基于Levenberg-Marquardt训练算法的BP网络Python实现

经过一个多月的努力，终于完成了BP网络，参考的资料为：

1、Training feed-forward networks with the Marquardt algorithm

2、The Levenberg-Marquardt method for nonlinear least squares curve-fitting problems

3、Neural Network Design

4、http://deeplearning.stanford.edu/wiki/index.php/UFLDL%E6%95%99%E7%A8%8B 中介绍的神经网络部分

以下给出Python脚本：

import numpy as np
from math import exp, pow
from mpl_toolkits.mplot3d import Axes3D
import matplotlib.pyplot as plt
import sys
import copy
from scipy.linalg import norm, pinv
class Layer:
    def __init__(self,w, b, neure_number, transfer_function, layer_index):
        self.transfer_function = transfer_function
        self.neure_number = neure_number
        self.layer_index = layer_index
        self.w = w
        self.b = b

class NetStruct:
    def __init__(self, x, y, hidden_layers, activ_fun_list, performance_function = 'mse'):
        if len(hidden_layers) == len(activ_fun_list):
            activ_fun_list.append('line')
        self.active_fun_list = activ_fun_list
        self.performance_function = performance_function
        x = np.array(x)
        y = np.array(y)
        if(x.shape[1] != y.shape[1]):
            print 'The dimension of x and y are not same.'
            sys.exit()
        self.x = x
        self.y = y
        input_eles = self.x.shape[0]
        output_eles = self.y.shape[0]
        tmp = []
        tmp.append(input_eles)
        tmp.extend(hidden_layers)
        tmp.append(output_eles)
        self.hidden_layers = np.array(tmp)
        self.layer_num = len(self.hidden_layers)
        self.layers = []
        for i in range(0, len(self.hidden_layers)):

            if i == 0:
                self.layers.append(Layer([],[],\
                                        self.hidden_layers[i], 'none', i))
                continue
            f = self.hidden_layers[i - 1]
            s = self.hidden_layers[i]
            self.layers.append(Layer(np.random.randn(s, f),np.random.randn(s, 1),\
                                        self.hidden_layers[i], activ_fun_list[i-1], i)) 

class Train:
    def __init__(self, net_struct, mu = 1e-3, beta = 10, iteration = 100, tol = 0.1):
        self.net_struct = net_struct
        self.mu = mu
        self.beta = beta
        self.iteration = iteration
        self.tol = tol
    def train(self, method = 'lm'):
        if(method == 'lm'):
            self.lm()
    def sim(self, x):
        self.net_struct.x = x
        self.forward()
        layer_num = len(self.net_struct.layers)
        predict = self.net_struct.layers[layer_num - 1].output_val
        return predict
    def actFun(self, z, activ_type = 'sigm'):
        if activ_type == 'sigm':
            f = 1.0 / (1.0 + np.exp(-z))
        elif activ_type == 'tanh':
            f = (np.exp(z) + np.exp(-z)) / (np.exp(z) + np.exp(-z))
        elif activ_type == 'radb':
            f = np.exp(-z * z)
        elif activ_type == 'line':
            f = z
        return f
    def actFunGrad(self, z, activ_type = 'sigm'):
        if activ_type == 'sigm':
            grad = self.actFun(z, activ_type) * (1.0 - self.actFun(z, activ_type))
        elif activ_type == 'tanh':
            grad = 1.0 - self.actFun(z, activ_type) * self.actFun(z, activ_type)
        elif activ_type == 'radb':
            grad = -2.0 * z * self.actFun(z, activ_type)
        elif activ_type == 'line':
            m = z.shape[0]
            n = z.shape[1]
            grad = np.ones((m, n))
        return grad
    def forward(self):
        layer_num = len(self.net_struct.layers)
        for i in range(0, layer_num):
            if i == 0:
                curr_layer = self.net_struct.layers[i]
                curr_layer.input_val = self.net_struct.x
                curr_layer.output_val = self.net_struct.x
                continue
            before_layer = self.net_struct.layers[i - 1]
            curr_layer = self.net_struct.layers[i]
            curr_layer.input_val = curr_layer.w.dot(before_layer.output_val) + curr_layer.b
            curr_layer.output_val = self.actFun(curr_layer.input_val,
                                                self.net_struct.active_fun_list[i - 1])
    def backward(self):
        layer_num = len(self.net_struct.layers)
        last_layer = self.net_struct.layers[layer_num - 1]
        last_layer.error = -self.actFunGrad(last_layer.input_val,
                                            self.net_struct.active_fun_list[layer_num - 2])
        layer_index = range(1, layer_num - 1)
        layer_index.reverse()
        for i in layer_index:
            curr_layer = self.net_struct.layers[i]
            curr_layer.error = (last_layer.w.transpose().dot(last_layer.error)) \
                      * self.actFunGrad(curr_layer.input_val,self.net_struct.active_fun_list[i - 1])
            last_layer = curr_layer
    def parDeriv(self):
        layer_num = len(self.net_struct.layers)
        for i in range(1, layer_num):
            befor_layer = self.net_struct.layers[i - 1]
            befor_input_val = befor_layer.output_val.transpose()
            curr_layer = self.net_struct.layers[i]
            curr_error = curr_layer.error
            curr_error = curr_error.reshape(curr_error.shape[0]*curr_error.shape[1], 1, order='F')
            row =  curr_error.shape[0]
            col = befor_input_val.shape[1]
            a = np.zeros((row, col))
            num = befor_input_val.shape[0]
            neure_number = curr_layer.neure_number
            for i in range(0, num):
                a[neure_number*i:neure_number*i + neure_number,:] = \
                 np.repeat([befor_input_val[i,:]],neure_number,axis = 0)
            tmp_w_par_deriv = curr_error * a
            curr_layer.w_par_deriv = np.zeros((num, befor_layer.neure_number * curr_layer.neure_number))
            for i in range(0, num):
                tmp = tmp_w_par_deriv[neure_number*i:neure_number*i + neure_number,:]
                tmp = tmp.reshape(tmp.shape[0] * tmp.shape[1], order='C')
                curr_layer.w_par_deriv[i, :] = tmp
            curr_layer.b_par_deriv = curr_layer.error.transpose()
    def jacobian(self):
        layers = self.net_struct.hidden_layers
        row = self.net_struct.x.shape[1]
        col = 0
        for i in range(0, len(layers) - 1):
            col = col + layers[i] * layers[i + 1] + layers[i + 1]
        j = np.zeros((row, col))
        layer_num = len(self.net_struct.layers)
        index = 0
        for i in range(1, layer_num):
            curr_layer = self.net_struct.layers[i]
            w_col = curr_layer.w_par_deriv.shape[1]
            b_col = curr_layer.b_par_deriv.shape[1]
            j[:, index : index + w_col] = curr_layer.w_par_deriv
            index = index + w_col
            j[:, index : index + b_col] = curr_layer.b_par_deriv
            index = index + b_col
        return j
    def gradCheck(self):
        W1 = self.net_struct.layers[1].w
        b1 = self.net_struct.layers[1].b
        n = self.net_struct.layers[1].neure_number
        W2 = self.net_struct.layers[2].w
        b2 = self.net_struct.layers[2].b
        x = self.net_struct.x
        p = []
        p.extend(W1.reshape(1,W1.shape[0]*W1.shape[1],order = 'C')[0])
        p.extend(b1.reshape(1,b1.shape[0]*b1.shape[1],order = 'C')[0])
        p.extend(W2.reshape(1,W2.shape[0]*W2.shape[1],order = 'C')[0])
        p.extend(b2.reshape(1,b2.shape[0]*b2.shape[1],order = 'C')[0])
        old_p = p
        jac = []
        for i in range(0, x.shape[1]):
            xi = np.array([x[:,i]])
            xi = xi.transpose()
            ji = []
            for j in range(0, len(p)):
                W1 = np.array(p[0:2*n]).reshape(n,2,order='C')
                b1 = np.array(p[2*n:2*n+n]).reshape(n,1,order='C')
                W2 = np.array(p[3*n:4*n]).reshape(1,n,order='C')
                b2 = np.array(p[4*n:4*n+1]).reshape(1,1,order='C')

                z2 = W1.dot(xi) + b1
                a2 = self.actFun(z2)
                z3 = W2.dot(a2) + b2
                h1 = self.actFun(z3)
                p[j] = p[j] + 0.00001
                W1 = np.array(p[0:2*n]).reshape(n,2,order='C')
                b1 = np.array(p[2*n:2*n+n]).reshape(n,1,order='C')
                W2 = np.array(p[3*n:4*n]).reshape(1,n,order='C')
                b2 = np.array(p[4*n:4*n+1]).reshape(1,1,order='C')

                z2 = W1.dot(xi) + b1
                a2 = self.actFun(z2)
                z3 = W2.dot(a2) + b2
                h = self.actFun(z3)
                g = (h[0][0]-h1[0][0])/0.00001
                ji.append(g)
            jac.append(ji)
            p = old_p
        return jac
    def jjje(self):
        layer_number = self.net_struct.layer_num
        e = self.net_struct.y - \
          self.net_struct.layers[layer_number - 1].output_val
        e = e.transpose()
        j = self.jacobian()
        #check gradient
        #j1 = -np.array(self.gradCheck())
        #jk = j.reshape(1,j.shape[0]*j.shape[1])
        #jk1 = j1.reshape(1,j1.shape[0]*j1.shape[1])
        #plt.plot(jk[0])
        #plt.plot(jk1[0],'.')
        #plt.show()
        jj = j.transpose().dot(j)
        je = -j.transpose().dot(e)
        return[jj, je]
    def lm(self):
        mu = self.mu
        beta = self.beta
        iteration = self.iteration
        tol = self.tol
        y = self.net_struct.y
        self.forward()
        pred =  self.net_struct.layers[self.net_struct.layer_num - 1].output_val
        pref = self.perfermance(y, pred)
        for i in range(0, iteration):
            print 'iter:',i, 'error:', pref
            #1) step 1:
            if(pref < tol):
                break
            #2) step 2:
            self.backward()
            self.parDeriv()
            [jj, je] = self.jjje()
            while(1):
                #3) step 3:
                A = jj + mu * np.diag(np.ones(jj.shape[0]))
                delta_w_b = pinv(A).dot(je)
                #4) step 4:
                old_net_struct = copy.deepcopy(self.net_struct)
                self.updataNetStruct(delta_w_b)
                self.forward()
                pred1 =  self.net_struct.layers[self.net_struct.layer_num - 1].output_val
                pref1 = self.perfermance(y, pred1)
                if (pref1 < pref):
                    mu = mu / beta
                    pref = pref1
                    break
                mu = mu * beta
                self.net_struct = copy.deepcopy(old_net_struct)
    def updataNetStruct(self, delta_w_b):
        layer_number = self.net_struct.layer_num
        index = 0
        for i in range(1, layer_number):
            before_layer = self.net_struct.layers[i - 1]
            curr_layer = self.net_struct.layers[i]
            w_num = before_layer.neure_number * curr_layer.neure_number
            b_num = curr_layer.neure_number
            w = delta_w_b[index : index + w_num]
            w = w.reshape(curr_layer.neure_number, before_layer.neure_number, order='C')
            index = index + w_num
            b = delta_w_b[index : index + b_num]
            index = index + b_num
            curr_layer.w += w
            curr_layer.b += b
    def perfermance(self, y, pred):
        error = y - pred
        return norm(error) / len(y)
    def plotSamples(self, n = 40):
        x = np.array([np.linspace(0, 3, n)])
        x = x.repeat(n, axis = 0)
        y = x.transpose()
        z = np.zeros((n, n))
        for i in range(0, x.shape[0]):
            for j in range(0, x.shape[1]):
                z[i][j] = self.sampleFun(x[i][j], y[i][j])
        fig = plt.figure()
        ax = fig.gca(projection='3d')
        surf = ax.plot_surface(x, y, z, cmap='autumn', cstride=2, rstride=2)
        ax.set_xlabel("X-Label")
        ax.set_ylabel("Y-Label")
        ax.set_zlabel("Z-Label")
        plt.show()
def sinSamples(n):
        x = np.array([np.linspace(-0.5, 0.5, n)])
        #x = x.repeat(n, axis = 0)
        y = x + 0.2
        z = np.zeros((n, 1))
        for i in range(0, x.shape[1]):
            z[i] = np.sin(x[0][i] * y[0][i])
        X = np.zeros((n, 2))
        n = 0
        for xi, yi in zip(x.transpose(), y.transpose()):
            X[n][0] = xi
            X[n][1] = yi
            n = n + 1
        return X,z
def peaksSamples(n):
    x = np.array([np.linspace(-3, 3, n)])
    x = x.repeat(n, axis = 0)
    y = x.transpose()
    z = np.zeros((n, n))
    for i in range(0, x.shape[0]):
        for j in range(0, x.shape[1]):
            z[i][j] = sampleFun(x[i][j], y[i][j])
    X = np.zeros((n*n, 2))
    x_list = x.reshape(n*n,1 )
    y_list = y.reshape(n*n,1)
    z_list = z.reshape(n*n,1)
    n = 0
    for xi, yi in zip(x_list, y_list):
        X[n][0] = xi
        X[n][1] = yi
        n = n + 1

    return X,z_list.transpose()
def sampleFun(x, y):
    z =  3*pow((1-x),2) * exp(-(pow(x,2)) - pow((y+1),2)) \
   - 10*(x/5 - pow(x, 3) - pow(y, 5)) * exp(-pow(x, 2) - pow(y, 2)) \
   - 1/3*exp(-pow((x+1), 2) - pow(y, 2))
    return z
if __name__ == '__main__':

    hidden_layers = [10,10] #设置网络层数，共两层，每层10个神经元
    activ_fun_list = ['sigm','sigm']#设置隐层的激活函数类型，可以设置为tanh,radb，tanh,line类型，如果不显式的设置最后一层为line 

    [X, z] = peaksSamples(20) #产生训练数据点
    X = X.transpose()
    bp = NetStruct(X, z, hidden_layers, activ_fun_list) #初始化网络信息
    tr = Train(bp) #初始化训练网络的类
    tr.train() #训练
    [XX, z0] = peaksSamples(40) #产生测试数据
    XX = XX.transpose()
    z1 = tr.sim(XX) #用训练好的神经网络预测数据，z1为预测结果

    fig  = plt.figure()
    ax = fig.add_subplot(111)
    ax.plot(z0[0]) #真值
    ax.plot(z1[0],'r.') #预测值
    plt.legend((r'real data', r'predict data'))
    plt.show()

以上代码计算的结果如下图，由于初始值等原因的影响偶尔收敛效果会变差，不过大多数时候都可以收敛到下图的结果，以后再改进，欢迎指正。