版权声明：王家林大咖2018年新书《SPARK大数据商业实战三部曲》清华大学出版，清华大学出版社官方旗舰店（天猫）https://qhdx.tmall.com/?spm=a220o.1000855.1997427721.d4918089.4b2a2e5dT6bUsM https://blog.csdn.net/duan_zhihua/article/details/82942709
cs231 神经网络初步之Forwardpropagation 及Backpropagation
神经网络的计算图：
from __future__ import print_function
import numpy as np
import matplotlib.pyplot as plt
class TwoLayerNet(object):
"""
A two-layer fully-connected neural network. The net has an input dimension of
N, a hidden layer dimension of H, and performs classification over C classes.
We train the network with a softmax loss function and L2 regularization on the
weight matrices. The network uses a ReLU nonlinearity after the first fully
connected layer.
In other words, the network has the following architecture:
input - fully connected layer - ReLU - fully connected layer - softmax
The outputs of the second fully-connected layer are the scores for each class.
"""
def __init__(self, input_size, hidden_size, output_size, std=1e-4):
"""
Initialize the model. Weights are initialized to small random values and
biases are initialized to zero. Weights and biases are stored in the
variable self.params, which is a dictionary with the following keys:
W1: First layer weights; has shape (D, H)
b1: First layer biases; has shape (H,)
W2: Second layer weights; has shape (H, C)
b2: Second layer biases; has shape (C,)
Inputs:
- input_size: The dimension D of the input data.
- hidden_size: The number of neurons H in the hidden layer.
- output_size: The number of classes C.
"""
self.params = {}
self.params['W1'] = std * np.random.randn(input_size, hidden_size)
self.params['b1'] = np.zeros(hidden_size)
self.params['W2'] = std * np.random.randn(hidden_size, output_size)
self.params['b2'] = np.zeros(output_size)
def loss(self, X, y=None, reg=0.0):
"""
Compute the loss and gradients for a two layer fully connected neural
network.
Inputs:
- X: Input data of shape (N, D). Each X[i] is a training sample.
- y: Vector of training labels. y[i] is the label for X[i], and each y[i] is
an integer in the range 0 <= y[i] < C. This parameter is optional; if it
is not passed then we only return scores, and if it is passed then we
instead return the loss and gradients.
- reg: Regularization strength.
Returns:
If y is None, return a matrix scores of shape (N, C) where scores[i, c] is
the score for class c on input X[i].
If y is not None, instead return a tuple of:
- loss: Loss (data loss and regularization loss) for this batch of training
samples.
- grads: Dictionary mapping parameter names to gradients of those parameters
with respect to the loss function; has the same keys as self.params.
"""
# Unpack variables from the params dictionary
W1, b1 = self.params['W1'], self.params['b1']
W2, b2 = self.params['W2'], self.params['b2']
N, D = X.shape
# Compute the forward pass
scores = None
#############################################################################
# TODO: Perform the forward pass, computing the class scores for the input. #
# Store the result in the scores variable, which should be an array of #
# shape (N, C). #
#############################################################################
#pass
s1=np.dot(X,W1) +b1 # (N,D) (D,H) ---->N,H
#reLU
s1_act =np.maximum(s1, 0)
scores=np.dot(s1_act ,W2) +b2 # (N,H) (H,C)----->N,C
#############################################################################
# END OF YOUR CODE #
#############################################################################
# If the targets are not given then jump out, we're done
if y is None:
return scores
# Compute the loss
loss = None
#############################################################################
# TODO: Finish the forward pass, and compute the loss. This should include #
# both the data loss and L2 regularization for W1 and W2. Store the result #
# in the variable loss, which should be a scalar. Use the Softmax #
# classifier loss. #
#############################################################################
#pass Softmax
scores -= np.max(scores, axis=1, keepdims=True)# 数值稳定性S
scores = np.exp(scores)
scores /= np.sum(scores, axis=1, keepdims=True)# softmax
loss = scores[np.arange(X.shape[0]), y]
loss = -np.log(loss).sum()
loss /= X.shape[0]
#w1,w2 regularization
loss += reg *(np.sum(W1*W1))
loss += reg *(np.sum(W2*W2))
#############################################################################
# END OF YOUR CODE #
#############################################################################
# Backward pass: compute gradients
grads = {}
#############################################################################
# TODO: Compute the backward pass, computing the derivatives of the weights #
# and biases. Store the results in the grads dictionary. For example, #
# grads['W1'] should store the gradient on W1, and be a matrix of same size #
#############################################################################
#Forward: X---->W1X+b1-->s1--->Relu--->s1_act (x)--->W2X+b2---->scores --->softmax---->scores ----> l
# w2,w1 ----> l
#l--> scores---> softmax --->scores
ds2 = np.copy(scores) #(N,C)
ds2[np.arange(X.shape[0]), y] -= 1 # softmax derivative
# scores---> W2X+b2
#W ---->z = Wx+b --->l #▽Wl = ▽zl X^T #▽bl = ▽zl * 1
dW2 = np.dot(s1_act .T, ds2) / X.shape[0] #(H,N) (N,C)--->(H,C)
# l--->w2 2 path merging
dW2 += 2 * reg * W2
db2 = np.sum(ds2, axis=0) / X.shape[0]
grads['W2']=dW2
grads['b2']=db2
#W2X+b2 ---->s1_act (x)---->s1
#x ---->z = Wx+b --->l #▽xl = ▽zl W^T
ds1_act =np.dot(ds2, W2.T) # (N,C) (C,H)--->(N,H)
ds1 = (s1 > 0) *ds1_act
#s1---->W1X+b1
dW1 = np.dot(X.T, ds1) / X.shape[0] # (D,H) (N,H) ---->D,H
db1 = np.sum(ds1, axis=0) / X.shape[0]
# l--->w1 2 path merging
dW1 += 2 * reg * W1
grads['W1']=dW1
grads['b1']=db1
#############################################################################
# END OF YOUR CODE #
#############################################################################
return loss, grads