cs231n作業：assignment1 - softmax
阿新 • • 發佈：2018-11-09
title: cs231n作業：assignment1 - softmax
id: cs231n-1h-3
tags:
- cs231n
- homework
categories: - AI
- Deep Learning
date: 2018-09-27 16:02:57
GitHub地址：https://github.com/ZJUFangzh/cs231n
個人部落格：fangzh.top
softmax是最常用的分類器之一。
softmax和svm都是常用的分類器，而softmax更為常用。
具體可以參考我這篇的最後，ng老師有講， softmax
前面資料集的都跟SVM的一樣。
直接進入loss和grads推導環節。
可以看到，計算的公式也就是cross-entropy，即
但是，這樣有一個缺點，就是指數 可能會特別大，這樣可能導致記憶體不足，計算不穩定等問題。那麼可以在分子分母同乘一個常數C，一般C取為
f = np.array([123, 456, 789]) # 例子中有3個分類，每個評分的數值都很大
p = np.exp(f) / np.sum(np.exp(f)) # 不妙：數值問題，可能導致數值爆炸
# 那麼將f中的值平移到最大值為0：
f -= np.max(f) # f becomes [-666, -333, 0]
p = np.exp(f) / np.sum(np.exp(f)) # 現在OK了，將給出正確結果
精確地說，SVM分類器使用的是折葉損失（hinge loss），有時候又被稱為最大邊界損失（max-margin loss）。Softmax分類器使用的是交叉熵損失（corss-entropy loss）。Softmax分類器的命名是從softmax函式那裡得來的，softmax函式將原始分類評分變成正的歸一化數值，所有數值和為1，這樣處理後交叉熵損失才能應用。注意從技術上說“softmax損失（softmax loss）”是沒有意義的，因為softmax只是一個壓縮數值的函式。但是在這個說法常常被用來做簡稱。
求導過程參考：cs231n softmax求導
最終得到的公式是：
softmax程式碼實現
編輯cs231n/classifiers/softmax.py,先寫一下softmax_loss_naive函式，依舊是迴圈：
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! #
#############################################################################
(N, D) = X.shape
C = W.shape[1]
#遍歷每個樣本
for i in range(N):
f_i = X[i].dot(W)
#進行公式的指數修正
f_i -= np.max(f_i)
sum_j = np.sum(np.exp(f_i))
#得到樣本中每個類別的概率
p = lambda k : np.exp(f_i[k]) / sum_j
loss += - np.log(p(y[i]))
#根據softmax求導公式
for k in range(C):
p_k = p(k)
dW[:, k] += (p_k - (k == y[i])) * X[i]
loss /= N
loss += 0.5 * reg * np.sum(W * W)
dW /= N
dW += reg*W
#############################################################################
# END OF YOUR CODE #
#############################################################################
return loss, dW
驗證一下loss和grad得到：
numerical: -0.621593 analytic: -0.621593, relative error: 7.693773e-09
numerical: -2.576505 analytic: -2.576505, relative error: 4.492083e-09
numerical: -1.527801 analytic: -1.527801, relative error: 4.264914e-08
numerical: 1.101379 analytic: 1.101379, relative error: 9.735173e-09
numerical: 2.375620 analytic: 2.375620, relative error: 3.791861e-08
numerical: 3.166961 analytic: 3.166960, relative error: 8.526285e-09
numerical: -1.440997 analytic: -1.440998, relative error: 4.728898e-08
numerical: 0.563304 analytic: 0.563304, relative error: 2.409996e-08
numerical: -2.057292 analytic: -2.057292, relative error: 1.820335e-08
numerical: -0.450338 analytic: -0.450338, relative error: 8.075985e-08
numerical: -0.233090 analytic: -0.233090, relative error: 4.136546e-08
numerical: 0.251391 analytic: 0.251391, relative error: 4.552523e-08
numerical: 0.787031 analytic: 0.787031, relative error: 5.036469e-08
numerical: -1.801593 analytic: -1.801594, relative error: 3.159903e-08
numerical: -0.294108 analytic: -0.294109, relative error: 1.792497e-07
numerical: -1.974307 analytic: -1.974307, relative error: 1.160708e-08
numerical: 2.986921 analytic: 2.986921, relative error: 2.788065e-08
numerical: -0.247281 analytic: -0.247281, relative error: 8.957573e-08
numerical: 0.569337 analytic: 0.569337, relative error: 2.384912e-08
numerical: -1.579298 analytic: -1.579298, relative error: 1.728733e-08
向量化softmax
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! #
#############################################################################
(N, D) = X.shape
C = W.shape[1]
f = X.dot(W)
#在列方向進行指數修正
f -= np.max(f,axis=1,keepdims=True)
#求得softmax各個類的概率
p = np.exp(f) / np.sum(np.exp(f),axis=1,keepdims=True)
y_lable = np.zeros((N,C))
#y_lable就是(N,C)維的矩陣，每一行中只有對應的那個正確類別 = 1，其他都是0
y_lable[np.arange(N),y] = 1
#cross entropy
loss = -1 * np.sum(np.multiply(np.log(p),y_lable)) / N
loss += 0.5 * reg * np.sum( W * W)
#求導公式，很清晰
dW = X.T.dot(p-y_lable)
dW /= N
dW += reg*W
#############################################################################
# END OF YOUR CODE #
#############################################################################
return loss, dW
檢驗一下向量化和非向量化的時間：
naive loss: 2.357905e+00 computed in 0.091724s
vectorized loss: 2.357905e+00 computed in 0.002995s
Loss difference: 0.000000
Gradient difference: 0.000000
softmax的函式已經編寫完成了，接下來調一下學習率和正則化兩個超引數：
# 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. #
################################################################################
for lr in learning_rates:
for reg in regularization_strengths:
softmax = Softmax()
loss_hist = softmax.train(X_train, y_train, learning_rate=lr, reg=reg,
num_iters=1500, verbose=True)
y_train_pred = softmax.predict(X_train)
y_val_pred = softmax.predict(X_val)
y_train_acc = np.mean(y_train_pred==y_train)
y_val_acc = np.mean(y_val_pred==y_val)
results[(lr,reg)] = [y_train_acc, y_val_acc]
if y_val_acc > best_val:
best_val = y_val_acc
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)
lr 1.000000e-07 reg 2.500000e+04 train accuracy: 0.350592 val accuracy: 0.354000
lr 1.000000e-07 reg 5.000000e+04 train accuracy: 0.329551 val accuracy: 0.342000
lr 5.000000e-07 reg 2.500000e+04 train accuracy: 0.347286 val accuracy: 0.359000
lr 5.000000e-07 reg 5.000000e+04 train accuracy: 0.328551 val accuracy: 0.337000
best validation accuracy achieved during cross-validation: 0.359000