.. _sec_scheduler:

学习率调度器
============


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
   :numref:`sec_momentum`\ ）。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。
   :numref:`sec_minibatch_sgd`\ 比较详细地讨论了这一点，在
   :numref:`sec_sgd`\ 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\ :math:`\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
   :numref:`sec_numerical_stability`\ ），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
   :cite:`Izmailov.Podoprikhin.Garipov.ea.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。

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
:numref:`chap_cnn`\ 进行复习。



.. raw:: html

    <div class="mdl-tabs mdl-js-tabs mdl-js-ripple-effect"><div class="mdl-tabs__tab-bar code"><a href="#mxnet-1-0" onclick="tagClick('mxnet'); return false;" class="mdl-tabs__tab is-active">mxnet</a><a href="#pytorch-1-1" onclick="tagClick('pytorch'); return false;" class="mdl-tabs__tab ">pytorch</a><a href="#tensorflow-1-2" onclick="tagClick('tensorflow'); return false;" class="mdl-tabs__tab ">tensorflow</a><a href="#paddle-1-3" onclick="tagClick('paddle'); return false;" class="mdl-tabs__tab ">paddle</a></div>



.. raw:: html

    <div class="mdl-tabs__panel is-active" id="mxnet-1-0">

.. raw:: latex

   \diilbookstyleinputcell

.. code:: python

    %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}')


.. raw:: latex

   \diilbookstyleoutputcell

.. parsed-literal::
    :class: output

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




.. raw:: html

    </div>



.. raw:: html

    <div class="mdl-tabs__panel " id="pytorch-1-1">

.. raw:: latex

   \diilbookstyleinputcell

.. code:: python

    %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}')



.. raw:: html

    </div>



.. raw:: html

    <div class="mdl-tabs__panel " id="tensorflow-1-2">

.. raw:: latex

   \diilbookstyleinputcell

.. code:: python

    %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


.. raw:: latex

   \diilbookstyleoutputcell

.. parsed-literal::
    :class: output

    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




.. raw:: html

    </div>



.. raw:: html

    <div class="mdl-tabs__panel " id="paddle-1-3">

.. raw:: latex

   \diilbookstyleinputcell

.. code:: python

    %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}')



.. raw:: html

    </div>



.. raw:: html

    </div>

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\ :math:`0.3`\ 并训练\ :math:`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。



.. raw:: html

    <div class="mdl-tabs mdl-js-tabs mdl-js-ripple-effect"><div class="mdl-tabs__tab-bar code"><a href="#mxnet-3-0" onclick="tagClick('mxnet'); return false;" class="mdl-tabs__tab is-active">mxnet</a><a href="#pytorch-3-1" onclick="tagClick('pytorch'); return false;" class="mdl-tabs__tab ">pytorch</a><a href="#tensorflow-3-2" onclick="tagClick('tensorflow'); return false;" class="mdl-tabs__tab ">tensorflow</a><a href="#paddle-3-3" onclick="tagClick('paddle'); return false;" class="mdl-tabs__tab ">paddle</a></div>



.. raw:: html

    <div class="mdl-tabs__panel is-active" id="mxnet-3-0">

.. raw:: latex

   \diilbookstyleinputcell

.. code:: python

    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)


.. raw:: latex

   \diilbookstyleoutputcell

.. parsed-literal::
    :class: output

    train loss 0.151, train acc 0.942, test acc 0.892



.. figure:: output_lr-scheduler_1dfeb6_18_1.svg




.. raw:: html

    </div>



.. raw:: html

    <div class="mdl-tabs__panel " id="pytorch-3-1">

.. raw:: latex

   \diilbookstyleinputcell

.. code:: python

    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)


.. raw:: latex

   \diilbookstyleoutputcell

.. parsed-literal::
    :class: output

    train loss 0.128, train acc 0.951, test acc 0.885



.. figure:: output_lr-scheduler_1dfeb6_21_1.svg




.. raw:: html

    </div>



.. raw:: html

    <div class="mdl-tabs__panel " id="tensorflow-3-2">

.. raw:: latex

   \diilbookstyleinputcell

.. code:: python

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


.. raw:: latex

   \diilbookstyleoutputcell

.. parsed-literal::
    :class: output

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




.. raw:: latex

   \diilbookstyleoutputcell

.. parsed-literal::
    :class: output

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




.. figure:: output_lr-scheduler_1dfeb6_24_2.svg




.. raw:: html

    </div>



.. raw:: html

    <div class="mdl-tabs__panel " id="paddle-3-3">

.. raw:: latex

   \diilbookstyleinputcell

.. code:: python

    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)


.. raw:: latex

   \diilbookstyleoutputcell

.. parsed-literal::
    :class: output

    train loss 0.179, train acc 0.932, test acc 0.879



.. figure:: output_lr-scheduler_1dfeb6_27_1.svg




.. raw:: html

    </div>



.. raw:: html

    </div>

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。



.. raw:: html

    <div class="mdl-tabs mdl-js-tabs mdl-js-ripple-effect"><div class="mdl-tabs__tab-bar code"><a href="#mxnet-5-0" onclick="tagClick('mxnet'); return false;" class="mdl-tabs__tab is-active">mxnet</a><a href="#pytorch-5-1" onclick="tagClick('pytorch'); return false;" class="mdl-tabs__tab ">pytorch</a><a href="#tensorflow-5-2" onclick="tagClick('tensorflow'); return false;" class="mdl-tabs__tab ">tensorflow</a><a href="#paddle-5-3" onclick="tagClick('paddle'); return false;" class="mdl-tabs__tab ">paddle</a></div>



.. raw:: html

    <div class="mdl-tabs__panel is-active" id="mxnet-5-0">

.. raw:: latex

   \diilbookstyleinputcell

.. code:: python

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


.. raw:: latex

   \diilbookstyleoutputcell

.. parsed-literal::
    :class: output

    learning rate is now 0.10




.. raw:: html

    </div>



.. raw:: html

    <div class="mdl-tabs__panel " id="pytorch-5-1">

.. raw:: latex

   \diilbookstyleinputcell

.. code:: python

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


.. raw:: latex

   \diilbookstyleoutputcell

.. parsed-literal::
    :class: output

    learning rate is now 0.10




.. raw:: html

    </div>



.. raw:: html

    <div class="mdl-tabs__panel " id="tensorflow-5-2">

.. raw:: latex

   \diilbookstyleinputcell

.. code:: python

    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())


.. raw:: latex

   \diilbookstyleoutputcell

.. parsed-literal::
    :class: output

    learning rate is now , 0.1




.. raw:: html

    </div>



.. raw:: html

    <div class="mdl-tabs__panel " id="paddle-5-3">

.. raw:: latex

   \diilbookstyleinputcell

.. code:: python

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


.. raw:: latex

   \diilbookstyleoutputcell

.. parsed-literal::
    :class: output

    learning rate is now 0.10




.. raw:: html

    </div>



.. raw:: html

    </div>

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\ :math:`\eta = \eta_0 (t + 1)^{-\frac{1}{2}}`\ 。

.. raw:: latex

   \diilbookstyleinputcell

.. code:: python

    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。



.. raw:: html

    <div class="mdl-tabs mdl-js-tabs mdl-js-ripple-effect"><div class="mdl-tabs__tab-bar code"><a href="#mxnet-9-0" onclick="tagClick('mxnet'); return false;" class="mdl-tabs__tab is-active">mxnet</a><a href="#pytorch-9-1" onclick="tagClick('pytorch'); return false;" class="mdl-tabs__tab ">pytorch</a><a href="#tensorflow-9-2" onclick="tagClick('tensorflow'); return false;" class="mdl-tabs__tab ">tensorflow</a><a href="#paddle-9-3" onclick="tagClick('paddle'); return false;" class="mdl-tabs__tab ">paddle</a></div>



.. raw:: html

    <div class="mdl-tabs__panel is-active" id="mxnet-9-0">

.. raw:: latex

   \diilbookstyleinputcell

.. code:: python

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



.. figure:: output_lr-scheduler_1dfeb6_50_0.svg




.. raw:: html

    </div>



.. raw:: html

    <div class="mdl-tabs__panel " id="pytorch-9-1">

.. raw:: latex

   \diilbookstyleinputcell

.. code:: python

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



.. figure:: output_lr-scheduler_1dfeb6_53_0.svg




.. raw:: html

    </div>



.. raw:: html

    <div class="mdl-tabs__panel " id="tensorflow-9-2">

.. raw:: latex

   \diilbookstyleinputcell

.. code:: python

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



.. figure:: output_lr-scheduler_1dfeb6_56_0.svg




.. raw:: html

    </div>



.. raw:: html

    <div class="mdl-tabs__panel " id="paddle-9-3">

.. raw:: latex

   \diilbookstyleinputcell

.. code:: python

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



.. figure:: output_lr-scheduler_1dfeb6_59_0.svg




.. raw:: html

    </div>



.. raw:: html

    </div>

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。



.. raw:: html

    <div class="mdl-tabs mdl-js-tabs mdl-js-ripple-effect"><div class="mdl-tabs__tab-bar code"><a href="#mxnet-11-0" onclick="tagClick('mxnet'); return false;" class="mdl-tabs__tab is-active">mxnet</a><a href="#pytorch-11-1" onclick="tagClick('pytorch'); return false;" class="mdl-tabs__tab ">pytorch</a><a href="#tensorflow-11-2" onclick="tagClick('tensorflow'); return false;" class="mdl-tabs__tab ">tensorflow</a><a href="#paddle-11-3" onclick="tagClick('paddle'); return false;" class="mdl-tabs__tab ">paddle</a></div>



.. raw:: html

    <div class="mdl-tabs__panel is-active" id="mxnet-11-0">

.. raw:: latex

   \diilbookstyleinputcell

.. code:: python

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


.. raw:: latex

   \diilbookstyleoutputcell

.. parsed-literal::
    :class: output

    train loss 0.521, train acc 0.812, test acc 0.805



.. figure:: output_lr-scheduler_1dfeb6_65_1.svg




.. raw:: html

    </div>



.. raw:: html

    <div class="mdl-tabs__panel " id="pytorch-11-1">

.. raw:: latex

   \diilbookstyleinputcell

.. code:: python

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


.. raw:: latex

   \diilbookstyleoutputcell

.. parsed-literal::
    :class: output

    train loss 0.270, train acc 0.901, test acc 0.876



.. figure:: output_lr-scheduler_1dfeb6_68_1.svg




.. raw:: html

    </div>



.. raw:: html

    <div class="mdl-tabs__panel " id="tensorflow-11-2">

.. raw:: latex

   \diilbookstyleinputcell

.. code:: python

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


.. raw:: latex

   \diilbookstyleoutputcell

.. parsed-literal::
    :class: output

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




.. raw:: latex

   \diilbookstyleoutputcell

.. parsed-literal::
    :class: output

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




.. figure:: output_lr-scheduler_1dfeb6_71_2.svg




.. raw:: html

    </div>



.. raw:: html

    <div class="mdl-tabs__panel " id="paddle-11-3">

.. raw:: latex

   \diilbookstyleinputcell

.. code:: python

    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)


.. raw:: latex

   \diilbookstyleoutputcell

.. parsed-literal::
    :class: output

    train loss 0.237, train acc 0.913, test acc 0.883



.. figure:: output_lr-scheduler_1dfeb6_74_1.svg




.. raw:: html

    </div>



.. raw:: html

    </div>

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。

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。

单因子调度器
~~~~~~~~~~~~

多项式衰减的一种替代方案是乘法衰减，即\ :math:`\eta_{t+1} \leftarrow \eta_t \cdot \alpha`\ 其中\ :math:`\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\ :math:`\eta_{t+1} \leftarrow \mathop{\mathrm{max}}(\eta_{\mathrm{min}}, \eta_t \cdot \alpha)`\ 。



.. raw:: html

    <div class="mdl-tabs mdl-js-tabs mdl-js-ripple-effect"><div class="mdl-tabs__tab-bar code"><a href="#mxnet-13-0" onclick="tagClick('mxnet'); return false;" class="mdl-tabs__tab is-active">mxnet</a><a href="#pytorch-13-1" onclick="tagClick('pytorch'); return false;" class="mdl-tabs__tab ">pytorch</a><a href="#tensorflow-13-2" onclick="tagClick('tensorflow'); return false;" class="mdl-tabs__tab ">tensorflow</a><a href="#paddle-13-3" onclick="tagClick('paddle'); return false;" class="mdl-tabs__tab ">paddle</a></div>



.. raw:: html

    <div class="mdl-tabs__panel is-active" id="mxnet-13-0">

.. raw:: latex

   \diilbookstyleinputcell

.. code:: python

    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)])



.. figure:: output_lr-scheduler_1dfeb6_80_0.svg




.. raw:: html

    </div>



.. raw:: html

    <div class="mdl-tabs__panel " id="pytorch-13-1">

.. raw:: latex

   \diilbookstyleinputcell

.. code:: python

    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)])



.. figure:: output_lr-scheduler_1dfeb6_83_0.svg




.. raw:: html

    </div>



.. raw:: html

    <div class="mdl-tabs__panel " id="tensorflow-13-2">

.. raw:: latex

   \diilbookstyleinputcell

.. code:: python

    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)])



.. figure:: output_lr-scheduler_1dfeb6_86_0.svg




.. raw:: html

    </div>



.. raw:: html

    <div class="mdl-tabs__panel " id="paddle-13-3">

.. raw:: latex

   \diilbookstyleinputcell

.. code:: python

    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)])



.. figure:: output_lr-scheduler_1dfeb6_89_0.svg




.. raw:: html

    </div>



.. raw:: html

    </div>

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。

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\ :math:`s = \{5, 10, 20\}`\ ，
每当\ :math:`t \in s`\ 时，降低\ :math:`\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。



.. raw:: html

    <div class="mdl-tabs mdl-js-tabs mdl-js-ripple-effect"><div class="mdl-tabs__tab-bar code"><a href="#mxnet-15-0" onclick="tagClick('mxnet'); return false;" class="mdl-tabs__tab is-active">mxnet</a><a href="#pytorch-15-1" onclick="tagClick('pytorch'); return false;" class="mdl-tabs__tab ">pytorch</a><a href="#tensorflow-15-2" onclick="tagClick('tensorflow'); return false;" class="mdl-tabs__tab ">tensorflow</a><a href="#paddle-15-3" onclick="tagClick('paddle'); return false;" class="mdl-tabs__tab ">paddle</a></div>



.. raw:: html

    <div class="mdl-tabs__panel is-active" id="mxnet-15-0">

.. raw:: latex

   \diilbookstyleinputcell

.. code:: python

    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)])



.. figure:: output_lr-scheduler_1dfeb6_95_0.svg




.. raw:: html

    </div>



.. raw:: html

    <div class="mdl-tabs__panel " id="pytorch-15-1">

.. raw:: latex

   \diilbookstyleinputcell

.. code:: python

    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)])



.. figure:: output_lr-scheduler_1dfeb6_98_0.svg




.. raw:: html

    </div>



.. raw:: html

    <div class="mdl-tabs__panel " id="tensorflow-15-2">

.. raw:: latex

   \diilbookstyleinputcell

.. code:: python

    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)])



.. figure:: output_lr-scheduler_1dfeb6_101_0.svg




.. raw:: html

    </div>



.. raw:: html

    <div class="mdl-tabs__panel " id="paddle-15-3">

.. raw:: latex

   \diilbookstyleinputcell

.. code:: python

    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)])



.. figure:: output_lr-scheduler_1dfeb6_104_0.svg




.. raw:: html

    </div>



.. raw:: html

    </div>

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。



.. raw:: html

    <div class="mdl-tabs mdl-js-tabs mdl-js-ripple-effect"><div class="mdl-tabs__tab-bar code"><a href="#mxnet-17-0" onclick="tagClick('mxnet'); return false;" class="mdl-tabs__tab is-active">mxnet</a><a href="#pytorch-17-1" onclick="tagClick('pytorch'); return false;" class="mdl-tabs__tab ">pytorch</a><a href="#tensorflow-17-2" onclick="tagClick('tensorflow'); return false;" class="mdl-tabs__tab ">tensorflow</a><a href="#paddle-17-3" onclick="tagClick('paddle'); return false;" class="mdl-tabs__tab ">paddle</a></div>



.. raw:: html

    <div class="mdl-tabs__panel is-active" id="mxnet-17-0">

.. raw:: latex

   \diilbookstyleinputcell

.. code:: python

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


.. raw:: latex

   \diilbookstyleoutputcell

.. parsed-literal::
    :class: output

    train loss 0.167, train acc 0.936, test acc 0.904



.. figure:: output_lr-scheduler_1dfeb6_110_1.svg




.. raw:: html

    </div>



.. raw:: html

    <div class="mdl-tabs__panel " id="pytorch-17-1">

.. raw:: latex

   \diilbookstyleinputcell

.. code:: python

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


.. raw:: latex

   \diilbookstyleoutputcell

.. parsed-literal::
    :class: output

    train loss 0.191, train acc 0.928, test acc 0.889



.. figure:: output_lr-scheduler_1dfeb6_113_1.svg




.. raw:: html

    </div>



.. raw:: html

    <div class="mdl-tabs__panel " id="tensorflow-17-2">

.. raw:: latex

   \diilbookstyleinputcell

.. code:: python

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


.. raw:: latex

   \diilbookstyleoutputcell

.. parsed-literal::
    :class: output

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




.. raw:: latex

   \diilbookstyleoutputcell

.. parsed-literal::
    :class: output

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




.. figure:: output_lr-scheduler_1dfeb6_116_2.svg




.. raw:: html

    </div>



.. raw:: html

    <div class="mdl-tabs__panel " id="paddle-17-3">

.. raw:: latex

   \diilbookstyleinputcell

.. code:: python

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


.. raw:: latex

   \diilbookstyleoutputcell

.. parsed-literal::
    :class: output

    train loss 0.157, train acc 0.942, test acc 0.899



.. figure:: output_lr-scheduler_1dfeb6_119_1.svg




.. raw:: html

    </div>



.. raw:: html

    </div>

余弦调度器
~~~~~~~~~~

余弦调度器是 :cite:`Loshchilov.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\ :math:`t \in [0, T]`\ 之间。

.. math:: \eta_t = \eta_T + \frac{\eta_0 - \eta_T}{2} \left(1 + \cos(\pi t/T)\right)

这里\ :math:`\eta_0`\ 是初始学习率，\ :math:`\eta_T`\ 是当\ :math:`T`\ 时的目标学习率。
此外，对于\ :math:`t > T`\ ，我们只需将值固定到\ :math:`\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\ :math:`T = 20`\ 。



.. raw:: html

    <div class="mdl-tabs mdl-js-tabs mdl-js-ripple-effect"><div class="mdl-tabs__tab-bar code"><a href="#mxnet-19-0" onclick="tagClick('mxnet'); return false;" class="mdl-tabs__tab is-active">mxnet</a><a href="#pytorch-19-1" onclick="tagClick('pytorch'); return false;" class="mdl-tabs__tab ">pytorch</a><a href="#tensorflow-19-2" onclick="tagClick('tensorflow'); return false;" class="mdl-tabs__tab ">tensorflow</a><a href="#paddle-19-3" onclick="tagClick('paddle'); return false;" class="mdl-tabs__tab ">paddle</a></div>



.. raw:: html

    <div class="mdl-tabs__panel is-active" id="mxnet-19-0">

.. raw:: latex

   \diilbookstyleinputcell

.. code:: python

    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)])



.. figure:: output_lr-scheduler_1dfeb6_125_0.svg




.. raw:: html

    </div>



.. raw:: html

    <div class="mdl-tabs__panel " id="pytorch-19-1">

.. raw:: latex

   \diilbookstyleinputcell

.. code:: python

    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)])



.. figure:: output_lr-scheduler_1dfeb6_128_0.svg




.. raw:: html

    </div>



.. raw:: html

    <div class="mdl-tabs__panel " id="tensorflow-19-2">

.. raw:: latex

   \diilbookstyleinputcell

.. code:: python

    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)])



.. figure:: output_lr-scheduler_1dfeb6_131_0.svg




.. raw:: html

    </div>



.. raw:: html

    <div class="mdl-tabs__panel " id="paddle-19-3">

.. raw:: latex

   \diilbookstyleinputcell

.. code:: python

    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)])



.. figure:: output_lr-scheduler_1dfeb6_134_0.svg




.. raw:: html

    </div>



.. raw:: html

    </div>

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。



.. raw:: html

    <div class="mdl-tabs mdl-js-tabs mdl-js-ripple-effect"><div class="mdl-tabs__tab-bar code"><a href="#mxnet-21-0" onclick="tagClick('mxnet'); return false;" class="mdl-tabs__tab is-active">mxnet</a><a href="#pytorch-21-1" onclick="tagClick('pytorch'); return false;" class="mdl-tabs__tab ">pytorch</a><a href="#tensorflow-21-2" onclick="tagClick('tensorflow'); return false;" class="mdl-tabs__tab ">tensorflow</a><a href="#paddle-21-3" onclick="tagClick('paddle'); return false;" class="mdl-tabs__tab ">paddle</a></div>



.. raw:: html

    <div class="mdl-tabs__panel is-active" id="mxnet-21-0">

.. raw:: latex

   \diilbookstyleinputcell

.. code:: python

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


.. raw:: latex

   \diilbookstyleoutputcell

.. parsed-literal::
    :class: output

    train loss 0.343, train acc 0.879, test acc 0.876



.. figure:: output_lr-scheduler_1dfeb6_140_1.svg




.. raw:: html

    </div>



.. raw:: html

    <div class="mdl-tabs__panel " id="pytorch-21-1">

.. raw:: latex

   \diilbookstyleinputcell

.. code:: python

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


.. raw:: latex

   \diilbookstyleoutputcell

.. parsed-literal::
    :class: output

    train loss 0.207, train acc 0.923, test acc 0.892



.. figure:: output_lr-scheduler_1dfeb6_143_1.svg




.. raw:: html

    </div>



.. raw:: html

    <div class="mdl-tabs__panel " id="tensorflow-21-2">

.. raw:: latex

   \diilbookstyleinputcell

.. code:: python

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


.. raw:: latex

   \diilbookstyleoutputcell

.. parsed-literal::
    :class: output

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




.. raw:: latex

   \diilbookstyleoutputcell

.. parsed-literal::
    :class: output

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




.. figure:: output_lr-scheduler_1dfeb6_146_2.svg




.. raw:: html

    </div>



.. raw:: html

    <div class="mdl-tabs__panel " id="paddle-21-3">

.. raw:: latex

   \diilbookstyleinputcell

.. code:: python

    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)


.. raw:: latex

   \diilbookstyleoutputcell

.. parsed-literal::
    :class: output

    train loss 0.142, train acc 0.948, test acc 0.904



.. figure:: output_lr-scheduler_1dfeb6_149_1.svg




.. raw:: html

    </div>



.. raw:: html

    </div>

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。



.. raw:: html

    <div class="mdl-tabs mdl-js-tabs mdl-js-ripple-effect"><div class="mdl-tabs__tab-bar code"><a href="#mxnet-23-0" onclick="tagClick('mxnet'); return false;" class="mdl-tabs__tab is-active">mxnet</a><a href="#pytorch-23-1" onclick="tagClick('pytorch'); return false;" class="mdl-tabs__tab ">pytorch</a><a href="#tensorflow-23-2" onclick="tagClick('tensorflow'); return false;" class="mdl-tabs__tab ">tensorflow</a><a href="#paddle-23-3" onclick="tagClick('paddle'); return false;" class="mdl-tabs__tab ">paddle</a></div>



.. raw:: html

    <div class="mdl-tabs__panel is-active" id="mxnet-23-0">

.. raw:: latex

   \diilbookstyleinputcell

.. code:: python

    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)])



.. figure:: output_lr-scheduler_1dfeb6_155_0.svg




.. raw:: html

    </div>



.. raw:: html

    <div class="mdl-tabs__panel " id="pytorch-23-1">

.. raw:: latex

   \diilbookstyleinputcell

.. code:: python

    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)])



.. figure:: output_lr-scheduler_1dfeb6_158_0.svg




.. raw:: html

    </div>



.. raw:: html

    <div class="mdl-tabs__panel " id="tensorflow-23-2">

.. raw:: latex

   \diilbookstyleinputcell

.. code:: python

    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)])



.. figure:: output_lr-scheduler_1dfeb6_161_0.svg




.. raw:: html

    </div>



.. raw:: html

    <div class="mdl-tabs__panel " id="paddle-23-3">

.. raw:: latex

   \diilbookstyleinputcell

.. code:: python

    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)])



.. figure:: output_lr-scheduler_1dfeb6_164_0.svg




.. raw:: html

    </div>



.. raw:: html

    </div>

注意，观察前5个迭代轮数的性能，网络最初收敛得更好。



.. raw:: html

    <div class="mdl-tabs mdl-js-tabs mdl-js-ripple-effect"><div class="mdl-tabs__tab-bar code"><a href="#mxnet-25-0" onclick="tagClick('mxnet'); return false;" class="mdl-tabs__tab is-active">mxnet</a><a href="#pytorch-25-1" onclick="tagClick('pytorch'); return false;" class="mdl-tabs__tab ">pytorch</a><a href="#tensorflow-25-2" onclick="tagClick('tensorflow'); return false;" class="mdl-tabs__tab ">tensorflow</a><a href="#paddle-25-3" onclick="tagClick('paddle'); return false;" class="mdl-tabs__tab ">paddle</a></div>



.. raw:: html

    <div class="mdl-tabs__panel is-active" id="mxnet-25-0">

.. raw:: latex

   \diilbookstyleinputcell

.. code:: python

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


.. raw:: latex

   \diilbookstyleoutputcell

.. parsed-literal::
    :class: output

    train loss 0.347, train acc 0.875, test acc 0.875



.. figure:: output_lr-scheduler_1dfeb6_170_1.svg




.. raw:: html

    </div>



.. raw:: html

    <div class="mdl-tabs__panel " id="pytorch-25-1">

.. raw:: latex

   \diilbookstyleinputcell

.. code:: python

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


.. raw:: latex

   \diilbookstyleoutputcell

.. parsed-literal::
    :class: output

    train loss 0.261, train acc 0.904, test acc 0.878



.. figure:: output_lr-scheduler_1dfeb6_173_1.svg




.. raw:: html

    </div>



.. raw:: html

    <div class="mdl-tabs__panel " id="tensorflow-25-2">

.. raw:: latex

   \diilbookstyleinputcell

.. code:: python

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


.. raw:: latex

   \diilbookstyleoutputcell

.. parsed-literal::
    :class: output

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




.. raw:: latex

   \diilbookstyleoutputcell

.. parsed-literal::
    :class: output

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




.. figure:: output_lr-scheduler_1dfeb6_176_2.svg




.. raw:: html

    </div>



.. raw:: html

    <div class="mdl-tabs__panel " id="paddle-25-3">

.. raw:: latex

   \diilbookstyleinputcell

.. code:: python

    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)


.. raw:: latex

   \diilbookstyleoutputcell

.. parsed-literal::
    :class: output

    train loss 0.143, train acc 0.948, test acc 0.902



.. figure:: output_lr-scheduler_1dfeb6_179_1.svg




.. raw:: html

    </div>



.. raw:: html

    </div>

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
:cite:`Gotmare.Keskar.Xiong.ea.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。

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。

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
   :cite:`Welling.Teh.2011`\ 的结果。



.. raw:: html

    <div class="mdl-tabs mdl-js-tabs mdl-js-ripple-effect"><div class="mdl-tabs__tab-bar text"><a href="#mxnet-27-0" onclick="tagClick('mxnet'); return false;" class="mdl-tabs__tab is-active">mxnet</a><a href="#pytorch-27-1" onclick="tagClick('pytorch'); return false;" class="mdl-tabs__tab ">pytorch</a><a href="#tensorflow-27-2" onclick="tagClick('tensorflow'); return false;" class="mdl-tabs__tab ">tensorflow</a><a href="#paddle-27-3" onclick="tagClick('paddle'); return false;" class="mdl-tabs__tab ">paddle</a></div>



.. raw:: html

    <div class="mdl-tabs__panel is-active" id="mxnet-27-0">

`Discussions <https://discuss.d2l.ai/t/4333>`__



.. raw:: html

    </div>



.. raw:: html

    <div class="mdl-tabs__panel " id="pytorch-27-1">

`Discussions <https://discuss.d2l.ai/t/4334>`__



.. raw:: html

    </div>



.. raw:: html

    <div class="mdl-tabs__panel " id="tensorflow-27-2">

`Discussions <https://discuss.d2l.ai/t/4335>`__



.. raw:: html

    </div>



.. raw:: html

    <div class="mdl-tabs__panel " id="paddle-27-3">

`Discussions <https://discuss.d2l.ai/t/11856>`__



.. raw:: html

    </div>



.. raw:: html

    </div>