L1与L2损失
这个脚本展示如何用TensorFlow求解线性回归。
在算法的收敛性中，理解损失函数的影响是很重要的。这里我们展示L2损失函数是如何影响线性回归的收敛性的。我们使用iris数据集,但是我们将改变损失函数和学习速率来看收敛性的改变。
L2-Loss
这里我们展示用L2损失的线性回归。
线性最小二乘的L2损失函数为
其中 N 是数据点数, yi 是第i个实际y值, ^yi^ 是第i个预测y值。
def loss2(x, y,w,b):
# Declare loss functions
loss_l2 = tf.reduce_mean(tf.square(y - model(x,w,b)))
return loss_l2
def grad2(x,y,w,b):
with tf.GradientTape() as tape:
loss_2 = loss2(x,y,w,b)
return tape.gradient(loss_2,[w,b])
batch_size = 25
learning_rate = 0.4 # Will not converge with learning rate at 0.4
iterations = 50
# Create variables for linear regression
w2 = tf.Variable(tf.random.normal(shape=[1,1]),tf.float32)
b2 = tf.Variable(tf.random.normal(shape=[1,1]),tf.float32)
optimizer = tf.optimizers.Adam(learning_rate)
# Training loop
loss_vec_l1=[]
loss_vec_l2=[]
for i in range(5000):
rand_index = np.random.choice(len(x_vals), size=batch_size)
rand_x = np.transpose([x_vals[rand_index]])
rand_y = np.transpose([y_vals[rand_index]])
x=tf.cast(rand_x,tf.float32)
y=tf.cast(rand_y,tf.float32)
grads2=grad2(x,y,w2,b2)
optimizer.apply_gradients(zip(grads2,[w2,b2]))
#sess.run(train_step, feed_dict={x_data: rand_x, y_target: rand_y})
temp_loss2 = loss2(x, y,w2,b2).numpy()
#sess.run(loss, feed_dict={x_data: rand_x, y_target: rand_y})
loss_vec_l2.append(temp_loss2)
if (i+1)%25==0:
print('Step #' + str(i+1) + ' A = ' + str(w2.numpy()) + ' b = ' + str(b2.numpy()))
print('Loss = ' + str(temp_loss2))
# Get the optimal coefficients
[slope] = w2.numpy()
[y_intercept] = b2.numpy()
# Get best fit line
best_fit2 = []
for i in x_vals:
best_fit2.append(slope*i+y_intercept)
# Plot the result
plt.plot(x_vals, y_vals, 'o', label='Data Points')
plt.plot(x_vals, best_fit2, 'r-', label='Best fit line', linewidth=3)
plt.legend(loc='upper left')
plt.title('Sepal Length vs Petal Width')
plt.xlabel('Petal Width')
plt.ylabel('Sepal Length')
plt.show()
# Plot loss over time
plt.plot(loss_vec_l2, 'k-')
plt.title('L2 Loss per Generation')
plt.xlabel('Generation')
plt.ylabel('L2 Loss')
plt.show()
# Plot loss over time
plt.plot(loss_vec_l1, 'k-', label='L1 Loss')
plt.plot(loss_vec_l2, 'r--', label='L2 Loss')
plt.title('L1 and L2 Loss per Generation')
plt.xlabel('Generation')
plt.ylabel('L1 Loss')
plt.legend(loc='upper right')
plt.show()
