Tutorial 1: Gradient Descent and AutoGrad

Tutorial 1: Gradient Descent and AutoGrad#

Week 1, Day 2: Linear Deep Learning

By Neuromatch Academy

Content creators: Saeed Salehi, Vladimir Haltakov, Andrew Saxe

Content reviewers: Polina Turishcheva, Antoine De Comite, Kelson Shilling-Scrivo

Content editors: Anoop Kulkarni, Spiros Chavlis

Production editors: Khalid Almubarak, Gagana B, Spiros Chavlis

Tutorial Objectives#

Day 2 Tutorial 1 will continue on buiding PyTorch skillset and motivate its core functionality: Autograd. In this notebook, we will cover the key concepts and ideas of:

Gradient descent
PyTorch Autograd
PyTorch nn module

Setup#

This a GPU-Free tutorial!

Install and import feedback gadget#

# Imports
import torch
import numpy as np
from torch import nn
from math import pi
import matplotlib.pyplot as plt

Figure settings#

Plotting functions#

Set random seed#

Executing set_seed(seed=seed) you are setting the seed

Set device (GPU or CPU). Execute `set_device()`#

SEED = 2021
set_seed(seed=SEED)
DEVICE = set_device()

Random seed 2021 has been set.
GPU is not enabled in this notebook. 
If you want to enable it, in the menu under `Runtime` -> 
`Hardware accelerator.` and select `GPU` from the dropdown menu

Section 0: Introduction#

Today, we will go through 3 tutorials.

Starting with Gradient Descent, the workhorse of deep learning algorithms.
The second tutorial will help us build a better intuition about neural networks and basic hyper-parameters.
Finally, in tutorial 3, we learn about the learning dynamics, what the (a good) deep network is learning, and why sometimes they may perform poorly.

Video 0: Introduction#

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Section 1: Gradient Descent Algorithm#

Time estimate: ~30-45 mins

Since the goal of most learning algorithms is minimizing the risk (also known as the cost or loss) function, optimization is often the core of most machine learning techniques! The gradient descent algorithm, along with its variations such as stochastic gradient descent, is one of the most powerful and popular optimization methods used for deep learning. Today we will introduce the basics, but you will learn much more about Optimization in the coming days (Week 1, Day 4).

Section 1.1: Gradients & Steepest Ascent#

Video 1: Gradient Descent#

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Before introducing the gradient descent algorithm, let’s review a very important property of gradients. The gradient of a function always points in the direction of the steepest ascent. The following exercise will help clarify this.

Analytical Exercise 1.1: Gradient vector (Optional)#

Given the following function:

(9)#\[\begin{equation} z = h(x, y) = \sin(x^2 + y^2) \end{equation}\]

find the gradient vector:

(10)#\[\begin{equation} \begin{bmatrix} \dfrac{\partial z}{\partial x} \\ \\ \dfrac{\partial z}{\partial y} \end{bmatrix} \end{equation}\]

Hint: Use the chain rule!

Chain rule: For a composite function \(F(x) = g(h(x)) \equiv (g \circ h)(x)\):

(11)#\[\begin{equation} F'(x) = g'(h(x)) \cdot h'(x) \end{equation}\]

or differently denoted:

(12)#\[\begin{equation} \frac{dF}{dx} = \frac{dg}{dh} ~ \frac{dh}{dx} \end{equation}\]

Click here for the solution

We can rewrite the function as a composite function:

(13)#\[\begin{equation} z = f\left( g(x,y) \right), ~~ f(u) = \sin(u), ~~ g(x, y) = x^2 + y^2 \end{equation}\]

Using the chain rule:

(14)#\[\begin{align} \dfrac{\partial z}{\partial x} &= \dfrac{\partial f}{\partial g} \dfrac{\partial g}{\partial x} = \cos(g(x,y)) ~ (2x) = \cos(x^2 + y^2) \cdot 2x \\ \\ \dfrac{\partial z}{\partial y} &= \dfrac{\partial f}{\partial g} \dfrac{\partial g}{\partial y} = \cos(g(x,y)) ~ (2y) = \cos(x^2 + y^2) \cdot 2y \end{align}\]

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Coding Exercise 1.1: Gradient Vector#

Implement (complete) the function which returns the gradient vector for \(z=\sin(x^2 + y^2)\).

def fun_z(x, y):
  """
  Implements function sin(x^2 + y^2)

  Args:
    x: (float, np.ndarray)
      Variable x
    y: (float, np.ndarray)
      Variable y

  Returns:
    z: (float, np.ndarray)
      sin(x^2 + y^2)
  """
  z = np.sin(x**2 + y**2)
  return z


def fun_dz(x, y):
  """
  Implements function sin(x^2 + y^2)

  Args:
    x: (float, np.ndarray)
      Variable x
    y: (float, np.ndarray)
      Variable y

  Returns:
    Tuple of gradient vector for sin(x^2 + y^2)
  """
  #################################################
  ## Implement the function which returns gradient vector
  ## Complete the partial derivatives dz_dx and dz_dy
  # Complete the function and remove or comment the line below
  raise NotImplementedError("Gradient function `fun_dz`")
  #################################################
  dz_dx = ...
  dz_dy = ...
  return (dz_dx, dz_dy)


## Uncomment to run
# ex1_plot(fun_z, fun_dz)

Click for solution

We can see from the plot that for any given \(x_0\) and \(y_0\), the gradient vector \(\left[ \dfrac{\partial z}{\partial x}, \dfrac{\partial z}{\partial y}\right]^{\top}_{(x_0, y_0)}\) points in the direction of \(x\) and \(y\) for which \(z\) increases the most. It is important to note that gradient vectors only see their local values, not the whole landscape! Also, length (size) of each vector, which indicates the steepness of the function, can be very small near local plateaus (i.e. minima or maxima).

Thus, we can simply use the aforementioned formula to find the local minima.

In 1847, Augustin-Louis Cauchy used negative of gradients to develop the Gradient Descent algorithm as an iterative method to minimize a continuous and (ideally) differentiable function of many variables.

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Video 2: Gradient Descent - Discussion#

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Section 1.2: Gradient Descent Algorithm#

Let \(f(\mathbf{w}): \mathbb{R}^d \rightarrow \mathbb{R}\) be a differentiable function. Gradient Descent is an iterative algorithm for minimizing the function \(f\), starting with an initial value for variables \(\mathbf{w}\), taking steps of size \(\eta\) (learning rate) in the direction of the negative gradient at the current point to update the variables \(\mathbf{w}\).

(15)#\[\begin{equation} \mathbf{w}^{(t+1)} = \mathbf{w}^{(t)} - \eta \nabla f \left( \mathbf{w}^{(t)} \right) \end{equation}\]

where \(\eta > 0\) and \(\nabla f (\mathbf{w})= \left( \frac{\partial f(\mathbf{w})}{\partial w_1}, ..., \frac{\partial f(\mathbf{w})}{\partial w_d} \right)\). Since negative gradients always point locally in the direction of steepest descent, the algorithm makes small steps at each point towards the minimum.

Vanilla Algorithm

Inputs: initial guess \(\mathbf{w}^{(0)}\), step size \(\eta > 0\), number of steps \(T\).

For \(t = 0, 1, 2, \dots , T-1\) do
\(\qquad\) \(\mathbf{w}^{(t+1)} = \mathbf{w}^{(t)} - \eta \nabla f \left( \mathbf{w}^{(t)} \right)\)
end

Return: \(\mathbf{w}^{(t+1)}\)

Hence, all we need is to calculate the gradient of the loss function with respect to the learnable parameters (i.e., weights):

(16)#\[\begin{equation} \dfrac{\partial Loss}{\partial \mathbf{w}} = \left[ \dfrac{\partial Loss}{\partial w_1}, \dfrac{\partial Loss}{\partial w_2} , \dots, \dfrac{\partial Loss}{\partial w_d} \right]^{\top} \end{equation}\]

Analytical Exercise 1.2: Gradients#

Given \(f(x, y, z) = \tanh \left( \ln \left[1 + z \frac{2x}{sin(y)} \right] \right)\), how easy is it to derive \(\dfrac{\partial f}{\partial x}\), \(\dfrac{\partial f}{\partial y}\) and \(\dfrac{\partial f}{\partial z}\)?

Hint: You don’t have to actually calculate them!

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Section 1.3: Computational Graphs and Backprop#

Video 3: Computational Graph#

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Exercise 1.2 is an example of how overwhelming the derivation of gradients can get, as the number of variables and nested functions increases. This function is still extraordinarily simple compared to the loss functions of modern neural networks. So how can we (as well as PyTorch and similar frameworks) approach such beasts?

Let’s look at the function again:

(17)#\[\begin{equation} f(x, y, z) = \tanh \left(\ln \left[1 + z \frac{2x}{sin(y)} \right] \right) \end{equation}\]

We can build a so-called computational graph (shown below) to break the original function into smaller and more approachable expressions.

Starting from \(x\), \(y\), and \(z\) and following the arrows and expressions, you would see that our graph returns the same function as \(f\). It does so by calculating intermediate variables \(a,b,c,d,\) and \(e\). This is called the forward pass.

Now, let’s start from \(f\), and work our way against the arrows while calculating the gradient of each expression as we go. This is called the backward pass, from which the backpropagation of errors algorithm gets its name.

By breaking the computation into simple operations on intermediate variables, we can use the chain rule to calculate any gradient:

(18)#\[\begin{equation} \dfrac{\partial f}{\partial x} = \dfrac{\partial f}{\partial e}~\dfrac{\partial e}{\partial d}~\dfrac{\partial d}{\partial c}~\dfrac{\partial c}{\partial a}~\dfrac{\partial a}{\partial x} = \left( 1-\tanh^2(e) \right) \cdot \frac{1}{d+1}\cdot z \cdot \frac{1}{b} \cdot 2 \end{equation}\]

Conveniently, the values for \(e\), \(b\), and \(d\) are available to us from when we did the forward pass through the graph. That is, the partial derivatives have simple expressions in terms of the intermediate variables \(a,b,c,d,e\) that we calculated and stored during the forward pass.

Analytical Exercise 1.3: Chain Rule (Optional)#

For the function above, calculate the \(\dfrac{\partial f}{\partial y}\) using the computational graph and chain rule.

Click here for the solution

(19)#\[\begin{equation} \dfrac{\partial f}{\partial y} = \dfrac{\partial f}{\partial e}~\dfrac{\partial e}{\partial d}~\dfrac{\partial d}{\partial c}~\dfrac{\partial c}{\partial b}~\dfrac{\partial b}{\partial y} = \left( 1-\tanh^2(e) \right) \cdot \frac{1}{d+1}\cdot z \cdot \frac{-a}{b^2} \cdot \cos(y) \end{equation}\]

For more: Calculus on Computational Graphs: Backpropagation

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Section 2: PyTorch AutoGrad#

Time estimate: ~30-45 mins

Video 4: Auto-Differentiation#

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Deep learning frameworks such as PyTorch, JAX, and TensorFlow come with a very efficient and sophisticated set of algorithms, commonly known as Automatic Differentiation. AutoGrad is PyTorch’s automatic differentiation engine. Here we start by covering the essentials of AutoGrad, and you will learn more in the coming days.

Section 2.1: Forward Propagation#

Everything starts with the forward propagation (pass). PyTorch tracks all the instructions, as we declare the variables and operations, and it builds the graph when we call the .backward() pass. PyTorch rebuilds the graph every time we iterate or change it (or simply put, PyTorch uses a dynamic graph).

For gradient descent, it is only required to have the gradients of cost function with respect to the variables we wish to learn. These variables are often called “learnable / trainable parameters” or simply “parameters” in PyTorch. In neural nets, weights and biases are often the learnable parameters.

Coding Exercise 2.1: Buiding a Computational Graph#

In PyTorch, to indicate that a certain tensor contains learnable parameters, we can set the optional argument requires_grad to True. PyTorch will then track every operation using this tensor while configuring the computational graph. For this exercise, use the provided tensors to build the following graph, which implements a single neuron with scalar input and output.

class SimpleGraph:
  """
  Implementing Simple Computational Graph
  """

  def __init__(self, w, b):
    """
    Initializing the SimpleGraph

    Args:
      w: float
        Initial value for weight
      b: float
        Initial value for bias

    Returns:
      Nothing
    """
    assert isinstance(w, float)
    assert isinstance(b, float)
    self.w = torch.tensor([w], requires_grad=True)
    self.b = torch.tensor([b], requires_grad=True)

  def forward(self, x):
    """
    Forward pass

    Args:
      x: torch.Tensor
        1D tensor of features

    Returns:
      prediction: torch.Tensor
        Model predictions
    """
    assert isinstance(x, torch.Tensor)
    #################################################
    ## Implement the the forward pass to calculate prediction
    ## Note that prediction is not the loss, but the value after `tanh`
    # Complete the function and remove or comment the line below
    raise NotImplementedError("Forward Pass `forward`")
    #################################################
    prediction = ...
    return prediction


def sq_loss(y_true, y_prediction):
  """
  L2 loss function

  Args:
    y_true: torch.Tensor
      1D tensor of target labels
    y_prediction: torch.Tensor
      1D tensor of predictions

  Returns:
    loss: torch.Tensor
      L2-loss (squared error)
  """
  assert isinstance(y_true, torch.Tensor)
  assert isinstance(y_prediction, torch.Tensor)
  #################################################
  ## Implement the L2-loss (squred error) given true label and prediction
  # Complete the function and remove or comment the line below
  raise NotImplementedError("Loss function `sq_loss`")
  #################################################
  loss = ...
  return loss



feature = torch.tensor([1])  # Input tensor
target = torch.tensor([7])  # Target tensor
## Uncomment to run
# simple_graph = SimpleGraph(-0.5, 0.5)
# print(f"initial weight = {simple_graph.w.item()}, "
#       f"\ninitial bias = {simple_graph.b.item()}")
# prediction = simple_graph.forward(feature)
# square_loss = sq_loss(target, prediction)
# print(f"for x={feature.item()} and y={target.item()}, "
#       f"prediction={prediction.item()}, and L2 Loss = {square_loss.item()}")

Click for solution

It is important to appreciate the fact that PyTorch can follow our operations as we arbitrarily go through classes and functions.

Submit your feedback#

Section 2.2: Backward Propagation#

Here is where all the magic lies. In PyTorch, Tensor and Function are interconnected and build up an acyclic graph, that encodes a complete history of computation. Each variable has a grad_fn attribute that references a function that has created the Tensor (except for Tensors created by the user - these have None as grad_fn). The example below shows that the tensor c = a + b is created by the Add operation and the gradient function is the object <AddBackward...>. Replace + with other single operations (e.g., c = a * b or c = torch.sin(a)) and examine the results.

a = torch.tensor([1.0], requires_grad=True)
b = torch.tensor([-1.0], requires_grad=True)
c = a + b
print(f'Gradient function = {c.grad_fn}')

Gradient function = <AddBackward0 object at 0x7f82b6364b50>

For more complex functions, printing the grad_fn would only show the last operation, even though the object tracks all the operations up to that point:

print(f'Gradient function for prediction = {prediction.grad_fn}')
print(f'Gradient function for loss = {square_loss.grad_fn}')

Now let’s kick off the backward pass to calculate the gradients by calling .backward() on the tensor we wish to initiate the backpropagation from. Often, .backward() is called on the loss, which is the last node on the graph. Before doing that, let’s calculate the loss gradients by hand:

\[\frac{\partial{loss}}{\partial{w}} = - 2 x (y_t - y_p)(1 - y_p^2)\]

\[\frac{\partial{loss}}{\partial{b}} = - 2 (y_t - y_p)(1 - y_p^2)\]

Where \(y_t\) is the target (true label), and \(y_p\) is the prediction (model output). We can then compare it to PyTorch gradients, which can be obtained by calling .grad on the relevant tensors.

Important Notes:

Learnable parameters (i.e. requires_grad tensors) are “contagious”. Let’s look at a simple example: Y = W @ X, where X is the feature tensors and W is the weight tensor (learnable parameters, requires_grad), the newly generated output tensor Y will be also requires_grad. So any operation that is applied to Y will be part of the computational graph. Therefore, if we need to plot or store a tensor that is requires_grad, we must first .detach() it from the graph by calling the .detach() method on that tensor.
.backward() accumulates gradients in the leaf nodes (i.e., the input nodes to the node of interest). We can call .zero_grad() on the loss or optimizer to zero out all .grad attributes (see autograd.backward for more information).
Recall that in python we can access variables and associated methods with .method_name. You can use the command dir(my_object) to observe all variables and associated methods to your object, e.g., dir(simple_graph.w).

# Analytical gradients (Remember detaching)
ana_dloss_dw = - 2 * feature * (target - prediction.detach())*(1 - prediction.detach()**2)
ana_dloss_db = - 2 * (target - prediction.detach())*(1 - prediction.detach()**2)

square_loss.backward()  # First we should call the backward to build the graph
autograd_dloss_dw = simple_graph.w.grad  # We calculate the derivative w.r.t weights
autograd_dloss_db = simple_graph.b.grad  # We calculate the derivative w.r.t bias

print(ana_dloss_dw == autograd_dloss_dw)
print(ana_dloss_db == autograd_dloss_db)

References and more#

Section 3: PyTorch’s Neural Net module (`nn.Module`)#

Time estimate: ~30 mins

Video 5: PyTorch `nn` module#

Submit your feedback#

PyTorch provides us with ready-to-use neural network building blocks, such as layers (e.g., linear, recurrent, etc.), different activation and loss functions, and much more, packed in the torch.nn module. If we build a neural network using torch.nn layers, the weights and biases are already in requires_grad mode and will be registered as model parameters.

For training, we need three things:

Model parameters: Model parameters refer to all the learnable parameters of the model, which are accessible by calling .parameters() on the model. Please note that NOT all the requires_grad tensors are seen as model parameters. To create a custom model parameter, we can use nn.Parameter (A kind of Tensor that is to be considered a module parameter).
Loss function: The loss that we are going to be optimizing, which is often combined with regularization terms (coming up in few days).
Optimizer: PyTorch provides us with many optimization methods (different versions of gradient descent). Optimizer holds the current state of the model and by calling the step() method, will update the parameters based on the computed gradients.

You will learn more details about choosing the right model architecture, loss function, and optimizer later in the course.

Section 3.1: Training loop in PyTorch#

We use a regression problem to study the training loop in PyTorch.

The task is to train a wide nonlinear (using \(\tanh\) activation function) neural net for a simple \(\sin\) regression task. Wide neural networks are thought to be really good at generalization.

Generate the sample dataset#

Random seed 2021 has been set.

../../../_images/2a86f0d5f59f4ce91f9955c72a6b23dc94ea876fc850976da4d611d7eaa13b7e.png

Let’s define a very wide (512 neurons) neural net with one hidden layer and nn.Tanh() activation function.

class WideNet(nn.Module):
  """
   A Wide neural network with a single hidden layer
   Structure is as follows:
   nn.Sequential(
        nn.Linear(1, n_cells) + nn.Tanh(), # Fully connected layer with tanh activation
        nn.Linear(n_cells, 1) # Final fully connected layer
    )
  """

  def __init__(self):
    """
    Initializing the parameters of WideNet

    Args:
      None

    Returns:
      Nothing
    """
    n_cells = 512
    super().__init__()
    self.layers = nn.Sequential(
        nn.Linear(1, n_cells),
        nn.Tanh(),
        nn.Linear(n_cells, 1),
    )

  def forward(self, x):
    """
    Forward pass of WideNet

    Args:
      x: torch.Tensor
        2D tensor of features

    Returns:
      Torch tensor of model predictions
    """
    return self.layers(x)

We can now create an instance of our neural net and print its parameters.

# Creating an instance
set_seed(seed=SEED)
wide_net = WideNet()
print(wide_net)

Random seed 2021 has been set.
WideNet(
  (layers): Sequential(
    (0): Linear(in_features=1, out_features=512, bias=True)
    (1): Tanh()
    (2): Linear(in_features=512, out_features=1, bias=True)
  )
)

# Create a mse loss function
loss_function = nn.MSELoss()

# Stochstic Gradient Descent optimizer (you will learn about momentum soon)
lr = 0.003  # Learning rate
sgd_optimizer = torch.optim.SGD(wide_net.parameters(), lr=lr, momentum=0.9)

The training process in PyTorch is interactive - you can perform training iterations as you wish and inspect the results after each iteration.

Let’s perform one training iteration. You can run the cell multiple times and see how the parameters are being updated and the loss is reducing. This code block is the core of everything to come: please make sure you go line-by-line through all the commands and discuss their purpose with your pod.

# Reset all gradients to zero
sgd_optimizer.zero_grad()

# Forward pass (Compute the output of the model on the features (inputs))
prediction = wide_net(inputs)

# Compute the loss
loss = loss_function(prediction, targets)
print(f'Loss: {loss.item()}')

# Perform backpropagation to build the graph and compute the gradients
loss.backward()

# Optimizer takes a tiny step in the steepest direction (negative of gradient)
# and "updates" the weights and biases of the network
sgd_optimizer.step()

Loss: 0.888942301273346

Coding Exercise 3.1: Training Loop#

Using everything we’ve learned so far, we ask you to complete the train function below.

def train(features, labels, model, loss_fun, optimizer, n_epochs):
  """
  Training function

  Args:
    features: torch.Tensor
      Features (input) with shape torch.Size([n_samples, 1])
    labels: torch.Tensor
      Labels (targets) with shape torch.Size([n_samples, 1])
    model: torch nn.Module
      The neural network
    loss_fun: function
      Loss function
    optimizer: function
      Optimizer
    n_epochs: int
      Number of training iterations

  Returns:
    loss_record: list
      Record (evolution) of training losses
  """
  loss_record = []  # Keeping recods of loss

  for i in range(n_epochs):
    #################################################
    ## Implement the missing parts of the training loop
    # Complete the function and remove or comment the line below
    raise NotImplementedError("Training loop `train`")
    #################################################
    ...  # Set gradients to 0
    predictions = ...  # Compute model prediction (output)
    loss = ...  # Compute the loss
    ...  # Compute gradients (backward pass)
    ...  # Update parameters (optimizer takes a step)

    loss_record.append(loss.item())
  return loss_record



set_seed(seed=2021)
epochs = 1847 # Cauchy, Exercices d'analyse et de physique mathematique (1847)
## Uncomment to run
# losses = train(inputs, targets, wide_net, loss_function, sgd_optimizer, epochs)
# ex3_plot(wide_net, inputs, targets, epochs, losses)

Random seed 2021 has been set.

Click for solution

Submit your feedback#

Summary#

In this tutorial, we covered one of the most basic concepts of deep learning; the computational graph and how a network learns via gradient descent and the backpropagation algorithm. We have seen all of these using PyTorch modules and we compared the analytical solutions with the ones provided directly by the PyTorch module.

Tutorial 1: Gradient Descent and AutoGrad

Contents

Tutorial 1: Gradient Descent and AutoGrad#

Tutorial Objectives#

Setup#

Install and import feedback gadget#

Figure settings#

Plotting functions#

Set random seed#

Set device (GPU or CPU). Execute set_device()#

Section 0: Introduction#

Video 0: Introduction#

Submit your feedback#

Section 1: Gradient Descent Algorithm#

Section 1.1: Gradients & Steepest Ascent#

Video 1: Gradient Descent#

Submit your feedback#

Analytical Exercise 1.1: Gradient vector (Optional)#

Submit your feedback#

Coding Exercise 1.1: Gradient Vector#

Submit your feedback#

Video 2: Gradient Descent - Discussion#

Submit your feedback#

Section 1.2: Gradient Descent Algorithm#

Analytical Exercise 1.2: Gradients#

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Section 1.3: Computational Graphs and Backprop#

Video 3: Computational Graph#

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Analytical Exercise 1.3: Chain Rule (Optional)#

Submit your feedback#

Section 2: PyTorch AutoGrad#

Video 4: Auto-Differentiation#

Submit your feedback#

Section 2.1: Forward Propagation#

Coding Exercise 2.1: Buiding a Computational Graph#

Submit your feedback#

Section 2.2: Backward Propagation#

References and more#

Section 3: PyTorch’s Neural Net module (nn.Module)#

Video 5: PyTorch nn module#

Submit your feedback#

Section 3.1: Training loop in PyTorch#

Generate the sample dataset#

Coding Exercise 3.1: Training Loop#

Submit your feedback#

Summary#

Video 6: Tutorial 1 Wrap-up#

Submit your feedback#

Set device (GPU or CPU). Execute `set_device()`#

Section 3: PyTorch’s Neural Net module (`nn.Module`)#

Video 5: PyTorch `nn` module#