We consider the problem of decentralized consensus optimization, where the
sum of $n$ smooth and strongly convex functions are minimized over $n$
distributed agents that form a connected network. In particular, we consider
the case that the communicated local decision variables among nodes are
quantized in order to alleviate the communication bottleneck in distributed
optimization. We propose the Quantized Decentralized Gradient Descent (QDGD)
algorithm, in which nodes update their local decision variables by combining
the quantized information received from their neighbors with their local
information. We prove that under standard strong convexity and smoothness
assumptions for the objective function, QDGD achieves a vanishing mean solution
error under customary conditions for quantizers. To the best of our knowledge,
this is the first algorithm that achieves vanishing consensus error in the
presence of quantization noise. Moreover, we provide simulation results that
show tight agreement between our derived theoretical convergence rate and the
numerical results.

研究去中心化共识优化中量化对优化带来的影响，并提出了一种基于梯度下降的算法，证明算法在标准强凸和平滑假设下可实现消失的均值解误差，并通过模拟结果验证了理论收敛速度与实际结果的紧密一致性。

一种精确量子化的分散梯度下降算法

An Exact Quantized Decentralized Gradient Descent Algorithm

This paper considers decentralized consensus optimization problems where
nodes of a network have access to different summands of a global objective
function. Nodes cooperate to minimize the global objective by exchanging
information with neighbors only. A decentralized version of the alternating
directions method of multipliers (DADMM) is a common method for solving this
category of problems. DADMM exhibits linear convergence rate to the optimal
objective but its implementation requires solving a convex optimization problem
at each iteration. This can be computationally costly and may result in large
overall convergence times. The decentralized quadratically approximated ADMM
algorithm (DQM), which minimizes a quadratic approximation of the objective
function that DADMM minimizes at each iteration, is proposed here. The
consequent reduction in computational time is shown to have minimal effect on
convergence properties. Convergence still proceeds at a linear rate with a
guaranteed constant that is asymptotically equivalent to the DADMM linear
convergence rate constant. Numerical results demonstrate advantages of DQM
relative to DADMM and other alternatives in a logistic regression problem.

本文提出了一种分散式的求解全局目标函数最小化问题的算法 —— 分散式二次逼近交替方向乘子法（DQM），通过在每次迭代中最小化 DADMM 最小化目标函数的二次逼近，可以减少计算成本并仍然保持收敛速度达到 DADMM 的线性收敛率常数，并在逻辑回归问题中展示了 DQM 相对于 DADMM 和其他替代算法的优势。

DQM: 分散的二次近似交替方向乘子方法

DQM: Decentralized Quadratically Approximated Alternating Direction  Method of Multipliers

In decentralized consensus optimization, a connected network of agents
collaboratively minimize the sum of their local objective functions over a
common decision variable, where their information exchange is restricted
between the neighbors. To this end, one can first obtain a problem
reformulation and then apply the alternating direction method of multipliers
(ADMM). The method applies iterative computation at the individual agents and
information exchange between the neighbors. This approach has been observed to
converge quickly and deemed powerful. This paper establishes its linear
convergence rate for decentralized consensus optimization problem with strongly
convex local objective functions. The theoretical convergence rate is
explicitly given in terms of the network topology, the properties of local
objective functions, and the algorithm parameter. This result is not only a
performance guarantee but also a guideline toward accelerating the ADMM
convergence.

本文研究了具有强凸局部目标函数的去中心化一致性优化问题，并建立了其线性收敛速率及加速的指导原则，其中采用了交替方向乘子方法 (ADMM) 来解决该问题，该方法可在单个代理处进行迭代计算并在邻居之间进行信息交换。