[论文解读] Fast Convergence Rates of Distributed Subgradient Methods with Adaptive Quantization

Thinh T. Doan, Siva Theja Maguluri+1 more|arXiv (Cornell University)|Oct 30, 2018

Stochastic Gradient Optimization Techniques被引用 4

一句话总结

该论文提出了一种用于分布式次梯度方法的自适应量化方法，在有限通信带宽下实现了与无量化方法相当的收敛速率。通过根据算法进展动态调整量化器分辨率，该方法在凸和强凸目标函数下均能保持最优收敛速率——其速率与量化器分辨率相关的常数因子内，与无量化方法相当。

ABSTRACT

We study distributed optimization problems over a network when the communication between the nodes is constrained, and so information that is exchanged between the nodes must be quantized. Recent advances using the distributed gradient algorithm with a quantization scheme at a fixed resolution have established convergence, but at rates significantly slower than when the communications are unquantized. In this paper, we introduce a novel quantization method, which we refer to as adaptive quantization, that allows us to match the convergence rates under perfect communications. Our approach adjusts the quantization scheme used by each node as the algorithm progresses: as we approach the solution, we become more certain about where the state variables are localized, and adapt the quantizer codebook accordingly. We bound the convergence rates of the proposed method as a function of the communication bandwidth, the underlying network topology, and structural properties of the constituent objective functions. In particular, we show that if the objective functions are convex or strongly convex, then using adaptive quantization does not affect the rate of convergence of the distributed subgradient methods when the communications are quantized, except for a constant that depends on the resolution of the quantizer. To the best of our knowledge, the rates achieved in this paper are better than any existing work in the literature for distributed gradient methods under finite communication bandwidths. We also provide numerical simulations that compare convergence properties of the distributed gradient methods with and without quantization for solving distributed regression problems for both quadratic and absolute loss functions.

研究动机与目标

• 解决带宽受限网络中因通信量化导致的分布式次梯度方法性能下降问题。
• 克服先前工作中固定分辨率量化方案收敛缓慢的缺陷。
• 设计一种能根据算法进展自适应调整的量化策略，以保持收敛速率。
• 在自适应量化下，为凸和强凸目标函数建立理论收敛边界。
• 通过仿真表明，8比特自适应量化可实现与无量化通信几乎完全相同的性能。

提出的方法

• 提出一种改进的分布式次梯度算法，在每次迭代中显式考虑量化误差。
• 设计一种自适应量化方案，其中量化器码书在每次迭代中根据当前步长和估计的解定位进行调整。
• 采用懒惰梅特罗波利斯矩阵作为混合矩阵，以确保网络连通性并满足谱隙条件。
• 应用一种量化规则，随着算法趋近解而缩小量化区间，从而在关键区域提高分辨率。
• 理论分析利用谱隙 $1 - ho_2$ 和量化器分辨率 $ riangle$ 来界定收敛速率。
• 结合使用次梯度下降与递减步长，以及在连通网络上的平均一致性，以确保收敛。

实验结果

研究问题

• RQ1 自适应量化能否消除量化与无量化分布式次梯度方法之间的性能差距？
• RQ2 在有限通信带宽下，自适应量化可实现何种收敛速率？
• RQ3 量化器分辨率的选择如何影响收敛速度和最终精度？
• RQ4 在量化条件下，能否保持分布式次梯度方法的收敛速率，使其与无量化情况相当？
• RQ5 与固定分辨率或随机量化相比，自适应量化在收敛速度和稳定性方面表现如何？

主要发现

• 对于凸目标函数，收敛速率被界定为 $ f( extbf{z}_i(k)) - f^* riangleq rac{ riangle^2}{(1 - ho_2)^2} rac{ ext{ln } k}{ ext{sqrt}(k)} $，与无量化 DSG 方法相比仅相差一个常数因子。
• 对于强凸目标函数，收敛速率为 $ ext{||} extbf{z}_i(k) - extbf{x}^* ext{||}^2 riangleq rac{ riangle^2}{(1 - ho_2)^2} rac{ ext{ln } k}{k} $，同样在常数因子内与无量化 DSG 方法一致。
• 数值仿真表明，8比特自适应量化可实现与无量化通信几乎无法区分的收敛性能。
• 仅使用4比特时，算法仍能有效收敛，且迭代次数近似与 $ 1/(2^b - 1)^2 $ 成比例，与理论上限边界一致。
• 自适应量化优于时变量化（需无穷比特）和随机量化（收敛缓慢）。
• 所提方法实现了两全其美：既保持最优收敛速率，又实现有限比特通信，而先前方法往往需在两者之间取舍。