Differential Evolution: A Survey of the State-of-the-Art

@

目录

• 概
• 主要内容
  • DE/rand/1/bin
  • DE/?/?/?
    • DE/rand/1/exp
    • DE/best/1
    • DE/best/2
    • DE/rand/2
  • 超参数的选择
    • $F$的选择
    • $NP$的选择
    • $Cr$的选择
  • 一些连续变体
    • A
    • B
    • C
    • D
    • E
    • G
  • 一些缺点
• 代码

Das S, Suganthan P N. Differential Evolution: A Survey of the State-of-the-Art[J]. IEEE Transactions on Evolutionary Computation, 2011, 15(1): 4-31.

@article{das2011differential,
title={Differential Evolution: A Survey of the State-of-the-Art},
author={Das, Swagatam and Suganthan, P N},
journal={IEEE Transactions on Evolutionary Computation},
volume={15},
number={1},
pages={4--31},
year={2011}}

概

这是一篇关于Differential Evolution (DE) 的综述, 由于对这类方法并不熟悉, 只能简单地做个记录.

主要内容

考虑如下问题,

$$\min \: f(X),$$

其中$X=(x_1,\ldots,x_D)$.

我所知的, 如梯度下降方法, 贝叶斯优化可以用来处理这类问题, 但是还有诸如 evolutionary algorithm (EA), evolutionary programming (EP), evolution strategies(ESs), genetic algorithm (GA), 以及本文介绍的 DE (后面的基本都不了解).

DE/rand/1/bin

先给出最初的形式, 称之为DE/rand/1/bin:

Input: scale factor $F$, crossover rate $Cr$, population size $NP$.
1: 令$G=0$, 并随机初始化$P_G=\{ X_{1,G},\ldots, X_{NP,G}\}$.
2: While the stopping criterion is not satisfied Do:

• For $i=1,\ldots, NP$ do:

1. Mutation step:

$$V_{i,G} = X_{r_1^i,G} + F \cdot (X_{r_2^i,G} - X_{r_3^i,G}).$$

2. Crossover step: 按照如下方式生成$U_{i,G}=(u_{1,i,G},\ldots, u_{D,i,G})$

$$u_{j,i,G} = \left \{ \begin{array}{ll} v_{j,i,G} & if \: \mathrm{rand}[0,1] \le Cr \: or \: j=j_{rand} \\ x_{j,i,G} & otherwise. \end{array} \right.$$

3. Selection step:

$$X_{i,G} = \left \{ \begin{array}{ll} U_{i,G} & if \: f(U_{i,G}) \le f(X_{i,G}) \\ X_{i,G} & otherwise. \end{array} \right.$$

• End For.
• $G=G+1$.
  End While.

其中$X_{i,G}=(x_{j,i,G}, \ldots, x_{D,i,G})$, $j_{rand}$是预先随机生成的一个属于$[1,D]$的整数, 以保证$U$相对于$X$至少有些许变化产生, $X_{r_1^i,G}, X_{r_2^i,G},X_{r_3^i,G}$是从$P_G$中随机抽取且互异的.

在接下来我们可以发现很多变种, 而这些变种往往是Mutation step 和 Crossover step的变体.

DE/?/?/?

DE/rand/1/exp

这是crossover step步的的一个变种:

随机从$[1, D]$中抽取整数$n$和$L$, 然后

$$u_{j,i,G} = \left \{ \begin{array}{ll} v_{j,i,G} & if \: n \le j \le n+L-1\\ x_{j,i,G} & otherwise. \end{array} \right.$$

$L$可以通过下面的步骤生成

• $L=0$
• while $\mathrm{rand}[0,1] \le Cr$ and $L\le D$:

$$L=L+1.$$

DE/best/1

$$V_{i,G}=X_{best,G} + F\cdot (X_{r_1^i,G} - X_{r_2^i,G}),$$

其中$X_{best,G}$是$P_{G}$中的最优的点.

DE/best/2

$$V_{i,G}=X_{best,G} + F\cdot (X_{r_1^i,G} - X_{r_2^i,G}) + F\cdot (X_{r_3^i,G} - X_{r_4^i,G}).$$

DE/rand/2

$$V_{i,G}=X_{r_1^i,G} + F\cdot (X_{r_2^i,G} - X_{r_3^i,G}) + F\cdot (X_{r_4^i,G} - X_{r_5^i,G}),$$

超参数的选择

真的没有细看, 文中粗略地介绍了几处, 还有很多需要查原文.

$F$的选择

有的推荐$[0.4, 1]$(最佳0.5), 有的推荐$0.6$, 有的推荐$[0.4, 0.95]$(最佳0.9).

还有一些自适应的选择, 如

$$F = \left \{ \begin{array}{ll} \max \{l_{\min}, 1- |\frac{f_{\max}}{f_{\min}}|\} & if : |\frac{f_{\max}}{f_{\min}}|<1 \\ \max \{l_{\min}, 1- |\frac{f_{\max}}{f_{\min}}|\} & otherwise, \end{array} \right.$$

我比较疑惑的是难道$|\frac{f_{\max}}{f_{\min}}|$不是大于等于1吗?

$$F_{i,G+1} = \left \{ \begin{array}{ll} \mathrm{rand}[F_l, F_{u}] & with \: probability \:\tau \\ F_{i,G} & else, \end{array} \right.$$

其中$F_l$, $F_u$分别为$F$取值的下界和上界.

$NP$的选择

有的推荐$[5D,10D]$, 有的推荐$[3D, 8D]$.

$Cr$的选择

有的推荐$[0.3, 0.9]$.

还有

$$Cr_{i,G+1} = \left \{ \begin{array}{ll} \mathrm{rand}[0, 1] & with \: probability \:\tau \\ Cr_{i,G} & else, \end{array} \right.$$

一些连续变体

A

$$p=f(X_{r_1})+f(X_{r_2}) + f(X_{r_3}) \\ p_1 = f(X_{r_1})/p \\ p_2 = f(X_{r_2}) / p \\ p_3 = f(X_{r_1}) / p.$$

如果$\mathrm{rand}[0,1] < \Gamma$($\Gamma$是给定的):

$$\begin{array}{ll} V_{i,G+1} = & (X_{r_1}+X_{r_2}+X_{r_3})/3 +(p_2-p_1)(X_{r_1}-X_{r_2}) \\ &+ (p_3-p_2)(X_{r_2} - X_{r_3}) + (p_1-p_3) (X_{r_3}- X_{r_1}), \end{array}$$

否则

$$V_{i,G+1} = X_{r_1} + F \cdot (X_{r_2}-X_{r_3}).$$

B

$$U_{i,G}=X_{i, G}+k_i \cdot (X_{r_1,G}-X_{i,G})+F' \cdot (X_{r_2,G}-X_{r_3, G}),$$

其中$k_i$给定, $F'=k_i \cdot F$.

C

D

即在考虑$x$的时候, 还需要考虑其反$a+b-x$, 假定$x \in [a, b]$, $[a,b]$为我们给定范围, $X$的反类似的构造.

E

其中$X_{n_{best},G}$表示在$X_{i,G}$的$n$的近邻中的最优点, $p, q\in [i-k,i+k]$.

其中$X_{g_{best},G}$为$P_G$中的最优点.

$$V_{i,G}= w \cdot g_{i, G} + (1-w) \cdot L_{i, G}.$$

G

剩下的在复杂环境下的应用就不记录了(只是单纯讲了该怎么做).

一些缺点

1. 高维问题不易处理;
2. 容易被一些问题欺骗, 而现如局部最优解;
3. 对不能分解的函数效果不是很好;
4. 路径往往不会太大(即探索不充分);
5. 缺少收敛性的理论的保证.

代码

$f(x,y)=x^2+50y^2$.

{
  "dim": 2,
  "F": 0.5,
  "NP": 5,
  "Cr": 0.35
}

"""
de.py
"""

import numpy as np
from scipy import stats
import random




class Parameter:

    def __init__(self, dim, xmin, xmax):
        self.dim = dim
        self.xmin = xmin
        self.xmax = xmax
        self.initial()


    def initial(self):
        self.para = stats.uniform.rvs(
            self.xmin, self.xmax - self.xmin
        )

    @property
    def data(self):
        return self.para

    def __getitem__(self, item):
        return self.para[item]

    def __setitem__(self, key, value):
        self.para[key] = value

    def __len__(self):
        return len(self.para)

    def __add__(self, other):
        return self.para + other

    def __mul__(self, other):
        return self.para * other

    def __pow__(self, power):
        return self.para ** power

    def __neg__(self):
        return -self.para

    def __sub__(self, other):
        return self.para - other

    def __truediv__(self, other):
        return self.para / other


class DE:

    def __init__(self, func, dim ,F=0.5, NP=50,
                 Cr=0.35, xmin=-10, xmax=10,
                 require_history=True):
        self.func = func
        self.dim = dim
        self.F = F
        self.NP = NP
        self.Cr = Cr
        self.xmin = np.array(xmin)
        self.xmax = np.array(xmax)
        assert all(self.xmin <= self.xmax), "Invalid xmin or xmax"
        self.require_history = require_history
        self.init_x()
        if self.require_history:
            self.build_history()


    def init_x(self):
        self.paras = [Parameter(self.dim, self.xmin, self.xmax)
                      for i in range(self.NP)]

    @property
    def data(self):
        return [para.data for para in self.paras]

    def build_history(self):
        self.paras_history = [self.data]

    def add_history(self):
        self.paras_history.append(self.data)

    def choose(self, size=3):
        return random.sample(self.paras, k=size)

    def mutation(self):
        x1, x2, x3 = self.choose(3)
        return x1 + self.F * (x2 - x3)

    def crossover(self, v, x):
        u = np.zeros_like(v)
        for i, _ in enumerate(v):
            jrand = random.randint(0, self.dim)
            if np.random.rand() < self.Cr or i is jrand:
                u[i] = v[i]
            else:
                u[i] = x[i]
            u[i] = v[i] if np.random.rand() < self.Cr else x[i]
        return u

    def selection(self, u, x):
        if self.func(u) < self.func(x):
            x.para = u
        else:
            pass

    def step(self):
        donors = [self.mutation()
                  for i in range(self.NP)]

        for i, donor in enumerate(donors):
            x = self.paras[i]
            u = self.crossover(donor, x)
            self.selection(u, x)
        if self.require_history:
            self.add_history()

    def multi_steps(self, times):
        for i in range(times):
            self.step()





class DEbest1(DE):

    def bestone(self):
        y = np.array([self.func(para)
             for para in self.paras])
        return self.paras[np.argmax(y)]

    def mutation(self, bestone):
        x1, x2 = self.choose(2)
        return bestone + self.F * (x1 - x2)

    def step(self):
        bestone = self.bestone()
        donors = [self.mutation(bestone)
                  for i in range(self.NP)]

        for i, donor in enumerate(donors):
            x = self.paras[i]
            u = self.crossover(donor, x)
            self.selection(u, x)
        if self.require_history:
            self.add_history()

class DEbest2(DEbest1):

    def mutation(self, bestone):
        x1, x2, x3, x4 = self.choose(4)
        return bestone + self.F * (x1 - x2) \
                + self.F * (x3 - x4)

class DErand2(DE):

    def mutation(self):
        x1, x2, x3, x4, x5 = self.choose(5)
        return x1 + self.F * (x2 - x3) \
                + self.F * (x4 - x5)


class DErandTM(DE):

    def mutation(self):
        x = self.choose(3)
        y = np.array(list(map(self.func, x)))
        p = y / y.sum()
        part1 = (x[0] + x[1] + x[2]) / 3
        part2 = (p[1] - p[0]) * (x[0] - x[1])
        part3 = (p[2] - p[1]) * (x[2] - x[1])
        part4 = (p[0] - p[2]) * (x[2] - x[0])
        return part1 + part2 + part3 + part4