step_size_policies.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Step size policies"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In this notebook I, Philip Blagoveschensky, tried out different step size policies of batch gradient descent. A lot of code is copied from https://github.com/stevenliuyi/information-bottleneck."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Data generation"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "First, we will generate a very simple dataset for the demonstration. The inputs are vectors of 10 binaries, and the outputs are just single binaries. The inputs could be represented by integers from 0 to 1023 ($=2^{10}-1$). The 1024 possible inputs are divided into 16 groups (each group has 64 numbers), and each integer input $n\\in[0,1023]$ belongs to group $i$ if $x\\equiv i \\pmod{16}$, where $i \\in [0,15]$. Each group $i$ is then associated with a random binary number (output)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "from random import randint, seed"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "n_train_samples = 50000 # number of train samples\n",
    "n_test_samples = 10000 # number of test samples\n",
    "batch_size = 100\n",
    "\n",
    "groups = np.append(np.zeros(8),np.ones(8)) # 16 groups\n",
    "np.random.seed(1234)\n",
    "np.random.shuffle(groups)\n",
    "p_noise_y = 0.1\n",
    "p_noise_x = 0.0\n",
    "\n",
    "def bin_noise(x, p):\n",
    "    return x if np.random.rand() > 2 * p else np.random.randint(0, 1)\n",
    "\n",
    "# generate samples\n",
    "seed(1234)\n",
    "def generate_samples(n_samples):\n",
    "    x_data = np.zeros((n_samples, 10)) # inputs\n",
    "    x_int = np.zeros(n_samples) # integers representing the inputs\n",
    "    y_data = np.zeros((n_samples, 2)) # outputs\n",
    "    \n",
    "    for i in range(n_samples):\n",
    "        random_int = randint(0, 1023)\n",
    "        x_data[i,:] = [bin_noise(int(b), p_noise_x) for b in list(\"{0:b}\".format(random_int).zfill(10))]\n",
    "        x_int[i] = np.sum([x_data[i, k] * (2 ** (9 - k)) for k in range(10)])\n",
    "        y_data[i,0] = bin_noise(groups[random_int % 16], p_noise_y)\n",
    "        y_data[i,1] = 1 - y_data[i,0]\n",
    "        \n",
    "    return x_data, y_data, x_int\n",
    "\n",
    "x_train, y_train, x_train_int = generate_samples(n_train_samples) # training dataset\n",
    "x_test, y_test, _ = generate_samples(n_test_samples) # testing dataset"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For our dataset, the theoritical mutual information between $X$ and $Y$ would be\n",
    "\\begin{align}\n",
    "I(X;Y) & = \\sum_{x\\in X, y\\in Y}P(x,y)\\log\\Big(\\frac{P(x,y)}{P(x)P(y)}\\Big) \\\\\n",
    "& = \\sum_{x\\in X}\\Big[P(x,y=0)\\log\\Big(\\frac{P(x,y=0)}{P(x)P(y=0)}\\Big) + P(x,y=1)\\log\\Big(\\frac{P(x,y=1)}{P(x)P(y=1)}\\Big)\\Big] \\\\\n",
    "& = 1024 \\Big[ \\frac{1}{1024}\\log\\Big(\\frac{1/1024}{0.5/1024}\\Big) + 0\\Big] \\\\\n",
    "& = 0.693.\n",
    "\\end{align}\n",
    "Note that terms with $P(x,y)=0$ are set to $0$ for entropy calculation."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Neural network"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "import tensorflow as tf\n",
    "from tensorflow.keras.optimizers import SGD\n",
    "\n",
    "def fully_conn(tensor, n_outputs, name):\n",
    "    size = tensor.get_shape().as_list()[1]\n",
    "    \n",
    "    # weight and bias tensor\n",
    "    weight = tf.Variable(tf.truncated_normal([size, n_outputs]))\n",
    "    bias = tf.Variable(tf.zeros([n_outputs]))\n",
    "    \n",
    "    # fully-connected layer\n",
    "    tensor = tf.matmul(tensor, weight)\n",
    "    bias = tf.nn.bias_add(tensor, bias)\n",
    "    \n",
    "    # hyperbolic tangent activation\n",
    "    tensor = tf.tanh(tensor, name=name)\n",
    "    \n",
    "    return tensor"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "def output(tensor, n_outputs):\n",
    "    size = tensor.get_shape().as_list()[1]\n",
    "    \n",
    "    # weight and bias tensor\n",
    "    weight = tf.Variable(tf.truncated_normal([size, n_outputs]))\n",
    "    bias = tf.Variable(tf.zeros([n_outputs]))\n",
    "    \n",
    "    # fully-connected layer\n",
    "    tensor = tf.matmul(tensor, weight)\n",
    "    bias = tf.nn.bias_add(tensor, bias)\n",
    "    \n",
    "    return tensor"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "def mlp(x, n_neurons): # x is the input layer  \n",
    "    # mlp stands for multilayer perceptron\n",
    "    # hidden layers\n",
    "    hidden = x\n",
    "    for n in range(len(n_neurons)):\n",
    "        hidden = fully_conn(hidden, n_neurons[n], 'hidden%s' % (n+1))\n",
    "    \n",
    "    # output layer\n",
    "    y = output(hidden, 2)\n",
    "    \n",
    "    return y"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "def build_network(hidden_layer_neurons):\n",
    "    global x, y, learning_rate, logits, cost, optimizer, accuracy\n",
    "    \n",
    "    tf.reset_default_graph()\n",
    "    x = tf.placeholder(tf.float32, (None, 10), name='x')\n",
    "    y = tf.placeholder(tf.float32, (None, 2), name='y')\n",
    "    learning_rate = tf.placeholder(tf.float32, shape=[], name=\"lr\")\n",
    "\n",
    "    tf.set_random_seed(12345)\n",
    "    logits = mlp(x, hidden_layer_neurons)\n",
    "    logits = tf.identity(logits, name='logits')\n",
    "\n",
    "    # cross-entropy coss function\n",
    "    cost = tf.reduce_mean(tf.nn.sigmoid_cross_entropy_with_logits(logits=logits, labels=y), name='cost')\n",
    "    \n",
    "    # optimizer\n",
    "    optimizer = tf.train.GradientDescentOptimizer(learning_rate).minimize(cost)\n",
    "    \n",
    "    # accuracy\n",
    "    correct_predictions = tf.equal(tf.argmax(logits, 1), tf.argmax(y, 1))\n",
    "    accuracy = tf.reduce_mean(tf.cast(correct_predictions, tf.float32), name='accuracy')\n",
    "    \n",
    "def print_out_summary(sess, epoch, lr=None):\n",
    "    acc, loss = sess.run([accuracy, cost], feed_dict={x: x_test, y: y_test})\n",
    "    print(\n",
    "        'Epoch {:>4}:  Testing accuracy {:>.4f} - Testing loss {:>.4f} - lr {:>.4f}'\n",
    "        .format(epoch, acc, loss, lr)\n",
    "    )"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Mutual information"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we are ready to explore the information bottleneck theory for our network. To estimate the mutual information between all the hidden layers and intput/output layers, we could binned the output activations as stated in the paper  (here we choose 30 bins, the same as in the paper), so that the hidden layer random variables $T_i$ (each $i$ corresponds to one hidden layer) would be discrete. Then, we will be able to estimate the joint distribution $P(X,T_i)$ and $P(T_i,Y)$, and use them to calculate the encoder mutual information (between input $X$ and hidden layer $T_i$)\n",
    "\\begin{equation}\n",
    "    I(X;T_i) = \\sum_{x\\in X, t\\in T_i}P(x,t)\\log\\Big(\\frac{P(x,t)}{P(x)P(t)}\\Big)\n",
    "\\end{equation}\n",
    "\n",
    "and decoder mutual information (between hidden layer $T_i$ and desired output $Y$, note that it is not the model output $\\widehat{Y}$)\n",
    "\\begin{equation}\n",
    "    I(T_i;Y) = \\sum_{t\\in T_i, y\\in Y}P(t,y)\\log\\Big(\\frac{P(t,y)}{P(t)P(y)}\\Big).\n",
    "\\end{equation}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "from collections import Counter\n",
    "\n",
    "def calc_mutual_information(hidden):\n",
    "    #print(\"hidden.shape: {}\".format(hidden.shape))\n",
    "    n_neurons = hidden.shape[1]\n",
    "  \n",
    "    # discretization \n",
    "    n_bins = 30\n",
    "    bins = np.linspace(-1, 1, n_bins+1)\n",
    "    indices = np.digitize(hidden, bins)\n",
    "    \n",
    "    # initialize pdfs\n",
    "    pdf_x = Counter(); pdf_y = Counter(); pdf_t = Counter(); pdf_xt = Counter(); pdf_yt = Counter()\n",
    "\n",
    "    for i in range(n_train_samples):\n",
    "        pdf_x[x_train_int[i]] += 1/float(n_train_samples)\n",
    "        pdf_y[y_train[i,0]] += 1/float(n_train_samples)      \n",
    "        pdf_xt[(x_train_int[i],)+tuple(indices[i,:])] += 1/float(n_train_samples)\n",
    "        pdf_yt[(y_train[i,0],)+tuple(indices[i,:])] += 1/float(n_train_samples)\n",
    "        pdf_t[tuple(indices[i,:])] += 1/float(n_train_samples)\n",
    "    \n",
    "    # calcuate encoder mutual information I(X;T)\n",
    "    mi_xt = 0\n",
    "    for i in pdf_xt:\n",
    "        # P(x,t), P(x) and P(t)\n",
    "        p_xt = pdf_xt[i]; p_x = pdf_x[i[0]]; p_t = pdf_t[i[1:]]\n",
    "        # I(X;T)\n",
    "        mi_xt += p_xt * np.log(p_xt / p_x / p_t)\n",
    " \n",
    "    # calculate decoder mutual information I(T;Y)\n",
    "    mi_ty = 0\n",
    "    for i in pdf_yt:\n",
    "        # P(t,y), P(t) and P(y)\n",
    "        p_yt = pdf_yt[i]; p_t = pdf_t[i[1:]]; p_y = pdf_y[i[0]]\n",
    "        # I(X;T)\n",
    "        mi_ty += p_yt * np.log(p_yt / p_t / p_y)\n",
    "            \n",
    "    return mi_xt, mi_ty\n",
    "\n",
    "# get mutual information for all hidden layers\n",
    "def get_mutual_information(hiddens):\n",
    "    mi_xt_list = []; mi_ty_list = []\n",
    "    for hidden in hiddens:\n",
    "        mi_xt, mi_ty = calc_mutual_information(hidden)\n",
    "        mi_xt_list.append(mi_xt)\n",
    "        mi_ty_list.append(mi_ty)\n",
    "    return mi_xt_list, mi_ty_list"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We are now able to estimate the mutual information while training the network. We'll save the mutual information for later use."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "def get_hidden_layers(names):\n",
    "    hidden_layers = []\n",
    "    for name in names:\n",
    "        hidden_layers.append(tf.get_default_graph().get_tensor_by_name(\"%s:0\" % name))\n",
    "    return hidden_layers\n",
    "\n",
    "# train the neural network and obtain mutual information\n",
    "def train_with_mi(n_epochs, n_hidden_layers, lr_callable):\n",
    "    mi_xt_all = []; mi_ty_all = []; epochs = []\n",
    "    \n",
    "    with tf.Session() as sess:\n",
    "        sess.run(tf.global_variables_initializer()) # initialization\n",
    "    \n",
    "        hidden_layer_names = ['hidden%s' % i for i in range(1,n_hidden_layers+1)]\n",
    "        for epoch in range(n_epochs):\n",
    "            # run one step of optimizer\n",
    "            batch_indices = np.random.randint(low=0, high=n_train_samples, size=batch_size)\n",
    "            sess.run(optimizer, feed_dict={\n",
    "                x: x_train[batch_indices],\n",
    "                y: y_train[batch_indices],\n",
    "                learning_rate: lr_callable(epoch)\n",
    "            })\n",
    "#                x: [x_train[epoch]],\n",
    "#                y: [y_train[epoch]]})\n",
    "\n",
    "            if epoch % print_summary_every_n == 0:\n",
    "                print_out_summary(sess, epoch, lr_callable(epoch))\n",
    "                \n",
    "            if epoch % calc_mi_every_n_epochs == 0:\n",
    "                #print(\"calculating mi for epoch {}\".format(epoch))\n",
    "                _, hidden_layers = sess.run(\n",
    "                    [cost, get_hidden_layers(hidden_layer_names)],\n",
    "                    feed_dict={x: x_train, y: y_train}\n",
    "                )\n",
    "                mi_xt, mi_ty = get_mutual_information(hidden_layers)\n",
    "                mi_xt_all.append(mi_xt)\n",
    "                mi_ty_all.append(mi_ty)\n",
    "                epochs.append(epoch)\n",
    "    \n",
    "    return np.array(mi_xt_all), np.array(mi_ty_all), np.array(epochs)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Visualization"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "import matplotlib.pyplot as plt\n",
    "from matplotlib import animation\n",
    "from IPython.display import HTML"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "cmap = plt.cm.get_cmap('cool')\n",
    "\n",
    "def animate(i):\n",
    "    i = i\n",
    "    title.set_text('Epoch %s' % str(epochs[i]).zfill(4))\n",
    "    ax.plot(mi_xt_all[i,:], mi_ty_all[i,:], 'k-',alpha=0.2)\n",
    "    if i > 0:\n",
    "        for j in range(n_hidden_layers):\n",
    "            ax.plot(mi_xt_all[(i-1):(i+1),j],mi_ty_all[(i-1):(i+1),j],'.-',c=cmap(j*.2))\n",
    "    return"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Epoch    0:  Testing accuracy 0.5659 - Testing loss 0.7423 - lr 0.1000\n",
      "Epoch 3000:  Testing accuracy 0.6940 - Testing loss 0.6043 - lr 0.0250\n",
      "Epoch 6000:  Testing accuracy 0.7191 - Testing loss 0.5815 - lr 0.0143\n",
      "Epoch 9000:  Testing accuracy 0.7240 - Testing loss 0.5637 - lr 0.0100\n",
      "Epoch 12000:  Testing accuracy 0.7317 - Testing loss 0.5406 - lr 0.0077\n",
      "Epoch 15000:  Testing accuracy 0.7453 - Testing loss 0.5215 - lr 0.0063\n",
      "Epoch 18000:  Testing accuracy 0.7550 - Testing loss 0.5014 - lr 0.0053\n",
      "Epoch 21000:  Testing accuracy 0.7867 - Testing loss 0.4772 - lr 0.0045\n",
      "Epoch 24000:  Testing accuracy 0.7994 - Testing loss 0.4507 - lr 0.0040\n",
      "Epoch 27000:  Testing accuracy 0.8036 - Testing loss 0.4358 - lr 0.0036\n",
      "Epoch 30000:  Testing accuracy 0.8085 - Testing loss 0.4253 - lr 0.0032\n"
     ]
    }
   ],
   "source": [
    "np.random.seed(600)\n",
    "seed(600)\n",
    "n_epochs = 30001\n",
    "calc_mi_every_n_epochs = n_epochs // 40\n",
    "print_summary_every_n = n_epochs // 10\n",
    "n_hidden_layers = 5\n",
    "batch_size = 100\n",
    "build_network([8,7,6,5,3]) # 5 hidden layers\n",
    "\n",
    "# you can use one of learning rate (step size policy) definitions below\n",
    "#lr_callable = lambda epoch: 3 * 0.5**(epoch // 4000)\n",
    "#lr_callable = lambda epoch: 0.03\n",
    "#lr_callable = lambda epoch: 0.1 * 0.5**(epoch/4000)\n",
    "lr_callable = lambda epoch: 0.1 / (1 + epoch/1000)\n",
    "\n",
    "mi_xt_all, mi_ty_all, epochs = train_with_mi(\n",
    "    n_epochs, n_hidden_layers,\n",
    "    lr_callable=lr_callable\n",
    ")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
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       "\">\n",
       "  Your browser does not support the video tag.\n",
       "</video>"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fig, ax = plt.subplots(figsize=(8,8))\n",
    "ax.set_xlim((0.5,7))\n",
    "ax.set_ylim((0.1,0.43))\n",
    "ax.set_xlabel('I(X;T)')\n",
    "ax.set_ylabel('I(T;Y)')\n",
    "title = ax.set_title('')\n",
    "plt.close(fig)\n",
    "\n",
    "#_ = list(map(animate, range(mi_ty_all.shape[0])))\n",
    "anim = animation.FuncAnimation(fig,\n",
    "                               animate,\n",
    "                               init_func=None,\n",
    "                               frames=len(epochs),\n",
    "                               interval=100)\n",
    "HTML(anim.to_html5_video())"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.6.6"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}