04_gradient_descent_practice.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Part 1: First, make a baseline model to determine between 3's and 7's"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "!pip install -Uqq fastbook\n",
    "import fastbook\n",
    "fastbook.setup_book()\n",
    "from fastai.vision.all import *\n",
    "from fastbook import *"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Path('/storage/data/mnist_sample')"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "path = untar_data(URLs.MNIST_SAMPLE)\n",
    "path"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Path.BASE_PATH = path"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(#2) [Path('/storage/data/mnist_sample/train/3'),Path('/storage/data/mnist_sample/train/7')]"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# look inside of the train folder from the path to see what is located inside\n",
    "(path/'train').ls()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(#6131) [Path('/storage/data/mnist_sample/train/3/55705.png'),Path('/storage/data/mnist_sample/train/3/32379.png'),Path('/storage/data/mnist_sample/train/3/36132.png'),Path('/storage/data/mnist_sample/train/3/50201.png'),Path('/storage/data/mnist_sample/train/3/39704.png'),Path('/storage/data/mnist_sample/train/3/8475.png'),Path('/storage/data/mnist_sample/train/3/5139.png'),Path('/storage/data/mnist_sample/train/3/32610.png'),Path('/storage/data/mnist_sample/train/3/7784.png'),Path('/storage/data/mnist_sample/train/3/45704.png')...]"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# get the 3s and 7s\n",
    "threes = (path/'train'/'3').ls()\n",
    "threes"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(#6265) [Path('/storage/data/mnist_sample/train/7/23645.png'),Path('/storage/data/mnist_sample/train/7/13644.png'),Path('/storage/data/mnist_sample/train/7/22228.png'),Path('/storage/data/mnist_sample/train/7/2229.png'),Path('/storage/data/mnist_sample/train/7/6064.png'),Path('/storage/data/mnist_sample/train/7/24505.png'),Path('/storage/data/mnist_sample/train/7/26356.png'),Path('/storage/data/mnist_sample/train/7/19987.png'),Path('/storage/data/mnist_sample/train/7/19592.png'),Path('/storage/data/mnist_sample/train/7/49413.png')...]"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sevens = (path/'train'/'7').ls()\n",
    "sevens"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABwAAAAcCAAAAABXZoBIAAAA/UlEQVR4nGNgGK5AQ5iBgYFBycB7+s1mBgYWZCnd0uDnrxgYGDT5GRgYVqPo0l3969+/F9///fv379+/2wZMDAyMcDmZq7w3d226LsPLwJ3h8Sp6L4rGzf8Wi0NYDf++WaI6JfnVOQkIK+bWiyYpFDnelz9LIazY73+Xourzev0vHaJo0t9/4Wj+6/h3KJCBgYGBQfHVv1WcaJJX/xUxMDAwMLDc/DeFAR1wmYgzMDAwSO75d0MCQxICOK//O49LTqr33zVccoJb//1rwSHHEPHvXzsPDjmtV/8/oHsCBtTW/HuN09Def38X8uKQU7/8bxEufQwMX3eJ45bEBgD04FWZE8STZwAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<PIL.PngImagePlugin.PngImageFile image mode=L size=28x28 at 0x7F2980410CD0>"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# exmaine one of the images\n",
    "example_seven = Image.open(sevens[0])\n",
    "example_seven"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Build the baseline model that makes the average 3/7"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "# convert the urls into a list of tensors\n",
    "threes_tensor_list = [tensor(Image.open(three_path)) for three_path in threes]\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "sevens_tensor_list = [tensor(Image.open(seven_path)) for seven_path in sevens]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "tensor(True)\n"
     ]
    }
   ],
   "source": [
    "# stack the tensors to create a single 3D tensor\n",
    "threes_tensor = torch.stack(threes_tensor_list)/255\n",
    "threes_tensor_2 = torch.stack(threes_tensor_list).float()/255\n",
    "\n",
    "print((threes_tensor == threes_tensor_2).all())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "# create single 3 and 7 tensors by stacking the list and convert the values into 0-1 range\n",
    "threes_tensor = torch.stack(threes_tensor_list)/255\n",
    "sevens_tensor = torch.stack(sevens_tensor_list)/255"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(torch.Size([6131, 28, 28]), 3)"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# experiment with shape and dimensions in pytorch\n",
    "threes_tensor.shape, threes_tensor.ndim"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<AxesSubplot:>"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 72x72 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# to calculte the 'averaged' 3, we must take a running average through the 1st dimension (over all of the 3s)\n",
    "av_3 = threes_tensor.mean(0)\n",
    "show_image(av_3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<AxesSubplot:>"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 72x72 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# do the same for the 'averaged' 7\n",
    "av_7 = sevens_tensor.mean(0)\n",
    "show_image(av_7)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### To predict between 3s and 7s, we must be able to compare between the current number and the ideal numbers\n",
    "- Attempt to implement MAE  and RMSE manually and also using the pytorch built in error metrics"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [],
   "source": [
    "# manually create a mae function\n",
    "def manual_mae(num_tensor, ideal_num):\n",
    "    # note that due to broadcasting, num_tensor can be a 2D tensor or a 3D tensor (a tensor containing multiple image tensors) and pytorch will be able to cotinue with the computations\n",
    "    # take the mean over the final 2 dimensions - the image dimensions\n",
    "    return torch.mean(abs(num_tensor-ideal_num), (-2, -1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(tensor(0.1397),\n",
       " tensor([0.1397, 0.1281, 0.1542,  ..., 0.1159, 0.1253, 0.1337]))"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# use one a single image, then use broadcasting to use on all images\n",
    "manual_mae(threes_tensor[0], av_3), manual_mae(threes_tensor, av_3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [],
   "source": [
    "# manually create rmse function\n",
    "def manual_rmse(num_tensor, ideal_num):\n",
    "    return torch.sqrt(torch.mean((num_tensor - ideal_num)**2, (-2, -1)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(tensor(0.2532), tensor(0.2373))"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "manual_rmse(threes_tensor[0], av_3), manual_rmse(threes_tensor, av_3).mean()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [],
   "source": [
    "pytorch_rmse = lambda num_tensor, ideal: F.mse_loss(num_tensor, ideal).sqrt()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "<ipython-input-19-fa4fa006808b>:1: UserWarning: Using a target size (torch.Size([28, 28])) that is different to the input size (torch.Size([6131, 28, 28])). This will likely lead to incorrect results due to broadcasting. Please ensure they have the same size.\n",
      "  pytorch_rmse = lambda num_tensor, ideal: F.mse_loss(num_tensor, ideal).sqrt()\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "(tensor(0.2532), tensor(0.2394))"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pytorch_rmse(threes_tensor[0], av_3), pytorch_rmse(threes_tensor, av_3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [],
   "source": [
    "def is_3(input_tensor):\n",
    "    return manual_rmse(input_tensor, av_3) < manual_rmse(input_tensor, av_7) "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "tensor(0.9584)"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# calculate the accuracy of all 3s and all 7s\n",
    "\n",
    "acc_3 = is_3(torch.stack([tensor(Image.open(valid_3_path)) for valid_3_path in (path/'valid'/'3').ls()])/255).float().mean()\n",
    "acc_3\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "tensor(0.9997)"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "acc_7 = (1 - is_3(torch.stack([tensor(Image.open(valid_7_path)) for valid_7_path in (path/'valid'/'7').ls()]))/255).float().mean()\n",
    "acc_7"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Part 2: Use Gradient Descent using Pytorch on a few mathematical functions\n",
    "- Write step by step what each example is doing in sgd\n",
    "- Start working with a linear function\n",
    "- Visualize the results"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "tensor([ 0.,  1.,  2.,  3.,  4.,  5.,  6.,  7.,  8.,  9., 10., 11., 12., 13., 14., 15., 16., 17., 18., 19.])"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x_points = torch.arange(0, 20).float()\n",
    "x_points"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "tensor([ 11.9269,  16.4873,  20.9007,  22.8945,  29.2419,  36.0783,  40.8008,  46.6806,  50.3559,  54.3134,  59.5066,  65.2415,  69.7684,  75.0418,  79.7484,  85.8599,  89.6903,  94.6043, 100.8034,\n",
       "        104.3784])"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# create a function similar to a line\n",
    "line_y = (5 * x_points) + 10 + torch.randn(20)\n",
    "line_y"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Requirement already satisfied: seaborn in /opt/conda/envs/fastai/lib/python3.8/site-packages (0.11.1)\n",
      "Requirement already satisfied: pandas>=0.23 in /opt/conda/envs/fastai/lib/python3.8/site-packages (from seaborn) (1.1.4)\n",
      "Requirement already satisfied: numpy>=1.15 in /opt/conda/envs/fastai/lib/python3.8/site-packages (from seaborn) (1.19.4)\n",
      "Requirement already satisfied: scipy>=1.0 in /opt/conda/envs/fastai/lib/python3.8/site-packages (from seaborn) (1.5.3)\n",
      "Requirement already satisfied: matplotlib>=2.2 in /opt/conda/envs/fastai/lib/python3.8/site-packages (from seaborn) (3.3.3)\n",
      "Requirement already satisfied: python-dateutil>=2.7.3 in /opt/conda/envs/fastai/lib/python3.8/site-packages (from pandas>=0.23->seaborn) (2.8.1)\n",
      "Requirement already satisfied: pytz>=2017.2 in /opt/conda/envs/fastai/lib/python3.8/site-packages (from pandas>=0.23->seaborn) (2020.4)\n",
      "Requirement already satisfied: pillow>=6.2.0 in /opt/conda/envs/fastai/lib/python3.8/site-packages (from matplotlib>=2.2->seaborn) (8.0.1)\n",
      "Requirement already satisfied: kiwisolver>=1.0.1 in /opt/conda/envs/fastai/lib/python3.8/site-packages (from matplotlib>=2.2->seaborn) (1.3.1)\n",
      "Requirement already satisfied: cycler>=0.10 in /opt/conda/envs/fastai/lib/python3.8/site-packages (from matplotlib>=2.2->seaborn) (0.10.0)\n",
      "Requirement already satisfied: pyparsing!=2.0.4,!=2.1.2,!=2.1.6,>=2.0.3 in /opt/conda/envs/fastai/lib/python3.8/site-packages (from matplotlib>=2.2->seaborn) (2.4.7)\n",
      "Requirement already satisfied: six>=1.5 in /opt/conda/envs/fastai/lib/python3.8/site-packages (from python-dateutil>=2.7.3->pandas>=0.23->seaborn) (1.15.0)\n"
     ]
    }
   ],
   "source": [
    "# make a function to visualize the models we create\n",
    "!pip install seaborn\n",
    "import seaborn as sns"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [],
   "source": [
    "def graph_function(function_vals, x_vals=x_points, ax=None):\n",
    "    if ax == None:\n",
    "        ax = sns.scatterplot(x=x_vals, y=function_vals)\n",
    "        ax.set_xlabel(\"X vals\")\n",
    "        ax.set_ylabel(\"Y vals\")\n",
    "    else:\n",
    "        sns.scatterplot(x=x_vals, y=function_vals, ax=ax)\n",
    "    return ax"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<AxesSubplot:xlabel='X vals', ylabel='Y vals'>"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "graph_function(line_y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Step 1: Initialize the parameters for the model**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [],
   "source": [
    "# randomly initialize the 2 parameters for the line model (slope and y intercept)\n",
    "line_model_params = torch.randn(2).requires_grad_()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Step 2: Calculate the current predictions given by the model**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [],
   "source": [
    "def get_curr_line_model(x_vals, params):\n",
    "    slope, y_int = params\n",
    "    curr_model_y = slope*x_vals + y_int\n",
    "    return curr_model_y"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 77,
   "metadata": {},
   "outputs": [],
   "source": [
    "def compare_models(curr_model, correct_model, x_vals, ax=None):\n",
    "    if ax == None:\n",
    "        fig, ax = plt.subplots()\n",
    "    graph_function(correct_model, x_vals=x_vals, ax=ax)\n",
    "    graph_function(curr_model, x_vals=x_vals, ax=ax)\n",
    "    return ax\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 78,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<AxesSubplot:>"
      ]
     },
     "execution_count": 78,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "init_model = get_curr_line_model(x_points, line_model_params)\n",
    "compare_models(init_model.detach().numpy(), line_y, x_points)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Step 3: Calculate the loss**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [],
   "source": [
    "# create a simple loss function using mae\n",
    "def get_mae(temp, model):\n",
    "    return (temp - model).abs().mean()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "tensor(65.7422, grad_fn=<MeanBackward0>)"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "loss = get_mae(init_model, line_y)\n",
    "loss"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Step 4: Calculate the gradients**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [],
   "source": [
    "# use pytorch to calculate the derivatives for the loss\n",
    "loss.backward()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "tensor([-9.5000, -1.0000])"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "line_model_params.grad"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "tensor([-0.7658, -0.7506], requires_grad=True)"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "line_model_params"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Step 5: Step the weights**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "tensor([-0.6708, -0.7406], requires_grad=True)"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "lr = 0.01\n",
    "line_model_params.data -= line_model_params.grad.data * lr\n",
    "# reset the gradients after changing the model since the new gradients have changed\n",
    "line_model_params.grad = None\n",
    "line_model_params"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<AxesSubplot:>"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "model_2 = get_curr_line_model(x_points, line_model_params)\n",
    "compare_models(model_2.detach().numpy(), line_y, x_points)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Convert the process into a single apply step function**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [],
   "source": [
    "def apply_step(params, input_data, lr=0.01):\n",
    "    # get the current model for the params\n",
    "    curr_model = get_curr_line_model(x_points, params)\n",
    "    # first calculate the loss\n",
    "    loss = get_mae(curr_model, input_data)\n",
    "    # find the gradients of the loss\n",
    "    loss.backward()\n",
    "    # step the loss\n",
    "    params.data -= lr * params.grad.data\n",
    "    \n",
    "    # set the new gradients to none\n",
    "    params.grad = None\n",
    "    \n",
    "    # return the model at the beginning of the function\n",
    "    return curr_model"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<AxesSubplot:>"
      ]
     },
     "execution_count": 41,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAXsAAAD7CAYAAACL+TRnAAAAOXRFWHRTb2Z0d2FyZQBNYXRwbG90bGliIHZlcnNpb24zLjMuMywgaHR0cHM6Ly9tYXRwbG90bGliLm9yZy/Il7ecAAAACXBIWXMAAAsTAAALEwEAmpwYAAAWP0lEQVR4nO3df5Dcd33f8ef7sAZRna5B1iETptZVCqAiGpnhOvaUcWDGpAFNOk2sTGpwE/NHxh4YN21IM810sKPY8aSUhJk6JgnKGDAO0xh3ZCbGgU5pTWsyQDgX7OQmqhvJNphE9kk4su7kA4d994/dhdV6tXvH97u7393v8zFzc3ffH/v56Kvvvb7f/Xzf3+9GZiJJmm4z4+6AJGn4DHtJqgHDXpJqwLCXpBow7CWpBi4adwd62blzZy4sLIy7G5I0UR5++OFTmTnfa14lw35hYYGlpaVxd0OSJkpEPHmheQ7jSFINGPaSVAOGvSTVgGEvSTVg2EtSDVSyGkeS6qbRSJ44vcbTz62za24rCxdvY2YmSnt9w16SxqzRSD67fJL3fvJrrL/QYOuWGT74s5fxtv2XlBb4DuNI0pg9cXrte0EPsP5Cg/d+8ms8cXqttDYMe0kas6efW/9e0Letv9DgmbPrpbVh2EvSmO2a28rWLefH8dYtM7xi+9bS2jDsJWnMFi7exgd/9rLvBX57zH7h4m2lteEFWkkqQZFqmpmZ4G37L2HfL17JM2fXecX28qtxNnRmHxE3RsRSRHw7Ij7WNe+qiDgWEeci4sGI2N0xLyLi/RFxuvX1nyKivN5LUgW0q2kO3v4Q7/iDL3Pw9of47PJJGo2Nf8b3zEywZ36WK/bsZM/8bKlBDxsfxvlr4DeAj3ROjIidwFHgJmAHsATc07HI9cBPAQeAHwV+ErihUI8lqWJGUU1T1IbCPjOPZuangNNds64GljPz3sxcBw4DByJiX2v+dcBvZ+ZTmflN4LeBd5XRcUmqilFU0xRV9ALtfuCR9i+ZuQYcb01/0fzWz/uRpCkyimqaooqG/SxwpmvaGWD7BeafAWZ7jdtHxPWt6wJLKysrBbslSaMzimqaoopW46wCc13T5oCzF5g/B6xm5ouuWmTmEeAIwOLi4savakjSmI2imqaoomG/THNcHoCI2AbsbU1vzz8A/Fnr9wMd8yRparSrafbMz467Kz1ttPTyoojYCrwEeElEbI2Ii4D7gNdHxKHW/JuBRzPzWGvVjwPvjYhXRcQPA78MfKz0f4Ukqa+Njtm/D3ge+FXgX7V+fl9mrgCHgNuAZ4HLgWs61vswcD/w58BfAA+0pkmSRih6DJ+P3eLiYi4tLY27G5I0USLi4cxc7DXPxyVIEsP/8JBxM+wl1d4oPjxk3HzqpaTam4THHRRl2EuqvUl43EFRhr2k2puExx0UZdhLqr1JeNxBUV6glVR7k/C4g6IMe0mi+o87KMqwlzQVpr1OvijDXtLEq0OdfFFeoJU08epQJ1+UYS9p4tWhTr4ow17SxKtDnXxRhr2kiVeHOvmivEArqRKKVNPUoU6+KMNe0tiVUU0z7XXyRTmMI2nsrKYZPsNe0thZTTN8hr2ksbOaZvgMe0ljZzXN8HmBVtLYWU0zfIa9pEqwmma4HMaRpBow7CWpBkoJ+4hYiIg/iYhnI+JkRNwRERe15l0VEcci4lxEPBgRu8toU5K0cWWd2f8u8AzwSuAy4M3AeyJiJ3AUuAnYASwB95TUpqQKaTSSEyurfPH4KU6srNJo5Li7pA5lXaD9h8AdmbkOnIyIzwL7gauB5cy8FyAiDgOnImJfZh4rqW1JY+aHh1RfWWf2/xm4JiL+XkS8Cng70A78R9oLZeYacLw1/TwRcX1ELEXE0srKSkndkjQKPu6g+soK+/9FM8CfA56iOVzzKWAWONO17Blge/cLZOaRzFzMzMX5+fmSuiVpFHzcQfUVDvuImAH+G82x+W3ATuDlwPuBVWCua5U54GzRdiVVh487qL4yzux3AP+A5pj9tzPzNPBR4CCwDBxoLxgR24C9remSpoSPO6i+whdoM/NURDwOvDsifovm0M11NMfq7wM+EBGHgAeAm4FHvTgrTRcfd1B9ZY3ZXw28DVgB/gr4O+CXMnMFOATcBjwLXA5cU1KbkkpUtHSy/biDK/bsZM/8rEFfMaWUXmbm14C3XGDe54B9ZbQjaTgsnZx+Pi5BkqWTNWDYS7J0sgYMe0mWTtaAYS/J0ska8MNLJFk6WQOGvSTAT4qadg7jSFINeGYvTYlGI3ni9BpPP7fOrjmHYXQ+w16aAt4UpUEcxpGmgDdFaRDDXpoC3hSlQQx7aQp4U5QGMeylKeBNURrEC7TSFPCmKA1i2EtTwpui1I9hL1WEdfIaJsNeqgDr5DVsXqCVKsA6eQ2bYS9VgHXyGjbDXqoA6+Q1bIa9VAHWyWvYvEArVYB18ho2w16qCOvkNUwO40hSDZQW9hFxTUT8ZUSsRcTxiLiyNf2qiDgWEeci4sGI2F1Wm1JVNBrJiZVVvnj8FCdWVmk0ctxdks5TyjBORPw48H7gXwJ/BryyNX0ncBT4BeB+4FbgHuCKMtqVqsAbojQJyjqz/3Xglsz8UmY2MvObmflN4GpgOTPvzcx14DBwICL2ldSuNHbeEKVJUDjsI+IlwCIwHxF/FRFPRcQdEfEyYD/wSHvZzFwDjremd7/O9RGxFBFLKysrRbsljYw3RGkSlHFmvwvYAvwMcCVwGfAG4H3ALHCma/kzwPbuF8nMI5m5mJmL8/PzJXRLGg1viNIkKCPsn299/53M/JvMPAV8EDgIrAJzXcvPAWdLaFeqBG+I0iQofIE2M5+NiKeAXuUHy8B17V8iYhuwtzVdmgreEKVJUNYF2o8C/zoiXhERLwf+LfBp4D7g9RFxKCK2AjcDj2bmsZLalSqhfUPUFXt2smd+1qBX5ZQV9rcCXwEeA/4S+CpwW2auAIeA24BngcuBa0pqU5K0QaXU2WfmC8B7Wl/d8z4HWGqpSvNTojTtfDaOas+bolQHPhtHtedNUaoDw161501RqgPDXrXnTVGqA8NetedNUaoDL9Cq9rwpSnVg2GsqFC2d9FOiNO0Me008SyelwRyz18SzdFIazLDXxLN0UhrMsNfEs3RSGsyw18SzdFIazAu0qoQi1TSWTkqDGfYauzKqaSydlPpzGEdjZzWNNHyGvcbOahpp+Ax7jZ3VNNLwGfYaO6tppOHzAq3GzmoaafgMe1WC1TTScDmMI0k1YNhLUg04jKNSFH2evKThKu3MPiJeHRHrEfGHHdOuiohjEXEuIh6MiN1ltafqaN8Be/D2h3jHH3yZg7c/xGeXT9Jo5Li7JqmlzGGcDwFfaf8SETuBo8BNwA5gCbinxPZUEd4BK1VfKWEfEdcAfwv8j47JVwPLmXlvZq4Dh4EDEbGvjDZVHd4BK1Vf4bCPiDngFuCXu2btBx5p/5KZa8Dx1vRer3N9RCxFxNLKykrRbmmEvANWqr4yzuxvBe7MzG90TZ8FznRNOwNs7/UimXkkMxczc3F+fr6EbmlUvANWqr5C1TgRcRnwVuANPWavAnNd0+aAs0XaVPV4B6xUfUVLL98CLABfjwhons2/JCJeB/w+cF17wYjYBuwFlgu2qQryDlip2oqG/RHgjzp+/3c0w//drd8/EBGHgAeAm4FHM/NYwTY1BNbJS9OtUNhn5jngXPv3iFgF1jNzpfX7IeAO4A+BLwPXFGlPw1HGJ0VJqrbIrN6NL4uLi7m0tDTubtTGiZVVDt7+0Hnlk1u3zPAnv3ilwzLSBImIhzNzsdc8n40j6+SlGjDsZZ28VAOGvayTl2rAp17KOnmpBgx7AdbJS9POYRxJqgHDXpJqwLCXpBpwzH5K+LgDSf0Y9lPAxx1IGsRhnCngxwJKGsSwnwI+7kDSIIb9FPBxB5IGMeyngI87kDSIF2ingI87kDSIYT8lfNyBpH4cxpGkGjDsJakGDHtJqgHH7CvCxx1IGibDvgJ83IGkYXMYpwJ83IGkYTPsS9JoJCdWVvni8VOcWFml0cgNr+vjDiQNm8M4JSg6DNN+3EFn4Pu4A0llKnxmHxEvjYg7I+LJiDgbEV+NiLd3zL8qIo5FxLmIeDAidhdts2qKDsP4uANJw1bGmf1FwDeANwNfBw4Cn4yIfwysAkeBXwDuB24F7gGuKKHdyug3DLORO1p93IGkYSsc9pm5BhzumPTpiHgceCNwMbCcmfcCRMRh4FRE7MvMY0XbrooyhmF83IGkYSr9Am1E7AJeAywD+4FH2vNaB4bjrend610fEUsRsbSyslJ2t4bKYRhJVVfqBdqI2AJ8ArgrM49FxCzQndxngO3d62bmEeAIwOLi4sZLWSrAYRhJVVda2EfEDHA38B3gxtbkVWCua9E54GxZ7VaFwzCSqqyUYZyICOBOYBdwKDNfaM1aBg50LLcN2NuaXilF6uQlqerKOrP/PeAfAW/NzOc7pt8HfCAiDgEPADcDj1bt4qyPK5A07cqos98N3ABcBpyMiNXW17WZuQIcAm4DngUuB64p2mbZfFyBpGlXRunlk8AFT38z83PAvqLtDFPROnlJqjqfjcP36+Q7+bgCSdPEsMc6eUnTzwehYZ28pOk3NWFf9JOerJOXNM2mIuwtnZSk/qZizN7SSUnqbyrC3k96kqT+piLsLZ2UpP6mIuwtnZSk/qbiAq2lk5LU31SEPVg6KUn9TMUwjiSpP8NekmrAsJekGjDsJakGDHtJqgHDXpJqwLCXpBow7CWpBgx7SaoBw16SasCwl6QaMOwlqQaGHvYRsSMi7ouItYh4MiLeOew2JUnnG8VTLz8EfAfYBVwGPBARj2Tm8gjaliQx5DP7iNgGHAJuyszVzPwC8MfAz5XeWKMBp/4fPP5Q83ujMXgdSaqJYZ/Zvwb4bmY+1jHtEeDN3QtGxPXA9QCXXnrp5lppNODY/XDfDfDC87DlZfDTH4Z9/xxmNng8azTgW8fh7EnYfgns2LvxdSWp4oadZrPAma5pZ4Dt3Qtm5pHMXMzMxfn5+c218q3j3w96aH6/74bm9I1oHyw+fCXc9ZPN78fu39y7A99ZSKqwYYf9KjDXNW0OOFtqK2dPfj/o2154HlZPbmx9DxaSptyww/4x4KKIeHXHtANAuRdnt1/SHLrptOVlMHvJxtb3YCFpyg017DNzDTgK3BIR2yLiTcC/AO4utaEde5tj9O3Ab4/Z79i7sfU9WHiwkKbcKEov3wN8BHgGOA28u/Syy5mZ5sXYG17XDNjZTV5gbR8sui/wbvZg0Rn4ZR0sdr669zqdLnSwuOF1G1vfC9zS1Bt62Gfmt4CfGnY7zMw0g20j4dZrXQ8WHiykKTaKM/vJ4MHi/GmTdrCQ1JdhXxYPFudPG+XBAnxnIA1g2FeFB4vzp23mYOEwkjSQYT8t6nywqMIwkgcLVZxhr6ZJPliMexiprGsOHjA0RIa9yjHOg8W4h5HKuubguwsNkXuCqqF9sFi4svl9MyE16TfVgTfWaeg8s9fkG/cwUtF3FjD+dxe+s5h6hr2mwyRfc4DJH4ryYFF5hr0E4z1YwPjfXUzDwUJ9GfZSGYocLNrrT/JQ1LgPFuA7gwEMe6kqJnkoatwHC4eRBjLspWlR54NFFYaRKn6wMOwlNU3ywWLcw0gTcLAw7CWVwxvrKn2BujrvMSTVW51vrCt6U90GeGYvafKNexhp3O8sNsCwlzQdJvmaQxl3YQ9g2EsSTPbBYgMMe0kqw7jvwh7AsJekKih6F/aglx/Kq0qSKsWwl6QaMOwlqQYMe0mqAcNekmogMnPcfXiRiFgBnvwBV98JnCqxO2Wzf8XYv+Kq3kf794PbnZnzvWZUMuyLiIilzFwcdz8uxP4VY/+Kq3of7d9wOIwjSTVg2EtSDUxj2B8ZdwcGsH/F2L/iqt5H+zcEUzdmL0l6sWk8s5ckdTHsJakGDHtJqoGJDPuI2BER90XEWkQ8GRHv7LPsL0XEyYg4ExEfiYiXDrlvL42IO1v9OhsRX42It19g2XdFxHcjYrXj6y3D7F+r3c9HxHpHm/+3z7Kj3n6rXV/fjYjfucCyI9l+EXFjRCxFxLcj4mNd866KiGMRcS4iHoyI3X1eZ8P7bRn9i4grIuK/R8S3ImIlIu6NiFf2eZ0N7xcl9W8hIrLr/++mPq8z6u13bVffzrX6+8YLvM5Qtl9ZJjLsgQ8B3wF2AdcCvxcR+7sXioifAH4VuApYAPYAvz7kvl0EfAN4M/D3gZuAT0bEwgWW/2JmznZ8fX7I/Wu7saPN1/ZaYBzbr3Nb0Pz/fR64t88qo9h+fw38BvCRzokRsRM4SvP/eAewBNzT53U2tN+W1T/g5TQrRxaA3cBZ4KMDXmvgflFi/9p+qKPNW/u8zki3X2Z+omt/fA9wAvg/fV5rGNuvFBMX9hGxDTgE3JSZq5n5BeCPgZ/rsfh1wJ2ZuZyZzwK3Au8aZv8ycy0zD2fmE5nZyMxPA48DPc8GKm7k26/LzwDPAA+NsM0Xycyjmfkp4HTXrKuB5cy8NzPXgcPAgYjY1/0am9xvS+lfZn6m1bfnMvMccAfwpqLtldW/zRjH9uvhOuDjOaEljBMX9sBrgO9m5mMd0x4Beh3h97fmdS63KyIuHmL/zhMRu2j2efkCi7whIk5FxGMRcVNEjOrTw36z1e6f9hn6GPf228gf17i2H3Rtn8xcA47Te1/czH47LD/GhffDto3sF2V7MiKeioiPtt4t9TLW7dcanvsx4OMDFh3H9tuQSQz7WeBM17QzwPYNLNv+udeypYuILcAngLsy81iPRf438HrgFTTPWt4B/MoIuvbvaQ7JvIrm2/z7I6LXJxuPbftFxKU0h8Lu6rPYuLZfW5F9sd+ypYuIHwVupv/22eh+UZZTwD+hOcT0Rprb4hMXWHas2w/4eeChzHy8zzKj3n6bMolhvwrMdU2bozkeOWjZ9s+9li1VRMwAd9McY7yx1zKZeSIzH28N9/w5cAvNoYuhyswvZ+bZzPx2Zt4F/ClwsMeiY9t+NP+4vtDvj2tc269DkX2x37KliogfAT4D/JvMvOCQ2Cb2i1K0hmOWMvPvMvNpmn8n/ywiurcTjHH7tfw8/U88Rr79NmsSw/4x4KKI6PxU3gP0fnu63JrXudzTmfkDjx1uREQEcCfNC0mHMvOFDa6aQAytY5tvdyzbr2XgH1cPo95+522f1rjyXnrvi5vZb0vTGn74HHBrZt69ydVHvT3bw3W92hzL9gOIiDcBPwz8102uOq6/594yc+K+gD8C/guwjeYFpzPA/h7LvQ04CbyOZmXC/wT+4wj69/vAl4DZAcu9HdjV+nkf8BfArw25bz8E/ASwlWbl0LXAGvDaCm2/f9rq0/YqbL/WdtoK/CbNd2vtbTff2vcOtaa9H/hS0f22xP69iuY1hF8pc78osX+XA6+ledJ5Mc1Kpgersv065h+hee1oLNuvtP143B34Af9zdgCfam3MrwPvbE2/lObbvUs7ln0v8DTwHM2ys5cOuW+7aR7R11t9aX9d290/4LdafVujWdJ1C7BlyP2bB75C8+3v39I8KP14VbZfq80PA3f3mD6W7Uezyia7vg635r0VOEazRPTzwELHev8B+Myg/XZY/QN+rfVz53642qt//faLIfbvHTQr1daAv6F58fOSqmy/1rytre1xVY/1RrL9yvryQWiSVAOTOGYvSdokw16SasCwl6QaMOwlqQYMe0mqAcNekmrAsJekGjDsJakG/j9tPT9oR4uQ0gAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "compare_models(apply_step(line_model_params, line_y).detach().numpy(), line_y, x_points)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Step 6: Repeat this process multiple times**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 864x216 with 6 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, axs = plt.subplots(1, 6, figsize=(12, 3))\n",
    "\n",
    "for ax in axs:\n",
    "    compare_models(apply_step(line_model_params, line_y, lr=0.1).detach().numpy(), line_y, x_points, ax=ax)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Part 3: Gradient Descent with a more complicated function"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 165,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "tensor([-20, -19, -18, -17, -16, -15, -14, -13, -12, -11, -10,  -9,  -8,  -7,  -6,  -5,  -4,  -3,  -2,  -1,   0,   1,   2,   3,   4,   5,   6,   7,   8,   9,  10,  11,  12,  13,  14,  15,  16,  17,\n",
       "         18,  19,  20])"
      ]
     },
     "execution_count": 165,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "location = torch.arange(-20, 21)\n",
    "location"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 166,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<AxesSubplot:xlabel='X vals', ylabel='Y vals'>"
      ]
     },
     "execution_count": 166,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "complex_model = 5*location**3 + 7*location**2 -1*location + torch.randn(41)*5000\n",
    "\n",
    "graph_function(complex_model, x_vals=location)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 167,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "tensor([-0.4700, -1.3253,  1.2051,  1.1527], requires_grad=True)"
      ]
     },
     "execution_count": 167,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# initialize the models parameters\n",
    "cubic_params = torch.randn(4).requires_grad_()\n",
    "cubic_params"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 168,
   "metadata": {},
   "outputs": [],
   "source": [
    "# create a function to get the results of the new model\n",
    "def get_new_cubic(loc, params):\n",
    "    co3, co2, co1, const = params\n",
    "    return co3*(loc**3) + co2*(loc**2) + (co1*loc) + const"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 169,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "tensor([ 3.2068e+03,  2.7235e+03,  2.2910e+03,  1.9067e+03,  1.5677e+03,  1.2711e+03,  1.0142e+03,  7.9407e+02,  6.0798e+02,  4.5309e+02,  3.2656e+02,  2.2558e+02,  1.4732e+02,  8.8982e+01,\n",
       "         4.7728e+01,  2.0743e+01,  5.2063e+00, -1.7008e+00, -2.7989e+00, -9.0774e-01,  1.1527e+00,  5.6250e-01, -5.4983e+00, -1.9850e+01, -4.5311e+01, -8.4703e+01, -1.4085e+02, -2.1656e+02,\n",
       "        -3.1466e+02, -4.3797e+02, -5.8932e+02, -7.7151e+02, -9.8737e+02, -1.2397e+03, -1.5314e+03, -1.8652e+03, -2.2439e+03, -2.6704e+03, -3.1475e+03, -3.6780e+03, -4.2648e+03],\n",
       "       grad_fn=<AddBackward0>)"
      ]
     },
     "execution_count": 169,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "get_new_cubic(location, cubic_params)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 170,
   "metadata": {},
   "outputs": [],
   "source": [
    "# create a new loss function\n",
    "def rmse(preds, actual):\n",
    "    return ((preds - actual)**2).mean().sqrt()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 171,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "tensor(18840.9609, grad_fn=<SqrtBackward>)"
      ]
     },
     "execution_count": 171,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "rmse(get_new_cubic(location, cubic_params), complex_model)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 172,
   "metadata": {},
   "outputs": [],
   "source": [
    "# create a new function to apply a step\n",
    "def apply_cubic_step(params, loc, lr=0.00001):\n",
    "    # get the current model\n",
    "    curr_cubic = get_new_cubic(loc, params)\n",
    "    \n",
    "    # compute the loss\n",
    "    loss = rmse(curr_cubic, complex_model)\n",
    "    \n",
    "    # apply back propagation (calculate the gradients)\n",
    "    loss.backward()\n",
    "    \n",
    "    # step the parameters\n",
    "    params.data -= lr * params.grad.data\n",
    "    \n",
    "    # remove the old gradients\n",
    "    params.grad = None\n",
    "    \n",
    "    print(loss)\n",
    "    \n",
    "    return curr_cubic"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 173,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "tensor(18840.9609, grad_fn=<SqrtBackward>)\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "tensor([ 3.2068e+03,  2.7235e+03,  2.2910e+03,  1.9067e+03,  1.5677e+03,  1.2711e+03,  1.0142e+03,  7.9407e+02,  6.0798e+02,  4.5309e+02,  3.2656e+02,  2.2558e+02,  1.4732e+02,  8.8982e+01,\n",
       "         4.7728e+01,  2.0743e+01,  5.2063e+00, -1.7008e+00, -2.7989e+00, -9.0774e-01,  1.1527e+00,  5.6250e-01, -5.4983e+00, -1.9850e+01, -4.5311e+01, -8.4703e+01, -1.4085e+02, -2.1656e+02,\n",
       "        -3.1466e+02, -4.3797e+02, -5.8932e+02, -7.7151e+02, -9.8737e+02, -1.2397e+03, -1.5314e+03, -1.8652e+03, -2.2439e+03, -2.6704e+03, -3.1475e+03, -3.6780e+03, -4.2648e+03],\n",
       "       grad_fn=<AddBackward0>)"
      ]
     },
     "execution_count": 173,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "apply_cubic_step(cubic_params, location)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 174,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "tensor(18745.0918, grad_fn=<SqrtBackward>)\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "<AxesSubplot:xlabel='X vals', ylabel='Y vals'>"
      ]
     },
     "execution_count": 174,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "graph_function(apply_cubic_step(cubic_params, location).detach().numpy(), x_vals=location)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 176,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "tensor(16757.3066, grad_fn=<SqrtBackward>)\n",
      "tensor(16664.0059, grad_fn=<SqrtBackward>)\n",
      "tensor(16570.8457, grad_fn=<SqrtBackward>)\n",
      "tensor(16477.8242, grad_fn=<SqrtBackward>)\n",
      "tensor(16384.9453, grad_fn=<SqrtBackward>)\n",
      "tensor(16292.2109, grad_fn=<SqrtBackward>)\n",
      "tensor(16199.6250, grad_fn=<SqrtBackward>)\n",
      "tensor(16107.1865, grad_fn=<SqrtBackward>)\n",
      "tensor(16014.9004, grad_fn=<SqrtBackward>)\n",
      "tensor(15922.7656, grad_fn=<SqrtBackward>)\n",
      "tensor(15830.7910, grad_fn=<SqrtBackward>)\n",
      "tensor(15738.9727, grad_fn=<SqrtBackward>)\n",
      "tensor(15647.3164, grad_fn=<SqrtBackward>)\n",
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      "tensor(15464.4971, grad_fn=<SqrtBackward>)\n",
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      "tensor(15282.3545, grad_fn=<SqrtBackward>)\n",
      "tensor(15191.5430, grad_fn=<SqrtBackward>)\n",
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      "tensor(8791.2461, grad_fn=<SqrtBackward>)\n",
      "tensor(8730.3076, grad_fn=<SqrtBackward>)\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 864x216 with 7 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import math\n",
    "fig, axs = plt.subplots(1, 7, figsize=(12, 3))\n",
    "ITERATIONS = 100\n",
    "ITER_DIV = math.floor(ITERATIONS/6)\n",
    "ax_index = 0\n",
    "\n",
    "for i in range(ITERATIONS):\n",
    "    # step the model using gradient descent\n",
    "    training_cubic = apply_cubic_step(cubic_params, location)\n",
    "    \n",
    "    # visualize some of the steps in between\n",
    "    if i % ITER_DIV == 0:\n",
    "#         print(i, ITER_DIV)\n",
    "#         print(training_quartic)\n",
    "#         print()\n",
    "        compare_models(training_cubic.detach().numpy(), complex_model, location, ax=axs[ax_index])\n",
    "        ax_index += 1\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "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": "3.8.6"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}