file: tensorflow/python/training/learning_rate_decay.py

参考：tensorflow中常用学习率更新策略

神经网络中通过超参数 learning rate，来控制每次参数更新的幅度。学习率太小会降低网络优化的速度，增加训练时间；学习率太大则可能导致可能导致参数在局部最优解两侧来回振荡，网络不能收敛。

tensorflow 定义了很多的 学习率衰减方式：

指数衰减 tf.train.exponential_decay()

指数衰减是比较常用的衰减方法，学习率是跟当前的训练轮次指数相关的。

tf.train.exponential_decay(

    learning_rate,     # 初始学习率

    global_step,       # 当前训练轮次

    decay_steps,       # 衰减周期

    decay_rate,        # 衰减率系数

    staircase=False,   # 定义是否是阶梯型衰减，还是连续衰减，默认是 False

    name=None

)

'''

decayed_learning_rate = learning_rate *

                      decay_rate ^ (global_step / decay_steps)

'''

示例代码：

import tensorflow as tf

import matplotlib.pyplot as plt

style1 = []

style2 = []

N = 200

with tf.Session() as sess:

  sess.run(tf.global_variables_initializer())

  for step in range(N):

    # 标准指数型衰减

    learing_rate1 = tf.train.exponential_decay(

      learning_rate=0.5, global_step=step, decay_steps=10, decay_rate=0.9, staircase=False)

    # 阶梯型衰减

    learing_rate2 = tf.train.exponential_decay(

      learning_rate=0.5, global_step=step, decay_steps=10, decay_rate=0.9, staircase=True)

    lr1 = sess.run([learing_rate1])

    lr2 = sess.run([learing_rate2])

    style1.append(lr1)

    style2.append(lr2)

step = range(N)

plt.plot(step, style1, 'g-', linewidth=2, label='exponential_decay')

plt.plot(step, style2, 'r--', linewidth=1, label='exponential_decay_staircase')

plt.title('exponential_decay')

plt.xlabel('step')

plt.ylabel('learing rate')

plt.legend(loc='upper right')

plt.tight_layout()

plt.show()

分段常数衰减 tf.train.piecewise_constant()

tf.train.piecewise_constant_decay(

    x,                # 当前训练轮次

    boundaries,       # 学习率应用区间

    values,           # 学习率常数列表

    name=None

)

'''

learning_rate value is `values[0]` when `x <= boundaries[0]`,

    `values[1]` when `x > boundaries[0]` and `x <= boundaries[1]`, ...,

    and values[-1] when `x > boundaries[-1]`.

'''

示例代码：

import tensorflow as tf

import matplotlib.pyplot as plt

boundaries = [10, 20, 30]

learing_rates = [0.1, 0.07, 0.025, 0.0125]

style = []

N = 40

with tf.Session() as sess:

  sess.run(tf.global_variables_initializer())

  for step in range(N):

    learing_rate = tf.train.piecewise_constant(step, boundaries=boundaries, values=learing_rates)

    lr = sess.run([learing_rate])

    style.append(lr)

step = range(N)

plt.plot(step, style, 'r-', linewidth=2)

plt.title('piecewise_constant')

plt.xlabel('step')

plt.ylabel('learing rate')

plt.tight_layout()

plt.show()

多项式衰减 tf.train.polynomial_decay()

tf.train.polynomial_decay(

    learning_rate,               # 初始学习率

    global_step,                 # 当前训练轮次

    decay_steps,                 # 大衰减周期

    end_learning_rate=0.0001,    # 最小的学习率

    power=1.0,                   # 多项式的幂

    cycle=False,                 # 学习率是否循环

    name=None)

'''

global_step = min(global_step, decay_steps)

decayed_learning_rate = (learning_rate - end_learning_rate) *

                        (1 - global_step / decay_steps) ^ (power) +

                         end_learning_rate

'''

示例代码：

import tensorflow as tf

import matplotlib.pyplot as plt

style1 = []

style2 = []

N = 200

with tf.Session() as sess:

  sess.run(tf.global_variables_initializer())

  for step in range(N):

    # cycle=False

    learing_rate1 = tf.train.polynomial_decay(

      learning_rate=0.1, global_step=step, decay_steps=50,

      end_learning_rate=0.01, power=0.5, cycle=False)

    # cycle=True

    learing_rate2 = tf.train.polynomial_decay(

      learning_rate=0.1, global_step=step, decay_steps=50,

      end_learning_rate=0.01, power=0.5, cycle=True)

    lr1 = sess.run([learing_rate1])

    lr2 = sess.run([learing_rate2])

    style1.append(lr1)

    style2.append(lr2)

steps = range(N)

plt.plot(steps, style1, 'g-', linewidth=2, label='polynomial_decay')

plt.plot(steps, style2, 'r--', linewidth=1, label='polynomial_decay_cycle')

plt.title('polynomial_decay')

plt.xlabel('step')

plt.ylabel('learing rate')

plt.legend(loc='upper right')

plt.tight_layout()

plt.show()

自然指数衰减 tf.train.natural_exp_decay()

tf.train.natural_exp_decay(

    learning_rate,        # 初始学习率

    global_step,          # 当前训练轮次

    decay_steps,          # 衰减周期

    decay_rate,           # 衰减率系数

    staircase=False,      # 定义是否是阶梯型衰减，还是连续衰减，默认是 False

    name=None

)

'''

decayed_learning_rate = learning_rate * exp(-decay_rate * global_step)

'''

示例代码：

import tensorflow as tf

import matplotlib.pyplot as plt

style1 = []

style2 = []

N = 200

with tf.Session() as sess:

  sess.run(tf.global_variables_initializer())

  for step in range(N):

    # 标准指数型衰减

    learing_rate1 = tf.train.natural_exp_decay(

      learning_rate=0.5, global_step=step, decay_steps=10, decay_rate=0.9, staircase=False)

    # 阶梯型衰减

    learing_rate2 = tf.train.natural_exp_decay(

      learning_rate=0.5, global_step=step, decay_steps=10, decay_rate=0.9, staircase=True)

    lr1 = sess.run([learing_rate1])

    lr2 = sess.run([learing_rate2])

    style1.append(lr1)

    style2.append(lr2)

step = range(N)

plt.plot(step, style1, 'g-', linewidth=2, label='natural_exp_decay')

plt.plot(step, style2, 'r--', linewidth=1, label='natural_exp_decay_staircase')

plt.title('natural_exp_decay')

plt.xlabel('step')

plt.ylabel('learing rate')

plt.legend(loc='upper right')

plt.tight_layout()

plt.show()

倒数衰减 tf.train.inverse_time_decay()

tf.train.inverse_time_decay(

    learning_rate,     # 初始学习率

    global_step,       # 当前训练轮次

    decay_steps,       # 衰减周期

    decay_rate,        # 衰减率系数

    staircase=False,   # 定义是否是阶梯型衰减，还是连续衰减，默认是 False

    name=None

)

'''

decayed_learning_rate = learning_rate / (1 + decay_rate * global_step / decay_step)

'''

示例代码：

import tensorflow as tf

import matplotlib.pyplot as plt

style1 = []

style2 = []

N = 200

with tf.Session() as sess:

  sess.run(tf.global_variables_initializer())

  for step in range(N):

    # 标准指数型衰减

    learing_rate1 = tf.train.inverse_time_decay(

      learning_rate=0.5, global_step=step, decay_steps=20, decay_rate=0.2, staircase=False)

    # 阶梯型衰减

    learing_rate2 = tf.train.inverse_time_decay(

      learning_rate=0.5, global_step=step, decay_steps=20, decay_rate=0.2, staircase=True)

    lr1 = sess.run([learing_rate1])

    lr2 = sess.run([learing_rate2])

    style1.append(lr1)

    style2.append(lr2)

step = range(N)

plt.plot(step, style1, 'g-', linewidth=2, label='inverse_time_decay')

plt.plot(step, style2, 'r--', linewidth=1, label='inverse_time_decay_staircase')

plt.title('inverse_time_decay')

plt.xlabel('step')

plt.ylabel('learing rate')

plt.legend(loc='upper right')

plt.tight_layout()

plt.show()

余弦衰减 tf.train.cosine_decay()

tf.train.cosine_decay(

    learning_rate,   # 初始学习率

    global_step,     # 当前训练轮次

    decay_steps,     # 衰减周期

    alpha=0.0,       # 最小的学习率

    name=None

)

'''

global_step = min(global_step, decay_steps)

cosine_decay = 0.5 * (1 + cos(pi * global_step / decay_steps))

decayed = (1 - alpha) * cosine_decay + alpha

decayed_learning_rate = learning_rate * decayed

'''

改进的余弦衰减方法还有：
线性余弦衰减，对应函数 tf.train.linear_cosine_decay()
噪声线性余弦衰减，对应函数 tf.train.noisy_linear_cosine_decay()

示例代码：

import tensorflow as tf

import matplotlib.pyplot as plt

style1 = []

style2 = []

style3 = []

N = 200

with tf.Session() as sess:

  sess.run(tf.global_variables_initializer())

  for step in range(N):

    # 余弦衰减

    learing_rate1 = tf.train.cosine_decay(

      learning_rate=0.1, global_step=step, decay_steps=50)

    # 线性余弦衰减

    learing_rate2 = tf.train.linear_cosine_decay(

      learning_rate=0.1, global_step=step, decay_steps=50)

    # 噪声线性余弦衰减

    learing_rate3 = tf.train.noisy_linear_cosine_decay(

      learning_rate=0.1, global_step=step, decay_steps=50,

      initial_variance=0.01, variance_decay=0.1, num_periods=0.2, alpha=0.5, beta=0.2)

    lr1 = sess.run([learing_rate1])

    lr2 = sess.run([learing_rate2])

    lr3 = sess.run([learing_rate3])

    style1.append(lr1)

    style2.append(lr2)

    style3.append(lr3)

step = range(N)

plt.plot(step, style1, 'g-', linewidth=2, label='cosine_decay')

plt.plot(step, style2, 'r--', linewidth=1, label='linear_cosine_decay')

plt.plot(step, style3, 'b--', linewidth=1, label='linear_cosine_decay')

plt.title('cosine_decay')

plt.xlabel('step')

plt.ylabel('learing rate')

plt.legend(loc='upper right')

plt.tight_layout()

plt.show()

循环余弦衰减 tf.train.cosine_decay_restarts()

这是在 fast.ai 中强推的衰减方式

tf.train.cosine_decay_restarts(

    learning_rate,      # 初始学习率

    global_step,        # 当前训练轮次

    first_decay_steps,  # 首次衰减周期

    t_mul=2.0,          # 随后每次衰减周期倍数

    m_mul=1.0,          # 随后每次初始学习率倍数

    alpha=0.0,          # 最小的学习率=alpha*learning_rate

    name=None

)

'''

See [Loshchilov & Hutter, ICLR2016], SGDR: Stochastic Gradient Descent

with Warm Restarts. https://arxiv.org/abs/1608.03983

The learning rate multiplier first decays

from 1 to `alpha` for `first_decay_steps` steps. Then, a warm

restart is performed. Each new warm restart runs for `t_mul` times more steps

and with `m_mul` times smaller initial learning rate.

'''

示例代码：

import tensorflow as tf

import matplotlib.pyplot as plt

style1 = []

style2 = []

N = 200

with tf.Session() as sess:

  sess.run(tf.global_variables_initializer())

  for step in range(N):

    # 循环余弦衰减

    learing_rate1 = tf.train.cosine_decay_restarts(

      learning_rate=0.1, global_step=step, first_decay_steps=50,

      )

    # 余弦衰减

    learing_rate2 = tf.train.cosine_decay(

      learning_rate=0.1, global_step=step, decay_steps=50)

    lr1 = sess.run([learing_rate1])

    lr2 = sess.run([learing_rate2])

    style1.append(lr1)

    style2.append(lr2)

step = range(N)

plt.plot(step, style1, 'g-', linewidth=2, label='cosine_decay_restarts')

plt.plot(step, style2, 'r--', linewidth=1, label='cosine_decay')

plt.title('cosine_decay_restarts')

plt.xlabel('step')

plt.ylabel('learing rate')

plt.legend(loc='upper right')

plt.tight_layout()

plt.show()

调用例子

import tensorflow as tf

def Swish(features):

    return features*tf.nn.sigmoid(features)

# 1. create data

from tensorflow.examples.tutorials.mnist import input_data

mnist = input_data.read_data_sets('../MNIST_data', one_hot=True)

X = tf.placeholder(tf.float32, shape=(None, 784), name='X')

y = tf.placeholder(tf.int32, shape=(None), name='y')

is_training = tf.placeholder(tf.bool, None, name='is_training')

# 2. define network

he_init = tf.contrib.layers.variance_scaling_initializer()

with tf.name_scope('dnn'):

    hidden1 = tf.layers.dense(X, 300, kernel_initializer=he_init, name='hidden1')

    hidden1 = tf.layers.batch_normalization(hidden1, momentum=0.9)

    hidden1 = tf.nn.relu(hidden1)

    hidden2 = tf.layers.dense(hidden1, 100, kernel_initializer=he_init, name='hidden2')

    hidden2 = tf.layers.batch_normalization(hidden2, training=is_training, momentum=0.9)

    hidden2 = tf.nn.relu(hidden2)

    logits = tf.layers.dense(hidden2, 10, kernel_initializer=he_init, name='output')

    # prob = tf.layers.dense(hidden2, 10, tf.nn.softmax, name='prob')

# 3. define loss

with tf.name_scope('loss'):

    # tf.losses.sparse_softmax_cross_entropy() label is not one_hot and dtype is int*

    # xentropy = tf.losses.sparse_softmax_cross_entropy(labels=tf.argmax(y, axis=1), logits=logits)

    # tf.nn.sparse_softmax_cross_entropy_with_logits() label is not one_hot and dtype is int*

    # xentropy = tf.nn.sparse_softmax_cross_entropy_with_logits(labels=tf.argmax(y, axis=1), logits=logits)

    # loss = tf.reduce_mean(xentropy)

    loss = tf.losses.softmax_cross_entropy(onehot_labels=y, logits=logits) # label is one_hot

# 4. define optimizer

learning_rate_init = 0.01

global_step = tf.Variable(0, trainable=False)

with tf.name_scope('train'):

    learning_rate = tf.train.polynomial_decay(  # 多项式衰减

        learning_rate=learning_rate_init,  # 初始学习率

        global_step=global_step,  # 当前迭代次数

        decay_steps=22000,  # 在迭代到该次数实际，学习率衰减为 learning_rate * dacay_rate

        end_learning_rate=learning_rate_init / 10,  # 最小的学习率

        power=0.9,

        cycle=False

    )

    update_ops = tf.get_collection(tf.GraphKeys.UPDATE_OPS)  # for batch normalization

    with tf.control_dependencies(update_ops):

        optimizer_op = tf.train.GradientDescentOptimizer(

            learning_rate=learning_rate).minimize(

            loss=loss,

            var_list=tf.trainable_variables(),

            global_step=global_step # 不指定的话学习率不更新

        )

        # ================= clip gradient

        # threshold = 1.0

        # optimizer = tf.train.GradientDescentOptimizer(learning_rate)

        # grads_and_vars = optimizer.compute_gradients(loss)

        # capped_gvs = [(tf.clip_by_value(grad, -threshold, threshold), var)

        #               for grad, var in grads_and_vars]

        # optimizer_op = optimizer.apply_gradients(capped_gvs)

        # =================

with tf.name_scope('eval'):

    correct = tf.nn.in_top_k(logits, tf.argmax(y, axis=1), 1) # 目标是否在前K个预测中, label's dtype is int*

    accuracy = tf.reduce_mean(tf.cast(correct, tf.float32))

# 5. initialize

init_op = tf.group(tf.global_variables_initializer(), tf.local_variables_initializer())

saver = tf.train.Saver()

# =================

print([v.name for v in tf.trainable_variables()])

print([v.name for v in tf.global_variables()])

# =================

# 6. train & test

n_epochs = 20

n_batches = 50

batch_size = 50

with tf.Session() as sess:

    sess.run(init_op)

    # saver.restore(sess, './my_model_final.ckpt')

    for epoch in range(n_epochs):

        for iteration in range(mnist.train.num_examples // batch_size):

            X_batch, y_batch = mnist.train.next_batch(batch_size)

            sess.run([optimizer_op, learning_rate], feed_dict={X: X_batch, y: y_batch, is_training:True})

            # ================= check gradient

            # for grad, var in grads_and_vars:

            #     grad = grad.eval(feed_dict={X: X_batch, y: y_batch, is_training:True})

            #     var = var.eval()

            # =================

        learning_rate_cur = learning_rate.eval()

        acc_train = accuracy.eval(feed_dict={X: X_batch, y: y_batch, is_training:False}) # 最后一个 batch 的 accuracy

        acc_test = accuracy.eval(feed_dict={X: mnist.test.images, y: mnist.test.labels, is_training:False})

        loss_test = loss.eval(feed_dict={X: mnist.test.images, y: mnist.test.labels, is_training:False})

        print(epoch, "Current learning rate:", learning_rate_cur, "Train accuracy:", acc_train, "Test accuracy:", acc_test, "Test loss:", loss_test)

    save_path = saver.save(sess, "./my_model_final.ckpt")

with tf.Session() as sess:

    sess.run(init_op)

    saver.restore(sess, "./my_model_final.ckpt")

    acc_test = accuracy.eval(feed_dict={X: mnist.test.images, y: mnist.test.labels, is_training:False})

    loss_test = loss.eval(feed_dict={X: mnist.test.images, y: mnist.test.labels, is_training:False})

    print("Test accuracy:", acc_test, ", Test loss:", loss_test)

Since AdaGrad, RMSProp, and Adam optimization automatically reduce the learning rate during training, it is not necessary to add an extra learning schedule. For other optimization algorithms, using exponential decay or performance scheduling can considerably speed up convergence.

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