Fundamentals of PyTorch – Introduction

Since it was introduced by the Facebook AI Research (FAIR) team, back in early 2017, PyTorch has become a highly popular and widely used Deep Learning (DL) framework. Since the humble beginning, it has caught the attention of serious AI researchers and practitioners around the world, both in industry and academia, and has matured significantly over the years.

Scores of DL enthusiasts and professionals started their journey with the Google TensorFlow (TF), but the learning curve with base TensorFlow has always been steep. On the other hand, PyTorch has approached DL programming in an intuitive fashion since the beginning, focusing on fundamental linear algebra and data flow operations in a manner that is easily understood and amenable to step-by-step learning.

Due to this modular approach, building and experimenting with complex DL architectures has been much easier with PyTorch than following the somewhat rigid framework of TF and TF-based tools. Moreover, PyTorch was built to integrate seamlessly with the numerical computing infrastructure of the Python ecosystem and Python being the lingua franca of data science and machine learning, it has ridden over that wave of increasing popularity.

PyTorch is a constantly developing DL framework with many exciting additions and features. In this article, we will go over some of the basic elements and show an example of building a simple Deep Neural Network (DNN) step-by-step.

Tensor Operations with PyTorch

Tensors are at the heart of any DL framework. PyTorch provides tremendous flexibility to a programmer about how to create, combine, and process tensors as they flow through a network (called computational graph) paired with a relatively high-level, object-oriented API.

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What are Tensors?

Representing data (e.g. about the physical world or some business process) for Machine Learning (ML), in particular for DNN, is accomplished via a data/mathematical structure known as the tensor. A tensor is a container which can house data in N dimensions. A tensor is often used interchangeably with another more familiar mathematical object matrix (which is specifically a 2-dimensional tensor). In fact, tensors are generalizations of 2-dimensional matrices to N-dimensional space.

In simplistic terms, one can think of scalar-vectors-matrices- tensors as a flow.

  • Scalar are 0-dimensional tensors.
  • Vectors are 1-dimensional tensors.
  • Matrices are 2-dimensional tensors
  • Tensors are generalized N-dimensional tensors. N can be anything from 3 to infinity…

 

Often, these dimensions are also called ranks.

Tensors of various dimensions | PyTorch | Exxact Corp

Fig 1: Tensors of various dimensions (ranks) (Image source)

 

Why Are Tensors Important for ML and DL?

Think of a supervised ML problem. You are given a table of data with some labels (could be numerical entities or binary classification such as Yes/No answers). For ML algorithms to process it, the data must be fed as a mathematical object. A table is naturally equivalent to a 2-D matrix where an individual row (or instance), or individual column (or feature) can be treated as 1-D vector.

Similarly, a black-and-white image can be treated as a 2-D matrix containing numbers 0 or 1. This can be fed into a neural network for image classification or segmentation tasks. 

A time-series or sequence data (e.g. ECG data from a monitoring machine or a stock market price tracking data stream) is another example of 2-D data where one dimension (time) is fixed.

These are examples of using 2-D tensors in classical ML (e.g. linear regression, support vector machines, decision trees, etc.) and DL algorithms.

Going beyond 2-D, a color or grayscale image can be treated as a 3-D tensor where each pixel is associated with a so-called ‘color-channel’ – a vector of 3 numbers representing intensities in the Red-Green-Blue (RGB) spectrum. This is an example of a 3-D tensor.

Similarly, videos can be thought of as sequences of color images (or frames) in time and can be thought of as 4-D tensors.

In short, all kinds of data from the physical word, sensors and instruments, business and finance, scientific or social experiments, can be easily represented by multi-dimensional tensors to make them amenable for processing by ML/DL algorithms inside a computing machine.

Let’s see how PyTorch defines and handles tensors.

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Creating and Converting Tensors in PyTorch

Tensors can be defined from a Python list as follows,

Converting Tensors in PyTorch | PyTorch | Exxact Corp

Actual elements can be accessed and indexed as follows,

Accessing Element and Data Type | PyTorch | Exxact Corp

Tensors with specific data types can be created easily (e.g. floating points),

Tensors with Specific Data Types | PyTorch | Exxact Corp

Size and dimensions can be read easily,

Size and Dimensions Reading | PyTorch | Exxact Corp 

We can change the view of a tensor. Let us start with a 1-dimensional tensor as follows,

Changing the view in the tensor | PyTorch | Exxact Corp

Then change the view to a 2-D tensor,

2-D Tensor | PyTorch | Exxact Corp
Changing back and forth between a PyTorch tensor and a NumPy array is easy and efficient.

Changing tensor and NumPy | PyTorch | Exxact Corp

Converting from a Pandas series object is also easy,

Conversion from a Panda Series | PyTorch | Exxact Corp
Finally, converting back to a Python list can be accomplished,

Converting back to a list | PyTorch | Exxact Corp

Deep Learning Computers

Vector and matrix mathematics with PyTorch tensors

PyTorch provides an easy-to-understand API and programmatic toolbox to manipulate tensors mathematically. We show basic operations with 1-D and 2-D tensors here.

Simple vector addition,

Vector addition | PyTorch | Exxact Corp
Vector multiplication with a scalar,

Vector multiplication | PyTorch | Exxact Corp

Linear combination,

Linear Combination | PyTorch | Exxact Corp

Element-wise product,

Product of two tensors | PyTorch | Exxact Corp
Dot product,

PyTorch | Exxact Corp

Adding a scalar to every element of a tensor, i.e. broadcasting

PyTorch | Exxact Corp
Creating 2-D tensor from list of lists,

PyTorch | Exxact Corp

Slicing and indexing of matrix elements,

PyTorch | Exxact Corp

Matrix multiplication,

PyTorch | Exxact Corp
Matrix transpose,

PyTorch | Exxact Corp
Matrix inverse and determinant,

PyTorch | Exxact Corp

Autograd: Automatic differentiation

Neural network training and prediction involves taking derivatives of various functions (tensor-valued) over and over. The Tensor object supports the magical Autograd feature i.e. automatic differentiation which is achieved by tracking and storing all the operations performed on the Tensor while it flows through a network. You can watch this wonderful tutorial video for a visual explanation.

https://youtu.be/MswxJw-8PvE

The Pytorch autograd official documentation is here.

We show simple examples to illustrate the autograd feature of PyTorch.

PyTorch | Exxact Corp

We define a generic function and a tensor variable x, then define another variable y assigning it to the function of x.

PyTorch | Exxact Corp

Then, we use a special backward() method on y to take the derivative and calculate the derivative value at the given value of x.

PyTorch | Exxact Corp

We can also deal with partial derivatives!

PyTorch | Exxact Corp

We can define u and v as tensor variables, define a function combining them, apply the backward() method and calculate the partial derivatives. See below,

PyTorch | Exxact Corp

PyTorch computes derivatives of scalar functions only but if we pass a vector then essentially it computes derivatives element wise and stores them in an array of the same dimension.

PyTorch | Exxact Corp

The following code will calculate the derivative with respect to the three constituent vectors.

PyTorch | Exxact Corp

We can show the plot of the derivative. Note, a derivative of a quadratic function is a straight-line, tangent to the parabolic curve.

Quadratic Function | PyTorch | Exxact Corp

Building a Full-Fledged Neural Network

Apart from the tensors and automatic differentiation ability, there are few more core components/features of PyTorch that come together for a deep neural network definition.

The core components of PyTorch that will be used for building the neural classifier are,

  • The Tensor (the central data structure in PyTorch)
  • The Autograd feature of the Tensor (automatic differentiation formula baked into the
  • The nn.Module class, that is used to build any other neural classifier class
  • The Optimizer (of course, there are many of them to choose from)
  • The Loss function (a big selection is available for your choice)

We have already described in detail the Tensor and the Autograd. Let us quickly discuss the other components,

The nn.Module Class

In PyTorch, we construct a neural network by defining it as a custom class. However, instead of deriving from the native Python object this class inherits from the nn.Module class. This imbues the neural net class with useful properties and powerful methods. This way, full power of the Object-Oriented-Programming (OOP) can be maintained while working with neural net models. We will see a full example of such a class definition in our article.

The Loss Function

In a neural network architecture and operation, the loss functions define how far the final prediction of the neural net is from the ground truth (given labels/classes or data for supervised training). The quantitative measure of loss helps drive the network to move closer to the configuration (the optimal settings of the weights of the neurons) which classifies the given dataset best or predicts the numerical output with least total error.

PyTorch offers all the usual loss functions for classification and regression tasks —

  • binary and multi-class cross-entropy,
  • mean squared and mean absolute errors,
  • smooth L1 loss,
  • neg log-likelihood loss, and even
  • Kullback-Leibler divergence.

A detailed discussion of these can be found in this article.

Loss Functions PyTorch | PyTorch | Exxact Corp

The Optimizer

Optimization of the weights to achieve the lowest loss is at the heart of the backpropagation algorithm for training a neural network. PyTorch offers a plethora of optimizers to do the job, exposed through the torch.optim module —

  • Stochastic gradient descent (SGD),
  • Adam, Adadelta, Adagrad, SpareAdam,
  • L-BFGS,
  • RMSprop, etc.

Look at this article to know more about activation functions and optimizers, used in modern deep neural networks.

Activation Functions | PyTorch | Exxact Corp

The Five-Step-Process

Using these components, we will build the classifier in five simple steps,

  • Construct our neural network as our custom class (inherited from the nn.Module class), complete with hidden layer tensors and forward method for propagating the input tensor through various layers and activation function
  • Propagate the feature (from a dataset) tensor through the network using this forward() method — say we get an output tensor as a result
  • Calculate the loss by comparing the output to the ground truth and using built-in loss functions
  • Propagate the gradient of the loss using the automatic differentiation ability (Autograd) with the backward method
  • Update the weights of the network using the gradient of the loss — this is accomplished by executing one step of the so-called optimizer — optimizer.step().

And that’s it. This five-step process constitutes one complete epoch of training. We just repeat it a bunch of times to drive down the loss and obtain high classification accuracy.

The idea looks like following,

Five Step Process | PyTorch | Exxact Corp

Hands-on example

Let’s suppose we want to build and train the following 2-layer neural network.

2-Layer Neural Network | PyTorch | Exxact Corp

We start with the class definition,

Class Definition | PyTorch | Exxact Corp

We can define a variable as an object belonging to this class and print the summary.

Print Model | PyTorch | Exxact Corp

We choose the Binary cross-entropy loss,

Binary Cross-Entropy | PyTorch | Exxact Corp

Let us run the input dataset through the neural net model we have defined i.e. forward pass once and compute the output probabilities. As the weights have been initialized as random, we will see random output probabilities (mostly close to 0.5). This network has not been trained yet.

Logits | PyTorch | Exxact Corp

Output Probabilities | PyTorch | Exxact Corp

We define the optimizer,

Optimizer | PyTorch | Exxact Corp

Next, we show how to do forward and backward passes with one step of optimizer. This set of code can be found at the heart of any PyTorch neural net model. We follow another five-step process,

  • reset the gradients to zero (to prevent the accumulation of grads)
  • forward pass the tensors through the layers
  • calculate the loss tensor
  • calculate the gradients of the loss
  • update the weights by incrementing the optimizer by one step (in the direction of the negative gradient)

The five steps above are exactly what you can observe and read about in all the theoretical discussion (and in the textbooks) on neural nets and deep learning. And, with PyTorch, you are able to implement this process with deceptively simple code, step-by-step.

The code is shown below,

PyTorch Fundamentals | Exxact Corp

When we run the same type of code over a loop (for multiple epochs), we can observe the familiar loss-curve going down i.e. the neural network getting trained gradually.

Loss over epochs | Exxact Corp

After training for 200 epochs, we can look at the probability distributions again directly to see how the neural network output probabilities are now different (trying to match with the true data distributions).

Output Probabilities | Exxact Corp

Summary of PyTorch Fundamentals

PyTorch is a great package for reaching out to the heart of a neural net and customizing it for your application or trying out bold new ideas with the architecture, optimization, and mechanics of the network.

You can easily build complex interconnected networks, try out novel activation functions, mix and match custom loss functions, etc. The core ideas of computation graphs, easy auto-differentiation, and forward and backward flow of tensors will come in handy for any of your neural network definitions and optimization.

In this article, we summarized a few key steps which can be followed to quickly build a neural network for classification or regression tasks. We also showed how neat ideas can be easily tried out with this framework.

All the code for this article can be found here in this Github repo.

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