7.8. Adam算法.ipynb

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        "<a href=\"https://colab.research.google.com/github/chongzicbo/Dive-into-Deep-Learning-tf.keras/blob/master/7.8.%20Adam%E7%AE%97%E6%B3%95.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>"
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        "##7.8. Adam算法\n",
        "Adam算法在RMSProp算法基础上对小批量随机梯度也做了指数加权移动平均 [1]。下面我们来介绍这个算法。\n",
        "\n",
        "###7.8.1. 算法\n",
        "Adam算法使用了动量变量 $v_t$ 和RMSProp算法中小批量随机梯度按元素平方的指数加权移动平均变量 $s_t$ ，并在时间步0将它们中每个元素初始化为0。给定超参数$0 \\leq \\beta_1 < 1$（算法作者建议设为0.9），时间步 $t$ 的动量变量 $v_t$ 即小批量随机梯度 $g_t $的指数加权移动平均：\n",
        "$$\n",
        "\\boldsymbol{v}_t \\leftarrow \\beta_1 \\boldsymbol{v}_{t-1} + (1 - \\beta_1) \\boldsymbol{g}_t.\n",
        "$$\n",
        "和RMSProp算法中一样，给定超参数$0 \\leq \\beta_2 < 1$（算法作者建议设为0.999）， 将小批量随机梯度按元素平方后的项 $\\boldsymbol{g}_t \\odot \\boldsymbol{g}_t$做指数加权移动平均得到 $s_t $：\n",
        "$$\n",
        "\\boldsymbol{s}_t \\leftarrow \\beta_2 \\boldsymbol{s}_{t-1} + (1 - \\beta_2) \\boldsymbol{g}_t \\odot \\boldsymbol{g}_t.\n",
        "$$\n",
        "由于我们将 $v_0$ 和 $s_0$ 中的元素都初始化为0， 在时间步 $t$ 我们得到\n",
        "$\\boldsymbol{v}_t = (1-\\beta_1) \\sum_{i=1}^t \\beta_1^{t-i} \\boldsymbol{g}_i$。将过去各时间步小批量随机梯度的权值相加，得到$(1-\\beta_1) \\sum_{i=1}^t \\beta_1^{t-i} = 1 - \\beta_1^t$。需要注意的是，当 $t$ 较小时，过去各时间步小批量随机梯度权值之和会较小。例如，当$\\beta_1 = 0.9$时，$\\boldsymbol{v}_1 = 0.1\\boldsymbol{g}_1$。为了消除这样的影响，对于任意时间步 $t $，我们可以将$ v_t$ 再除以 $1 - \\beta_1^t$，从而使过去各时间步小批量随机梯度权值之和为1。这也叫作偏差修正。在Adam算法中，我们对变量 $v_t$ 和 $s_t$ 均作偏差修正：\n",
        "$$\n",
        "\\hat{\\boldsymbol{v}}_t \\leftarrow \\frac{\\boldsymbol{v}_t}{1 - \\beta_1^t},\n",
        "$$\n",
        "$$\n",
        "\\hat{\\boldsymbol{s}}_t \\leftarrow \\frac{\\boldsymbol{s}_t}{1 - \\beta_2^t}.\n",
        "$$\n",
        "接下来，Adam算法使用以上偏差修正后的变量$\\hat{\\boldsymbol{v}}_t$和$\\hat{\\boldsymbol{s}}_t$，将模型参数中每个元素的学习率通过按元素运算重新调整：\n",
        "$$\n",
        "\\boldsymbol{g}_t' \\leftarrow \\frac{\\eta \\hat{\\boldsymbol{v}}_t}{\\sqrt{\\hat{\\boldsymbol{s}}_t} + \\epsilon},\n",
        "$$\n",
        "其中 $\\eta$是学习率， $\\epsilon$ 是为了维持数值稳定性而添加的常数，如 $10^{-8}$ 。和AdaGrad算法、RMSProp算法以及AdaDelta算法一样，目标函数自变量中每个元素都分别拥有自己的学习率。最后，使用 $\\boldsymbol{g}_t'$ 迭代自变量：\n",
        "$$\n",
        "\\boldsymbol{x}_t \\leftarrow \\boldsymbol{x}_{t-1} - \\boldsymbol{g}_t'.\n",
        "$$\n",
        "\n",
        "\n",
        "###7.8.2. 从零开始实现\n",
        "我们按照Adam算法中的公式实现该算法。其中时间步 t 通过hyperparams参数传入adam函数。"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "CAGyt6oSSDhO",
        "colab_type": "code",
        "outputId": "75124706-782b-4d93-a87a-816370e199a2",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 64
        }
      },
      "source": [
        "%matplotlib inline\n",
        "import math\n",
        "import tensorflow as tf\n",
        "import numpy as np\n",
        "from IPython import display\n",
        "import matplotlib.pyplot as plt\n",
        "from tensorflow import keras\n",
        "from tensorflow.keras import losses\n",
        "from tensorflow.data import Dataset\n",
        "import time,"
      ],
      "execution_count": 0,
      "outputs": [
        {
          "output_type": "display_data",
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              "<p style=\"color: red;\">\n",
              "The default version of TensorFlow in Colab will soon switch to TensorFlow 2.x.<br>\n",
              "We recommend you <a href=\"https://www.tensorflow.org/guide/migrate\" target=\"_blank\">upgrade</a> now \n",
              "or ensure your notebook will continue to use TensorFlow 1.x via the <code>%tensorflow_version 1.x</code> magic:\n",
              "<a href=\"https://colab.research.google.com/notebooks/tensorflow_version.ipynb\" target=\"_blank\">more info</a>.</p>\n"
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              "<IPython.core.display.HTML object>"
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    {
      "cell_type": "code",
      "metadata": {
        "id": "47h7vdmuSLSS",
        "colab_type": "code",
        "colab": {}
      },
      "source": [
        "tf.enable_eager_execution()"
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "tTCEk5bYSOd5",
        "colab_type": "code",
        "outputId": "cd6e1aba-eb4a-4363-c7b3-aeb4e4194757",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 127
        }
      },
      "source": [
        "from google.colab import drive\n",
        "drive.mount('/content/drive')\n",
        "def get_data_ch7():\n",
        "  data=np.genfromtxt('/content/drive/My Drive/data/d2l-zh-tensoflow/airfoil_self_noise.dat',delimiter='\\t')\n",
        "  data=(data-data.mean(axis=0))/data.std(axis=0)\n",
        "  return tf.constant(data[:1500,:-1]),tf.constant(data[:1500,-1])\n",
        "features,labels=get_data_ch7() "
      ],
      "execution_count": 0,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "Go to this URL in a browser: https://accounts.google.com/o/oauth2/auth?client_id=947318989803-6bn6qk8qdgf4n4g3pfee6491hc0brc4i.apps.googleusercontent.com&redirect_uri=urn%3aietf%3awg%3aoauth%3a2.0%3aoob&response_type=code&scope=email%20https%3a%2f%2fwww.googleapis.com%2fauth%2fdocs.test%20https%3a%2f%2fwww.googleapis.com%2fauth%2fdrive%20https%3a%2f%2fwww.googleapis.com%2fauth%2fdrive.photos.readonly%20https%3a%2f%2fwww.googleapis.com%2fauth%2fpeopleapi.readonly\n",
            "\n",
            "Enter your authorization code:\n",
            "··········\n",
            "Mounted at /content/drive\n"
          ],
          "name": "stdout"
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "sqpsRV-7SSQu",
        "colab_type": "code",
        "colab": {}
      },
      "source": [
        "def init_adam_states():\n",
        "  v_w,v_b=tf.Variable(tf.zeros(shape=(features.shape[1],1),dtype=tf.double)),tf.Variable(tf.zeros(1,tf.double))\n",
        "  s_w,s_b=tf.Variable(tf.zeros(shape=(features.shape[1],1),dtype=tf.double)),tf.Variable(tf.zeros(1,tf.double))\n",
        "  return ((v_w,s_w),(v_b,s_b))"
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "6nNttgGBS_hx",
        "colab_type": "code",
        "colab": {}
      },
      "source": [
        "def adadelta(params,states,hyperparams,loss,t):\n",
        "  rho,eps=hyperparams['rho'],1e-5\n",
        "  for p,(s,delta) in zip(params,states):\n",
        "    dl_dp=t.gradient(loss,p)\n",
        "    s.assign(rho*s+(1-rho)*tf.square(dl_dp))\n",
        "    g=(tf.sqrt((delta+eps))/tf.sqrt((s+eps)))*dl_dp\n",
        "    p.assign_sub(g)\n",
        "    delta.assign(rho*delta+(1-rho)*g*g)\n",
        "def adam(params,states,hyperparams,loss,t):\n",
        "  beta1,beta2,eps=0.9,0.999,1e-6\n",
        "  for p,(v,s) in zip(params,states):\n",
        "    dl_dp=t.gradient(loss,p)\n",
        "    v.assign(beta1*v+(1-beta1)*dl_dp)\n",
        "    s.assign(beta2*s+(1-beta2)*tf.square(dl_dp))\n",
        "    v_bias_corr=v/(1-beta1**hyperparams['t'])\n",
        "    s_bias_corr=s/(1-beta2**hyperparams['t'])\n",
        "    p.assign_sub(hyperparams['lr']*v_bias_corr/(tf.sqrt(s_bias_corr)+eps))\n",
        "  hyperparams['t']+=1"
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "MN5xMXFLUUOW",
        "colab_type": "code",
        "colab": {}
      },
      "source": [
        "def train_ch7(trainer_fn,states,hyperparams,features,labels,batch_size=10,num_epochs=2):\n",
        "  net,loss=linreg,squared_loss\n",
        "  w=tf.Variable(tf.random.normal(stddev=0.01,shape=(features.shape[1],1),dtype=tf.double))\n",
        "  b=tf.Variable(tf.zeros(1,dtype=tf.double))\n",
        "\n",
        "  def eval_loss():\n",
        "    return tf.reduce_mean(loss(net(features,w,b),labels)).numpy()\n",
        "\n",
        "  ls=[eval_loss()]\n",
        "\n",
        "  data_iter=Dataset.from_tensor_slices((features,labels)).shuffle(buffer_size=1000).batch(batch_size)\n",
        "\n",
        "  for _ in range(num_epochs):\n",
        "    start=time.time()\n",
        "    for batch_i,(X,y) in enumerate(data_iter):\n",
        "      with tf.GradientTape(persistent=True) as t:\n",
        "        t.watch([w,b])\n",
        "        l=tf.reduce_mean(loss(net(X,w,b),y)) #使用平均损失\n",
        "\n",
        "      trainer_fn([w,b],states,hyperparams,l,t) #迭代模型参数\n",
        "\n",
        "      if(batch_i+1)*batch_size %100==0:\n",
        "        ls.append(eval_loss()) #每100个样本记录下当前训练误差\n",
        "  # 打印结果和作图\n",
        "  print('loss: %f, %f sec per epoch' % (ls[-1], time.time() - start))\n",
        "  set_figsize()\n",
        "  plt.plot(np.linspace(0, num_epochs, len(ls)), ls)\n",
        "  plt.xlabel('epoch')\n",
        "  plt.ylabel('loss')  \n",
        "\n",
        "\n",
        "def use_svg_display():\n",
        "  #用矢量图显示\n",
        "  display.set_matplotlib_formats('svg')\n",
        "def set_figsize(figsize=(3.5,2.5)):\n",
        "  use_svg_display()\n",
        "  #设置图的尺寸\n",
        "  plt.rcParams['figure.figsize']=figsize\n",
        "  \n",
        "  \n",
        "def linreg(X,w,b):\n",
        "  return tf.matmul(X,w)+b \n",
        "  \n",
        "def squared_loss(y_hat,y):\n",
        "  return (y_hat-tf.reshape(y,shape=y_hat.shape))**2/2    "
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "VF7AtvakrboB",
        "colab_type": "text"
      },
      "source": [
        "使用学习率为0.01的Adam算法来训练模型。"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "Xwx_rojjUWmT",
        "colab_type": "code",
        "outputId": "bcf772b3-409a-4965-cf92-c7635fbbd997",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 280
        }
      },
      "source": [
        "train_ch7(adam,init_adam_states(),{'lr':0.01,'t':1},features,labels)"
      ],
      "execution_count": 0,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "loss: 0.246332, 0.595903 sec per epoch\n"
          ],
          "name": "stdout"
        },
        {
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          },
          "metadata": {
            "tags": []
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "Kjx-dDDWregj",
        "colab_type": "text"
      },
      "source": [
        "###7.8.3. 简洁实现"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "9jHJdSflUflz",
        "colab_type": "code",
        "colab": {}
      },
      "source": [
        "def train_keras_ch7(trainer_name,features,labels,batch_size=10,num_epochs=2):\n",
        "  net=keras.Sequential(keras.layers.Dense(1))\n",
        "  loss=losses.MeanSquaredError()\n",
        "  data_iter=Dataset.from_tensor_slices((features,labels)).shuffle(buffer_size=1000).batch(batch_size)\n",
        "  net.compile(optimizer=trainer_name,loss=loss,metrics=['mse'])\n",
        "  history=net.fit_generator(data_iter,epochs=num_epochs)\n",
        "  loss_ls=history.history['loss']\n",
        "  set_figsize()\n",
        "  plt.plot(np.linspace(0,num_epochs,len(loss_ls)),loss_ls)\n",
        "  plt.xlabel('epoch')\n",
        "  plt.ylabel('loss')\n",
        "  plt.show()\n",
        "  return history"
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "a_JvwVzfUqYy",
        "colab_type": "code",
        "outputId": "d01fc13c-63f2-4d30-cde3-84af03afd18a",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 1000
        }
      },
      "source": [
        "train_keras_ch7(keras.optimizers.Adam(learning_rate=0.001),features,labels,batch_size=10,num_epochs=40)"
      ],
      "execution_count": 0,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "Epoch 1/40\n",
            "150/150 [==============================] - 1s 7ms/step - loss: 2.2897 - mean_squared_error: 2.3970\n",
            "Epoch 2/40\n",
            "150/150 [==============================] - 1s 7ms/step - loss: 1.8866 - mean_squared_error: 1.9458\n",
            "Epoch 3/40\n",
            "150/150 [==============================] - 1s 7ms/step - loss: 1.6162 - mean_squared_error: 1.6487\n",
            "Epoch 4/40\n",
            "150/150 [==============================] - 1s 7ms/step - loss: 1.4144 - mean_squared_error: 1.4315\n",
            "Epoch 5/40\n",
            "150/150 [==============================] - 1s 7ms/step - loss: 1.2512 - mean_squared_error: 1.2586\n",
            "Epoch 6/40\n",
            "150/150 [==============================] - 1s 7ms/step - loss: 1.1144 - mean_squared_error: 1.1146\n",
            "Epoch 7/40\n",
            "150/150 [==============================] - 1s 7ms/step - loss: 0.9986 - mean_squared_error: 0.9929\n",
            "Epoch 8/40\n",
            "150/150 [==============================] - 1s 7ms/step - loss: 0.9009 - mean_squared_error: 0.8901\n",
            "Epoch 9/40\n",
            "150/150 [==============================] - 1s 7ms/step - loss: 0.8190 - mean_squared_error: 0.8039\n",
            "Epoch 10/40\n",
            "150/150 [==============================] - 1s 7ms/step - loss: 0.7511 - mean_squared_error: 0.7322\n",
            "Epoch 11/40\n",
            "150/150 [==============================] - 1s 7ms/step - loss: 0.6954 - mean_squared_error: 0.6732\n",
            "Epoch 12/40\n",
            "150/150 [==============================] - 1s 7ms/step - loss: 0.6501 - mean_squared_error: 0.6254\n",
            "Epoch 13/40\n",
            "150/150 [==============================] - 1s 7ms/step - loss: 0.6138 - mean_squared_error: 0.5871\n",
            "Epoch 14/40\n",
            "150/150 [==============================] - 1s 7ms/step - loss: 0.5851 - mean_squared_error: 0.5569\n",
            "Epoch 15/40\n",
            "150/150 [==============================] - 1s 7ms/step - loss: 0.5625 - mean_squared_error: 0.5333\n",
            "Epoch 16/40\n",
            "150/150 [==============================] - 1s 7ms/step - loss: 0.5449 - mean_squared_error: 0.5151\n",
            "Epoch 17/40\n",
            "150/150 [==============================] - 1s 7ms/step - loss: 0.5314 - mean_squared_error: 0.5013\n",
            "Epoch 18/40\n",
            "150/150 [==============================] - 1s 7ms/step - loss: 0.5209 - mean_squared_error: 0.4909\n",
            "Epoch 19/40\n",
            "150/150 [==============================] - 1s 7ms/step - loss: 0.5129 - mean_squared_error: 0.4832\n",
            "Epoch 20/40\n",
            "150/150 [==============================] - 1s 7ms/step - loss: 0.5067 - mean_squared_error: 0.4775\n",
            "Epoch 21/40\n",
            "150/150 [==============================] - 1s 7ms/step - loss: 0.5019 - mean_squared_error: 0.4732\n",
            "Epoch 22/40\n",
            "150/150 [==============================] - 1s 7ms/step - loss: 0.4982 - mean_squared_error: 0.4701\n",
            "Epoch 23/40\n",
            "150/150 [==============================] - 1s 7ms/step - loss: 0.4953 - mean_squared_error: 0.4679\n",
            "Epoch 24/40\n",
            "150/150 [==============================] - 1s 7ms/step - loss: 0.4930 - mean_squared_error: 0.4662\n",
            "Epoch 25/40\n",
            "150/150 [==============================] - 1s 7ms/step - loss: 0.4912 - mean_squared_error: 0.4651\n",
            "Epoch 26/40\n",
            "150/150 [==============================] - 1s 7ms/step - loss: 0.4898 - mean_squared_error: 0.4642\n",
            "Epoch 27/40\n",
            "150/150 [==============================] - 1s 7ms/step - loss: 0.4886 - mean_squared_error: 0.4637\n",
            "Epoch 28/40\n",
            "150/150 [==============================] - 1s 7ms/step - loss: 0.4877 - mean_squared_error: 0.4633\n",
            "Epoch 29/40\n",
            "150/150 [==============================] - 1s 7ms/step - loss: 0.4870 - mean_squared_error: 0.4630\n",
            "Epoch 30/40\n",
            "150/150 [==============================] - 1s 7ms/step - loss: 0.4865 - mean_squared_error: 0.4629\n",
            "Epoch 31/40\n",
            "150/150 [==============================] - 1s 7ms/step - loss: 0.4860 - mean_squared_error: 0.4629\n",
            "Epoch 32/40\n",
            "150/150 [==============================] - 1s 7ms/step - loss: 0.4857 - mean_squared_error: 0.4629\n",
            "Epoch 33/40\n",
            "150/150 [==============================] - 1s 7ms/step - loss: 0.4854 - mean_squared_error: 0.4629\n",
            "Epoch 34/40\n",
            "150/150 [==============================] - 1s 7ms/step - loss: 0.4852 - mean_squared_error: 0.4630\n",
            "Epoch 35/40\n",
            "150/150 [==============================] - 1s 7ms/step - loss: 0.4850 - mean_squared_error: 0.4630\n",
            "Epoch 36/40\n",
            "150/150 [==============================] - 1s 7ms/step - loss: 0.4849 - mean_squared_error: 0.4631\n",
            "Epoch 37/40\n",
            "150/150 [==============================] - 1s 7ms/step - loss: 0.4848 - mean_squared_error: 0.4632\n",
            "Epoch 38/40\n",
            "150/150 [==============================] - 1s 7ms/step - loss: 0.4847 - mean_squared_error: 0.4633\n",
            "Epoch 39/40\n",
            "150/150 [==============================] - 1s 7ms/step - loss: 0.4846 - mean_squared_error: 0.4634\n",
            "Epoch 40/40\n",
            "150/150 [==============================] - 1s 7ms/step - loss: 0.4846 - mean_squared_error: 0.4635\n"
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          "name": "stdout"
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            "tags": []
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        },
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "<tensorflow.python.keras.callbacks.History at 0x7f94ed2e99e8>"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 16
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    },
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      "cell_type": "markdown",
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      "source": [
        "###7.8.4. 小结\n",
        "* Adam算法在RMSProp算法的基础上对小批量随机梯度也做了指数加权移动平均。\n",
        "* Adam算法使用了偏差修正。"
      ]
    }
  ]
}