在神經(jīng)網(wǎng)絡(luò)中實現(xiàn)反向傳播
建立神經(jīng)網(wǎng)絡(luò)時，需要采取幾個步驟。其中兩個最重要的步驟是實現(xiàn)正向和反向傳播。這兩個詞聽起來真的很沉重，并且總是讓初學者感到恐懼。但實際上，如果將這些技術(shù)分解為各自的步驟，則可以正確理解它們。在本文中，我們將專注于反向傳播及其每個步驟的直觀知識。
什么是反向傳播?
這只是實現(xiàn)神經(jīng)網(wǎng)絡(luò)的一項簡單技術(shù)，允許我們計算參數(shù)的梯度，以執(zhí)行梯度下降并使成本函數(shù)最小化。許多學者將反向傳播描述為神經(jīng)網(wǎng)絡(luò)中數(shù)學上最密集的部分。不過請放輕松，因為在本文中我們將完全解密反向傳播的每個部分。
實施反向傳播
假設(shè)一個簡單的兩層神經(jīng)網(wǎng)絡(luò)-一個隱藏層和一個輸出層。我們可以如下執(zhí)行反向傳播初始化要用于神經(jīng)網(wǎng)絡(luò)的權(quán)重和偏差：這涉及隨機初始化神經(jīng)網(wǎng)絡(luò)的權(quán)重和偏差。這些參數(shù)的梯度將從反向傳播中獲得，并用于更新梯度下降。
#Import Numpy library
import numpy as np
#set seed for reproducability
np.random.seed(100)
#We will first initialize the weights and bias needed and store them in a dictionary called W_B
def initialize(num_f, num_h, num_out):
'''
Description: This function randomly initializes the weights and biases of each layer of the neural network
Input Arguments:
num_f - number of training features
num_h -the number of nodes in the hidden layers
num_out - the number of nodes in the output
Output:
W_B - A dictionary of the initialized parameters.
'''
#randomly initialize weights and biases, and proceed to store in a dictionary
W_B = {
'W1': np.random.randn(num_h, num_f),
'b1': np.zeros((num_h, 1)),
'W2': np.random.randn(num_out, num_h),
'b2': np.zeros((num_out, 1))
}
return W_B執(zhí)行前向傳播：這涉及到計算隱藏層和輸出層的線性和激活輸出。
對于隱藏層：我們將使用如下所示的relu激活功能：
#We will now proceed to create functions for each of our activation functions
def relu (Z):
'''
Description: This function performs the relu activation function on a given number or matrix.
Input Arguments:
Z - matrix or integer
Output:
relu_Z - matrix or integer with relu performed on it
'''
relu_Z = np.maximum(Z,0)
return relu_Z對于輸出層：
我們將使用S型激活函數(shù)，如下所示：
def sigmoid (Z):
'''
Description: This function performs the sigmoid activation function on a given number or matrix.
Input Arguments:
Z - matrix or integer
Output:
sigmoid_Z - matrix or integer with sigmoid performed on it
'''
sigmoid_Z = 1 / (1 + (np.exp(-Z)))
return sigmoid_Z執(zhí)行前向傳播：
#We will now proceed to perform forward propagation
def forward_propagation(X, W_B):
'''
Description: This function performs the forward propagation in a vectorized form
Input Arguments:
X - input training examples
W_B - initialized weights and biases
Output:
forward_results - A dictionary containing the linear and activation outputs
'''
#Calculate the linear Z for the hidden layer
Z1 = np.dot(X, W_B['W1'].T) + W_B['b1']
#Calculate the activation ouput for the hidden layer
A = relu(Z1)
#Calculate the linear Z for the output layer
Z2 = np.dot(A, W_B['W2'].T) + W_B['b2']
#Calculate the activation ouput for the ouptu layer
Y_pred = sigmoid(Z2)
#Save all ina dictionary
forward_results = {"Z1": Z1,
"A": A,
"Z2": Z2,
"Y_pred": Y_pred}
return forward_results執(zhí)行向后傳播：相對于與梯度下降相關(guān)的參數(shù)，計算成本的梯度。在這種情況下，為dLdZ2，dLdW2，dLdb2，dLdZ1，dLdW1和dLdb1。這些參數(shù)將與學習率結(jié)合起來執(zhí)行梯度下降。我們將為許多訓練樣本(no_examples)實現(xiàn)反向傳播的矢量化版本。
分步指南如下：
- 從傳遞中獲取結(jié)果，如下所示：
forward_results = forward_propagation(X, W_B)
Z1 = forward_results['Z1']
A = forward_results['A']
Z2 = forward_results['Z2']
Y_pred = forward_results['Y_pred']- 獲得訓練樣本的數(shù)量，如下所示：
no_examples = X.shape[1]- 計算函數(shù)的損失：
L = (1/no_examples) * np.sum(-Y_true * np.log(Y_pred) - (1 - Y_true) * np.log(1 - Y_pred))- 計算每個參數(shù)的梯度，如下所示：
dLdZ2= Y_pred - Y_true
dLdW2 = (1/no_examples) * np.dot(dLdZ2, A.T)
dLdb2 = (1/no_examples) * np.sum(dLdZ2, axis=1, keepdims=True)
dLdZ1 = np.multiply(np.dot(W_B['W2'].T, dLdZ2), (1 - np.power(A, 2)))
dLdW1 = (1/no_examples) * np.dot(dLdZ1, X.T)
dLdb1 = (1/no_examples) * np.sum(dLdZ1, axis=1, keepdims=True)- 將梯度下降所需的計算梯度存儲在字典中：
gradients = {"dLdW1": dLdW1,
"dLdb1": dLdb1,
"dLdW2": dLdW2,
"dLdb2": dLdb2}- 返回損耗和存儲的梯度：
return gradients, L這是完整的向后傳播功能：
def backward_propagation(X, W_B, Y_true):
'''Description: This function performs the backward propagation in a vectorized form
Input Arguments:
X - input training examples
W_B - initialized weights and biases
Y_True - the true target values of the training examples
Output:
gradients - the calculated gradients of each parameter
L - the loss function
'''
# Obtain the forward results from the forward propagation
forward_results = forward_propagation(X, W_B)
Z1 = forward_results['Z1']
A = forward_results['A']
Z2 = forward_results['Z2']
Y_pred = forward_results['Y_pred']
#Obtain the number of training samples
no_examples = X.shape[1]
# Calculate loss
L = (1/no_examples) * np.sum(-Y_true * np.log(Y_pred) - (1 - Y_true) * np.log(1 - Y_pred))
#Calculate the gradients of each parameter needed for gradient descent
dLdZ2= Y_pred - Y_true
dLdW2 = (1/no_examples) * np.dot(dLdZ2, A.T)
dLdb2 = (1/no_examples) * np.sum(dLdZ2, axis=1, keepdims=True)
dLdZ1 = np.multiply(np.dot(W_B['W2'].T, dLdZ2), (1 - np.power(A, 2)))
dLdW1 = (1/no_examples) * np.dot(dLdZ1, X.T)
dLdb1 = (1/no_examples) * np.sum(dLdZ1, axis=1, keepdims=True)
#Store gradients for gradient descent in a dictionary
gradients = {"dLdW1": dLdW1,
"dLdb1": dLdb1,
"dLdW2": dLdW2,
"dLdb2": dLdb2}
return gradients, L許多人總是認為反向傳播很困難，但是正如本文中介紹的情形，事實并非如此。必須掌握每個步驟，才能掌握整個反向傳播技術(shù)。另外，有必要掌握線性代數(shù)和微積分等數(shù)學知識，以了解如何計算每個函數(shù)的各個梯度。使用這些工具，反向傳播應(yīng)該是小菜一碟!實際上，反向傳播通常由使用的深度學習框架來處理。但是，了解這種技術(shù)的內(nèi)在作用是值得的，因為它有時可以幫助我們理解神經(jīng)網(wǎng)絡(luò)為何訓練得不好。
本文轉(zhuǎn)載 ??小白遇見AI?? ，作者：小煩
