import numpy as np
from random import shuffle

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!                                                           #
  #############################################################################
  num_classes = W.shape[1]
  num_train = X.shape[0]

  for i in xrange(num_train):
     scores = X[i].dot(W)
     shift_scores = scores - max(scores)
     loss_i = - shift_scores[y[i]] + np.log(sum(np.exp(shift_scores)))
     loss += loss_i
     for j in xrange(num_classes):
         softmax_output = np.exp(shift_scores[j])/sum(np.exp(shift_scores))
         if j == y[i]:
             dW[:,j] += (softmax_output-1) *X[i]   #预测正确时，L对W的梯度为：xi*(Pm-1)
         else: 
             dW[:,j] += softmax_output *X[i]     #预测正确时，L对W的梯度为：xi*Pm

  loss /= num_train 
  loss +=  0.5* reg * np.sum(W * W)   #1/N*loss
  dW = dW/num_train + reg* W     #加正则化项
  #pass
  #############################################################################
  #                          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_classes = W.shape[1]
  num_train = X.shape[0]
  scores = X.dot(W)
  shift_scores = scores - np.max(scores, axis = 1).reshape(-1,1)
  softmax_output = np.exp(shift_scores)/np.sum(np.exp(shift_scores), axis = 1).reshape(-1,1)
  loss = -np.sum(np.log(softmax_output[range(num_train), list(y)]))
  loss /= num_train 
  loss +=  0.5* reg * np.sum(W * W)
  
  dS = softmax_output.copy()
  dS[range(num_train), list(y)] += -1
  dW = (X.T).dot(dS)
  dW = dW/num_train + reg* W 
  #pass
  #############################################################################
  #                          END OF YOUR CODE                                 #
  #############################################################################

  return loss, dW

最难的是softmax对w的梯度求导，下面给出自己的推导过程：