{
 "cells": [
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   "cell_type": "raw",
   "metadata": {},
   "source": [
    "---\n",
    "execute:\n",
    "  enabled: false\n",
    "---"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<!-- d2l:prose id=linear-regression-concise-md-1 fw=all -->\n",
    "\n",
    "# Concise Implementation of Linear Regression\n",
    "\n",
    "In that section we implemented every piece of linear regression\n",
    "by hand: we initialized the weights, coded the forward pass, wrote out the squared\n",
    "error, and ran the parameter update ourselves.\n",
    "You *should* know how to do this, and doing it once is instructive.\n",
    "But because data iterators, loss functions, optimizers, and neural network layers\n",
    "are so common, modern deep learning frameworks package all of them as reusable,\n",
    "heavily optimized, well-tested components, freeing us to focus on the model\n",
    "rather than on low-level bookkeeping.\n",
    "In this section we rebuild the very same model from that section\n",
    "using these high-level APIs, showing exactly which hand-rolled piece each\n",
    "framework primitive replaces."
   ]
  },
  {
   "cell_type": "code",
   "metadata": {
    "id": "linear-regression-concise-concise-implementation-of-linear-regression"
   },
   "source": [
    "from d2l import torch as d2l\n",
    "import numpy as np\n",
    "import torch\n",
    "from torch import nn"
   ],
   "execution_count": 1,
   "outputs": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<!-- d2l:prose id=linear-regression-concise-md-2 fw=all -->\n",
    "\n",
    "## Defining the Model\n",
    "\n",
    "Each component from that section has a direct counterpart here.\n",
    "The hand-rolled weight vector $\\mathbf{w}$ and bias $b$ are replaced by a single\n",
    "*layer*; our manual squared-error computation is replaced by a built-in *loss*;\n",
    "and our explicit parameter-update loop is replaced by an *optimizer* object.\n",
    "The situation is similar to coding up your own blog from scratch.\n",
    "Doing it once or twice is rewarding and instructive,\n",
    "but you would be a lousy web developer\n",
    "if you spent a month reinventing the wheel.\n",
    "\n",
    "For standard operations,\n",
    "we can use a framework's predefined layers,\n",
    "which allow us to focus\n",
    "on the layers used to construct the model\n",
    "rather than worrying about their implementation.\n",
    "Recall the architecture of a single-layer network\n",
    "as described in the figure.\n",
    "The layer is called *fully connected*,\n",
    "since each of its inputs is connected\n",
    "to each of its outputs\n",
    "by means of a matrix--vector multiplication."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<!-- d2l:prose id=linear-regression-concise-md-3 fw=pytorch -->\n",
    "\n",
    "In PyTorch, the fully connected layer is defined in `Linear` and `LazyLinear` classes (available since version 1.8.0). \n",
    "The latter\n",
    "allows users to specify *merely*\n",
    "the output dimension,\n",
    "while the former\n",
    "additionally asks for\n",
    "how many inputs go into this layer.\n",
    "Specifying input shapes is inconvenient and may require nontrivial calculations\n",
    "(such as in convolutional layers).\n",
    "Thus, for simplicity, we will use such \"lazy\" layers\n",
    "whenever we can."
   ]
  },
  {
   "cell_type": "code",
   "metadata": {
    "id": "linear-regression-concise-defining-the-model-1"
   },
   "source": [
    "class LinearRegression(d2l.Module):\n",
    "    \"\"\"The linear regression model implemented with high-level APIs.\"\"\"\n",
    "    def __init__(self, lr):\n",
    "        super().__init__()\n",
    "        self.save_hyperparameters()\n",
    "        self.net = nn.LazyLinear(1)"
   ],
   "execution_count": 2,
   "outputs": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<!-- d2l:prose id=linear-regression-concise-md-4 fw=all -->\n",
    "\n",
    "In the `forward` method we just invoke the built-in `__call__` method of the predefined layers to compute the outputs."
   ]
  },
  {
   "cell_type": "code",
   "metadata": {
    "id": "linear-regression-concise-defining-the-model-2"
   },
   "source": [
    "@d2l.add_to_class(LinearRegression)\n",
    "def forward(self, X):\n",
    "    return self.net(X)"
   ],
   "execution_count": 3,
   "outputs": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<!-- d2l:prose id=linear-regression-concise-md-5 fw=all -->\n",
    "\n",
    "## Defining the Loss Function"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<!-- d2l:prose id=linear-regression-concise-md-6 fw=pytorch -->\n",
    "\n",
    "The `MSELoss` class computes the mean squared error (without the $1/2$ factor in the equation).\n",
    "By default, `MSELoss` returns the average loss over examples.\n",
    "Using the library primitive also gives us its tested reduction and\n",
    "automatic-differentiation behavior; performance should be measured rather\n",
    "than assumed."
   ]
  },
  {
   "cell_type": "code",
   "metadata": {
    "id": "linear-regression-concise-defining-the-loss-function"
   },
   "source": [
    "@d2l.add_to_class(LinearRegression)\n",
    "def loss(self, y_hat, y):\n",
    "    fn = nn.MSELoss()\n",
    "    return fn(y_hat, y)"
   ],
   "execution_count": 4,
   "outputs": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<!-- d2l:prose id=linear-regression-concise-md-7 fw=all -->\n",
    "\n",
    "## Defining the Optimization Algorithm"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<!-- d2l:prose id=linear-regression-concise-md-8 fw=pytorch -->\n",
    "\n",
    "Minibatch SGD is a standard tool\n",
    "for optimizing neural networks\n",
    "and thus PyTorch supports it alongside a number of\n",
    "variations on this algorithm in the `optim` module.\n",
    "When we instantiate an `SGD` instance,\n",
    "we specify the parameters to optimize over,\n",
    "obtainable from our model via `self.parameters()`,\n",
    "and the learning rate (`self.lr`)\n",
    "required by our optimization algorithm."
   ]
  },
  {
   "cell_type": "code",
   "metadata": {
    "id": "linear-regression-concise-defining-the-optimization-algorithm"
   },
   "source": [
    "@d2l.add_to_class(LinearRegression)\n",
    "def configure_optimizers(self):\n",
    "    return torch.optim.SGD(self.parameters(), self.lr)"
   ],
   "execution_count": 5,
   "outputs": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<!-- d2l:prose id=linear-regression-concise-md-9 fw=all -->\n",
    "\n",
    "## Training\n",
    "\n",
    "You might have noticed that expressing our model through\n",
    "high-level APIs of a deep learning framework\n",
    "requires fewer lines of code.\n",
    "We did not have to allocate parameters individually,\n",
    "define our loss function, or implement minibatch SGD.\n",
    "Once we start working with much more complex models,\n",
    "the advantages of the high-level API will grow considerably.\n",
    "\n",
    "Now that we have all the basic pieces in place,\n",
    "the training loop itself is the same\n",
    "as the one we implemented from scratch.\n",
    "So we just call the `fit` method (introduced in that section),\n",
    "which relies on the implementation of the `fit_epoch` method\n",
    "in that section,\n",
    "to train our model."
   ]
  },
  {
   "cell_type": "code",
   "metadata": {
    "id": "linear-regression-concise-training-1"
   },
   "source": [
    "model = LinearRegression(lr=0.03)\n",
    "data = d2l.SyntheticRegressionData(w=d2l.tensor([2, -3.4]), b=4.2)\n",
    "# Materialize lazy parameters before replacing their default initialization.\n",
    "model(data.X[:1])\n",
    "with torch.no_grad():\n",
    "    model.net.weight.normal_(0, 0.01)\n",
    "    model.net.bias.fill_(0)\n",
    "trainer = d2l.Trainer(max_epochs=10)\n",
    "trainer.fit(model, data)"
   ],
   "execution_count": 6,
   "outputs": [
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  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<!-- d2l:prose id=linear-regression-concise-md-10 fw=all -->\n",
    "\n",
    "Below, we\n",
    "compare the model parameters learned\n",
    "by training on finite data\n",
    "and the actual parameters\n",
    "that generated our dataset.\n",
    "This is where the concise version differs conceptually from the scratch one:\n",
    "the parameters no longer hang off our class as `self.w` and `self.b` but live\n",
    "*inside* the layer object, so `get_w_b` reaches through `net` to find them.\n",
    "As in our implementation from scratch,\n",
    "note that our estimated parameters\n",
    "are close to their true counterparts."
   ]
  },
  {
   "cell_type": "code",
   "metadata": {
    "id": "linear-regression-concise-training-2"
   },
   "source": [
    "@d2l.add_to_class(LinearRegression)\n",
    "def get_w_b(self):\n",
    "    return (self.net.weight.detach(), self.net.bias.detach())\n",
    "w, b = model.get_w_b()"
   ],
   "execution_count": 7,
   "outputs": []
  },
  {
   "cell_type": "code",
   "metadata": {
    "id": "linear-regression-concise-training-3"
   },
   "source": [
    "print(f'error in estimating w: {data.w - d2l.reshape(w, data.w.shape)}')\n",
    "print(f'error in estimating b: {data.b - b}')"
   ],
   "execution_count": 8,
   "outputs": [
    {
     "output_type": "stream",
     "name": "stdout",
     "text": [
      "error in estimating w: tensor([-0.0003,  0.0008])\nerror in estimating b: tensor([0.0004])\n"
     ]
    }
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<!-- d2l:prose id=linear-regression-concise-md-11 fw=all -->\n",
    "\n",
    "## Summary\n",
    "\n",
    "This section contains the first\n",
    "implementation of a deep network (in this book)\n",
    "to tap into the conveniences afforded\n",
    "by modern deep learning frameworks,\n",
    "such as MXNet [@Chen.Li.Li.ea.2015], \n",
    "JAX [@Frostig.Johnson.Leary.2018], \n",
    "PyTorch [@Paszke.Gross.Massa.ea.2019], \n",
    "and Tensorflow [@Abadi.Barham.Chen.ea.2016].\n",
    "We used framework defaults for loading data, defining a layer,\n",
    "a loss function, an optimizer and a training loop.\n",
    "Whenever the framework provides all necessary features,\n",
    "it is generally a good idea to use them,\n",
    "since the library implementations of these components\n",
    "tend to be heavily optimized for performance\n",
    "and properly tested for reliability.\n",
    "At the same time, try not to forget\n",
    "that these modules *can* be implemented directly.\n",
    "This is especially important for aspiring researchers\n",
    "who wish to live on the leading edge of model development,\n",
    "where you will be inventing new components\n",
    "that cannot possibly exist in any current library."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<!-- d2l:prose id=linear-regression-concise-md-12 fw=pytorch -->\n",
    "\n",
    "In PyTorch, the `data` module provides tools for data processing,\n",
    "the `nn` module defines a large number of neural network layers and common loss functions.\n",
    "Because we used `nn.LazyLinear`, the input dimensions are inferred automatically\n",
    "on the first forward pass. Before that pass, the layer contains\n",
    "`UninitializedParameter` placeholders: writing to them does not initialize the\n",
    "eventual weights. The training cell therefore makes a one-example dry run and\n",
    "then initializes the materialized parameters inside `torch.no_grad()`.\n",
    "This lazy shape inference pays off in deeper networks (convolutional layers,\n",
    "variable-length sequences), where computing the input size of each layer by hand\n",
    "would be tedious and error-prone."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<!-- d2l:prose id=linear-regression-concise-md-13 fw=all -->\n",
    "\n",
    "## Exercises\n",
    "\n",
    "1. The framework loss functions used above (e.g., `nn.MSELoss`) return the mean loss over the minibatch by default. How would you need to change the learning rate if instead you replaced this average with the *sum* of the losses over the minibatch (e.g., by passing `reduction='sum'`)?\n",
    "1. Review the framework documentation to see which loss functions are provided. In particular,\n",
    "   replace the squared loss with Huber's robust loss function. That is, use the loss function\n",
    "   $$l(y,y') = \\begin{cases}|y-y'| -\\frac{\\sigma}{2} & \\textrm{ if } |y-y'| > \\sigma \\\\ \\frac{1}{2 \\sigma} (y-y')^2 & \\textrm{ otherwise}\\end{cases}$$\n",
    "   Rerun the outlier demonstration of that section (one corrupted label) with it: does Huber's loss recover the robust estimate, the least-squares one, or something in between? (Compare the penalty curves in the figure.)\n",
    "1. How do you access the gradient of the weights of the model?\n",
    "1. What is the effect on the solution if you change the learning rate and the number of epochs? Does it keep on improving?\n",
    "1. How does the solution change as you vary the amount of data generated?\n",
    "    1. Plot the estimation error for $\\hat{\\mathbf{w}} - \\mathbf{w}$ and $\\hat{b} - b$ as a function of the amount of data. Hint: increase the amount of data logarithmically rather than linearly, i.e., 5, 10, 20, 50, ..., 10,000 rather than 1000, 2000, ..., 10,000.\n",
    "    2. Why is the suggestion in the hint appropriate?\n",
    "1. Time the from-scratch implementation of that section against the concise one here, training each for 10, 100, and 1,000 epochs on the same synthetic dataset. Which is faster, and does the gap grow with the number of epochs? What does this tell you about the overhead of Python-level parameter bookkeeping versus framework-optimized operations?"
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    "<!-- d2l:prose id=linear-regression-concise-md-14 fw=pytorch -->\n",
    "\n",
    "[Discussions](https://d2l.discourse.group/t/45)"
   ]
  }
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