1、思想：
與SVM不同，SVM直接利用線性對映的結果進行計算損失值，而softmax需要對線性對映得到的值進行指數歸一化,然後在進行損失值計算。在SVM損失函式中使用的是折葉函式，而在softmax中使用的交叉熵函式。
2、損失函式公式：
每個測試樣本的損失值計算
所有測試樣本的損失值計算

4、程式碼實現：
softmax.py

import numpy as np
from random import shuffle
#from past.builtins import xrange

def softmax_loss_naive(W, X, y, reg):
  """
  Softmax loss function, naive implementation (with loops)

  Inputs have dimension D, there are C classes, and we operate on minibatches
  of N examples.

  Inputs:
  - W: A numpy array of shape (D, C) containing weights.
  - X: A numpy array of shape (N, D) containing a minibatch of data.
  - y: A numpy array of shape (N,) containing training labels; y[i] = c means
    that X[i] has label c, where 0 <= c < C.
  - reg: (float) regularization strength

  Returns a tuple of:
  - loss as single float
  - gradient with respect to weights W; an array of same shape as W
  """
 

  # Initialize the loss and gradient to zero.
  loss = 0.0
  dW = np.zeros_like(W)

  #############################################################################
  # TODO: Compute the softmax loss and its gradient using explicit loops.     #
  # Store the loss in loss and the gradient in dW. If you are not careful     #
  # here, it is easy to run into numeric instability. Don't forget the        #
  # regularization!                                                           #
  ###
 
##########################################################################
  dW_each = np.zeros_like(W)
  num_train,dim = X.shape
  num_class = W.shape[1]
  f = X.dot(W)

  f_max = np.reshape(np.max(f,axis=1),(num_train,1))
  prob = np.exp(f-f_max)/np.sum(np.exp(f-f_max),axis=1,keepdims=True)
  y_trueClass = np.zeros_like(prob)
  y_trueClass[np.arange(num_train),y] = 1.0
 



  for i in xrange(num_train):
    for j in xrange(num_class):
      loss += -(y_trueClass[i,j]*np.log(prob[i,j]))
      dW_each[:,j] = -(y_trueClass[i,j]-prob[i,j])*X[i,:]
    dW +=dW_each
  loss /=num_train
  loss +=0.5*reg*np.sum(W*W)
  dW /=num_train
  W += reg*W


  #############################################################################
  #                          END OF YOUR CODE                                 #
  #############################################################################

  return loss, dW

def softmax_loss_vectorized(W, X, y, reg):
  """
  Softmax loss function, vectorized version.

  Inputs and outputs are the same as softmax_loss_naive.
  """
  # Initialize the loss and gradient to zero.
  loss = 0.0
  dW = np.zeros_like(W)

  #############################################################################
  # TODO: Compute the softmax loss and its gradient using no explicit loops.  #
  # Store the loss in loss and the gradient in dW. If you are not careful     #
  # here, it is easy to run into numeric instability. Don't forget the        #
  # regularization!                                                           #
  #############################################################################
  num_train,dim = X.shape

  f = X.dot(W)
  f_max = np.reshape(np.max(f,axis=1),(num_train,1))
  prob = np.exp(f-f_max)/np.sum(np.exp(f-f_max),axis=1,keepdims=True)
  y_trueClass = np.zeros_like(prob)
  y_trueClass[range(num_train),y] = 1.0
  loss += -np.sum(y_trueClass*np.log(prob))/num_train+0.5*reg*np.sum(W*W)
  dW += -np.dot(X.T,y_trueClass - prob)/num_train + reg*W
  #############################################################################
  #                          END OF YOUR CODE                                 #
  #############################################################################

  return loss, dW

softmax.ipynb

# Use the validation set to tune hyperparameters (regularization strength and
# learning rate). You should experiment with different ranges for the learning
# rates and regularization strengths; if you are careful you should be able to
# get a classification accuracy of over 0.35 on the validation set.
from cs231n.classifiers import Softmax
results = {}
best_val = -1
best_softmax = None
learning_rates = [1e-7, 5e-7]
regularization_strengths = [2.5e4, 5e4]

################################################################################
# TODO:                                                                        #
# Use the validation set to set the learning rate and regularization strength. #
# This should be identical to the validation that you did for the SVM; save    #
# the best trained softmax classifer in best_softmax.                          #
################################################################################
num_splt_lr = 3
num_splt_rs = 8
for i in xrange(num_splt_lr):
    for j in xrange(num_splt_rs):
        learning_rate_ij = learning_rates[0] + i * (learning_rates[1] - learning_rates[0]) / num_splt_lr
        reg_ij = regularization_strengths[0] + j * (regularization_strengths[1] - regularization_strengths[0])/ num_splt_rs
        softmax = Softmax()
        loss_hist = softmax.train(X_train, y_train, learning_rate=learning_rate_ij, reg=reg_ij,
                              num_iters=1500, verbose=False)

        y_train_pred = softmax.predict(X_train)
        accuracy_train = np.mean(y_train == y_train_pred)
        y_val_pred = softmax.predict(X_val)
        accuracy_val = np.mean(y_val == y_val_pred)

        results[(learning_rate_ij, reg_ij)] = (accuracy_train, accuracy_val)

        if accuracy_val > best_val:
            best_val = accuracy_val
            best_softmax = softmax
################################################################################
#                              END OF YOUR CODE                                #
################################################################################

# Print out results.
for lr, reg in sorted(results):
    train_accuracy, val_accuracy = results[(lr, reg)]
    print('lr %e reg %e train accuracy: %f val accuracy: %f' % (
                lr, reg, train_accuracy, val_accuracy))

print('best validation accuracy achieved during cross-validation: %f' % best_val)