Learning neural networks using the genetic algorithm (GA)
Introduction
f(x,y)=\frac{(\frac{\sin x^2}{cos y}+x^2-5y+30)}{80}
The function of is trained in a neural network using a genetic algorithm. (I couldn't do it because of my undergraduate assignment, so I tried to revenge)
What are Genetic Algorithms (GA)?
GA is a program developed by Holland to search for optimal solutions that mimic the evolutionary mechanism of living organisms. In the process of evolution of sexually reproducing organisms, individuals who can adapt to the environment can leave their genes to the next generation, and the crossing of two individuals produces offspring, and rarely mutations occur. I am paying attention to. A definitive and excellent exact solution has not been found, and it is effective for problems with a vast solution space that makes full search impossible, and it can be applied to various optimization problems in the future. Further development is expected.
GA algorithm
The GA algorithm is shown below.
- Initial generation of individuals: Randomly generate early individuals.
- Calculation of goodness of fit (evaluation value) Determine the goodness of fit (evaluation value) of each individual.
- Play Regeneration of individuals depends on the goodness of fit of each individual. Individuals with a high degree of fit proliferate, and individuals with a low degree of fit are eliminated.
- Crossing A new individual is generated from a pair of individuals randomly selected from the population selected in Step 3. Repeat this a predetermined number of times.
- Mutation A part of the gene of each individual is randomly rewritten based on the mutation probability.
- Judgment of end conditions If the end condition is satisfied, the process ends, otherwise the process returns to Step 2.
Properties of GA algorithm
In GA, particles with high evaluation values are searched intensively by regeneration, and at the same time, solutions are searched in a wide range by crossing and mutation, so if these genetic manipulations work effectively, good solutions can be found. Be expected. However, due to genetic manipulation, information on the best solution may be lost, leading to a local solution, and it is often the case that the optimal solution cannot be found successfully. In particular, since there are many local solutions in the solution space of control parameters, it can be said that the crossing operation is not suitable. GA is a solution search method that assumes that the coefficient of determination is a discrete value, but it may not be optimal for finding a solution because the solution information for optimizing control parameters is a continuous value. Conceivable.
Implementation
Assuming that the gene is composed of the weight and threshold of the neural network, we designed one neural network to correspond to one individual, and updated it using the above GA method. The code is here
Set parameters etc.
#generation
GEN = 100
#Number of NNs
In = 2
Hidden = 2
Out = 1
#NN population
Number = 1000
#Number of teacher signals
Num = 1000
#Crossing probability
kousa = 0.8
#Mutation probability
change = 0.05
#Function to learn
def kansu(x):
return((math.sin(x[0])*math.sin(x[0])/math.cos(x[1]))+x[0]*x[0]-5*x[1]+30)/80
#Sigmoid function
def sigmoid(x):
return 1/(1+np.exp(-x))
#plot
def plot(data,name):
fig = plt.figure()
plt.plot(data)
fig.show()
fig.savefig(str(name)+'.pdf')
Teacher signal creation
class Kyoshi:
def __init__(self):
self.input = np.random.rand(Num, In)*10-5
self.output = np.zeros(Num)
def make_teacher(self):
for count in range(Num):
self.output[count]=kansu(self.input[count])
Create a neural network (NN)
class NN:
def __init__(self):
self.u = np.random.rand(In, Hidden)*2-1 #Input layer-Hidden layer weight
self.v = np.random.rand(Hidden, Out)*2-1 #Hidden layer-Output layer weight
self.bias_h = np.random.rand(Hidden)*2-1 #Hidden layer bias
self.bias_o = np.random.rand(Out)*2-1 #Output layer bias
self.Output = 0 #output
##For GA
self.gosa = 0 #Error with teacher data
self.F = 0 #Goodness of fit
#Calculate the output given the input
def calOutput(self, x): #x is the input
hidden_node = np.zeros(Hidden)
for j in range(Hidden):
for i in range(In):
hidden_node[j]+=self.u[i][j]*x[i]
hidden_node[j]-=self.bias_h[j]
self.Output+=sigmoid(self.v[j]*hidden_node[j])
self.Output-=self.bias_o
Optimize NN with GA
class NNGA:
def __init__(self):
self.nn = [NN()]*Number
self.aveE = 0 #Overall error average
#Error and goodness of fit calculation
def error(self, x,y): #x is teacher input, y is teacher output
self.aveE = 0
for count in range(Number):
self.nn[count].gosa = 0
#Input and output to each NN
self.nn[count].calOutput(x[count])
#Calculate error
self.nn[count].gosa = abs(self.nn[count].Output - y[count])
#################################
# for i in range(Num):
# #Input and output to each NN
# self.nn[count].calOutput(x[i])
# #Calculate error
# self.nn[count].gosa = abs(self.nn[count].Output - y[i])/Num
#################################
self.aveE += self.nn[count].gosa/Num
#Goodness of fit calculation
for count in range(Number):
self.nn[count].F= 1/ self.nn[count].gosa
#Genetic algorithm(GA)
def GA(self):
#Population/Do it twice
for _ in range(int(Number/2)):
F_sum=0 #Total goodness of fit for each individual
for count in range(Number):
F_sum+=self.nn[count].F
#Choice
p = [0,0] #Record the index selected
#Roulette selection
for i in range(2):
F_temp=0
j = -1
for count in range(Number):
j +=1
F_temp+=self.nn[count].F
if F_temp > random.random()*F_sum:
break
p[i]=j
#Create a child candidate
child = [NN()]*2
#Uniform crossing
if random.random() < kousa:
if random.random() < 0.5:
child[0].u = self.nn[p[0]].u
child[1].u = self.nn[p[1]].u
else:
child[0].u = self.nn[p[1]].u
child[1].u = self.nn[p[0]].u
if random.random() < 0.5:
child[0].v = self.nn[p[0]].v
child[1].v = self.nn[p[1]].v
else:
child[0].v = self.nn[p[1]].v
child[1].v = self.nn[p[0]].v
if random.random() < 0.5:
child[0].bias_h = self.nn[p[0]].bias_h
child[1].bias_h = self.nn[p[1]].bias_h
else:
child[0].bias_h = self.nn[p[1]].bias_h
child[1].bias_h = self.nn[p[0]].bias_h
if random.random() < 0.5:
child[0].bias_o = self.nn[p[0]].bias_o
child[1].bias_o = self.nn[p[1]].bias_o
else:
child[0].bias_o = self.nn[p[1]].bias_o
child[1].bias_o = self.nn[p[0]].bias_o
else:
child[0] = self.nn[p[0]]
child[1] = self.nn[p[1]]
#Inherit the average goodness of fit of parents
child[0].F = (self.nn[p[0]].F+self.nn[p[1]].F)/2
child[1].F = (self.nn[p[0]].F+self.nn[p[1]].F)/2
#mutation
for count in range(2):
for j in range(Hidden):
for i in range(In):
if random.random() < change:
child[count].u[i][j] = random.random()*2-1
if random.random() < change:
child[count].bias_h[j] = random.random()*2-1
if random.random() < change:
child[count].v[j] = random.random()*2-1
if random.random() < change:
child[count].bias_o = random.random()*2-1
#Add children to the population
rm1=0
rm2=0
min_F=100000
#Replace with the individual with the minimum goodness of fit
rm1 = np.argmin(self.nn[count].F)
self.nn[rm1]=child[0]
#Replaced with the second lowest goodness of fit individual
for count in range(Number):
if count==rm1:
pass
elif min_F > self.nn[count].F:
min_F = self.nn[count].F
rm2 = count
self.nn[rm2]=child[1]
main function
def main():
#Counting the number of generations
generation=0
#Generate an early individual
nnga = NNGA()
#Determines the input / output of the teacher signal
teacher = Kyoshi()
teacher.make_teacher()
#test data
testTeacher = Kyoshi()
testTeacher.make_teacher()
#Goodness of fit calculation
nnga.error(teacher.input, teacher.output)
#Recording function
kiroku = []
eliteKiroku = []
minEKiroku = []
#Start learning
while(True):
generation += 1
#GA optimization
nnga.GA()
#Calculate goodness of fit
nnga.error(teacher.input,teacher.output)
#Find the least error elite
min_E = 100000
elite = 0
for count in range(Number):
if min_E > nnga.nn[count].gosa:
min_E = nnga.nn[count].gosa
elite = count
#Check the elite with test data
sumE = 0
for i in range(Num):
nnga.nn[elite].calOutput(testTeacher.input[i])
sumE += abs(nnga.nn[elite].Output - testTeacher.output[i])/Num
#Shuffle teacher data
# np.random.shuffle(teacher.input)
# teacher.make_teacher()
#Record
kiroku.append(nnga.aveE)
eliteKiroku.append(sumE)
minEKiroku.append(min_E)
print("generation:",generation,"average",nnga.aveE, "elite",min_E,"test", sumE)
# if min_E < 0.06:
# break
if generation==GEN:
break
# plot
plot(kiroku,"Mean error")
plot(minEKiroku,"Elite individual error")
plot(eliteKiroku,"Elite individual error(test data)")
if __name__ == '__main__':
main()
Experimental result
The graph below shows the average error between the elite individuals of each generation and the test data. The horizontal axis is the number of generations, and the vertical axis is the error.
The error became smaller at once up to the first 5 generations. However, there was almost no change after that.
I tried changing the condition of various parameters and changing the design, but the performance did not improve. Hmmm. I would like to update it when my knowledge increases. (If you have any advice, please do not hesitate to contact us.)
Recommended Posts