Run this notebook online:\ |Binder| or Colab: |Colab|

.. |Binder| image:: https://mybinder.org/badge_logo.svg
   :target: https://mybinder.org/v2/gh/deepjavalibrary/d2l-java/master?filepath=chapter_optimization/lr-scheduler.ipynb
.. |Colab| image:: https://colab.research.google.com/assets/colab-badge.svg
   :target: https://colab.research.google.com/github/deepjavalibrary/d2l-java/blob/colab/chapter_optimization/lr-scheduler.ipynb

.. _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。

.. code:: java

    %load ../utils/djl-imports
    %load ../utils/plot-utils
    %load ../utils/Functions.java
    %load ../utils/GradDescUtils.java
    %load ../utils/Accumulator.java
    %load ../utils/StopWatch.java
    
    %load ../utils/Training.java
    %load ../utils/TrainingChapter11.java

.. code:: java

    import ai.djl.basicdataset.cv.classification.*;
    import org.apache.commons.lang3.ArrayUtils;

.. code:: java

    SequentialBlock net = new SequentialBlock();
    
    net.add(Conv2d.builder()
            .setKernelShape(new Shape(5, 5))
            .optPadding(new Shape(2, 2))
            .setFilters(1)
            .build());
    net.add(Activation.reluBlock());
    net.add(Pool.maxPool2dBlock(new Shape(2, 2), new Shape(2, 2)));
    net.add(Conv2d.builder()
            .setKernelShape(new Shape(5, 5))
            .setFilters(1)
            .build());
    net.add(Blocks.batchFlattenBlock());
    net.add(Activation.reluBlock());
    net.add(Linear.builder().setUnits(120).build());
    net.add(Activation.reluBlock());
    net.add(Linear.builder().setUnits(84).build());
    net.add(Activation.reluBlock());
    net.add(Linear.builder().setUnits(10).build());




.. parsed-literal::
    :class: output

    SequentialBlock {
    	Conv2d
    	ReLU
    	maxPool2d
    	Conv2d
    	batchFlatten
    	ReLU
    	Linear
    	ReLU
    	Linear
    	ReLU
    	Linear
    }



.. code:: java

    int batchSize = 256;
    RandomAccessDataset trainDataset = FashionMnist.builder()
            .optUsage(Dataset.Usage.TRAIN)
            .setSampling(batchSize, false)
            .optLimit(Long.getLong("DATASET_LIMIT", Long.MAX_VALUE))
            .build();
    
    RandomAccessDataset testDataset = FashionMnist.builder()
            .optUsage(Dataset.Usage.TEST)
            .setSampling(batchSize, false)
            .optLimit(Long.getLong("DATASET_LIMIT", Long.MAX_VALUE))
            .build();

.. code:: java

    double[] trainLoss;
    double[] testAccuracy;
    double[] epochCount;
    double[] trainAccuracy;
    
    public static void train(RandomAccessDataset trainIter, RandomAccessDataset testIter,
                                 int numEpochs, Trainer trainer) throws IOException, TranslateException {
        epochCount = new double[numEpochs];
    
        for (int i = 0; i < epochCount.length; i++) {
            epochCount[i] = (i + 1);
        }
    
        double avgTrainTimePerEpoch = 0;
        Map<String, double[]> evaluatorMetrics = new HashMap<>();
    
        trainer.setMetrics(new Metrics());
    
        EasyTrain.fit(trainer, numEpochs, trainIter, testIter);
    
        Metrics metrics = trainer.getMetrics();
    
        trainer.getEvaluators().stream()
                .forEach(evaluator -> {
                    evaluatorMetrics.put("train_epoch_" + evaluator.getName(), metrics.getMetric("train_epoch_" + evaluator.getName()).stream()
                            .mapToDouble(x -> x.getValue().doubleValue()).toArray());
                    evaluatorMetrics.put("validate_epoch_" + evaluator.getName(), metrics.getMetric("validate_epoch_" + evaluator.getName()).stream()
                            .mapToDouble(x -> x.getValue().doubleValue()).toArray());
                });
    
        avgTrainTimePerEpoch = metrics.mean("epoch");
    
        trainLoss = evaluatorMetrics.get("train_epoch_SoftmaxCrossEntropyLoss");
        trainAccuracy = evaluatorMetrics.get("train_epoch_Accuracy");
        testAccuracy = evaluatorMetrics.get("validate_epoch_Accuracy");
    
        System.out.printf("loss %.3f," , trainLoss[numEpochs-1]);
        System.out.printf(" train acc %.3f," , trainAccuracy[numEpochs-1]);
        System.out.printf(" test acc %.3f\n" , testAccuracy[numEpochs-1]);
        System.out.printf("%.1f examples/sec \n", trainIter.size() / (avgTrainTimePerEpoch / Math.pow(10, 9)));
    }

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。

.. code:: java

    float lr = 0.3f;
    int numEpochs = Integer.getInteger("MAX_EPOCH", 10);
    
    Model model = Model.newInstance("Modern LeNet");
    model.setBlock(net);
    
    Loss loss = Loss.softmaxCrossEntropyLoss();
    Tracker lrt = Tracker.fixed(lr);
    Optimizer sgd = Optimizer.sgd().setLearningRateTracker(lrt).build();
    
    DefaultTrainingConfig config = new DefaultTrainingConfig(loss)
            .optOptimizer(sgd) // Optimizer
            .addEvaluator(new Accuracy()) // Model Accuracy
            .addTrainingListeners(TrainingListener.Defaults.logging()); // Logging
    
    Trainer trainer = model.newTrainer(config);
    trainer.initialize(new Shape(1, 1, 28, 28));
    
    train(trainDataset, testDataset, numEpochs, trainer);


.. parsed-literal::
    :class: output

    INFO Training on: 4 GPUs.
    INFO Load MXNet Engine Version 1.9.0 in 0.057 ms.


.. parsed-literal::
    :class: output

    Training:    100% |████████████████████████████████████████| Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30
    Validating:  100% |████████████████████████████████████████|


.. parsed-literal::
    :class: output

    INFO Epoch 1 finished.
    INFO Train: Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30
    INFO Validate: Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30


.. parsed-literal::
    :class: output

    Training:    100% |████████████████████████████████████████| Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30
    Validating:  100% |████████████████████████████████████████|


.. parsed-literal::
    :class: output

    INFO Epoch 2 finished.
    INFO Train: Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30
    INFO Validate: Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30


.. parsed-literal::
    :class: output

    Training:    100% |████████████████████████████████████████| Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30
    Validating:  100% |████████████████████████████████████████|


.. parsed-literal::
    :class: output

    INFO Epoch 3 finished.
    INFO Train: Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30
    INFO Validate: Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30


.. parsed-literal::
    :class: output

    Training:    100% |████████████████████████████████████████| Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30
    Validating:  100% |████████████████████████████████████████|


.. parsed-literal::
    :class: output

    INFO Epoch 4 finished.
    INFO Train: Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30
    INFO Validate: Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30


.. parsed-literal::
    :class: output

    Training:    100% |████████████████████████████████████████| Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30
    Validating:  100% |████████████████████████████████████████|


.. parsed-literal::
    :class: output

    INFO Epoch 5 finished.
    INFO Train: Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30
    INFO Validate: Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30


.. parsed-literal::
    :class: output

    Training:    100% |████████████████████████████████████████| Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30
    Validating:  100% |████████████████████████████████████████|


.. parsed-literal::
    :class: output

    INFO Epoch 6 finished.
    INFO Train: Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30
    INFO Validate: Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30


.. parsed-literal::
    :class: output

    Training:    100% |████████████████████████████████████████| Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30
    Validating:  100% |████████████████████████████████████████|


.. parsed-literal::
    :class: output

    INFO Epoch 7 finished.
    INFO Train: Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30
    INFO Validate: Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30


.. parsed-literal::
    :class: output

    Training:    100% |████████████████████████████████████████| Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30
    Validating:  100% |████████████████████████████████████████|


.. parsed-literal::
    :class: output

    INFO Epoch 8 finished.
    INFO Train: Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30
    INFO Validate: Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30


.. parsed-literal::
    :class: output

    Training:    100% |████████████████████████████████████████| Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30
    Validating:  100% |████████████████████████████████████████|


.. parsed-literal::
    :class: output

    INFO Epoch 9 finished.
    INFO Train: Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30
    INFO Validate: Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30


.. parsed-literal::
    :class: output

    Training:    100% |████████████████████████████████████████| Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30
    Validating:  100% |████████████████████████████████████████|


.. parsed-literal::
    :class: output

    INFO Epoch 10 finished.
    INFO Train: Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30
    INFO Validate: Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30


.. parsed-literal::
    :class: output

    loss 2.304, train acc 0.100, test acc 0.100
    8559.3 examples/sec 


.. code:: java

    public void plotMetrics() {
        String[] lossLabel = new String[trainLoss.length + testAccuracy.length + trainAccuracy.length];
    
        Arrays.fill(lossLabel, 0, trainLoss.length, "train loss");
        Arrays.fill(lossLabel, trainAccuracy.length, trainLoss.length + trainAccuracy.length, "train acc");
        Arrays.fill(lossLabel, trainLoss.length + trainAccuracy.length,
                        trainLoss.length + testAccuracy.length + trainAccuracy.length, "test acc");
    
        Table data = Table.create("Data").addColumns(
            DoubleColumn.create("epoch", ArrayUtils.addAll(epochCount, ArrayUtils.addAll(epochCount, epochCount))),
            DoubleColumn.create("metrics", ArrayUtils.addAll(trainLoss, ArrayUtils.addAll(trainAccuracy, testAccuracy))),
            StringColumn.create("lossLabel", lossLabel)
        );
    
        display(LinePlot.create("", data, "epoch", "metrics", "lossLabel"));
    }
    
    plotMetrics();



.. raw:: html

    <img id="1296043962a445808238e162015ceffe_img"></img>
    <div id="1296043962a445808238e162015ceffe"></div>
    <script>require(['https://cdn.plot.ly/plotly-1.57.0.min.js'], Plotly => {
    var target_1296043962a445808238e162015ceffe = document.getElementById('1296043962a445808238e162015ceffe');
    var layout = {
        height: 600,
        width: 800,
        showlegend: true,
        xaxis: {
        title: 'epoch',
        },
    
        yaxis: {
        title: 'metrics',
        },
    
    };
    
    var trace0 =
    {
    x: ["1.0","2.0","3.0","4.0","5.0","6.0","7.0","8.0","9.0","10.0"],
    y: ["2.3041308","2.3038523","2.3038523","2.3038523","2.3038523","2.3038523","2.3038523","2.3038523","2.3038523","2.3038523"],
    showlegend: true,
    mode: 'lines',
    xaxis: 'x',
    yaxis: 'y',
    type: 'scatter',
    name: 'train loss',
    };
    var trace1 =
    {
    x: ["1.0","2.0","3.0","4.0","5.0","6.0","7.0","8.0","9.0","10.0"],
    y: ["0.10036667","0.0998","0.0998","0.0998","0.0998","0.0998","0.0998","0.0998","0.0998","0.0998"],
    showlegend: true,
    mode: 'lines',
    xaxis: 'x',
    yaxis: 'y',
    type: 'scatter',
    name: 'train acc',
    };
    var trace2 =
    {
    x: ["1.0","2.0","3.0","4.0","5.0","6.0","7.0","8.0","9.0","10.0"],
    y: ["0.1","0.1","0.1","0.1","0.1","0.1","0.1","0.1","0.1","0.1"],
    showlegend: true,
    mode: 'lines',
    xaxis: 'x',
    yaxis: 'y',
    type: 'scatter',
    name: 'test acc',
    };
    
    
    var data = [ trace0, trace1, trace2];
    Plotly.newPlot(target_1296043962a445808238e162015ceffe, data, layout);
    })</script>



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。
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}}`\ 。

.. code:: java

    public class SquareRootTracker {
        float lr;
        public SquareRootTracker() {
            this(0.1f);
        }
        public SquareRootTracker(float learningRate) {
            this.lr = learningRate;
        }
        public float getNewLearningRate(int numUpdate) {
            return lr * (float) Math.pow(numUpdate + 1, -0.5);
        }
    }

让我们在一系列值上绘制它的行为。

.. code:: java

    public Figure plotLearningRate(int[] epochs, float[] learningRates) {
        
        String[] lossLabel = new String[trainLoss.length + testAccuracy.length + trainAccuracy.length];
    
        Arrays.fill(lossLabel, 0, trainLoss.length, "train loss");
        Arrays.fill(lossLabel, trainAccuracy.length, trainLoss.length + trainAccuracy.length, "train acc");
        Arrays.fill(lossLabel, trainLoss.length + trainAccuracy.length,
                        trainLoss.length + testAccuracy.length + trainAccuracy.length, "test acc");
    
        Table data = Table.create("Data").addColumns(
                    IntColumn.create("epoch", epochs),
                    DoubleColumn.create("learning rate", learningRates)
        );
    
        return LinePlot.create("Learning Rate vs. Epoch", data, "epoch", "learning rate");
    }

.. code:: java

    SquareRootTracker tracker = new SquareRootTracker();
    
    int[] epochs = new int[numEpochs];
    float[] learningRates = new float[numEpochs];
    for (int i = 0; i < numEpochs; i++) {
        epochs[i] = i;
        learningRates[i] = tracker.getNewLearningRate(i);
    }
    
    plotLearningRate(epochs, learningRates);




.. raw:: html

    <img id="2b6aac103c7b4cf2a090940eb9ab4fbc_img"></img>
    <div id="2b6aac103c7b4cf2a090940eb9ab4fbc"></div>
    <script>require(['https://cdn.plot.ly/plotly-1.57.0.min.js'], Plotly => {
    var target_2b6aac103c7b4cf2a090940eb9ab4fbc = document.getElementById('2b6aac103c7b4cf2a090940eb9ab4fbc');
    var layout = {
        title: 'Learning Rate vs. Epoch',
        height: 600,
        width: 800,
        xaxis: {
        title: 'epoch',
        },
    
        yaxis: {
        title: 'learning rate',
        },
    
    };
    
    var trace0 =
    {
    x: ["0","1","2","3","4","5","6","7","8","9"],
    y: ["0.10000000149011612","0.0707106813788414","0.05773502588272095","0.05000000074505806","0.04472136124968529","0.040824830532073975","0.03779644891619682","0.0353553406894207","0.03333333507180214","0.03162277862429619"],
    mode: 'lines',
    xaxis: 'x',
    yaxis: 'y',
    type: 'scatter',
    name: '',
    };
    
    
    var data = [ trace0];
    Plotly.newPlot(target_2b6aac103c7b4cf2a090940eb9ab4fbc, data, layout);
    })</script>




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。

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。

多因子调度器
~~~~~~~~~~~~

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超出合理的下限，更新方程经常修改为\ :math:`\eta_{t+1} \leftarrow \mathop{\mathrm{max}}(\eta_{\mathrm{min}}, \eta_t \cdot \alpha)`\ 。

.. code:: java

    public class DemoFactorTracker {
        float baseLr;
        float stopFactorLr;
        float factor;
        public DemoFactorTracker(float factor, float stopFactorLr, float baseLr) {
            this.factor = factor;
            this.stopFactorLr = stopFactorLr;
            this.baseLr = baseLr;
        }
        public DemoFactorTracker() {
            this(1f, (float) 1e-7, 0.1f);
        }
        public float getNewLearningRate(int numUpdate) {
            return lr * (float) Math.pow(numUpdate + 1, -0.5);
        }
    }

.. code:: java

    DemoFactorTracker tracker = new DemoFactorTracker(0.9f, (float) 1e-2, 2);
    
    numEpochs = 50;
    int[] epochs = new int[numEpochs];
    float[] learningRates = new float[numEpochs];
    for (int i = 0; i < numEpochs; i++) {
        epochs[i] = i;
        learningRates[i] = tracker.getNewLearningRate(i);
    }
    
    plotLearningRate(epochs, learningRates);




.. raw:: html

    <img id="e5999156ad94414c8b057db65261e7bf_img"></img>
    <div id="e5999156ad94414c8b057db65261e7bf"></div>
    <script>require(['https://cdn.plot.ly/plotly-1.57.0.min.js'], Plotly => {
    var target_e5999156ad94414c8b057db65261e7bf = document.getElementById('e5999156ad94414c8b057db65261e7bf');
    var layout = {
        title: 'Learning Rate vs. Epoch',
        height: 600,
        width: 800,
        xaxis: {
        title: 'epoch',
        },
    
        yaxis: {
        title: 'learning rate',
        },
    
    };
    
    var trace0 =
    {
    x: ["0","1","2","3","4","5","6","7","8","9","10","11","12","13","14","15","16","17","18","19","20","21","22","23","24","25","26","27","28","29","30","31","32","33","34","35","36","37","38","39","40","41","42","43","44","45","46","47","48","49"],
    y: ["0.30000001192092896","0.2121320366859436","0.17320507764816284","0.15000000596046448","0.13416408002376556","0.12247449904680252","0.11338934302330017","0.1060660183429718","0.10000000894069672","0.09486833214759827","0.09045340865850449","0.08660253882408142","0.08320502936840057","0.08017837256193161","0.0774596706032753","0.07500000298023224","0.0727606862783432","0.0707106813788414","0.06882472336292267","0.06708204001188278","0.06546536833047867","0.06396021693944931","0.06255432218313217","0.06123724952340126","0.06000000238418579","0.05883484333753586","0.057735029608011246","0.056694671511650085","0.05570860207080841","0.054772257804870605","0.05388159304857254","0.0530330091714859","0.05222329869866371","0.05144957825541496","0.050709255039691925","0.05000000447034836","0.04931969568133354","0.04866642877459526","0.04803844541311264","0.04743416607379913","0.04685213044285774","0.046291008591651917","0.04574957489967346","0.04522670432925224","0.04472136124968529","0.04423258826136589","0.04375949874520302","0.04330126941204071","0.04285714775323868","0.04242641106247902"],
    mode: 'lines',
    xaxis: 'x',
    yaxis: 'y',
    type: 'scatter',
    name: '',
    };
    
    
    var data = [ trace0];
    Plotly.newPlot(target_e5999156ad94414c8b057db65261e7bf, data, layout);
    })</script>




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分段稳定的学习率，并且每隔一段时间就一定程度学习率降低。
具体地说，给定一组降低学习率的时间，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。

.. code:: java

    MultiFactorTracker tracker = Tracker.multiFactor()
            .setSteps(new int[]{5, 30})
            .optFactor(0.5f)
            .setBaseValue(0.5f)
            .build();
    
    numEpochs = 10;
    int[] epochs = new int[numEpochs];
    float[] learningRates = new float[numEpochs];
    for (int i = 0; i < numEpochs; i++) {
        epochs[i] = i;
        learningRates[i] = tracker.getNewValue(i);
    }
    
    plotLearningRate(epochs, learningRates);




.. raw:: html

    <img id="a7783471b95147459bd2e35fd01d4787_img"></img>
    <div id="a7783471b95147459bd2e35fd01d4787"></div>
    <script>require(['https://cdn.plot.ly/plotly-1.57.0.min.js'], Plotly => {
    var target_a7783471b95147459bd2e35fd01d4787 = document.getElementById('a7783471b95147459bd2e35fd01d4787');
    var layout = {
        title: 'Learning Rate vs. Epoch',
        height: 600,
        width: 800,
        xaxis: {
        title: 'epoch',
        },
    
        yaxis: {
        title: 'learning rate',
        },
    
    };
    
    var trace0 =
    {
    x: ["0","1","2","3","4","5","6","7","8","9"],
    y: ["0.5","0.5","0.5","0.5","0.5","0.5","0.25","0.25","0.25","0.25"],
    mode: 'lines',
    xaxis: 'x',
    yaxis: 'y',
    type: 'scatter',
    name: '',
    };
    
    
    var data = [ trace0];
    Plotly.newPlot(target_a7783471b95147459bd2e35fd01d4787, data, layout);
    })</script>




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。

.. code:: java

    int numEpochs = Integer.getInteger("MAX_EPOCH", 10);
    
    Model model = Model.newInstance("Modern LeNet");
    model.setBlock(net);
    
    Loss loss = Loss.softmaxCrossEntropyLoss();
    Optimizer sgd = Optimizer.sgd().setLearningRateTracker(tracker).build();
    
    DefaultTrainingConfig config = new DefaultTrainingConfig(loss)
            .optOptimizer(sgd) // Optimizer
            .addEvaluator(new Accuracy()) // Model Accuracy
            .addTrainingListeners(TrainingListener.Defaults.logging()); // Logging
    
    Trainer trainer = model.newTrainer(config);
    trainer.initialize(new Shape(1, 1, 28, 28));
    
    train(trainDataset, testDataset, numEpochs, trainer);
    plotMetrics();


.. parsed-literal::
    :class: output

    INFO Training on: 4 GPUs.
    INFO Load MXNet Engine Version 1.9.0 in 0.020 ms.


.. parsed-literal::
    :class: output

    Training:    100% |████████████████████████████████████████| Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30
    Validating:  100% |████████████████████████████████████████|


.. parsed-literal::
    :class: output

    INFO Epoch 1 finished.
    INFO Train: Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30
    INFO Validate: Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30


.. parsed-literal::
    :class: output

    Training:    100% |████████████████████████████████████████| Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30
    Validating:  100% |████████████████████████████████████████|


.. parsed-literal::
    :class: output

    INFO Epoch 2 finished.
    INFO Train: Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30
    INFO Validate: Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30


.. parsed-literal::
    :class: output

    Training:    100% |████████████████████████████████████████| Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30
    Validating:  100% |████████████████████████████████████████|


.. parsed-literal::
    :class: output

    INFO Epoch 3 finished.
    INFO Train: Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30
    INFO Validate: Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30


.. parsed-literal::
    :class: output

    Training:    100% |████████████████████████████████████████| Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30
    Validating:  100% |████████████████████████████████████████|


.. parsed-literal::
    :class: output

    INFO Epoch 4 finished.
    INFO Train: Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30
    INFO Validate: Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30


.. parsed-literal::
    :class: output

    Training:    100% |████████████████████████████████████████| Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30
    Validating:  100% |████████████████████████████████████████|


.. parsed-literal::
    :class: output

    INFO Epoch 5 finished.
    INFO Train: Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30
    INFO Validate: Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30


.. parsed-literal::
    :class: output

    Training:    100% |████████████████████████████████████████| Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30
    Validating:  100% |████████████████████████████████████████|


.. parsed-literal::
    :class: output

    INFO Epoch 6 finished.
    INFO Train: Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30
    INFO Validate: Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30


.. parsed-literal::
    :class: output

    Training:    100% |████████████████████████████████████████| Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30
    Validating:  100% |████████████████████████████████████████|


.. parsed-literal::
    :class: output

    INFO Epoch 7 finished.
    INFO Train: Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30
    INFO Validate: Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30


.. parsed-literal::
    :class: output

    Training:    100% |████████████████████████████████████████| Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30
    Validating:  100% |████████████████████████████████████████|


.. parsed-literal::
    :class: output

    INFO Epoch 8 finished.
    INFO Train: Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30
    INFO Validate: Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30


.. parsed-literal::
    :class: output

    Training:    100% |████████████████████████████████████████| Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30
    Validating:  100% |████████████████████████████████████████|


.. parsed-literal::
    :class: output

    INFO Epoch 9 finished.
    INFO Train: Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30
    INFO Validate: Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30


.. parsed-literal::
    :class: output

    Training:    100% |████████████████████████████████████████| Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30
    Validating:  100% |████████████████████████████████████████|


.. parsed-literal::
    :class: output

    INFO Epoch 10 finished.
    INFO Train: Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30
    INFO Validate: Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30


.. parsed-literal::
    :class: output

    loss 2.303, train acc 0.100, test acc 0.100
    10782.6 examples/sec 



.. raw:: html

    <img id="4abb9d6444c741a2b1ac3023c20eff8d_img"></img>
    <div id="4abb9d6444c741a2b1ac3023c20eff8d"></div>
    <script>require(['https://cdn.plot.ly/plotly-1.57.0.min.js'], Plotly => {
    var target_4abb9d6444c741a2b1ac3023c20eff8d = document.getElementById('4abb9d6444c741a2b1ac3023c20eff8d');
    var layout = {
        height: 600,
        width: 800,
        showlegend: true,
        xaxis: {
        title: 'epoch',
        },
    
        yaxis: {
        title: 'metrics',
        },
    
    };
    
    var trace0 =
    {
    x: ["1.0","2.0","3.0","4.0","5.0","6.0","7.0","8.0","9.0","10.0"],
    y: ["2.3032598","2.3031666","2.3031666","2.3031666","2.3031666","2.3031666","2.3031666","2.3031666","2.3031666","2.3031666"],
    showlegend: true,
    mode: 'lines',
    xaxis: 'x',
    yaxis: 'y',
    type: 'scatter',
    name: 'train loss',
    };
    var trace1 =
    {
    x: ["1.0","2.0","3.0","4.0","5.0","6.0","7.0","8.0","9.0","10.0"],
    y: ["0.09975","0.0995","0.0995","0.0995","0.0995","0.0995","0.0995","0.0995","0.0995","0.0995"],
    showlegend: true,
    mode: 'lines',
    xaxis: 'x',
    yaxis: 'y',
    type: 'scatter',
    name: 'train acc',
    };
    var trace2 =
    {
    x: ["1.0","2.0","3.0","4.0","5.0","6.0","7.0","8.0","9.0","10.0"],
    y: ["0.1","0.1","0.1","0.1","0.1","0.1","0.1","0.1","0.1","0.1"],
    showlegend: true,
    mode: 'lines',
    xaxis: 'x',
    yaxis: 'y',
    type: 'scatter',
    name: 'test acc',
    };
    
    
    var data = [ trace0, trace1, trace2];
    Plotly.newPlot(target_4abb9d6444c741a2b1ac3023c20eff8d, data, layout);
    })</script>



.. code:: java

    String[] lossLabel = new String[trainLoss.length + testAccuracy.length + trainAccuracy.length];
    
    Arrays.fill(lossLabel, 0, trainLoss.length, "train loss");
    Arrays.fill(lossLabel, trainAccuracy.length, trainLoss.length + trainAccuracy.length, "train acc");
    Arrays.fill(lossLabel, trainLoss.length + trainAccuracy.length,
                    trainLoss.length + testAccuracy.length + trainAccuracy.length, "test acc");
    
    Table data = Table.create("Data").addColumns(
                DoubleColumn.create("epoch", ArrayUtils.addAll(epochCount, ArrayUtils.addAll(epochCount, epochCount))),
                DoubleColumn.create("metrics", ArrayUtils.addAll(trainLoss, ArrayUtils.addAll(trainAccuracy, testAccuracy))),
                StringColumn.create("lossLabel", lossLabel)
    );
    
    LinePlot.create("", data, "epoch", "metrics", "lossLabel");




.. raw:: html

    <img id="238cf1b368c9408cbe274006343c47a1_img"></img>
    <div id="238cf1b368c9408cbe274006343c47a1"></div>
    <script>require(['https://cdn.plot.ly/plotly-1.57.0.min.js'], Plotly => {
    var target_238cf1b368c9408cbe274006343c47a1 = document.getElementById('238cf1b368c9408cbe274006343c47a1');
    var layout = {
        height: 600,
        width: 800,
        showlegend: true,
        xaxis: {
        title: 'epoch',
        },
    
        yaxis: {
        title: 'metrics',
        },
    
    };
    
    var trace0 =
    {
    x: ["1.0","2.0","3.0","4.0","5.0","6.0","7.0","8.0","9.0","10.0"],
    y: ["2.3032598","2.3031666","2.3031666","2.3031666","2.3031666","2.3031666","2.3031666","2.3031666","2.3031666","2.3031666"],
    showlegend: true,
    mode: 'lines',
    xaxis: 'x',
    yaxis: 'y',
    type: 'scatter',
    name: 'train loss',
    };
    var trace1 =
    {
    x: ["1.0","2.0","3.0","4.0","5.0","6.0","7.0","8.0","9.0","10.0"],
    y: ["0.09975","0.0995","0.0995","0.0995","0.0995","0.0995","0.0995","0.0995","0.0995","0.0995"],
    showlegend: true,
    mode: 'lines',
    xaxis: 'x',
    yaxis: 'y',
    type: 'scatter',
    name: 'train acc',
    };
    var trace2 =
    {
    x: ["1.0","2.0","3.0","4.0","5.0","6.0","7.0","8.0","9.0","10.0"],
    y: ["0.1","0.1","0.1","0.1","0.1","0.1","0.1","0.1","0.1","0.1"],
    showlegend: true,
    mode: 'lines',
    xaxis: 'x',
    yaxis: 'y',
    type: 'scatter',
    name: 'test acc',
    };
    
    
    var data = [ trace0, trace1, trace2];
    Plotly.newPlot(target_238cf1b368c9408cbe274006343c47a1, data, layout);
    })</script>




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`\ 。

.. code:: java

    public class DemoCosineTracker {
        float baseLr;
        float finalLr;
        int maxUpdate;
        public DemoCosineTracker() {
            this(0.5f, 0.01f, 20);
        }
        public DemoCosineTracker(float baseLr, float finalLr, int maxUpdate) {
            this.baseLr = baseLr;
            this.finalLr = finalLr;
            this.maxUpdate = maxUpdate;
        }
        public float getNewLearningRate(int numUpdate) {
            if (numUpdate > maxUpdate) {
                return finalLr;
            }
            // Scale the curve to smoothly transition
            float step = (baseLr - finalLr) / 2 * (1 + (float) Math.cos(Math.PI * numUpdate / maxUpdate));
            return finalLr + step;
        }
    }

.. code:: java

    DemoCosineTracker tracker = new DemoCosineTracker(0.5f, 0.01f, 20);
    
    int[] epochs = new int[numEpochs];
    float[] learningRates = new float[numEpochs];
    for (int i = 0; i < numEpochs; i++) {
        epochs[i] = i;
        learningRates[i] = tracker.getNewLearningRate(i);
    }
    
    plotLearningRate(epochs, learningRates);




.. raw:: html

    <img id="bccabfe2302c4be6b971679c8d14f789_img"></img>
    <div id="bccabfe2302c4be6b971679c8d14f789"></div>
    <script>require(['https://cdn.plot.ly/plotly-1.57.0.min.js'], Plotly => {
    var target_bccabfe2302c4be6b971679c8d14f789 = document.getElementById('bccabfe2302c4be6b971679c8d14f789');
    var layout = {
        title: 'Learning Rate vs. Epoch',
        height: 600,
        width: 800,
        xaxis: {
        title: 'epoch',
        },
    
        yaxis: {
        title: 'learning rate',
        },
    
    };
    
    var trace0 =
    {
    x: ["0","1","2","3","4","5","6","7","8","9"],
    y: ["0.5","0.4969836473464966","0.4880088269710541","0.473296582698822","0.45320916175842285","0.42824116349220276","0.3990073800086975","0.3662276566028595","0.33070915937423706","0.2933264374732971"],
    mode: 'lines',
    xaxis: 'x',
    yaxis: 'y',
    type: 'scatter',
    name: '',
    };
    
    
    var data = [ trace0];
    Plotly.newPlot(target_bccabfe2302c4be6b971679c8d14f789, data, layout);
    })</script>




在计算机视觉中，这个调度可以引出改进的结果。
但请注意，如下所示，这种改进并不能保证。

.. code:: java

    CosineTracker cosineTracker = Tracker.cosine()
                                .setBaseValue(0.5f)
                                .optFinalValue(0.01f)
                                .setMaxUpdates(20)
                                .build();
    
    Loss loss = Loss.softmaxCrossEntropyLoss();
    Optimizer sgd = Optimizer.sgd().setLearningRateTracker(cosineTracker).build();
    
    DefaultTrainingConfig config = new DefaultTrainingConfig(loss)
            .optOptimizer(sgd) // Optimizer
            .addEvaluator(new Accuracy()) // Model Accuracy
            .addTrainingListeners(TrainingListener.Defaults.logging()); // Logging
    
    Trainer trainer = model.newTrainer(config);
    trainer.initialize(new Shape(1, 1, 28, 28));
    
    train(trainDataset, testDataset, numEpochs, trainer);


.. parsed-literal::
    :class: output

    INFO Training on: 4 GPUs.
    INFO Load MXNet Engine Version 1.9.0 in 0.022 ms.


.. parsed-literal::
    :class: output

    Training:    100% |████████████████████████████████████████| Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30
    Validating:  100% |████████████████████████████████████████|


.. parsed-literal::
    :class: output

    INFO Epoch 1 finished.
    INFO Train: Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30
    INFO Validate: Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30


.. parsed-literal::
    :class: output

    Training:    100% |████████████████████████████████████████| Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30
    Validating:  100% |████████████████████████████████████████|


.. parsed-literal::
    :class: output

    INFO Epoch 2 finished.
    INFO Train: Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30
    INFO Validate: Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30


.. parsed-literal::
    :class: output

    Training:    100% |████████████████████████████████████████| Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30
    Validating:  100% |████████████████████████████████████████|


.. parsed-literal::
    :class: output

    INFO Epoch 3 finished.
    INFO Train: Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30
    INFO Validate: Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30


.. parsed-literal::
    :class: output

    Training:    100% |████████████████████████████████████████| Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30
    Validating:  100% |████████████████████████████████████████|


.. parsed-literal::
    :class: output

    INFO Epoch 4 finished.
    INFO Train: Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30
    INFO Validate: Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30


.. parsed-literal::
    :class: output

    Training:    100% |████████████████████████████████████████| Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30
    Validating:  100% |████████████████████████████████████████|


.. parsed-literal::
    :class: output

    INFO Epoch 5 finished.
    INFO Train: Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30
    INFO Validate: Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30


.. parsed-literal::
    :class: output

    Training:    100% |████████████████████████████████████████| Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30
    Validating:  100% |████████████████████████████████████████|


.. parsed-literal::
    :class: output

    INFO Epoch 6 finished.
    INFO Train: Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30
    INFO Validate: Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30


.. parsed-literal::
    :class: output

    Training:    100% |████████████████████████████████████████| Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30
    Validating:  100% |████████████████████████████████████████|


.. parsed-literal::
    :class: output

    INFO Epoch 7 finished.
    INFO Train: Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30
    INFO Validate: Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30


.. parsed-literal::
    :class: output

    Training:    100% |████████████████████████████████████████| Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30
    Validating:  100% |████████████████████████████████████████|


.. parsed-literal::
    :class: output

    INFO Epoch 8 finished.
    INFO Train: Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30
    INFO Validate: Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30


.. parsed-literal::
    :class: output

    Training:    100% |████████████████████████████████████████| Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30
    Validating:  100% |████████████████████████████████████████|


.. parsed-literal::
    :class: output

    INFO Epoch 9 finished.
    INFO Train: Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30
    INFO Validate: Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30


.. parsed-literal::
    :class: output

    Training:    100% |████████████████████████████████████████| Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30
    Validating:  100% |████████████████████████████████████████|


.. parsed-literal::
    :class: output

    INFO Epoch 10 finished.
    INFO Train: Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30
    INFO Validate: Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30


.. parsed-literal::
    :class: output

    loss 2.303, train acc 0.096, test acc 0.100
    10674.0 examples/sec 


预热
~~~~

在某些情况下，初始化参数不足以得到良好的解。
这对于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。

.. code:: java

    public class CosineWarmupTracker {
        float baseLr;
        float finalLr;
        int maxUpdate;
        int warmUpSteps;
        float warmUpBeginValue;
        float warmUpFinalValue;
        
        public CosineWarmupTracker() {
            this(0.5f, 0.01f, 20, 5);
        }
        
        public CosineWarmupTracker(float baseLr, float finalLr, int maxUpdate, int warmUpSteps) {
            this.baseLr = baseLr;
            this.finalLr = finalLr;
            this.maxUpdate = maxUpdate;
            this.warmUpSteps = 5;
            this.warmUpBeginValue = 0f;
        }
        
        public float getNewLearningRate(int numUpdate) {
            if (numUpdate <= warmUpSteps) {
                return getWarmUpValue(numUpdate);
            }
            if (numUpdate > maxUpdate) {
                return finalLr;
            }
            // Scale the cosine curve to fit smoothly with the warmup steps
            float step = (baseLr - finalLr) / 2 * (1 + 
                (float) Math.cos(Math.PI * (numUpdate - warmUpSteps) / (maxUpdate - warmUpSteps)));
            return finalLr + step;
        }
        
        public float getWarmUpValue(int numUpdate) {
            // Linear warmup
            return warmUpBeginValue + (baseLr - warmUpBeginValue) * numUpdate / warmUpSteps;
        }
    }

.. code:: java

    CosineWarmupTracker tracker = new CosineWarmupTracker(0.5f, 0.01f, 20, 5);
    
    int[] epochs = new int[numEpochs];
    float[] learningRates = new float[numEpochs];
    for (int i = 0; i < numEpochs; i++) {
        epochs[i] = i;
        learningRates[i] = tracker.getNewLearningRate(i);
    }
    
    plotLearningRate(epochs, learningRates);




.. raw:: html

    <img id="5fad131955e4489197502be2ebafa66d_img"></img>
    <div id="5fad131955e4489197502be2ebafa66d"></div>
    <script>require(['https://cdn.plot.ly/plotly-1.57.0.min.js'], Plotly => {
    var target_5fad131955e4489197502be2ebafa66d = document.getElementById('5fad131955e4489197502be2ebafa66d');
    var layout = {
        title: 'Learning Rate vs. Epoch',
        height: 600,
        width: 800,
        xaxis: {
        title: 'epoch',
        },
    
        yaxis: {
        title: 'learning rate',
        },
    
    };
    
    var trace0 =
    {
    x: ["0","1","2","3","4","5","6","7","8","9"],
    y: ["0.0","0.10000000149011612","0.20000000298023224","0.30000001192092896","0.4000000059604645","0.5","0.4946461617946625","0.4788186252117157","0.45320916175842285","0.41893699765205383"],
    mode: 'lines',
    xaxis: 'x',
    yaxis: 'y',
    type: 'scatter',
    name: '',
    };
    
    
    var data = [ trace0];
    Plotly.newPlot(target_5fad131955e4489197502be2ebafa66d, data, layout);
    })</script>




注意，观察前5个迭代轮数的性能，网络最初收敛得更好。

.. code:: java

    CosineTracker cosineTracker = Tracker.cosine()
            .setBaseValue(0.5f)
            .optFinalValue(0.01f)
            .setMaxUpdates(15)
            .build();
    
    WarmUpTracker warmupCosine = Tracker.warmUp()
            .optWarmUpSteps(5)
            .setMainTracker(cosineTracker)
            .build();
    
    Loss loss = Loss.softmaxCrossEntropyLoss();
    Optimizer sgd = Optimizer.sgd().setLearningRateTracker(warmupCosine).build();
    
    DefaultTrainingConfig config = new DefaultTrainingConfig(loss)
            .optOptimizer(sgd) // Optimizer
            .addEvaluator(new Accuracy()) // Model Accuracy
            .addTrainingListeners(TrainingListener.Defaults.logging()); // Logging
    
    Trainer trainer = model.newTrainer(config);
    trainer.initialize(new Shape(1, 1, 28, 28));
    
    train(trainDataset, testDataset, numEpochs, trainer);
    plotMetrics();


.. parsed-literal::
    :class: output

    INFO Training on: 4 GPUs.
    INFO Load MXNet Engine Version 1.9.0 in 0.020 ms.


.. parsed-literal::
    :class: output

    Training:    100% |████████████████████████████████████████| Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30
    Validating:  100% |████████████████████████████████████████|


.. parsed-literal::
    :class: output

    INFO Epoch 1 finished.
    INFO Train: Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30
    INFO Validate: Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30


.. parsed-literal::
    :class: output

    Training:    100% |████████████████████████████████████████| Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30
    Validating:  100% |████████████████████████████████████████|


.. parsed-literal::
    :class: output

    INFO Epoch 2 finished.
    INFO Train: Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30
    INFO Validate: Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30


.. parsed-literal::
    :class: output

    Training:    100% |████████████████████████████████████████| Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30
    Validating:  100% |████████████████████████████████████████|


.. parsed-literal::
    :class: output

    INFO Epoch 3 finished.
    INFO Train: Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30
    INFO Validate: Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30


.. parsed-literal::
    :class: output

    Training:    100% |████████████████████████████████████████| Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30
    Validating:  100% |████████████████████████████████████████|


.. parsed-literal::
    :class: output

    INFO Epoch 4 finished.
    INFO Train: Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30
    INFO Validate: Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30


.. parsed-literal::
    :class: output

    Training:    100% |████████████████████████████████████████| Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30
    Validating:  100% |████████████████████████████████████████|


.. parsed-literal::
    :class: output

    INFO Epoch 5 finished.
    INFO Train: Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30
    INFO Validate: Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30


.. parsed-literal::
    :class: output

    Training:    100% |████████████████████████████████████████| Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30
    Validating:  100% |████████████████████████████████████████|


.. parsed-literal::
    :class: output

    INFO Epoch 6 finished.
    INFO Train: Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30
    INFO Validate: Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30


.. parsed-literal::
    :class: output

    Training:    100% |████████████████████████████████████████| Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30
    Validating:  100% |████████████████████████████████████████|


.. parsed-literal::
    :class: output

    INFO Epoch 7 finished.
    INFO Train: Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30
    INFO Validate: Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30


.. parsed-literal::
    :class: output

    Training:    100% |████████████████████████████████████████| Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30
    Validating:  100% |████████████████████████████████████████|


.. parsed-literal::
    :class: output

    INFO Epoch 8 finished.
    INFO Train: Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30
    INFO Validate: Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30


.. parsed-literal::
    :class: output

    Training:    100% |████████████████████████████████████████| Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30
    Validating:  100% |████████████████████████████████████████|


.. parsed-literal::
    :class: output

    INFO Epoch 9 finished.
    INFO Train: Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30
    INFO Validate: Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30


.. parsed-literal::
    :class: output

    Training:    100% |████████████████████████████████████████| Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30
    Validating:  100% |████████████████████████████████████████|


.. parsed-literal::
    :class: output

    INFO Epoch 10 finished.
    INFO Train: Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30
    INFO Validate: Accuracy: 0.10, SoftmaxCrossEntropyLoss: 2.30


.. parsed-literal::
    :class: output

    loss 2.303, train acc 0.096, test acc 0.100
    10864.1 examples/sec 



.. raw:: html

    <img id="67f1798be22c414ca249765792c56e3e_img"></img>
    <div id="67f1798be22c414ca249765792c56e3e"></div>
    <script>require(['https://cdn.plot.ly/plotly-1.57.0.min.js'], Plotly => {
    var target_67f1798be22c414ca249765792c56e3e = document.getElementById('67f1798be22c414ca249765792c56e3e');
    var layout = {
        height: 600,
        width: 800,
        showlegend: true,
        xaxis: {
        title: 'epoch',
        },
    
        yaxis: {
        title: 'metrics',
        },
    
    };
    
    var trace0 =
    {
    x: ["1.0","2.0","3.0","4.0","5.0","6.0","7.0","8.0","9.0","10.0"],
    y: ["2.3032575","2.3027117","2.302654","2.3026478","2.3026521","2.3026528","2.302652","2.3026526","2.3026528","2.3026528"],
    showlegend: true,
    mode: 'lines',
    xaxis: 'x',
    yaxis: 'y',
    type: 'scatter',
    name: 'train loss',
    };
    var trace1 =
    {
    x: ["1.0","2.0","3.0","4.0","5.0","6.0","7.0","8.0","9.0","10.0"],
    y: ["0.0997","0.1","0.09865","0.09745","0.09616667","0.097","0.09661666","0.09621666","0.0964","0.096366666"],
    showlegend: true,
    mode: 'lines',
    xaxis: 'x',
    yaxis: 'y',
    type: 'scatter',
    name: 'train acc',
    };
    var trace2 =
    {
    x: ["1.0","2.0","3.0","4.0","5.0","6.0","7.0","8.0","9.0","10.0"],
    y: ["0.1","0.1","0.1","0.1","0.1","0.1","0.1","0.1","0.1","0.1"],
    showlegend: true,
    mode: 'lines',
    xaxis: 'x',
    yaxis: 'y',
    type: 'scatter',
    name: 'test acc',
    };
    
    
    var data = [ trace0, trace1, trace2];
    Plotly.newPlot(target_67f1798be22c414ca249765792c56e3e, data, layout);
    })</script>



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量。
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。这种情况下你可以获得的最佳模型是什么？
2. 如果你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. 你能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。