斯坦福CS231n項目實戰（二）：線性支持向量機SVM

02-16

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支持向量機(Support Vector Machine, SVM)的目標是希望正確類別樣本的分數( $W^TX$ )比錯誤類別的分數越大越好。兩者之間的最小距離(margin)我們用 $Delta$ 來表示，一般令 $Delta$ =1。

對於單個樣本，SVM的Loss function可表示為：

$L_i=sum_{j eq y_i}max(0,s_j-s_{y_i}+Delta)$

將 $s_j=W_j^Tx_i$ ， $s_{y_i}=W_{y_i}^Tx_i$ 帶入上式：

$L_i=sum_{j eq y_i}max(0,W_j^Tx_i-W_{y_i}^Tx_i+Delta)$

其中， $(x_i,y_i)$ 表示正確類別， $s_{y_i}$ 表示正確類別的分數score， $s_j$ 表示錯誤類別的分數score。從 $L_i$ 表達式來看， $s_j$ 不僅要比 $s_{y_i}$ 小，而且距離至少是 $Delta$ ，才能保證 $L_i=0$ 。若 $s_j>s_{y_i}+Delta$ ，則 $L_i>0$ 。也就是說SVM希望 $s_j$ 與 $s_{y_i}$ 至少相差一個Δ的距離。

該Loss function我們稱之為Hinge Loss:

舉個簡單的例子，假如一個三分類的輸出分數為：[10, 20, -10]，正確的類別是第0類，則該樣本的Loss function為：

$L_i=max(0, 20-10+1)+max(0, -10-10+1)=11$

若正確的類別是第1類，則Loss function為：

$L_i=max(0, 10-20+1)+max(0, -10-20+1)=0$

值得一提的是，還可以對hinge loss進行平方處理，也稱為L2-SVM。其Loss function為：

$L_i=sum_{j eq y_i}max(0,W_j^Tx_i-W_{y_i}^Tx_i+Delta)^2$

這種平方處理的目的是增大對正類別與負類別之間距離的懲罰。

為了防止過擬合，限制權重Ｗ的大小，引入正則項：

$L_i=sum_{j eq y_i}max(0,W_j^Tx_i-W_{y_i}^Tx_i+Delta)+lambda sum_ksum_l W_{k,l}^2$

L2正則項作用是限制權重W過大，且使得權重W分布均勻。而L1正則項傾向於得到離散的W，各W之間差距較大。

下面是Linear SVM的實例代碼，本文詳細代碼請見我的：

Github
碼雲

1. Load the CIFAR10 data

# Load the raw CIFAR-10 data.cifar10_dir = CIFAR10/datasets/cifar-10-batches-pyX_train, y_train, X_test, y_test = load_CIFAR10(cifar10_dir)# As a sanity check, we print out the size of the training and test data.print(Training data shape: , X_train.shape)print(Training labels shape: , y_train.shape)print(Test data shape: , X_test.shape)print(Test labels shape: , y_test.shape)

Training data shape: (50000, 32, 32, 3)
Training labels shape: (50000,)
Test data shape: (10000, 32, 32, 3)

Test labels shape: (10000,)

Show some CIFAR10 images

classes = [plane, car, bird, cat, dear, dog, frog, horse, ship, truck]num_classes = len(classes)num_each_class = 7for y, cls in enumerate(classes): idxs = np.flatnonzero(y_train == y) idxs = np.random.choice(idxs, num_each_class, replace=False) for i, idx in enumerate(idxs): plt_idx = i * num_classes + (y + 1) plt.subplot(num_each_class, num_classes, plt_idx) plt.imshow(X_train[idx].astype(uint8)) plt.axis(off) if i == 0: plt.title(cls)plt.show()

Subsample the data for more efficient code execution

# Split the data into train, val, and test setsnum_train = 49000num_val = 1000num_test = 1000# Validation setmask = range(num_train, num_train + num_val)X_val = X_train[mask]y_val = y_train[mask]# Train setmask = range(num_train)X_train = X_train[mask]y_train = y_train[mask]# Test setmask = range(num_test)X_test = X_test[mask]y_test = y_test[mask]print(Train data shape: , X_train.shape)print(Train labels shape: , y_train.shape)print(Validation data shape: , X_val.shape)print(Validation labels shape , y_val.shape)print(Test data shape: , X_test.shape)print(Test labels shape: , y_test.shape)

Train data shape: (49000, 32, 32, 3)
Train labels shape: (49000,)
Validation data shape: (1000, 32, 32, 3)
Validation labels shape (1000,)
Test data shape: (1000, 32, 32, 3)
Test labels shape: (1000,)

2. Preprocessing

Reshape the images data into rows

# Preprocessing: reshape the images data into rowsX_train = np.reshape(X_train, (X_train.shape[0], -1))X_val = np.reshape(X_val, (X_val.shape[0], -1))X_test = np.reshape(X_test, (X_test.shape[0], -1))print(Train data shape: , X_train.shape)print(Validation data shape: , X_val.shape)print(Test data shape: , X_test.shape)

Train data shape: (49000, 3072)
Validation data shape: (1000, 3072)
Test data shape: (1000, 3072)

Subtract the mean images

# Processing: subtract the mean imagesmean_image = np.mean(X_train, axis=0)plt.figure(figsize=(4,4))plt.imshow(mean_image.reshape((32,32,3)).astype(uint8))plt.show()

X_train -= mean_imageX_val -= mean_imageX_test -= mean_image

Append the bias dimension of ones

# append the bias dimension of ones (i.e. bias trick)X_train = np.hstack([X_train, np.ones((X_train.shape[0], 1))])X_val = np.hstack([X_val, np.ones((X_val.shape[0], 1))])X_test = np.hstack([X_test, np.ones((X_test.shape[0], 1))])print(Train data shape: , X_train.shape)print(Validation data shape: , X_val.shape)print(Test data shape: , X_test.shape)

Train data shape: (49000, 3073)

Validation data shape: (1000, 3073)
Test data shape: (1000, 3073)

3. Define a linear SVM classifier

class LinearSVM(object): """ A subclass that uses the Multiclass SVM loss function """ def __init__(self): self.W = None def loss_naive(self, X, y, reg): """ Structured SVM loss function, naive implementation (with loops). Inputs: - X: A numpy array of shape (num_train, D) contain the training data consisting of num_train samples each of dimension D - y: A numpy array of shape (num_train,) contain the training labels, where y[i] is the label of X[i] - reg: float, regularization strength Return: - loss: the loss value between predict value and ground truth - dW: gradient of W """ # Initialize loss and dW loss = 0.0 dW = np.zeros(self.W.shape) # Compute the loss and dW num_train = X.shape[0] num_classes = self.W.shape[1] for i in range(num_train): scores = np.dot(X[i], self.W) for j in range(num_classes): if j == y[i]: margin = 0 else: margin = scores[j] - scores[y[i]] + 1 # delta = 1 if margin > 0: loss += margin dW[:,j] += X[i].T dW[:,y[i]] += -X[i].T # Divided by num_train loss /= num_train dW /= num_train # Add regularization loss += 0.5 * reg * np.sum(self.W * self.W) dW += reg * self.W return loss, dW def loss_vectorized(self, X, y, reg): """ Structured SVM loss function, naive implementation (with loops). Inputs: - X: A numpy array of shape (num_train, D) contain the training data consisting of num_train samples each of dimension D - y: A numpy array of shape (num_train,) contain the training labels, where y[i] is the label of X[i] - reg: (float) regularization strength Outputs: - loss: the loss value between predict value and ground truth - dW: gradient of W """ # Initialize loss and dW loss = 0.0 dW = np.zeros(self.W.shape) # Compute the loss num_train = X.shape[0] scores = np.dot(X, self.W) correct_score = scores[range(num_train), list(y)].reshape(-1, 1) # delta = -1 margin = np.maximum(0, scores - correct_score + 1) margin[range(num_train), list(y)] = 0 loss = np.sum(margin) / num_train + 0.5 * reg * np.sum(self.W * self.W) # Compute the dW num_classes = self.W.shape[1] mask = np.zeros((num_train, num_classes)) mask[margin > 0] = 1 mask[range(num_train), list(y)] = 0 mask[range(num_train), list(y)] = -np.sum(mask, axis=1) dW = np.dot(X.T, mask) dW = dW / num_train + reg * self.W return loss, dW def train(self, X, y, learning_rate = 1e-3, reg = 1e-5, num_iters = 100, batch_size = 200, print_flag = False): """ Train linear SVM classifier using SGD Inputs: - X: A numpy array of shape (num_train, D) contain the training data consisting of num_train samples each of dimension D - y: A numpy array of shape (num_train,) contain the training labels, where y[i] is the label of X[i], y[i] = c, 0 <= c <= C - learning rate: (float) learning rate for optimization - reg: (float) regularization strength - num_iters: (integer) numbers of steps to take when optimization - batch_size: (integer) number of training examples to use at each step - print_flag: (boolean) If true, print the progress during optimization Outputs: - loss_history: A list containing the loss at each training iteration """ loss_history = [] num_train = X.shape[0] dim = X.shape[1] num_classes = np.max(y) + 1 # Initialize W if self.W == None: self.W = 0.001 * np.random.randn(dim, num_classes) # iteration and optimization for t in range(num_iters): idx_batch = np.random.choice(num_train, batch_size, replace=True) X_batch = X[idx_batch] y_batch = y[idx_batch] loss, dW = self.loss_vectorized(X_batch, y_batch, reg) loss_history.append(loss) self.W += -learning_rate * dW if print_flag and t%100 == 0: print(iteration %d / %d: loss %f % (t, num_iters, loss)) return loss_history def predict(self, X): """ Use the trained weights of linear SVM to predict data labels Inputs: - X: A numpy array of shape (num_train, D) contain the training data Outputs: - y_pred: A numpy array, predicted labels for the data in X """ y_pred = np.zeros(X.shape[0]) scores = np.dot(X, self.W) y_pred = np.argmax(scores, axis=1) return y_pred

4. Gradient Check

Define loss function

def loss_naive1(X, y, W, reg): """ Structured SVM loss function, naive implementation (with loops). Inputs: - X: A numpy array of shape (num_train, D) contain the training data consisting of num_train samples each of dimension D - y: A numpy array of shape (num_train,) contain the training labels, where y[i] is the label of X[i] - W: A numpy array of shape (D, C) contain the weights - reg: float, regularization strength Return: - loss: the loss value between predict value and ground truth - dW: gradient of W """ # Initialize loss and dW loss = 0.0 dW = np.zeros(W.shape) # Compute the loss and dW num_train = X.shape[0] num_classes = W.shape[1] for i in range(num_train): scores = np.dot(X[i], W) for j in range(num_classes): if j == y[i]: margin = 0 else: margin = scores[j] - scores[y[i]] + 1 # delta = 1 if margin > 0: loss += margin dW[:,j] += X[i].T dW[:,y[i]] += -X[i].T # Divided by num_train loss /= num_train dW /= num_train # Add regularization loss += 0.5 * reg * np.sum(W * W) dW += reg * W return loss, dWdef loss_vectorized1(X, y, W, reg): """ Structured SVM loss function, naive implementation (with loops). Inputs: - X: A numpy array of shape (num_train, D) contain the training data consisting of num_train samples each of dimension D - y: A numpy array of shape (num_train,) contain the training labels, where y[i] is the label of X[i] - W: A numpy array of shape (D, C) contain the weights - reg: (float) regularization strength Outputs: - loss: the loss value between predict value and ground truth - dW: gradient of W """ # Initialize loss and dW loss = 0.0 dW = np.zeros(W.shape) # Compute the loss num_train = X.shape[0] scores = np.dot(X, W) correct_score = scores[range(num_train), list(y)].reshape(-1, 1) # delta = -1 margin = np.maximum(0, scores - correct_score + 1) margin[range(num_train), list(y)] = 0 loss = np.sum(margin) / num_train + 0.5 * reg * np.sum(W * W) # Compute the dW num_classes = W.shape[1] mask = np.zeros((num_train, num_classes)) mask[margin > 0] = 1 mask[range(num_train), list(y)] = 0 mask[range(num_train), list(y)] = -np.sum(mask, axis=1) dW = np.dot(X.T, mask) dW = dW / num_train + reg * W return loss, dW

numerical: -8.059958 analytic: -8.059958, relative error: 6.130237e-11
numerical: -7.522645 analytic: -7.522645, relative error: 3.601909e-11
numerical: 14.561062 analytic: 14.561062, relative error: 1.571510e-11
numerical: -0.636243 analytic: -0.636243, relative error: 7.796694e-10
numerical: -11.414171 analytic: -11.414171, relative error: 1.604323e-11
numerical: 12.628817 analytic: 12.628817, relative error: 1.141476e-11
numerical: -9.642228 analytic: -9.642228, relative error: 2.188900e-11
numerical: 9.577850 analytic: 9.577850, relative error: 6.228243e-11
numerical: -5.397272 analytic: -5.397272, relative error: 4.498183e-11

numerical: 12.226704 analytic: 12.226704, relative error: 5.457544e-11
numerical: 14.054682 analytic: 14.054682, relative error: 2.879899e-12
numerical: 0.444995 analytic: 0.444995, relative error: 4.021959e-10
numerical: 0.838312 analytic: 0.838312, relative error: 6.444258e-10
numerical: -1.160105 analytic: -1.160105, relative error: 5.096445e-10
numerical: -3.007970 analytic: -3.007970, relative error: 2.017297e-10
numerical: -2.135929 analytic: -2.135929, relative error: 2.708692e-10
numerical: -16.032463 analytic: -16.032463, relative error: 1.920198e-11
numerical: 5.949340 analytic: 5.949340, relative error: 2.138613e-11
numerical: -2.278258 analytic: -2.278258, relative error: 6.415350e-11

numerical: 8.316738 analytic: 8.316738, relative error: 1.901469e-11

5. Stochastic Gradient Descent

svm = LinearSVM()loss_history = svm.train(X_train, y_train, learning_rate = 1e-7, reg = 2.5e4, num_iters = 2000, batch_size = 200, print_flag = True)

iteration 0 / 2000: loss 403.810828
iteration 100 / 2000: loss 239.004354
iteration 200 / 2000: loss 145.934813
iteration 300 / 2000: loss 90.564682
iteration 400 / 2000: loss 56.126912
iteration 500 / 2000: loss 36.482452
iteration 600 / 2000: loss 23.327738
iteration 700 / 2000: loss 15.934542
iteration 800 / 2000: loss 11.508418
iteration 900 / 2000: loss 8.614351

iteration 1000 / 2000: loss 7.845596
iteration 1100 / 2000: loss 6.068847
iteration 1200 / 2000: loss 6.017030
iteration 1300 / 2000: loss 5.407498
iteration 1400 / 2000: loss 5.282425
iteration 1500 / 2000: loss 5.760450
iteration 1600 / 2000: loss 4.764250
iteration 1700 / 2000: loss 5.395108
iteration 1800 / 2000: loss 5.025213
iteration 1900 / 2000: loss 4.858321

# Plot the loss_historyplt.plot(loss_history)plt.xlabel(Iteration number)plt.ylabel(Loss value)plt.show()

# Use svm to predict# Training sety_pred = svm.predict(X_train)num_correct = np.sum(y_pred == y_train)accuracy = np.mean(y_pred == y_train)print(Training correct %d/%d: The accuracy is %f % (num_correct, X_train.shape[0], accuracy))# Test sety_pred = svm.predict(X_test)num_correct = np.sum(y_pred == y_test)accuracy = np.mean(y_pred == y_test)print(Test correct %d/%d: The accuracy is %f % (num_correct, X_test.shape[0], accuracy))

Training correct 18789/49000: The accuracy is 0.383449
Test correct 375/1000: The accuracy is 0.375000

6. Validation and Test

Cross-validation

learning_rates = [1.4e-7, 1.5e-7, 1.6e-7]regularization_strengths = [8000.0, 9000.0, 10000.0, 11000.0, 18000.0, 19000.0, 20000.0, 21000.0]results = {}best_lr = Nonebest_reg = Nonebest_val = -1 # The highest validation accuracy that we have seen so far.best_svm = None # The LinearSVM object that achieved the highest validation rate.for lr in learning_rates: for reg in regularization_strengths: svm = LinearSVM() loss_history = svm.train(X_train, y_train, learning_rate = lr, reg = reg, num_iters = 2000) y_train_pred = svm.predict(X_train) accuracy_train = np.mean(y_train_pred == y_train) y_val_pred = svm.predict(X_val) accuracy_val = np.mean(y_val_pred == y_val) if accuracy_val > best_val: best_lr = lr best_reg = reg best_val = accuracy_val best_svm = svm results[(lr, reg)] = accuracy_train, accuracy_val print(lr: %e reg: %e train accuracy: %f val accuracy: %f % (lr, reg, results[(lr, reg)][0], results[(lr, reg)][1]))print(Best validation accuracy during cross-validation: lr = %e, reg = %e, best_val = %f % (best_lr, best_reg, best_val))

lr: 1.400000e-07 reg: 8.000000e+03 train accuracy: 0.388633 val accuracy: 0.412000
lr: 1.400000e-07 reg: 9.000000e+03 train accuracy: 0.394918 val accuracy: 0.396000
lr: 1.400000e-07 reg: 1.000000e+04 train accuracy: 0.392388 val accuracy: 0.396000
lr: 1.400000e-07 reg: 1.100000e+04 train accuracy: 0.388265 val accuracy: 0.379000
lr: 1.400000e-07 reg: 1.800000e+04 train accuracy: 0.387408 val accuracy: 0.386000
lr: 1.400000e-07 reg: 1.900000e+04 train accuracy: 0.381673 val accuracy: 0.372000
lr: 1.400000e-07 reg: 2.000000e+04 train accuracy: 0.377531 val accuracy: 0.394000
lr: 1.400000e-07 reg: 2.100000e+04 train accuracy: 0.372735 val accuracy: 0.370000
lr: 1.500000e-07 reg: 8.000000e+03 train accuracy: 0.393837 val accuracy: 0.400000

lr: 1.500000e-07 reg: 9.000000e+03 train accuracy: 0.393735 val accuracy: 0.382000
lr: 1.500000e-07 reg: 1.000000e+04 train accuracy: 0.395735 val accuracy: 0.381000
lr: 1.500000e-07 reg: 1.100000e+04 train accuracy: 0.396469 val accuracy: 0.398000
lr: 1.500000e-07 reg: 1.800000e+04 train accuracy: 0.382694 val accuracy: 0.392000
lr: 1.500000e-07 reg: 1.900000e+04 train accuracy: 0.382429 val accuracy: 0.395000
lr: 1.500000e-07 reg: 2.000000e+04 train accuracy: 0.374265 val accuracy: 0.390000
lr: 1.500000e-07 reg: 2.100000e+04 train accuracy: 0.378327 val accuracy: 0.377000
lr: 1.600000e-07 reg: 8.000000e+03 train accuracy: 0.392551 val accuracy: 0.382000
lr: 1.600000e-07 reg: 9.000000e+03 train accuracy: 0.391184 val accuracy: 0.378000
lr: 1.600000e-07 reg: 1.000000e+04 train accuracy: 0.387939 val accuracy: 0.410000
lr: 1.600000e-07 reg: 1.100000e+04 train accuracy: 0.388224 val accuracy: 0.389000
lr: 1.600000e-07 reg: 1.800000e+04 train accuracy: 0.378102 val accuracy: 0.383000
lr: 1.600000e-07 reg: 1.900000e+04 train accuracy: 0.380918 val accuracy: 0.383000
lr: 1.600000e-07 reg: 2.000000e+04 train accuracy: 0.378224 val accuracy: 0.383000
lr: 1.600000e-07 reg: 2.100000e+04 train accuracy: 0.376204 val accuracy: 0.380000
Best validation accuracy during cross-validation:
lr = 1.400000e-07, reg = 8.000000e+03, best_val = 0.412000

# Visualize the cross-validation resultsimport mathx_scatter = [math.log10(x[0]) for x in results]y_scatter = [math.log10(x[1]) for x in results]# Plot training accuracyplt.figure(figsize=(10,10))make_size = 100colors = [results[x][0] for x in results]plt.subplot(2, 1, 1)plt.scatter(x_scatter, y_scatter, make_size, c = colors)plt.colorbar()plt.xlabel(log learning rate)plt.ylabel(log regularization strength)plt.title(Training accuracy)# Plot validation accuracycolors = [results[x][1] for x in results]plt.subplot(2, 1, 2)plt.scatter(x_scatter, y_scatter, make_size, c = colors)plt.colorbar()plt.xlabel(log learning rate)plt.ylabel(log regularization strength)plt.title(Validation accuracy)plt.show()

Test

# Use the best svm to testy_test_pred = best_svm.predict(X_test)num_correct = np.sum(y_test_pred == y_test)accuracy = np.mean(y_test_pred == y_test)print(Test correct %d/%d: The accuracy is %f % (num_correct, X_test.shape[0], accuracy))

Test correct 369/1000: The accuracy is 0.369000

Visualize the weights for each class

W = best_svm.W[:-1, :] # delete the biasW = W.reshape(32, 32, 3, 10)W_min, W_max = np.min(W), np.max(W)classes = [plane, car, bird, cat, deer, dog, frog, horse, ship, truck]for i in range(10): plt.subplot(2, 5, i+1) imgW = 255.0 * ((W[:, :, :, i].squeeze() - W_min) / (W_max - W_min)) plt.imshow(imgW.astype(uint8)) plt.axis(off) plt.title(classes[i])plt.show()

參考文獻：

linear classification notes

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