【论文考古】分布式优化 Communication Complexity of Convex Optimization

J. N. Tsitsiklis and Z.-Q. Luo, “Communication complexity of convex optimization,” Journal of Complexity, vol. 3, no. 3, pp. 231–243, Sep. 1987, doi: 10.1016/0885-064x(87)90013-6.

问题描述

两个用户各自有一个凸函数\(f_i\),相互交互最少的二进制消息,从而找到\(f_i+f_2\)的最优点

基本定义

  • \(\mathscr{F}\):定义域\([0,1]^n\)上凸函数的一个集合

  • \(I(f;\epsilon)\in[0,1]^n\):定义域上,给定误差\(\epsilon\)\(f\)最小值对应的自变量集合(\(f(x) \leq f(y)+\varepsilon, \forall y \in[0,1]^{n}\)

  • \(C(f_1,f_2;\epsilon,\pi)\):在协议\(\pi\)和精度\(\epsilon\)下,两个函数通过交换信息找到集合\(I\left(f_{1}+f_{2} ; \varepsilon\right)\)中元素所需的消息数目

  • \(C(\mathscr{F} ; \varepsilon, \pi)\):该协议在最坏情况下找到目标所需交换的消息数量

    \[C(\mathscr{F} ; \varepsilon, \pi)=\sup _{f_{1}, f_{2} \in \mathscr{F}} C\left(f_{1}, f_{2} ; \varepsilon, \pi\right) \]

  • \(C(\mathscr{F} ; \varepsilon)\):最优协议下所需的交换消息的数量,又称为\(\epsilon\)-communication complexity

    \[C(\mathscr{F} ; \varepsilon)=\inf _{\pi \in \mathrm{I}(\varepsilon)} C(\mathscr{F} ; \varepsilon, \pi) \]

  • 消息传输的模式,通信\(T\)

    • 每次传播信息的计算

      \[m_{i}(t)=M_{i, t}\left(f_{i}, m_{j}(0), \ldots, m_{j}(t-1)\right) \]

    • 最终最优点的确定

      \[x=Q\left(f_{1}, m_{2}(0), \ldots ., m_{2}(T-1)\right) \]

Straightforward Lower Bound

Lemma 1:\(\text { If } \mathscr{F} \subset \mathscr{G} \mathscr{\text { then }} C(\mathscr{F} ; \varepsilon) \leq C(\mathscr{G}; \varepsilon)\)

简单函数所需传输的消息数量更少

Proposition:\(C\left(\mathcal{F}_{Q} ; \varepsilon\right) \geq O(n(\log n+\log (1 / \varepsilon)))\)

其中\(\mathcal{F}_{Q}\)表示带有\(f(x)=\|x-x^\star\|^2\)形式的二次函数的集合,其中\(x^\star\in [0,1]^n\)。根据Lemma知道,选择最简单的函数能找到下界。考虑\(f_1=0\),所以\(f_2\)的最小值需要控制在\(\epsilon^{1/2}\)的精度内,因此至少需要\(\left(A n / \varepsilon^{1 / 2}\right)^{B n}\)个半径为\(\epsilon^{1/2}\)Euclidean ball来覆盖中\([0,1]^n\)。因此最终\(Q\)的解集的势至少就是\(\left(A n / \varepsilon^{1 / 2}\right)^{B n}\)。由于函数的值域的势不会超过定义域的势,所以\(Q\)的解集的势不超过\(2^T\),也就有\(T \geq O(n(\log n+\log (1 / \varepsilon))\)

Naive Upper Bounds

The method of the centers of gravity (MCG) 在求解凸函数势需要最小次数的梯度计算。将MCG方法扩展到了分布式的场景,得到上界。

一维下的最优算法

算法核心在于用消息指示不同的计算步骤,而不是传递数据

算法首先定义两个区间,分别表示

  • \([a,b]\)\(f_1+f_2\)最优点所在的区间,\(x^\star \in [a,b]\)
  • \([c,d]\)\(f'(x^{\star})\)\(f'_1(\frac{a+b}{2})\)\(f'_2(\frac{a+b}{2})\)所在的区间

以区间\([c,d]\)为基准,分别计算消息\(m_1,m_2\)

  • \(f'_1(\frac{a+b}{2})\in [c,\frac{c+d}{2}]\)\(m_1=0\),否则\(m_1=1\)
  • \(-f'_2(\frac{a+b}{2})\in[c,\frac{c+d}{2}]\)\(m_2=0\),否则\(m_2=1\)

根据消息\(m_1,m_2\)的不同组合,分别缩减区间\([a,b]\)或者\([c,d]\)。缩减的设计总从两个原则

  1. \((f_1+f_2)'=f'_1+f'_2\),导值的正负性来找最小值
  2. 通过压缩\((f_1+f_2)'(\frac{a+b}{2})\)趋于零,从而确定\(\frac{a+b}{2}\)就是最小值

代码:

import numpy as np
import matplotlib.pyplot as plt


def f1(x):
    return (x - 2) ** 2


def df1(x):
    return 2 * (x - 2)


def f2(x):
    return (x + 1) ** 2


def df2(x):
    return 2 * (x + 1)


a, b, c, d = -1, 1, -3, 3
eps = 0.1

while b - a > eps and d - c > eps:
    if df1((a + b) / 2) <= (c + d) / 2:
        m1 = 0
    else:
        m1 = 1

    if -df2((a + b) / 2) <= (c + d) / 2:
        m2 = 0
    else:
        m2 = 1

    if m1 == 0 and m2 == 1:
        a = (a + b) / 2
    elif m1 == 1 and m2 == 0:
        b = (a + b) / 2
    elif m1 == 1 and m2 == 1:
        c = (c + d) / 2
    elif m1 == 0 and m2 == 0:
        d = (c + d) / 2

    print('传输消息+2')
    print(a, b, c, d)

if b - a <= eps:
    optimum = a + eps
else:
    optimum = f1((a + b) / 2) + f2((a + b) / 2)

print(optimum)
print(f1(0.5) + f2(0.5))
# 直观画图结果
x = np.linspace(-1, 2, 100)
y = f1(x) + f2(x)
plt.plot(x, y)
plt.show()

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