我们能够建立例如以下的loss function：

Li=−log(pyi)=−log⎛⎝efyi∑jefj⎞⎠

L=1N∑iLi+12λ∑k∑lW2k,l

以下我们推导loss对W,b的偏导数，我们能够先计算loss对f的偏导数，利用链式法则。我们能够得到：

∂Li∂fk=∂Li∂pk∂pk∂fk∂pi∂fk=pi(1−pk)i=k∂pi∂fk=−pipki≠k∂Li∂fk=−1pyi∂pyi∂fk=(pk−1{

yi=k})

进一步，由f=XW+b，可知∂f∂W=XT,∂f∂b=1，我们能够得到：

ΔW=∂L∂W=1N∂Li∂W+λW=1N∂Li∂p∂p∂f∂f∂W+λWΔb=∂L∂b=1N∂Li∂b=1N∂Li∂p∂p∂f∂f∂bW=W−αΔWb=b−αΔb

以下是用Python实现的soft max 分类器，基于Python 2.7.9, numpy, matplotlib.

代码来源于斯坦福大学的课程：

基本是照搬过来，通过这个程序有助于了解python的语法。

import numpy as npimport matplotlib.pyplot as pltN = 100  # number of points per classD = 2    # dimensionalityK = 3    # number of classesX = np.zeros((N*K,D))    #data matrix (each row = single example)y = np.zeros(N*K, dtype='uint8')  # class labelsfor j in xrange(K):  ix = range(N*j,N*(j+1))  r = np.linspace(0.0,1,N)            # radius  t = np.linspace(j*4,(j+1)*4,N) + np.random.randn(N)*0.2 # theta  X[ix] = np.c_[r*np.sin(t), r*np.cos(t)]  y[ix] = j# print y# lets visualize the data:plt.scatter(X[:,0], X[:,1], s=40, c=y, alpha=0.5)plt.show()#Train a Linear Classifier# initialize parameters randomlyW = 0.01 * np.random.randn(D,K)b = np.zeros((1,K))# some hyperparametersstep_size = 1e-0reg = 1e-3 # regularization strength# gradient descent loopnum_examples = X.shape[0]for i in xrange(200):  # evaluate class scores, [N x K]  scores = np.dot(X, W) + b   # compute the class probabilities  exp_scores = np.exp(scores)  probs = exp_scores / np.sum(exp_scores, axis=1, keepdims=True) # [N x K]  # compute the loss: average cross-entropy loss and regularization  corect_logprobs = -np.log(probs[range(num_examples),y])  data_loss = np.sum(corect_logprobs)/num_examples  reg_loss = 0.5*reg*np.sum(W*W)  loss = data_loss + reg_loss  if i % 10 == 0:    print "iteration %d: loss %f" % (i, loss)  # compute the gradient on scores  dscores = probs  dscores[range(num_examples),y] -= 1  dscores /= num_examples  # backpropate the gradient to the parameters (W,b)  dW = np.dot(X.T, dscores)  db = np.sum(dscores, axis=0, keepdims=True)  dW += reg*W     #regularization gradient  # perform a parameter update  W += -step_size * dW  b += -step_size * db# evaluate training set accuracyscores = np.dot(X, W) + bpredicted_class = np.argmax(scores, axis=1)print 'training accuracy: %.2f' % (np.mean(predicted_class == y))

生成的随机数据

执行结果