Synthetic_case.ipynb

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    "colab": {
      "name": "Synthetic.ipynb",
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      "collapsed_sections": []
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    "kernelspec": {
      "name": "python3",
      "display_name": "Python 3"
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    "accelerator": "GPU"
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  "cells": [
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "4pAe-dGkpYkf",
        "colab_type": "text"
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      "source": [
        "# **Synthetic case: A convex setting**\n",
        "$$f_t(x) =  \\begin{cases} \n",
        "      1010x & \\text{for } t \\text{ mod 101} = 1\\\\\n",
        "      -10x & \\text{otherwise}\n",
        "   \\end{cases} $$\n",
        "\n",
        "With $\\mathcal{F}$ = [−1; 1]. \\\\\n",
        "The optimal solution is x = −1, providing the minimum regret. Thus, for convergence, we expect the algorithms to converge to x = −1. \n",
        "\n",
        "However, as we see in the paper, the ADAM method fails to converge to −1 and instead converges to the suboptimal solution of x = 1. This behavior is due to the exponential moving averages of squared past gradients that causes the rare but informative gradients to die out quickly. The gradient that is useful for optimal convergence will rarely appear (with probability of 0.01) and hence the ADAM method will not be able to effectively use\n",
        "this information. AMSGRAD does not suffer from the same problem, hence\n",
        "reaching the optimal solution.\n",
        "\n",
        "**Algorithm 1: ADAM**\n",
        "\n",
        "![alt text](https://i.imgur.com/vQpTtcE.png)\n",
        "![alt text](https://i.imgur.com/mmi1Fse.png)\n",
        "\n",
        "(According to proof of theorem 1 on the paper) Without loss of generality, assume that the initial point is $x_1$. This can\n",
        "be assumed without any loss of generality because for any choice of initial point, we can always\n",
        "translate the coordinate system such that the initial point is $x_1 $ in the new coordinate system\n",
        "and then choose the sequence of functions as above in the new coordinate system. Hence, we set $\\epsilon = 0$.\n",
        "\n",
        "Hyper-parameter: $\\beta_1 = 0.9, \\beta_2 = 0.99, \\alpha = 0.001 $"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "mViPGKALpj0x",
        "colab_type": "code",
        "colab": {}
      },
      "source": [
        "%tensorflow_version 1.x \n",
        "import matplotlib.pyplot as plt\n",
        "import tensorflow as tf"
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "dj6T09AepnwJ",
        "colab_type": "code",
        "colab": {}
      },
      "source": [
        "class AdamOptimizer(tf.train.Optimizer):\n",
        "  def __init__(self, learning_rate=0.001, decay=False, beta1=0.9, beta2=0.99,\n",
        "               epsilon=0.0, var_list=[]):\n",
        "    self.learning_rate = learning_rate\n",
        "    self.decay = decay\n",
        "    self.beta1 = beta1\n",
        "    self.beta2 = beta2\n",
        "    self.epsilon = epsilon\n",
        "    \n",
        "    self.var_list = var_list\n",
        "    self.m = {}\n",
        "    self.v = {}\n",
        "    self.beta1_t = tf.Variable(1.0, trainable=False)\n",
        "    self.beta2_t = tf.Variable(1.0, trainable=False)\n",
        "    self.t = tf.Variable(0.0, trainable=False)\n",
        "    \n",
        "    for var in self.var_list:\n",
        "      self.m[var] = tf.Variable(tf.zeros(tf.shape(var.initial_value)), trainable=False)\n",
        "      self.v[var] = tf.Variable(tf.zeros(tf.shape(var.initial_value)), trainable=False)\n",
        "    \n",
        "  def apply_gradients(self, gradient_variables):\n",
        "    beta1_t = self.beta1_t.assign(self.beta1_t * self.beta1)\n",
        "    beta2_t = self.beta2_t.assign(self.beta2_t * self.beta2)\n",
        "    \n",
        "    with tf.control_dependencies([self.t.assign_add(1.0), beta1_t, beta2_t]):\n",
        "      learning_rate = self.learning_rate\n",
        "      # if self.decay:\n",
        "      #   learning_rate /= tf.sqrt(self.t)\n",
        "      update_ops = []\n",
        "\n",
        "      for (g, var) in gradient_variables:\n",
        "        m = self.m[var].assign(self.beta1 * self.m[var] + (1 - self.beta1) * g)\n",
        "        v = self.v[var].assign(self.beta2 * self.v[var] + (1 - self.beta2) * g * g)\n",
        "        m_hat = m / (1 - beta1_t)\n",
        "        v_hat = v / (1 - beta2_t)\n",
        "\n",
        "        update = -learning_rate * m_hat / (self.epsilon + tf.sqrt(v_hat))\n",
        "        update_ops.append(var.assign_add(update))\n",
        "\n",
        "      return tf.group(*update_ops)"
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "e-3NfXZ3ppcY",
        "colab_type": "code",
        "colab": {}
      },
      "source": [
        "class AMSGradOptimizer(tf.train.Optimizer):\n",
        "  def __init__(self, learning_rate=0.001, decay=False, beta1=0.9, beta2=0.99,\n",
        "               epsilon=0.0, var_list=[]):\n",
        "    self.learning_rate = learning_rate\n",
        "    self.decay = decay\n",
        "    self.beta1 = beta1\n",
        "    self.beta2 = beta2\n",
        "    self.epsilon = epsilon\n",
        "    \n",
        "    self.var_list = var_list\n",
        "    self.m = {}\n",
        "    self.v = {}\n",
        "    self.v_hat = {}\n",
        "    self.t = tf.Variable(0.0, trainable=False)\n",
        "    \n",
        "    for var in self.var_list:\n",
        "      self.m[var] = tf.Variable(tf.zeros(tf.shape(var.initial_value)), trainable=False)\n",
        "      self.v[var] = tf.Variable(tf.zeros(tf.shape(var.initial_value)), trainable=False)\n",
        "      self.v_hat[var] = tf.Variable(tf.zeros(tf.shape(var.initial_value)), trainable=False)\n",
        "    \n",
        "  def apply_gradients(self, gradient_variables):\n",
        "    with tf.control_dependencies([self.t.assign_add(1.0)]):\n",
        "      learning_rate = self.learning_rate\n",
        "      # if self.decay:\n",
        "      #   learning_rate /= tf.sqrt(self.t)\n",
        "      update_ops = []\n",
        "\n",
        "      for (g, var) in gradient_variables:\n",
        "        m = self.m[var].assign(self.beta1 * self.m[var] + (1 - self.beta1) * g)\n",
        "        v = self.v[var].assign(self.beta2 * self.v[var] + (1 - self.beta2) * g * g)\n",
        "        v_hat = self.v_hat[var].assign(tf.maximum(self.v_hat[var], v))\n",
        "\n",
        "        update = -learning_rate * m / (self.epsilon + tf.sqrt(v_hat))\n",
        "        update_ops.append(var.assign_add(update))\n",
        "\n",
        "      return tf.group(*update_ops)"
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "4hyo9Wq1prwA",
        "colab_type": "code",
        "colab": {}
      },
      "source": [
        "def test_optimizer(optimizer_name, iterations=1000000, learning_rate=0.001,\n",
        "                   decay=False):\n",
        "  tf.reset_default_graph()\n",
        "  \n",
        "  with tf.Session() as sess:\n",
        "    r = tf.distributions.Bernoulli(probs=0.01, dtype=tf.float32).sample()\n",
        "    x = tf.Variable(0.0)\n",
        "    loss = (r * 1010 - (1 - r) * 10) * x\n",
        "    var_list = [x]\n",
        "    gradient = tf.gradients(loss, var_list)\n",
        "\n",
        "    if optimizer_name == \"adam\":\n",
        "      optimizer = AdamOptimizer(var_list=var_list, learning_rate=learning_rate,\n",
        "                                decay=decay)\n",
        "    elif optimizer_name == \"amsgrad\":\n",
        "      optimizer = AMSGradOptimizer(var_list=var_list, learning_rate=learning_rate,\n",
        "                                   decay=decay)\n",
        "    else:\n",
        "      raise Exception(\"Unknown optimizer\")\n",
        "\n",
        "    update_op = optimizer.apply_gradients([(grad, var) for grad, var in zip(gradient, var_list)])\n",
        "    with tf.control_dependencies([update_op]):\n",
        "      clip_op = x.assign(tf.clip_by_value(x, -1.0, 1.0))\n",
        "\n",
        "    sess.run(tf.global_variables_initializer())\n",
        "\n",
        "    results = []\n",
        "\n",
        "    for i in range(iterations):\n",
        "      _, cur_x = sess.run([clip_op, x])\n",
        "      results.append(cur_x)\n",
        "      \n",
        "      if i % 50000 == 0:\n",
        "        print(cur_x)\n",
        "\n",
        "    return results"
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "sgKPIUTxptXu",
        "colab_type": "code",
        "outputId": "ad7760b0-9d4d-4ff6-ba06-cf8bb4794886",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 479
        }
      },
      "source": [
        "results_adam = test_optimizer(\"adam\", iterations=1000000, learning_rate=0.001)"
      ],
      "execution_count": 0,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "WARNING:tensorflow:From <ipython-input-4-b0d56b5ce2e9>:6: Bernoulli.__init__ (from tensorflow.python.ops.distributions.bernoulli) is deprecated and will be removed after 2019-01-01.\n",
            "Instructions for updating:\n",
            "The TensorFlow Distributions library has moved to TensorFlow Probability (https://github.com/tensorflow/probability). You should update all references to use `tfp.distributions` instead of `tf.distributions`.\n",
            "WARNING:tensorflow:From /tensorflow-1.15.0/python3.6/tensorflow_core/python/ops/distributions/bernoulli.py:97: Distribution.__init__ (from tensorflow.python.ops.distributions.distribution) is deprecated and will be removed after 2019-01-01.\n",
            "Instructions for updating:\n",
            "The TensorFlow Distributions library has moved to TensorFlow Probability (https://github.com/tensorflow/probability). You should update all references to use `tfp.distributions` instead of `tf.distributions`.\n",
            "0.0\n",
            "1.0\n",
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            "0.97701055\n",
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            "1.0\n",
            "1.0\n",
            "1.0\n",
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            "0.95851386\n",
            "0.98906547\n",
            "0.9775075\n",
            "1.0\n"
          ],
          "name": "stdout"
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "KJrjvLNMpwgT",
        "colab_type": "code",
        "outputId": "66f0afd3-85c4-4a2c-caff-7f602d665af1",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 697
        }
      },
      "source": [
        "results_amsgrad = test_optimizer(\"amsgrad\", iterations=2000000, learning_rate=0.001)"
      ],
      "execution_count": 0,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "0.0\n",
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          "name": "stdout"
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      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "_picbU6ipx_E",
        "colab_type": "code",
        "outputId": "e7d62ee7-82e8-4c44-b04d-2f65fd31623b",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 295
        }
      },
      "source": [
        "plt.plot(results_adam, label=\"Adam\", color=(0,0,1))\n",
        "plt.plot(results_amsgrad, label=\"AMSGRAD\",color=(1,0,0))\n",
        "plt.legend(bbox_to_anchor=(0.8, 0.9), loc=2, borderaxespad=0.)\n",
        "\n",
        "\n",
        "plt.title('Model Loss for the defined function')\n",
        "plt.ylabel('Loss')\n",
        "plt.xlabel('Epochs')\n",
        "plt.legend()\n",
        "plt.show()"
      ],
      "execution_count": 0,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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0R/IsxlRC77wDe2Th32PZsoppe++d+Xah4pDD99zjrpJGjnStvsAV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\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": []
          }
        }
      ]
    }
  ]
}