11.11. 学习率调度器

Colab [mxnet]

Open the notebook in Colab

Colab [pytorch]

Open the notebook in Colab

Colab [tensorflow]

Open the notebook in Colab

Colab [paddle]

Open the notebook in Colab

SageMaker Studio Lab

Open the notebook in SageMaker Studio Lab

到目前为止，我们主要关注如何更新权重向量的优化算法，而不是它们的更新速率。 然而，调整学习率通常与实际算法同样重要，有如下几方面需要考虑：

• 首先，学习率的大小很重要。如果它太大，优化就会发散；如果它太小，训练就会需要过长时间，或者我们最终只能得到次优的结果。我们之前看到问题的条件数很重要（有关详细信息，请参见 11.6节）。直观地说，这是最不敏感与最敏感方向的变化量的比率。

• 其次，衰减速率同样很重要。如果学习率持续过高，我们可能最终会在最小值附近弹跳，从而无法达到最优解。 11.5节比较详细地讨论了这一点，在 11.4节中我们则分析了性能保证。简而言之，我们希望速率衰减，但要比\(\mathcal{O}(t^{-\frac{1}{2}})\)慢，这样能成为解决凸问题的不错选择。

• 另一个同样重要的方面是初始化。这既涉及参数最初的设置方式（详情请参阅 4.8节），又关系到它们最初的演变方式。这被戏称为预热（warmup），即我们最初开始向着解决方案迈进的速度有多快。一开始的大步可能没有好处，特别是因为最初的参数集是随机的。最初的更新方向可能也是毫无意义的。

• 最后，还有许多优化变体可以执行周期性学习率调整。这超出了本章的范围，我们建议读者阅读 (Izmailov et al., 2018)来了解个中细节。例如，如何通过对整个路径参数求平均值来获得更好的解。

鉴于管理学习率需要很多细节，因此大多数深度学习框架都有自动应对这个问题的工具。 在本章中，我们将梳理不同的调度策略对准确性的影响，并展示如何通过学习率调度器（learning rate scheduler）来有效管理。

11.11.1. 一个简单的问题

我们从一个简单的问题开始，这个问题可以轻松计算，但足以说明要义。 为此，我们选择了一个稍微现代化的LeNet版本（激活函数使用relu而不是sigmoid，汇聚层使用最大汇聚层而不是平均汇聚层），并应用于Fashion-MNIST数据集。 此外，我们混合网络以提高性能。 由于大多数代码都是标准的，我们只介绍基础知识，而不做进一步的详细讨论。如果需要，请参阅 6节进行复习。

mxnetpytorchtensorflowpaddle

%matplotlib inline
from mxnet import autograd, gluon, init, lr_scheduler, np, npx
from mxnet.gluon import nn
from d2l import mxnet as d2l

npx.set_np()

net = nn.HybridSequential()
net.add(nn.Conv2D(channels=6, kernel_size=5, padding=2, activation='relu'),
        nn.MaxPool2D(pool_size=2, strides=2),
        nn.Conv2D(channels=16, kernel_size=5, activation='relu'),
        nn.MaxPool2D(pool_size=2, strides=2),
        nn.Dense(120, activation='relu'),
        nn.Dense(84, activation='relu'),
        nn.Dense(10))
net.hybridize()
loss = gluon.loss.SoftmaxCrossEntropyLoss()
device = d2l.try_gpu()

batch_size = 256
train_iter, test_iter = d2l.load_data_fashion_mnist(batch_size=batch_size)

# 代码几乎与d2l.train_ch6定义在卷积神经网络一章LeNet一节中的相同
def train(net, train_iter, test_iter, num_epochs, loss, trainer, device):
    net.initialize(force_reinit=True, ctx=device, init=init.Xavier())
    animator = d2l.Animator(xlabel='epoch', xlim=[0, num_epochs],
                            legend=['train loss', 'train acc', 'test acc'])
    for epoch in range(num_epochs):
        metric = d2l.Accumulator(3)  # train_loss,train_acc,num_examples
        for i, (X, y) in enumerate(train_iter):
            X, y = X.as_in_ctx(device), y.as_in_ctx(device)
            with autograd.record():
                y_hat = net(X)
                l = loss(y_hat, y)
            l.backward()
            trainer.step(X.shape[0])
            metric.add(l.sum(), d2l.accuracy(y_hat, y), X.shape[0])
            train_loss = metric[0] / metric[2]
            train_acc = metric[1] / metric[2]
            if (i + 1) % 50 == 0:
                animator.add(epoch + i / len(train_iter),
                             (train_loss, train_acc, None))
        test_acc = d2l.evaluate_accuracy_gpu(net, test_iter)
        animator.add(epoch + 1, (None, None, test_acc))
    print(f'train loss {train_loss:.3f}, train acc {train_acc:.3f}, '
          f'test acc {test_acc:.3f}')

[07:46:33] ../src/storage/storage.cc:196: Using Pooled (Naive) StorageManager for CPU

%matplotlib inline
import math
import torch
from torch import nn
from torch.optim import lr_scheduler
from d2l import torch as d2l


def net_fn():
    model = nn.Sequential(
        nn.Conv2d(1, 6, kernel_size=5, padding=2), nn.ReLU(),
        nn.MaxPool2d(kernel_size=2, stride=2),
        nn.Conv2d(6, 16, kernel_size=5), nn.ReLU(),
        nn.MaxPool2d(kernel_size=2, stride=2),
        nn.Flatten(),
        nn.Linear(16 * 5 * 5, 120), nn.ReLU(),
        nn.Linear(120, 84), nn.ReLU(),
        nn.Linear(84, 10))

    return model

loss = nn.CrossEntropyLoss()
device = d2l.try_gpu()

batch_size = 256
train_iter, test_iter = d2l.load_data_fashion_mnist(batch_size=batch_size)

# 代码几乎与d2l.train_ch6定义在卷积神经网络一章LeNet一节中的相同
def train(net, train_iter, test_iter, num_epochs, loss, trainer, device,
          scheduler=None):
    net.to(device)
    animator = d2l.Animator(xlabel='epoch', xlim=[0, num_epochs],
                            legend=['train loss', 'train acc', 'test acc'])

    for epoch in range(num_epochs):
        metric = d2l.Accumulator(3)  # train_loss,train_acc,num_examples
        for i, (X, y) in enumerate(train_iter):
            net.train()
            trainer.zero_grad()
            X, y = X.to(device), y.to(device)
            y_hat = net(X)
            l = loss(y_hat, y)
            l.backward()
            trainer.step()
            with torch.no_grad():
                metric.add(l * X.shape[0], d2l.accuracy(y_hat, y), X.shape[0])
            train_loss = metric[0] / metric[2]
            train_acc = metric[1] / metric[2]
            if (i + 1) % 50 == 0:
                animator.add(epoch + i / len(train_iter),
                             (train_loss, train_acc, None))

        test_acc = d2l.evaluate_accuracy_gpu(net, test_iter)
        animator.add(epoch+1, (None, None, test_acc))

        if scheduler:
            if scheduler.__module__ == lr_scheduler.__name__:
                # UsingPyTorchIn-Builtscheduler
                scheduler.step()
            else:
                # Usingcustomdefinedscheduler
                for param_group in trainer.param_groups:
                    param_group['lr'] = scheduler(epoch)

    print(f'train loss {train_loss:.3f}, train acc {train_acc:.3f}, '
          f'test acc {test_acc:.3f}')

%matplotlib inline
import math
import tensorflow as tf
from tensorflow.keras.callbacks import LearningRateScheduler
from d2l import tensorflow as d2l


def net():
    return tf.keras.models.Sequential([
        tf.keras.layers.Conv2D(filters=6, kernel_size=5, activation='relu',
                               padding='same'),
        tf.keras.layers.AvgPool2D(pool_size=2, strides=2),
        tf.keras.layers.Conv2D(filters=16, kernel_size=5,
                               activation='relu'),
        tf.keras.layers.AvgPool2D(pool_size=2, strides=2),
        tf.keras.layers.Flatten(),
        tf.keras.layers.Dense(120, activation='relu'),
        tf.keras.layers.Dense(84, activation='sigmoid'),
        tf.keras.layers.Dense(10)])


batch_size = 256
train_iter, test_iter = d2l.load_data_fashion_mnist(batch_size=batch_size)

# 代码几乎与d2l.train_ch6定义在卷积神经网络一章LeNet一节中的相同
def train(net_fn, train_iter, test_iter, num_epochs, lr,
              device=d2l.try_gpu(), custom_callback = False):
    device_name = device._device_name
    strategy = tf.distribute.OneDeviceStrategy(device_name)
    with strategy.scope():
        optimizer = tf.keras.optimizers.SGD(learning_rate=lr)
        loss = tf.keras.losses.SparseCategoricalCrossentropy(from_logits=True)
        net = net_fn()
        net.compile(optimizer=optimizer, loss=loss, metrics=['accuracy'])
    callback = d2l.TrainCallback(net, train_iter, test_iter, num_epochs,
                             device_name)
    if custom_callback is False:
        net.fit(train_iter, epochs=num_epochs, verbose=0,
                callbacks=[callback])
    else:
         net.fit(train_iter, epochs=num_epochs, verbose=0,
                 callbacks=[callback, custom_callback])
    return net

Downloading data from https://storage.googleapis.com/tensorflow/tf-keras-datasets/train-labels-idx1-ubyte.gz
29515/29515 [==============================] - 0s 0us/step
Downloading data from https://storage.googleapis.com/tensorflow/tf-keras-datasets/train-images-idx3-ubyte.gz
26421880/26421880 [==============================] - 0s 0us/step
Downloading data from https://storage.googleapis.com/tensorflow/tf-keras-datasets/t10k-labels-idx1-ubyte.gz
5148/5148 [==============================] - 0s 0us/step
Downloading data from https://storage.googleapis.com/tensorflow/tf-keras-datasets/t10k-images-idx3-ubyte.gz
4422102/4422102 [==============================] - 0s 0us/step

%matplotlib inline
import warnings
from d2l import paddle as d2l

warnings.filterwarnings("ignore")
import math
import paddle
from paddle import nn
from paddle.optimizer import lr as lr_scheduler


def net_fn():
    model = nn.Sequential(
        nn.Conv2D(1, 6, kernel_size=5, padding=2), nn.ReLU(),
        nn.MaxPool2D(kernel_size=2, stride=2),
        nn.Conv2D(6, 16, kernel_size=5), nn.ReLU(),
        nn.MaxPool2D(kernel_size=2, stride=2),
        nn.Flatten(),
        nn.Linear(16 * 5 * 5, 120), nn.ReLU(),
        nn.Linear(120, 84), nn.ReLU(),
        nn.Linear(84, 10))

    return model

loss = nn.CrossEntropyLoss()
device = d2l.try_gpu()

batch_size = 256
train_iter, test_iter = d2l.load_data_fashion_mnist(batch_size=batch_size)

# 代码几乎与d2l.train_ch6定义在卷积神经网络一章LeNet一节中的相同
def train(net, train_iter, test_iter, num_epochs, loss, trainer, device,
          scheduler=None):
    animator = d2l.Animator(xlabel='epoch', xlim=[0, num_epochs],
                            legend=['train loss', 'train acc', 'test acc'])

    for epoch in range(num_epochs):
        metric = d2l.Accumulator(3)  # train_loss,train_acc,num_examples
        for i, (X, y) in enumerate(train_iter):
            net.train()
            trainer.clear_grad()
            y_hat = net(X)
            l = loss(y_hat, y)
            l.backward()
            trainer.step()
            with paddle.no_grad():
                metric.add(l * X.shape[0], d2l.accuracy(y_hat,y), X.shape[0])
            train_loss = metric[0] / metric[2]
            train_acc = metric[1] / metric[2]
            if (i + 1) % 50 == 0:
                animator.add(epoch + i / len(train_iter),
                             (train_loss, train_acc, None))

        test_acc = d2l.evaluate_accuracy_gpu(net, test_iter)
        animator.add(epoch+1, (None, None, test_acc))

        if scheduler:
            if scheduler.__module__ == lr_scheduler.__name__:
                # UsingPaddleIn-Builtscheduler
                scheduler.step()
            else:
                # Usingcustomdefinedscheduler
                trainer.set_lr(scheduler(epoch))
    print(f'train loss {train_loss:.3f}, train acc {train_acc:.3f}, 'f'test acc {test_acc:.3f}')

让我们来看看如果使用默认设置，调用此算法会发生什么。 例如设学习率为\(0.3\)并训练\(30\)次迭代。 留意在超过了某点、测试准确度方面的进展停滞时，训练准确度将如何继续提高。 两条曲线之间的间隙表示过拟合。

mxnetpytorchtensorflowpaddle

lr, num_epochs = 0.3, 30
net.initialize(force_reinit=True, ctx=device, init=init.Xavier())
trainer = gluon.Trainer(net.collect_params(), 'sgd', {'learning_rate': lr})
train(net, train_iter, test_iter, num_epochs, loss, trainer, device)

train loss 0.151, train acc 0.942, test acc 0.892

lr, num_epochs = 0.3, 30
net = net_fn()
trainer = torch.optim.SGD(net.parameters(), lr=lr)
train(net, train_iter, test_iter, num_epochs, loss, trainer, device)

train loss 0.128, train acc 0.951, test acc 0.885

lr, num_epochs = 0.3, 30
train(net, train_iter, test_iter, num_epochs, lr)

loss 0.208, train acc 0.922, test acc 0.877
61653.1 examples/sec on /GPU:0

<keras.engine.sequential.Sequential at 0x7fb0b435f2b0>

lr, num_epochs = 0.3, 30
net = net_fn()
trainer = paddle.optimizer.SGD(learning_rate=lr, parameters=net.parameters())
train(net, train_iter, test_iter, num_epochs, loss, trainer, device)

train loss 0.179, train acc 0.932, test acc 0.879

11.11.2. 学习率调度器

我们可以在每个迭代轮数（甚至在每个小批量）之后向下调整学习率。 例如，以动态的方式来响应优化的进展情况。

mxnetpytorchtensorflowpaddle

trainer.set_learning_rate(0.1)
print(f'learning rate is now {trainer.learning_rate:.2f}')

learning rate is now 0.10

lr = 0.1
trainer.param_groups[0]["lr"] = lr
print(f'learning rate is now {trainer.param_groups[0]["lr"]:.2f}')

learning rate is now 0.10

lr = 0.1
dummy_model = tf.keras.models.Sequential([tf.keras.layers.Dense(10)])
dummy_model.compile(tf.keras.optimizers.SGD(learning_rate=lr), loss='mse')
print(f'learning rate is now ,', dummy_model.optimizer.lr.numpy())

learning rate is now , 0.1

lr = 0.1
trainer.set_lr(lr)
print(f'learning rate is now {trainer.get_lr():.2f}')

learning rate is now 0.10

更通常而言，我们应该定义一个调度器。 当调用更新次数时，它将返回学习率的适当值。 让我们定义一个简单的方法，将学习率设置为\(\eta = \eta_0 (t + 1)^{-\frac{1}{2}}\)。

class SquareRootScheduler:
    def __init__(self, lr=0.1):
        self.lr = lr

    def __call__(self, num_update):
        return self.lr * pow(num_update + 1.0, -0.5)

让我们在一系列值上绘制它的行为。

mxnetpytorchtensorflowpaddle

scheduler = SquareRootScheduler(lr=0.1)
d2l.plot(np.arange(num_epochs), [scheduler(t) for t in range(num_epochs)])

scheduler = SquareRootScheduler(lr=0.1)
d2l.plot(torch.arange(num_epochs), [scheduler(t) for t in range(num_epochs)])

scheduler = SquareRootScheduler(lr=0.1)
d2l.plot(tf.range(num_epochs), [scheduler(t) for t in range(num_epochs)])

scheduler = SquareRootScheduler(lr=0.1)
d2l.plot(paddle.arange(num_epochs), [scheduler(t) for t in range(num_epochs)])

现在让我们来看看这对在Fashion-MNIST数据集上的训练有何影响。 我们只是提供调度器作为训练算法的额外参数。

mxnetpytorchtensorflowpaddle

trainer = gluon.Trainer(net.collect_params(), 'sgd',
                        {'lr_scheduler': scheduler})
train(net, train_iter, test_iter, num_epochs, loss, trainer, device)

train loss 0.521, train acc 0.812, test acc 0.805

net = net_fn()
trainer = torch.optim.SGD(net.parameters(), lr)
train(net, train_iter, test_iter, num_epochs, loss, trainer, device,
      scheduler)

train loss 0.270, train acc 0.901, test acc 0.876

train(net, train_iter, test_iter, num_epochs, lr,
      custom_callback=LearningRateScheduler(scheduler))

loss 0.377, train acc 0.862, test acc 0.850
61307.4 examples/sec on /GPU:0

<keras.engine.sequential.Sequential at 0x7faf33eb4940>

net = net_fn()
trainer = paddle.optimizer.SGD(learning_rate=lr , parameters=net.parameters())
train(net, train_iter, test_iter, num_epochs, loss, trainer, device,
      scheduler)

train loss 0.237, train acc 0.913, test acc 0.883

这比以前好一些：曲线比以前更加平滑，并且过拟合更小了。 遗憾的是，关于为什么在理论上某些策略会导致较轻的过拟合，有一些观点认为，较小的步长将导致参数更接近零，因此更简单。 但是，这并不能完全解释这种现象，因为我们并没有真正地提前停止，而只是轻柔地降低了学习率。

11.11.3. 策略

虽然我们不可能涵盖所有类型的学习率调度器，但我们会尝试在下面简要概述常用的策略：多项式衰减和分段常数表。 此外，余弦学习率调度在实践中的一些问题上运行效果很好。 在某些问题上，最好在使用较高的学习率之前预热优化器。

11.11.3.1. 单因子调度器

多项式衰减的一种替代方案是乘法衰减，即\(\eta_{t+1} \leftarrow \eta_t \cdot \alpha\)其中\(\alpha \in (0, 1)\)。 为了防止学习率衰减到一个合理的下界之下， 更新方程经常修改为\(\eta_{t+1} \leftarrow \mathop{\mathrm{max}}(\eta_{\mathrm{min}}, \eta_t \cdot \alpha)\)。

mxnetpytorchtensorflowpaddle

class FactorScheduler:
    def __init__(self, factor=1, stop_factor_lr=1e-7, base_lr=0.1):
        self.factor = factor
        self.stop_factor_lr = stop_factor_lr
        self.base_lr = base_lr

    def __call__(self, num_update):
        self.base_lr = max(self.stop_factor_lr, self.base_lr * self.factor)
        return self.base_lr

scheduler = FactorScheduler(factor=0.9, stop_factor_lr=1e-2, base_lr=2.0)
d2l.plot(np.arange(50), [scheduler(t) for t in range(50)])

class FactorScheduler:
    def __init__(self, factor=1, stop_factor_lr=1e-7, base_lr=0.1):
        self.factor = factor
        self.stop_factor_lr = stop_factor_lr
        self.base_lr = base_lr

    def __call__(self, num_update):
        self.base_lr = max(self.stop_factor_lr, self.base_lr * self.factor)
        return self.base_lr

scheduler = FactorScheduler(factor=0.9, stop_factor_lr=1e-2, base_lr=2.0)
d2l.plot(torch.arange(50), [scheduler(t) for t in range(50)])

class FactorScheduler:
    def __init__(self, factor=1, stop_factor_lr=1e-7, base_lr=0.1):
        self.factor = factor
        self.stop_factor_lr = stop_factor_lr
        self.base_lr = base_lr

    def __call__(self, num_update):
        self.base_lr = max(self.stop_factor_lr, self.base_lr * self.factor)
        return self.base_lr

scheduler = FactorScheduler(factor=0.9, stop_factor_lr=1e-2, base_lr=2.0)
d2l.plot(tf.range(50), [scheduler(t) for t in range(50)])

class FactorScheduler:
    def __init__(self, factor=1, stop_factor_lr=1e-7, base_lr=0.1):
        self.factor = factor
        self.stop_factor_lr = stop_factor_lr
        self.base_lr = base_lr

    def __call__(self, num_update):
        self.base_lr = max(self.stop_factor_lr, self.base_lr * self.factor)
        return self.base_lr

scheduler = FactorScheduler(factor=0.9, stop_factor_lr=1e-2, base_lr=2.0)
d2l.plot(paddle.arange(50), [scheduler(t) for t in range(50)])

接下来，我们将使用内置的调度器，但在这里仅解释它们的功能。

11.11.3.2. 多因子调度器

训练深度网络的常见策略之一是保持学习率为一组分段的常量，并且不时地按给定的参数对学习率做乘法衰减。 具体地说，给定一组降低学习率的时间点，例如\(s = \{5, 10, 20\}\)， 每当\(t \in s\)时，降低\(\eta_{t+1} \leftarrow \eta_t \cdot \alpha\)。 假设每步中的值减半，我们可以按如下方式实现这一点。

mxnetpytorchtensorflowpaddle

scheduler = lr_scheduler.MultiFactorScheduler(step=[15, 30], factor=0.5,
                                              base_lr=0.5)
d2l.plot(np.arange(num_epochs), [scheduler(t) for t in range(num_epochs)])

net = net_fn()
trainer = torch.optim.SGD(net.parameters(), lr=0.5)
scheduler = lr_scheduler.MultiStepLR(trainer, milestones=[15, 30], gamma=0.5)

def get_lr(trainer, scheduler):
    lr = scheduler.get_last_lr()[0]
    trainer.step()
    scheduler.step()
    return lr

d2l.plot(torch.arange(num_epochs), [get_lr(trainer, scheduler)
                                  for t in range(num_epochs)])

class MultiFactorScheduler:
    def __init__(self, step, factor, base_lr):
        self.step = step
        self.factor = factor
        self.base_lr = base_lr

    def __call__(self, epoch):
        if epoch in self.step:
            self.base_lr = self.base_lr * self.factor
            return self.base_lr
        else:
            return self.base_lr

scheduler = MultiFactorScheduler(step=[15, 30], factor=0.5, base_lr=0.5)
d2l.plot(tf.range(num_epochs), [scheduler(t) for t in range(num_epochs)])

net = net_fn()
scheduler =paddle.optimizer.lr.MultiStepDecay(learning_rate=0.5, milestones=[15,30], gamma=0.5)
trainer = paddle.optimizer.SGD(learning_rate=scheduler, parameters=net.parameters())
def get_lr(trainer, scheduler):
    lr=trainer.state_dict()['LR_Scheduler']['last_lr']
    trainer.step()
    scheduler.step()
    return lr

d2l.plot(paddle.arange(num_epochs), [get_lr(trainer, scheduler)
                                  for t in range(num_epochs)])

这种分段恒定学习率调度背后的直觉是，让优化持续进行，直到权重向量的分布达到一个驻点。 此时，我们才将学习率降低，以获得更高质量的代理来达到一个良好的局部最小值。 下面的例子展示了如何使用这种方法产生更好的解决方案。

mxnetpytorchtensorflowpaddle

trainer = gluon.Trainer(net.collect_params(), 'sgd',
                        {'lr_scheduler': scheduler})
train(net, train_iter, test_iter, num_epochs, loss, trainer, device)

train loss 0.167, train acc 0.936, test acc 0.904

train(net, train_iter, test_iter, num_epochs, loss, trainer, device,
      scheduler)

train loss 0.191, train acc 0.928, test acc 0.889

train(net, train_iter, test_iter, num_epochs, lr,
      custom_callback=LearningRateScheduler(scheduler))

loss 0.236, train acc 0.912, test acc 0.893
63056.2 examples/sec on /GPU:0

<keras.engine.sequential.Sequential at 0x7faf33cb96a0>

train(net, train_iter, test_iter, num_epochs, loss, trainer, device,
      scheduler)

train loss 0.157, train acc 0.942, test acc 0.899

11.11.3.3. 余弦调度器

余弦调度器是 (Loshchilov and Hutter, 2016)提出的一种启发式算法。 它所依据的观点是：我们可能不想在一开始就太大地降低学习率，而且可能希望最终能用非常小的学习率来“改进”解决方案。 这产生了一个类似于余弦的调度，函数形式如下所示，学习率的值在\(t \in [0, T]\)之间。

(11.11.1)\[\eta_t = \eta_T + \frac{\eta_0 - \eta_T}{2} \left(1 + \cos(\pi t/T)\right)\]

这里\(\eta_0\)是初始学习率，\(\eta_T\)是当\(T\)时的目标学习率。 此外，对于\(t > T\)，我们只需将值固定到\(\eta_T\)而不再增加它。 在下面的示例中，我们设置了最大更新步数\(T = 20\)。

mxnetpytorchtensorflowpaddle

scheduler = lr_scheduler.CosineScheduler(max_update=20, base_lr=0.3,
                                         final_lr=0.01)
d2l.plot(np.arange(num_epochs), [scheduler(t) for t in range(num_epochs)])

class CosineScheduler:
    def __init__(self, max_update, base_lr=0.01, final_lr=0,
               warmup_steps=0, warmup_begin_lr=0):
        self.base_lr_orig = base_lr
        self.max_update = max_update
        self.final_lr = final_lr
        self.warmup_steps = warmup_steps
        self.warmup_begin_lr = warmup_begin_lr
        self.max_steps = self.max_update - self.warmup_steps

    def get_warmup_lr(self, epoch):
        increase = (self.base_lr_orig - self.warmup_begin_lr) \
                       * float(epoch) / float(self.warmup_steps)
        return self.warmup_begin_lr + increase

    def __call__(self, epoch):
        if epoch < self.warmup_steps:
            return self.get_warmup_lr(epoch)
        if epoch <= self.max_update:
            self.base_lr = self.final_lr + (
                self.base_lr_orig - self.final_lr) * (1 + math.cos(
                math.pi * (epoch - self.warmup_steps) / self.max_steps)) / 2
        return self.base_lr

scheduler = CosineScheduler(max_update=20, base_lr=0.3, final_lr=0.01)
d2l.plot(torch.arange(num_epochs), [scheduler(t) for t in range(num_epochs)])

class CosineScheduler:
    def __init__(self, max_update, base_lr=0.01, final_lr=0,
               warmup_steps=0, warmup_begin_lr=0):
        self.base_lr_orig = base_lr
        self.max_update = max_update
        self.final_lr = final_lr
        self.warmup_steps = warmup_steps
        self.warmup_begin_lr = warmup_begin_lr
        self.max_steps = self.max_update - self.warmup_steps

    def get_warmup_lr(self, epoch):
        increase = (self.base_lr_orig - self.warmup_begin_lr) \
                       * float(epoch) / float(self.warmup_steps)
        return self.warmup_begin_lr + increase

    def __call__(self, epoch):
        if epoch < self.warmup_steps:
            return self.get_warmup_lr(epoch)
        if epoch <= self.max_update:
            self.base_lr = self.final_lr + (
                self.base_lr_orig - self.final_lr) * (1 + math.cos(
                math.pi * (epoch - self.warmup_steps) / self.max_steps)) / 2
        return self.base_lr

scheduler = CosineScheduler(max_update=20, base_lr=0.3, final_lr=0.01)
d2l.plot(tf.range(num_epochs), [scheduler(t) for t in range(num_epochs)])

class CosineScheduler:
    def __init__(self, max_update, base_lr=0.01, final_lr=0,
               warmup_steps=0, warmup_begin_lr=0):
        self.base_lr_orig = base_lr
        self.max_update = max_update
        self.final_lr = final_lr
        self.warmup_steps = warmup_steps
        self.warmup_begin_lr = warmup_begin_lr
        self.max_steps = self.max_update - self.warmup_steps

    def get_warmup_lr(self, epoch):
        increase = (self.base_lr_orig - self.warmup_begin_lr) \
                       * float(epoch) / float(self.warmup_steps)
        return self.warmup_begin_lr + increase

    def __call__(self, epoch):
        if epoch < self.warmup_steps:
            return self.get_warmup_lr(epoch)
        if epoch <= self.max_update:
            self.base_lr = self.final_lr + (
                self.base_lr_orig - self.final_lr) * (1 + math.cos(
                math.pi * (epoch - self.warmup_steps) / self.max_steps)) / 2
        return self.base_lr

scheduler = CosineScheduler(max_update=20, base_lr=0.3, final_lr=0.01)
d2l.plot(paddle.arange(num_epochs), [scheduler(t) for t in range(num_epochs)])

在计算机视觉的背景下，这个调度方式可能产生改进的结果。 但请注意，如下所示，这种改进并不一定成立。

mxnetpytorchtensorflowpaddle

trainer = gluon.Trainer(net.collect_params(), 'sgd',
                        {'lr_scheduler': scheduler})
train(net, train_iter, test_iter, num_epochs, loss, trainer, device)

train loss 0.343, train acc 0.879, test acc 0.876

net = net_fn()
trainer = torch.optim.SGD(net.parameters(), lr=0.3)
train(net, train_iter, test_iter, num_epochs, loss, trainer, device,
      scheduler)

train loss 0.207, train acc 0.923, test acc 0.892

train(net, train_iter, test_iter, num_epochs, lr,
      custom_callback=LearningRateScheduler(scheduler))

loss 0.259, train acc 0.906, test acc 0.883
63143.2 examples/sec on /GPU:0

<keras.engine.sequential.Sequential at 0x7faf33c5cfa0>

net = net_fn()
trainer = paddle.optimizer.SGD(learning_rate=0.3, parameters=net.parameters())
train(net, train_iter, test_iter, num_epochs, loss, trainer, device,
      scheduler)

train loss 0.142, train acc 0.948, test acc 0.904

11.11.3.4. 预热

在某些情况下，初始化参数不足以得到良好的解。 这对某些高级网络设计来说尤其棘手，可能导致不稳定的优化结果。 对此，一方面，我们可以选择一个足够小的学习率， 从而防止一开始发散，然而这样进展太缓慢。 另一方面，较高的学习率最初就会导致发散。

解决这种困境的一个相当简单的解决方法是使用预热期，在此期间学习率将增加至初始最大值，然后冷却直到优化过程结束。 为了简单起见，通常使用线性递增。 这引出了如下表所示的时间表。

mxnetpytorchtensorflowpaddle

scheduler = lr_scheduler.CosineScheduler(20, warmup_steps=5, base_lr=0.3,
                                         final_lr=0.01)
d2l.plot(np.arange(num_epochs), [scheduler(t) for t in range(num_epochs)])

scheduler = CosineScheduler(20, warmup_steps=5, base_lr=0.3, final_lr=0.01)
d2l.plot(torch.arange(num_epochs), [scheduler(t) for t in range(num_epochs)])

scheduler = CosineScheduler(20, warmup_steps=5, base_lr=0.3, final_lr=0.01)
d2l.plot(tf.range(num_epochs), [scheduler(t) for t in range(num_epochs)])

scheduler = CosineScheduler(20, warmup_steps=5, base_lr=0.3, final_lr=0.01)
d2l.plot(paddle.arange(num_epochs), [scheduler(t) for t in range(num_epochs)])

注意，观察前5个迭代轮数的性能，网络最初收敛得更好。

mxnetpytorchtensorflowpaddle

trainer = gluon.Trainer(net.collect_params(), 'sgd',
                        {'lr_scheduler': scheduler})
train(net, train_iter, test_iter, num_epochs, loss, trainer, device)

train loss 0.347, train acc 0.875, test acc 0.875

net = net_fn()
trainer = torch.optim.SGD(net.parameters(), lr=0.3)
train(net, train_iter, test_iter, num_epochs, loss, trainer, device,
      scheduler)

train loss 0.261, train acc 0.904, test acc 0.878

train(net, train_iter, test_iter, num_epochs, lr,
      custom_callback=LearningRateScheduler(scheduler))

loss 0.276, train acc 0.899, test acc 0.877
61250.8 examples/sec on /GPU:0

<keras.engine.sequential.Sequential at 0x7faf33802d00>

net = net_fn()
trainer = paddle.optimizer.SGD(learning_rate=0.3, parameters=net.parameters())
train(net, train_iter, test_iter, num_epochs, loss, trainer, device,
      scheduler)

train loss 0.143, train acc 0.948, test acc 0.902

预热可以应用于任何调度器，而不仅仅是余弦。 有关学习率调度的更多实验和更详细讨论，请参阅 (Gotmare et al., 2018)。 其中，这篇论文的点睛之笔的发现：预热阶段限制了非常深的网络中参数的发散程度 。 这在直觉上是有道理的：在网络中那些一开始花费最多时间取得进展的部分，随机初始化会产生巨大的发散。

11.11.4. 小结

• 在训练期间逐步降低学习率可以提高准确性，并且减少模型的过拟合。

• 在实验中，每当进展趋于稳定时就降低学习率，这是很有效的。从本质上说，这可以确保我们有效地收敛到一个适当的解，也只有这样才能通过降低学习率来减小参数的固有方差。

• 余弦调度器在某些计算机视觉问题中很受欢迎。

• 优化之前的预热期可以防止发散。

• 优化在深度学习中有多种用途。对于同样的训练误差而言，选择不同的优化算法和学习率调度，除了最大限度地减少训练时间，可以导致测试集上不同的泛化和过拟合量。

11.11.5. 练习

1. 试验给定固定学习率的优化行为。这种情况下可以获得的最佳模型是什么？

2. 如果改变学习率下降的指数，收敛性会如何改变？在实验中方便起见，使用PolyScheduler。

3. 将余弦调度器应用于大型计算机视觉问题，例如训练ImageNet数据集。与其他调度器相比，它如何影响性能？

4. 预热应该持续多长时间？

5. 可以试着把优化和采样联系起来吗？首先，在随机梯度朗之万动力学上使用 (Welling and Teh, 2011)的结果。

mxnetpytorchtensorflowpaddle

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