examples/lmu_fft_psmnist.ipynb

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    },
    "accelerator": "GPU"
  },
  "cells": [
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "jw-5Z8mZEarj"
      },
      "source": [
        "# Legendre Memory Units"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "aKUmF66mGX24"
      },
      "source": [
        "## Start\n",
        "To reset the notebook, run from this point"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "2xxn9gdeh6xK"
      },
      "source": [
        "%reset -f"
      ],
      "execution_count": 26,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "2hnjfSQRGdmL"
      },
      "source": [
        "## Imports"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "d-InHVnPgJt4"
      },
      "source": [
        "import numpy as np\n",
        "\n",
        "import torch\n",
        "from torch import nn\n",
        "from torch import fft\n",
        "from torch.nn import init\n",
        "from torch.nn import functional as F\n",
        "\n",
        "from scipy.signal import cont2discrete"
      ],
      "execution_count": 27,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "2ZOMK1kXNmPx"
      },
      "source": [
        "## Functions"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "hXEbgA0rsqKE"
      },
      "source": [
        "def leCunUniform(tensor):\n",
        "    \"\"\" \n",
        "        LeCun Uniform Initializer\n",
        "        References: \n",
        "        [1] https://keras.rstudio.com/reference/initializer_lecun_uniform.html\n",
        "        [2] Source code of _calculate_correct_fan can be found in https://pytorch.org/docs/stable/_modules/torch/nn/init.html\n",
        "        [3] Yann A LeCun, Léon Bottou, Genevieve B Orr, and Klaus-Robert Müller. Efficient backprop. In Neural networks: Tricks of the trade, pages 9–48. Springer, 2012\n",
        "    \"\"\"\n",
        "    fan_in = init._calculate_correct_fan(tensor, \"fan_in\")\n",
        "    limit = np.sqrt(3. / fan_in)\n",
        "    init.uniform_(tensor, -limit, limit) # fills the tensor with values sampled from U(-limit, limit)"
      ],
      "execution_count": 28,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "hgPkjNFiu8Qg"
      },
      "source": [
        "## Classes"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "uezRg-TvsINg"
      },
      "source": [
        "### Single LMU Cell/Unit"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "VwirJqmwfYOB"
      },
      "source": [
        "class LMUCell(nn.Module):\n",
        "    \"\"\" A single LMU Cell \"\"\"\n",
        "\n",
        "\n",
        "    def __init__(self, input_size, hidden_size, memory_size, theta, learn_a = False, learn_b = False, psmnist = False):\n",
        "        \"\"\"\n",
        "        Parameters:\n",
        "            input_size (int) : \n",
        "                Size of the input vector (x_t)\n",
        "            hidden_size (int) : \n",
        "                Size of the hidden vector (h_t)\n",
        "            memory_size (int) :\n",
        "                Size of the memory vector (m_t)\n",
        "            theta (int) :\n",
        "                The number of timesteps in the sliding window that is represented using the LTI system\n",
        "            learn_a (boolean) :\n",
        "                Whether to learn the matrix A (default = False)\n",
        "            learn_b (boolean) :\n",
        "                Whether to learn the matrix B (default = False)\n",
        "            psmnist (boolean) :\n",
        "                Uses different parameter initializers when training on psMNIST (as specified in the paper)\n",
        "        \"\"\"\n",
        "        \n",
        "        super(LMUCell, self).__init__()\n",
        "\n",
        "        self.hidden_size = hidden_size\n",
        "        self.memory_size = memory_size\n",
        "        self.f = nn.Tanh()\n",
        "\n",
        "        A, B = self.stateSpaceMatrices(memory_size, theta)\n",
        "        A = torch.from_numpy(A).float()\n",
        "        B = torch.from_numpy(B).float()\n",
        "\n",
        "        if learn_a:\n",
        "            self.A = nn.Parameter(A)\n",
        "        else:\n",
        "            self.register_buffer(\"A\", A)\n",
        "    \n",
        "        if learn_b:\n",
        "            self.B = nn.Parameter(B)\n",
        "        else:\n",
        "            self.register_buffer(\"B\", B)\n",
        "\n",
        "        # Declare Model parameters:\n",
        "        ## Encoding vectors\n",
        "        self.e_x = nn.Parameter(torch.empty(1, input_size))\n",
        "        self.e_h = nn.Parameter(torch.empty(1, hidden_size))\n",
        "        self.e_m = nn.Parameter(torch.empty(1, memory_size))\n",
        "        ## Kernels\n",
        "        self.W_x = nn.Parameter(torch.empty(hidden_size, input_size))\n",
        "        self.W_h = nn.Parameter(torch.empty(hidden_size, hidden_size))\n",
        "        self.W_m = nn.Parameter(torch.empty(hidden_size, memory_size))\n",
        "\n",
        "        self.initParameters(psmnist)\n",
        "\n",
        "\n",
        "    def initParameters(self, psmnist = False):\n",
        "        \"\"\" Initialize the cell's parameters \"\"\"\n",
        "\n",
        "        if psmnist:\n",
        "            # Initialize encoders\n",
        "            leCunUniform(self.e_x)\n",
        "            init.constant_(self.e_h, 0)\n",
        "            init.constant_(self.e_m, 0)\n",
        "            # Initialize kernels\n",
        "            init.constant_(self.W_x, 0)\n",
        "            init.constant_(self.W_h, 0)\n",
        "            init.xavier_normal_(self.W_m)\n",
        "        else:\n",
        "            # Initialize encoders\n",
        "            leCunUniform(self.e_x)\n",
        "            leCunUniform(self.e_h)\n",
        "            init.constant_(self.e_m, 0)\n",
        "            # Initialize kernels\n",
        "            init.xavier_normal_(self.W_x)\n",
        "            init.xavier_normal_(self.W_h)\n",
        "            init.xavier_normal_(self.W_m)\n",
        "\n",
        "\n",
        "    def stateSpaceMatrices(self, memory_size, theta):\n",
        "        \"\"\" Returns the discretized state space matrices A and B \"\"\"\n",
        "\n",
        "        Q = np.arange(memory_size, dtype = np.float64).reshape(-1, 1)\n",
        "        R = (2*Q + 1) / theta\n",
        "        i, j = np.meshgrid(Q, Q, indexing = \"ij\")\n",
        "\n",
        "        # Continuous\n",
        "        A = R * np.where(i < j, -1, (-1.0)**(i - j + 1))\n",
        "        B = R * ((-1.0)**Q)\n",
        "        C = np.ones((1, memory_size))\n",
        "        D = np.zeros((1,))\n",
        "\n",
        "        # Convert to discrete\n",
        "        A, B, C, D, dt = cont2discrete(\n",
        "            system = (A, B, C, D), \n",
        "            dt = 1.0, \n",
        "            method = \"zoh\"\n",
        "        )\n",
        "        \n",
        "        return A, B\n",
        "\n",
        "\n",
        "    def forward(self, x, state):\n",
        "        \"\"\"\n",
        "        Parameters:\n",
        "            x (torch.tensor): \n",
        "                Input of size [batch_size, input_size]\n",
        "            state (tuple): \n",
        "                h (torch.tensor) : [batch_size, hidden_size]\n",
        "                m (torch.tensor) : [batch_size, memory_size]\n",
        "        \"\"\"\n",
        "\n",
        "        h, m = state\n",
        "\n",
        "        # Equation (7) of the paper\n",
        "        u = F.linear(x, self.e_x) + F.linear(h, self.e_h) + F.linear(m, self.e_m) # [batch_size, 1]\n",
        "\n",
        "        # Equation (4) of the paper\n",
        "        m = F.linear(m, self.A) + F.linear(u, self.B) # [batch_size, memory_size]\n",
        "\n",
        "        # Equation (6) of the paper\n",
        "        h = self.f(\n",
        "            F.linear(x, self.W_x) +\n",
        "            F.linear(h, self.W_h) + \n",
        "            F.linear(m, self.W_m)\n",
        "        ) # [batch_size, hidden_size]\n",
        "\n",
        "        return h, m"
      ],
      "execution_count": 29,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "iFQpJVRHsFTw"
      },
      "source": [
        "### LMU Layer"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "JcUfChhhgQYX"
      },
      "source": [
        "class LMU(nn.Module):\n",
        "    \"\"\" An LMU layer \"\"\"\n",
        "\n",
        "    def __init__(self, input_size, hidden_size, memory_size, theta, learn_a = False, learn_b= False, psmnist = False):\n",
        "        \"\"\"\n",
        "        Parameters:\n",
        "            input_size (int) : \n",
        "                Size of the input vector (x_t)\n",
        "            hidden_size (int) : \n",
        "                Size of the hidden vector (h_t)\n",
        "            memory_size (int) :\n",
        "                Size of the memory vector (m_t)\n",
        "            theta (int) :\n",
        "                The number of timesteps in the sliding window that is represented using the LTI system\n",
        "            learn_a (boolean) :\n",
        "                Whether to learn the matrix A (default = False)\n",
        "            learn_b (boolean) :\n",
        "                Whether to learn the matrix B (default = False)\n",
        "            psmnist (boolean) :\n",
        "                Uses different parameter initializers when training on psMNIST (as specified in the paper)\n",
        "        \"\"\"\n",
        "\n",
        "        super(LMU, self).__init__()\n",
        "        self.hidden_size = hidden_size\n",
        "        self.memory_size = memory_size\n",
        "        self.cell = LMUCell(input_size, hidden_size, memory_size, theta, learn_a, learn_b, psmnist)\n",
        "\n",
        "\n",
        "    def forward(self, x, state = None):\n",
        "        \"\"\"\n",
        "        Parameters:\n",
        "            x (torch.tensor): \n",
        "                Input of size [batch_size, seq_len, input_size]\n",
        "            state (tuple) : (default = None) \n",
        "                h (torch.tensor) : [batch_size, hidden_size]\n",
        "                m (torch.tensor) : [batch_size, memory_size]\n",
        "        \"\"\"\n",
        "        \n",
        "        # Assuming batch dimension is always first, followed by seq. length as the second dimension\n",
        "        batch_size = x.size(0)\n",
        "        seq_len = x.size(1)\n",
        "\n",
        "        # Initial state (h_0, m_0)\n",
        "        if state == None:\n",
        "            h_0 = torch.zeros(batch_size, self.hidden_size)\n",
        "            m_0 = torch.zeros(batch_size, self.memory_size)\n",
        "            if x.is_cuda:\n",
        "                h_0 = h_0.cuda()\n",
        "                m_0 = m_0.cuda()\n",
        "            state = (h_0, m_0)\n",
        "\n",
        "        # Iterate over the timesteps\n",
        "        output = []\n",
        "        for t in range(seq_len):\n",
        "            x_t = x[:, t, :] # [batch_size, input_size]\n",
        "            h_t, m_t = self.cell(x_t, state)\n",
        "            state = (h_t, m_t)\n",
        "            output.append(h_t)\n",
        "\n",
        "        output = torch.stack(output) # [seq_len, batch_size, hidden_size]\n",
        "        output = output.permute(1, 0, 2) # [batch_size, seq_len, hidden_size]\n",
        "\n",
        "        return output, state # state is (h_n, m_n) where n = seq_len"
      ],
      "execution_count": 30,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "jNL2sWt2dUW8"
      },
      "source": [
        "### LMU - FFT"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "zcuvqDVQda45"
      },
      "source": [
        "class LMUFFT(nn.Module):\n",
        "\n",
        "    def __init__(self, input_size, hidden_size, memory_size, seq_len, theta):\n",
        "\n",
        "        super(LMUFFT, self).__init__()\n",
        "\n",
        "        self.hidden_size = hidden_size\n",
        "        self.memory_size = memory_size\n",
        "        self.seq_len = seq_len\n",
        "        self.theta = theta\n",
        "\n",
        "        self.W_u = nn.Linear(in_features = input_size, out_features = 1)\n",
        "        self.f_u = nn.ReLU()\n",
        "        self.W_h = nn.Linear(in_features = memory_size + input_size, out_features = hidden_size)\n",
        "        self.f_h = nn.ReLU()\n",
        "\n",
        "        A, B = self.stateSpaceMatrices()\n",
        "        self.register_buffer(\"A\", A) # [memory_size, memory_size]\n",
        "        self.register_buffer(\"B\", B) # [memory_size, 1]\n",
        "\n",
        "        H, fft_H = self.impulse()\n",
        "        self.register_buffer(\"H\", H) # [memory_size, seq_len]\n",
        "        self.register_buffer(\"fft_H\", fft_H) # [memory_size, seq_len + 1]\n",
        "\n",
        "\n",
        "    def stateSpaceMatrices(self):\n",
        "        \"\"\" Returns the discretized state space matrices A and B \"\"\"\n",
        "\n",
        "        Q = np.arange(self.memory_size, dtype = np.float64).reshape(-1, 1)\n",
        "        R = (2*Q + 1) / self.theta\n",
        "        i, j = np.meshgrid(Q, Q, indexing = \"ij\")\n",
        "\n",
        "        # Continuous\n",
        "        A = R * np.where(i < j, -1, (-1.0)**(i - j + 1))\n",
        "        B = R * ((-1.0)**Q)\n",
        "        C = np.ones((1, self.memory_size))\n",
        "        D = np.zeros((1,))\n",
        "\n",
        "        # Convert to discrete\n",
        "        A, B, C, D, dt = cont2discrete(\n",
        "            system = (A, B, C, D), \n",
        "            dt = 1.0, \n",
        "            method = \"zoh\"\n",
        "        )\n",
        "\n",
        "        # To torch.tensor\n",
        "        A = torch.from_numpy(A).float() # [memory_size, memory_size]\n",
        "        B = torch.from_numpy(B).float() # [memory_size, 1]\n",
        "        \n",
        "        return A, B\n",
        "\n",
        "\n",
        "    def impulse(self):\n",
        "        \"\"\" Returns the matrices H and the 1D Fourier transform of H (Equations 23, 26 of the paper) \"\"\"\n",
        "\n",
        "        H = []\n",
        "        A_i = torch.eye(self.memory_size)\n",
        "        for t in range(self.seq_len):\n",
        "            H.append(A_i @ self.B)\n",
        "            A_i = self.A @ A_i\n",
        "\n",
        "        H = torch.cat(H, dim = -1) # [memory_size, seq_len]\n",
        "        fft_H = fft.rfft(H, n = 2*self.seq_len, dim = -1) # [memory_size, seq_len + 1]\n",
        "\n",
        "        return H, fft_H\n",
        "\n",
        "\n",
        "    def forward(self, x):\n",
        "        \"\"\"\n",
        "        Parameters:\n",
        "            x (torch.tensor): \n",
        "                Input of size [batch_size, seq_len, input_size]\n",
        "        \"\"\"\n",
        "\n",
        "        batch_size, seq_len, input_size = x.shape\n",
        "\n",
        "        # Equation 18 of the paper\n",
        "        u = self.f_u(self.W_u(x)) # [batch_size, seq_len, 1]\n",
        "\n",
        "        # Equation 26 of the paper\n",
        "        fft_input = u.permute(0, 2, 1) # [batch_size, 1, seq_len]\n",
        "        fft_u = fft.rfft(fft_input, n = 2*seq_len, dim = -1) # [batch_size, seq_len, seq_len+1]\n",
        "\n",
        "        # Element-wise multiplication (uses broadcasting)\n",
        "        # [batch_size, 1, seq_len+1] * [1, memory_size, seq_len+1]\n",
        "        temp = fft_u * self.fft_H.unsqueeze(0) # [batch_size, memory_size, seq_len+1]\n",
        "\n",
        "        m = fft.irfft(temp, n = 2*seq_len, dim = -1) # [batch_size, memory_size, seq_len+1]\n",
        "        m = m[:, :, :seq_len] # [batch_size, memory_size, seq_len]\n",
        "        m = m.permute(0, 2, 1) # [batch_size, seq_len, memory_size]\n",
        "\n",
        "        # Equation 20 of the paper (W_m@m + W_x@x  W@[m;x])\n",
        "        input_h = torch.cat((m, x), dim = -1) # [batch_size, seq_len, memory_size + input_size]\n",
        "        h = self.f_h(self.W_h(input_h)) # [batch_size, seq_len, hidden_size]\n",
        "\n",
        "        h_n = h[:, -1, :] # [batch_size, hidden_size]\n",
        "\n",
        "        return h, h_n"
      ],
      "execution_count": 31,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "t0wK6z81GrIV"
      },
      "source": [
        "## Example: psMNIST"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "f9eO84qyG4IB"
      },
      "source": [
        "### Imports and Constants"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "lecfGg0yG7al"
      },
      "source": [
        "import random\n",
        "from tqdm.notebook import tqdm\n",
        "from matplotlib import pyplot as plt\n",
        "from sklearn.metrics import accuracy_score\n",
        "\n",
        "from torch import optim\n",
        "from torch.utils.data import Dataset, DataLoader\n",
        "from torchvision import datasets, transforms"
      ],
      "execution_count": 32,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "Avun8G-tJALx",
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "ef23daad-2ab2-441b-9770-6eaeb8df7e7e"
      },
      "source": [
        "# Connect to GPU\n",
        "if torch.cuda.is_available():\n",
        "    DEVICE = \"cuda\"\n",
        "    # Clear cache if non-empty\n",
        "    torch.cuda.empty_cache()\n",
        "    # See which GPU has been allotted \n",
        "    print(torch.cuda.get_device_name(torch.cuda.current_device()))\n",
        "else:\n",
        "    DEVICE = \"cpu\""
      ],
      "execution_count": 33,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "Tesla T4\n"
          ],
          "name": "stdout"
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "ucVH7x7GGqb9"
      },
      "source": [
        "SEED = 0\n",
        "\n",
        "def setSeed(seed):\n",
        "    \"\"\" Set all seeds to ensure reproducibility \"\"\"\n",
        "    random.seed(seed)\n",
        "    np.random.seed(seed)\n",
        "    torch.manual_seed(seed)\n",
        "    torch.cuda.manual_seed(seed)\n",
        "    torch.cuda.manual_seed_all(seed)\n",
        "    torch.backends.cudnn.deterministic = True\n",
        "    torch.backends.cudnn.benchmark = False\n",
        "\n",
        "setSeed(SEED)"
      ],
      "execution_count": 34,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "OGnp3ysSH0re"
      },
      "source": [
        "N_x = 1 # dimension of the input, a single pixel\n",
        "N_t = 784\n",
        "N_h = 346 # dimension of the hidden state\n",
        "N_m = 468 # dimension of the memory\n",
        "N_c = 10 # number of classes \n",
        "THETA = 784\n",
        "N_b = 100 # batch size\n",
        "N_epochs = 15"
      ],
      "execution_count": 35,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "tXL5Cxkj51b7"
      },
      "source": [
        "### Data"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "3viEQhDj-7ht"
      },
      "source": [
        "class psMNIST(Dataset):\n",
        "    \"\"\" Dataset that defines the psMNIST dataset, given the MNIST data and a fixed permutation \"\"\"\n",
        "\n",
        "    def __init__(self, mnist, perm):\n",
        "        self.mnist = mnist # also a torch.data.Dataset object\n",
        "        self.perm  = perm\n",
        "\n",
        "    def __len__(self):\n",
        "        return len(self.mnist)\n",
        "\n",
        "    def __getitem__(self, idx):\n",
        "        img, label = self.mnist[idx]\n",
        "        unrolled = img.reshape(-1)\n",
        "        permuted = unrolled[self.perm]\n",
        "        permuted = permuted.reshape(-1, 1)\n",
        "        return permuted, label"
      ],
      "execution_count": 36,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "NTict3geBurz"
      },
      "source": [
        "### Model"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "0aDZaByeBwSa"
      },
      "source": [
        "class Model(nn.Module):\n",
        "    \"\"\" A simple model for the psMNIST dataset consisting of a single LMUFFT layer and a single dense classifier \"\"\"\n",
        "\n",
        "    def __init__(self, input_size, output_size, hidden_size, memory_size, seq_len, theta):\n",
        "        super(Model, self).__init__()\n",
        "        self.lmu_fft = LMUFFT(input_size, hidden_size, memory_size, seq_len, theta)\n",
        "        self.dropout = nn.Dropout(p = 0.5)\n",
        "        self.classifier = nn.Linear(hidden_size, output_size)\n",
        "\n",
        "    def forward(self, x):\n",
        "        _, h_n = self.lmu_fft(x) # [batch_size, hidden_size]\n",
        "        x = self.dropout(h_n)\n",
        "        output = self.classifier(x)\n",
        "        return output # [batch_size, output_size]"
      ],
      "execution_count": 37,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "Nib69vnXH9G0"
      },
      "source": [
        "### Functions"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "xxZlPS-jKrr7"
      },
      "source": [
        "#### Utils"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "rmsbEVZa73hq"
      },
      "source": [
        "def disp(img):\n",
        "    \"\"\" Displays an image \"\"\"\n",
        "    if len(img.shape) == 3:\n",
        "        img = img.squeeze(0)\n",
        "    plt.imshow(img, cmap = \"gray\")\n",
        "    plt.axis(\"off\")\n",
        "    plt.tight_layout()\n",
        "    plt.show()"
      ],
      "execution_count": 38,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "-t9txMBdBAO7"
      },
      "source": [
        "def dispSeq(seq, rows = 8):\n",
        "    \"\"\" Displays a sequence of pixels \"\"\"\n",
        "    seq = seq.reshape(rows, -1) # divide the 1D sequence into `rows` rows for easy visualization\n",
        "    disp(seq)"
      ],
      "execution_count": 39,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "hRfe0sLPKt71"
      },
      "source": [
        "#### Model"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "fSZjCtSIH8kK"
      },
      "source": [
        "def countParameters(model):\n",
        "    \"\"\" Counts and prints the number of trainable and non-trainable parameters of a model \"\"\"\n",
        "    trainable = sum(p.numel() for p in model.parameters() if p.requires_grad)\n",
        "    frozen = sum(p.numel() for p in model.parameters() if not p.requires_grad)\n",
        "    print(f\"The model has {trainable:,} trainable parameters and {frozen:,} frozen parameters\")"
      ],
      "execution_count": 40,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "vPKKUMAVIvXt"
      },
      "source": [
        "def train(model, loader, optimizer, criterion):\n",
        "    \"\"\" A single training epoch on the psMNIST data \"\"\"\n",
        "\n",
        "    epoch_loss = 0\n",
        "    y_pred = []\n",
        "    y_true = []\n",
        "    \n",
        "    model.train()\n",
        "    for batch, labels in tqdm(loader):\n",
        "\n",
        "        torch.cuda.empty_cache()\n",
        "\n",
        "        batch = batch.to(DEVICE)\n",
        "        labels = labels.long().to(DEVICE)\n",
        "\n",
        "        optimizer.zero_grad()\n",
        "\n",
        "        output = model(batch)\n",
        "        loss = criterion(output, labels)\n",
        "        \n",
        "        loss.backward()\n",
        "        optimizer.step()\n",
        "\n",
        "        preds  = output.argmax(dim = 1)\n",
        "        y_pred += preds.tolist()\n",
        "        y_true += labels.tolist()\n",
        "        epoch_loss += loss.item()\n",
        "\n",
        "    # Loss\n",
        "    avg_epoch_loss = epoch_loss / len(loader)\n",
        "\n",
        "    # Accuracy\n",
        "    epoch_acc = accuracy_score(y_true, y_pred)\n",
        "\n",
        "    return avg_epoch_loss, epoch_acc"
      ],
      "execution_count": 41,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "lVQzoKwbIfWd"
      },
      "source": [
        "def validate(model, loader, criterion):\n",
        "    \"\"\" A single validation epoch on the psMNIST data \"\"\"\n",
        "\n",
        "    epoch_loss = 0\n",
        "    y_pred = []\n",
        "    y_true = []\n",
        "    \n",
        "    model.eval()\n",
        "    with torch.no_grad():\n",
        "        for batch, labels in tqdm(loader):\n",
        "\n",
        "            torch.cuda.empty_cache()\n",
        "\n",
        "            batch = batch.to(DEVICE)\n",
        "            labels = labels.long().to(DEVICE)\n",
        "\n",
        "            output = model(batch)\n",
        "            loss = criterion(output, labels)\n",
        "            \n",
        "            preds  = output.argmax(dim = 1)\n",
        "            y_pred += preds.tolist()\n",
        "            y_true += labels.tolist()\n",
        "            epoch_loss += loss.item()\n",
        "            \n",
        "    # Loss\n",
        "    avg_epoch_loss = epoch_loss / len(loader)\n",
        "\n",
        "    # Accuracy\n",
        "    epoch_acc = accuracy_score(y_true, y_pred)\n",
        "\n",
        "    return avg_epoch_loss, epoch_acc"
      ],
      "execution_count": 42,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "GKr_-Ix9u6xj"
      },
      "source": [
        "### Main"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "sCnH5uFoK2bZ"
      },
      "source": [
        "#### Data"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "u4KKlCN7KkaX"
      },
      "source": [
        "transform = transforms.ToTensor()\n",
        "mnist_train = datasets.MNIST(\"/content/\", train = True, download = True, transform = transform)\n",
        "mnist_val   = datasets.MNIST(\"/content/\", train = False, download = True, transform = transform)\n",
        "\n",
        "perm = torch.load(\"permutation.pt\").long() # created using torch.randperm(784)\n",
        "ds_train = psMNIST(mnist_train, perm)\n",
        "ds_val   = psMNIST(mnist_val, perm)\n",
        "\n",
        "dl_train = DataLoader(ds_train, batch_size = N_b, shuffle = True, num_workers = 2, pin_memory = True)\n",
        "dl_val   = DataLoader(ds_val, batch_size = N_b, shuffle = True, num_workers = 2, pin_memory = True)"
      ],
      "execution_count": 43,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 0
        },
        "id": "ByTnp5ClC8u3",
        "outputId": "84cbc621-f819-45f6-8e80-c5ba2dfdb667"
      },
      "source": [
        "# Example of the data\n",
        "eg_img, eg_label = ds_train[0]\n",
        "print(\"Label:\", eg_label)\n",
        "dispSeq(eg_img)"
      ],
      "execution_count": 44,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "Label: 5\n"
          ],
          "name": "stdout"
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": "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\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": [],
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "TCP8vPm8K398"
      },
      "source": [
        "#### Model"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "yMN5f_togfwy"
      },
      "source": [
        "model = Model(\n",
        "    input_size = N_x, \n",
        "    output_size = N_c, \n",
        "    hidden_size = N_h, \n",
        "    memory_size = N_m, \n",
        "    seq_len = N_t, \n",
        "    theta = THETA\n",
        ")\n",
        "model = model.to(DEVICE)"
      ],
      "execution_count": 45,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "Aw9LPaePtx9R",
        "outputId": "16c84c88-4ea6-4adc-845f-c5377d9ceb91"
      },
      "source": [
        "countParameters(model)"
      ],
      "execution_count": 46,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "The model has 166,092 trainable parameters and 0 frozen parameters\n"
          ],
          "name": "stdout"
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "MZzMA2EgK548"
      },
      "source": [
        "#### Optimization"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "FIqC4oxTKD2H"
      },
      "source": [
        "optimizer = optim.Adam(params = model.parameters())"
      ],
      "execution_count": 47,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "G_eBf5PZKHVu"
      },
      "source": [
        "criterion = nn.CrossEntropyLoss()\n",
        "criterion = criterion.to(DEVICE)"
      ],
      "execution_count": 48,
      "outputs": []
    },
    {
      "cell_type": "code",
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        "colab": {
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          "height": 1000,
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          ]
        },
        "outputId": "33def28e-f03f-44d0-9aea-ca6c9c152da7"
      },
      "source": [
        "train_losses = []\n",
        "train_accs = []\n",
        "val_losses = []\n",
        "val_accs = []\n",
        "\n",
        "for epoch in range(N_epochs):\n",
        "   \n",
        "    print(f\"Epoch: {epoch+1:02}/{N_epochs:02}\")\n",
        "\n",
        "    train_loss, train_acc = train(model, dl_train, optimizer, criterion)\n",
        "    val_loss, val_acc = validate(model, dl_val, criterion)\n",
        "\n",
        "    train_losses.append(train_loss)\n",
        "    train_accs.append(train_acc)\n",
        "    val_losses.append(val_loss)\n",
        "    val_accs.append(val_acc)\n",
        "\n",
        "    print(f\"Train Loss: {train_loss:.3f} | Train Acc: {train_acc*100:.2f}%\")\n",
        "    print(f\"Val. Loss: {val_loss:.3f} |  Val. Acc: {val_acc*100:.2f}%\")\n",
        "    print()"
      ],
      "execution_count": 49,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "Epoch: 01/15\n"
          ],
          "name": "stdout"
        },
        {
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        {
          "output_type": "stream",
          "text": [
            "\n"
          ],
          "name": "stdout"
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          },
          "metadata": {
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        },
        {
          "output_type": "stream",
          "text": [
            "\n",
            "Train Loss: 0.706 | Train Acc: 80.22%\n",
            "Val. Loss: 0.271 |  Val. Acc: 92.42%\n",
            "\n",
            "Epoch: 02/15\n"
          ],
          "name": "stdout"
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        {
          "output_type": "stream",
          "text": [
            "\n"
          ],
          "name": "stdout"
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        },
        {
          "output_type": "stream",
          "text": [
            "\n",
            "Train Loss: 0.268 | Train Acc: 92.18%\n",
            "Val. Loss: 0.180 |  Val. Acc: 95.02%\n",
            "\n",
            "Epoch: 03/15\n"
          ],
          "name": "stdout"
        },
        {
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          "text": [
            "\n",
            "Train Loss: 0.196 | Train Acc: 94.30%\n",
            "Val. Loss: 0.135 |  Val. Acc: 96.02%\n",
            "\n",
            "Epoch: 04/15\n"
          ],
          "name": "stdout"
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            "\n",
            "Train Loss: 0.157 | Train Acc: 95.42%\n",
            "Val. Loss: 0.111 |  Val. Acc: 96.65%\n",
            "\n",
            "Epoch: 05/15\n"
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        {
          "output_type": "stream",
          "text": [
            "\n",
            "Train Loss: 0.134 | Train Acc: 96.05%\n",
            "Val. Loss: 0.095 |  Val. Acc: 97.21%\n",
            "\n",
            "Epoch: 06/15\n"
          ],
          "name": "stdout"
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            "\n",
            "Train Loss: 0.117 | Train Acc: 96.53%\n",
            "Val. Loss: 0.086 |  Val. Acc: 97.34%\n",
            "\n",
            "Epoch: 07/15\n"
          ],
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            "Train Loss: 0.105 | Train Acc: 96.85%\n",
            "Val. Loss: 0.081 |  Val. Acc: 97.47%\n",
            "\n",
            "Epoch: 08/15\n"
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            "Train Loss: 0.096 | Train Acc: 97.12%\n",
            "Val. Loss: 0.075 |  Val. Acc: 97.68%\n",
            "\n",
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            "Train Loss: 0.086 | Train Acc: 97.38%\n",
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            "\n",
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            "Train Loss: 0.079 | Train Acc: 97.54%\n",
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            "\n",
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            "Train Loss: 0.075 | Train Acc: 97.75%\n",
            "Val. Loss: 0.067 |  Val. Acc: 97.90%\n",
            "\n",
            "Epoch: 12/15\n"
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            "Train Loss: 0.071 | Train Acc: 97.79%\n",
            "Val. Loss: 0.066 |  Val. Acc: 97.97%\n",
            "\n",
            "Epoch: 13/15\n"
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            "Train Loss: 0.066 | Train Acc: 97.91%\n",
            "Val. Loss: 0.063 |  Val. Acc: 98.12%\n",
            "\n",
            "Epoch: 14/15\n"
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          "text": [
            "\n",
            "Train Loss: 0.062 | Train Acc: 97.99%\n",
            "Val. Loss: 0.062 |  Val. Acc: 98.21%\n",
            "\n",
            "Epoch: 15/15\n"
          ],
          "name": "stdout"
        },
        {
          "output_type": "display_data",
          "data": {
            "application/vnd.jupyter.widget-view+json": {
              "model_id": "e34bfff86ccc4a81806f3da8460cc8ee",
              "version_minor": 0,
              "version_major": 2
            },
            "text/plain": [
              "HBox(children=(FloatProgress(value=0.0, max=600.0), HTML(value='')))"
            ]
          },
          "metadata": {
            "tags": []
          }
        },
        {
          "output_type": "stream",
          "text": [
            "\n"
          ],
          "name": "stdout"
        },
        {
          "output_type": "display_data",
          "data": {
            "application/vnd.jupyter.widget-view+json": {
              "model_id": "cfa7377a24464b5a9e889b98807a7bfc",
              "version_minor": 0,
              "version_major": 2
            },
            "text/plain": [
              "HBox(children=(FloatProgress(value=0.0), HTML(value='')))"
            ]
          },
          "metadata": {
            "tags": []
          }
        },
        {
          "output_type": "stream",
          "text": [
            "\n",
            "Train Loss: 0.058 | Train Acc: 98.11%\n",
            "Val. Loss: 0.064 |  Val. Acc: 98.07%\n",
            "\n"
          ],
          "name": "stdout"
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "eSfBhYRffalF",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 541
        },
        "outputId": "c1410448-c9f0-4ab4-c040-4f2bb8b09497"
      },
      "source": [
        "# Learning curves\n",
        "\n",
        "plt.plot(range(N_epochs), train_losses)\n",
        "plt.plot(range(N_epochs), val_losses)\n",
        "plt.ylabel(\"Loss\")\n",
        "plt.xlabel(\"Epochs\")\n",
        "plt.legend([\"Train\", \"Val.\"])\n",
        "plt.show()\n",
        "\n",
        "plt.plot(range(N_epochs), train_accs)\n",
        "plt.plot(range(N_epochs), val_accs)\n",
        "plt.ylabel(\"Accuracy\")\n",
        "plt.xlabel(\"Epochs\")\n",
        "plt.legend([\"Train\", \"Val.\"])\n",
        "plt.show()"
      ],
      "execution_count": 50,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": [],
            "needs_background": "light"
          }
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": [],
            "needs_background": "light"
          }
        }
      ]
    }
  ]
}