RQ5 如何设计凸松弛，使其通过可分的、基于边缘的约束来捕捉p-次幂目标？

主要发现

• 本文实现了对随机ep负载均衡问题（StochLoadBalp）的常数因子近似，显著优于先前的O(p/ln p)-近似。
• L-函数方法实现了仅基于单个随机变量边缘分布的精确、可分的p-次幂目标表征。
• 证明了结构性结果，将随机向量的期望p-范数与各坐标边缘分布的p-矩联系起来，且能适应不同的负载尺度。
• 从初始的指数规模线性规划中构建出一个简化线性规划，实现了高效舍入与分配，同时保持近似保证。
• 该方法通过利用尺度不变的矩不等式，成功应对了跨机器的未知分配尺度挑战。
• 该方法表明，通过基于边缘的凸松弛，可以有效捕捉并近似随机优化中的复杂p-范数目标。