The spelled-out intro to neural networks and backpropagation: building micrograd

In this comprehensive lecture, Andre, an experienced deep neural network trainer, guides viewers through the intricacies of neural network training. He introduces Micrograd, a library he created that simplifies the complex process of backpropagation, a fundamental algorithm in neural network training. Andre demonstrates how to build mathematical expressions using Micrograd and visualizes these expressions as computation graphs. He then delves into the concept of derivatives, explaining their significance in adjusting neural network weights to minimize loss functions. The lecture includes a step-by-step implementation of the backward pass for backpropagation, highlighting the importance of the chain rule in calculus. Andre also discusses the efficiency gains from using tensors in neural network libraries like PyTorch, contrasting them with Micrograd's scalar-focused approach. The lecture concludes with a practical example of training a neural network using a binary classification dataset, emphasizing the iterative process of forward pass, loss calculation, backpropagation, and parameter updates in gradient descent.

Takeaways

• 📚 Andre, an experienced deep neural network trainer, introduces the concept of neural network training and the inner workings of backpropagation through a step-by-step guide.
• 🌟 The lecture demonstrates building Micrograd, a simplified autograd engine that implements backpropagation, which is fundamental to training modern neural networks.
• 🔧 Micrograd allows for the construction of mathematical expressions and the computation of their derivatives, which is crucial for understanding how changes in inputs affect the output.
• 💡 Through examples, Andre illustrates how to perform a forward pass to compute the output of a neural network and a backward pass to calculate the gradient of the loss function with respect to the network's weights.
• 📈 The process of backpropagation is shown to be a recursive application of the chain rule from calculus, which is used to evaluate the gradient of the loss function throughout the network.
• 🔎 Andre emphasizes the importance of understanding the mathematical principles behind neural network training, which remains unchanged even when using more efficient tensor operations in production environments.
• 🤖 Micrograd is used as an educational tool to build a simple neural network module by module, starting from individual neurons up to a multi-layer perceptron (MLP).
• 📉 The concept of loss functions is introduced as a measure of a neural network's performance, with the mean squared error loss being used to guide the network's training process.
• 🚀 The training loop, consisting of forward pass, backward pass (backpropagation), and parameter updates, is shown to iteratively improve the neural network's predictions by minimizing the loss.
• 🧠 Andre discusses the challenge of choosing the right learning rate for gradient descent, highlighting the need for a balance between fast convergence and stability in the training process.
• ✅ The final demonstration shows how the trained neural network can successfully classify a more complex dataset, visualizing the decision surface that separates different classes.

Q & A

What is the main focus of Andre's lecture?

The main focus of Andre's lecture is to demonstrate the inner workings of neural network training, specifically by building and explaining the Micrograd library, which implements backpropagation for neural networks.

What is Micrograd?

Micrograd is a lightweight, educational library for automatic gradient calculation, which is at the core of the backpropagation algorithm used in training neural networks. It was created to provide a clear understanding of how neural network training works under the hood.

How does backpropagation work?

Backpropagation is an algorithm that efficiently evaluates the gradient of a loss function with respect to the weights of a neural network. It does this by starting at the output and recursively applying the chain rule from calculus to compute the derivative of the loss with respect to all the weights, allowing for iterative tuning of the weights to minimize the loss.

Why is understanding the derivative important in the context of neural networks?

Understanding the derivative is crucial because it provides the information on how a small change in the input (weights in the case of neural networks) affects the output (loss function). This information is used to adjust the weights in a way that improves the network's predictions by minimizing the loss.

What is the significance of the 'Value' object in Micrograd?

The 'Value' object in Micrograd wraps a scalar value and tracks the mathematical operations performed on it. It is essential for building the expression graph that represents the neural network's computations and is used to compute and store the gradients during backpropagation.

How does Andre visualize the expression graph in Micrograd?

Andre uses a graph visualization library to create a visual representation of the expression graph. This graph shows how each value is derived from the inputs through a series of operations, which is helpful for understanding the flow of data and gradients within the neural network.

What is the role of the 'backward' function in the context of backpropagation?

The 'backward' function is used to initiate the backpropagation process. It takes the gradient of the loss with respect to the output of a node and recursively distributes these gradients to the node's children (previous operations) by applying the chain rule, thus computing the gradients with respect to all the weights in the network.

Why is it necessary to use a non-linear activation function like tanh in a neural network?

Non-linear activation functions like tanh are essential in neural networks because they introduce non-linearity into the model, allowing it to capture and learn complex patterns in the data. Without non-linearity, the network would essentially be limited to linear relationships, severely limiting its expressive power.

What is the purpose of the 'zero_grad' method in neural network training?

The 'zero_grad' method is used to reset the gradients of all parameters to zero before each backward pass. This is necessary to prevent the gradients from accumulating across multiple iterations, which would lead to incorrect updates to the parameters during the optimization step.

How does Andre demonstrate the effectiveness of the neural network training process?

Andre demonstrates the effectiveness of the neural network training process by showing how the network's predictions improve over time as the loss decreases. He uses a simple binary classification problem and iteratively updates the network's weights using gradient descent until the network accurately classifies the given examples.

What is the importance of the chain rule in calculus for backpropagation?

The chain rule in calculus is fundamental to backpropagation as it allows the computation of the derivative of the loss with respect to any parameter in the network by successively multiplying the local derivatives of the operations that led to the parameter. This enables the efficient calculation of gradients throughout the entire network.

Outlines

00:00

👋 Introduction to Neural Network Training with Micrograd

Andre introduces himself as a deep neural network trainer with over a decade of experience. He outlines the goal of the lecture: to provide an understanding of neural network training by building and training a neural network from scratch using a Jupyter notebook. Andre also mentions Micrograd, a library he created that implements backpropagation for neural networks, allowing for the efficient evaluation of the gradient of a loss function with respect to the network's weights. The lecture aims to demystify the inner workings of neural network training through building and understanding Micrograd.

05:01

🌟 Understanding Micrograd and Automatic Gradient

Andre explains that Micrograd is an autograd engine, which stands for automatic gradient. It is designed to implement backpropagation, a fundamental algorithm in training neural networks that efficiently computes the gradient of a loss function concerning the network's weights. This enables the iterative tuning of weights to minimize the loss function and enhance the network's accuracy. Micrograd allows users to construct mathematical expressions and maintain a graph of these expressions to facilitate backpropagation. Andre demonstrates how to use Micrograd with a simple mathematical expression involving two inputs, a and b, and shows how to perform forward and backward passes to calculate the gradient.

10:01

📚 Derivatives and Numerical Computation

The lecture delves into the concept of derivatives, emphasizing their importance in understanding how small changes in inputs affect the output of a function. Andre discusses the definition of a derivative and how it can be computed numerically using a small change in the input value (h). He illustrates this with a quadratic function and shows how to estimate the slope at different points, which corresponds to the derivative. The numerical approach is contrasted with symbolic differentiation, which is not practical for complex neural networks due to the large number of terms involved.

15:03

🔢 Working with Scalar-Valued Autograd Engines

Andre points out that Micrograd operates on scalar values, breaking down neural networks to their most basic mathematical components. This approach, while inefficient for production, is pedagogically valuable as it simplifies the understanding of backpropagation and the chain rule. He contrasts this with modern deep neural network libraries that use n-dimensional tensors, which allow for parallel operations and increased efficiency. The core mathematical principles remain the same, but the implementation is optimized for speed.

20:03

🚀 Implementing Micrograd and Neural Networks

The lecture moves on to the implementation of Micrograd, starting with basic imports and the definition of a scalar-valued function. Andre demonstrates how to plot this function and numerically approximate its derivatives. He then introduces the concept of a 'Value' object in Micrograd, which wraps scalar values and tracks their mathematical operations. The lecture shows how to define operations like addition and multiplication for these Value objects and how to visualize the resulting expression graph.

25:04

🤖 Building a Simple Neural Network Model

Andre constructs a simple mathematical model of a neuron, which is a fundamental building block of neural networks. The neuron takes multiple inputs, weights them, adds a bias, and passes the result through an activation function. The lecture uses the hyperbolic tangent (tanh) function as the activation function and demonstrates its squashing effect on the input values. The model is kept simple to focus on the core concepts of neural network operation.

30:06

🔄 Backpropagation Through a Neuron

The lecture continues with a detailed example of manually performing backpropagation through a single neuron. Andre calculates the gradient of the output with respect to each input and weight, illustrating how the chain rule is applied at each step. He emphasizes the importance of understanding this process, as it is central to training neural networks. The lecture also covers the need to accumulate gradients when a variable is used more than once in an expression.

35:08

🧠 Complexifying the Model: From Neurons to Neural Networks

Andre expands the discussion to multi-layer perceptrons (MLPs), which are composed of multiple neurons arranged in layers. He outlines the structure of an MLP and demonstrates how to implement a single neuron, a layer of neurons, and finally an entire MLP in code. The lecture also covers how to perform a forward pass through the MLP using the defined classes and methods.

40:11

📉 Defining Loss and Optimizing Neural Network Performance

The concept of loss is introduced as a measure of the neural network's performance. Andre chooses the mean squared error loss for the binary classification task and shows how to calculate the loss for a simple dataset. The lecture then demonstrates how to perform a backward pass through the loss to calculate the gradients needed for updating the network's weights and biases.

45:12

🔧 Gradient Descent and Updating Network Parameters

Andre explains the process of gradient descent, where the network's parameters are updated based on the gradients calculated from the backward pass. He emphasizes the importance of choosing the right learning rate for the update step to avoid overshooting or slow convergence. The lecture also highlights a common mistake of not resetting gradients before the backward pass, which can lead to incorrect updates.

50:13

🔄 Iterative Optimization and Training Loop

The lecture concludes with the implementation of an iterative training loop that performs forward and backward passes, updates the network parameters, and prints the loss at each step. Andre discusses the importance of iterating this process to optimize the neural network's performance. He also touches on the concept of learning rate decay, which involves gradually reducing the learning rate as the optimization progresses.

55:15

🔬 Micrograd Overview and PyTorch Comparison

Andre provides a comprehensive overview of the Micrograd library, which they have developed alongside the lecture. He walks through the code, highlighting the various components and their functionalities. The lecture also includes a comparison with PyTorch, a popular deep learning framework, to show how similar the principles and operations are. Andre demonstrates how to find the backward pass implementation for the tanh function in PyTorch and discusses the complexity of production-grade libraries.

00:19

📝 Wrapping Up and Future Directions

In the final part of the lecture, Andre summarizes the key points covered, including the fundamentals of neural networks, the process of backpropagation, and the implementation of a training loop. He encourages viewers to explore the Micrograd library and provides guidance on where to find additional resources and discuss the content further. Andre also hints at potential future updates to Micrograd and offers to answer questions in a follow-up video or discussion forum.

Keywords

💡Neural Networks

Neural networks are a set of algorithms designed to recognize patterns. They are inspired by the human brain and are used in a variety of applications, from image recognition to machine translation. In the video, Andre demonstrates how to build and train a neural network using the micrograd library, starting from basic mathematical expressions and moving towards a simple binary classifier.

💡Backpropagation

Backpropagation is a method used to calculate the gradient of the loss function concerning the weights of the network. It is fundamental to the training of neural networks, as it allows for the iterative tuning of weights to minimize the loss function. Andre explains backpropagation in detail, showing how it is implemented in micrograd and its importance in neural network training.

💡Micrograd

Micrograd is a simplified automatic differentiation engine created by Andre. It is designed for educational purposes to illustrate the core concepts of neural network training without the complexity of full-fledged libraries. Andre walks through building micrograd from scratch, demonstrating how it can be used to define and train neural networks.

💡Gradient

In the context of the video, a gradient refers to the derivative of the loss function with respect to the weights of the neural network. It is a vector that points in the direction of the steepest increase of the loss function. By adjusting the weights in the opposite direction of the gradient, the neural network can be trained to minimize the loss. Andre discusses the calculation and use of gradients during backpropagation.

💡Loss Function

A loss function is a measure of how well the neural network is performing. It calculates the difference between the predicted outputs and the actual targets. The goal of training a neural network is to minimize this function. In the video, Andre uses the mean squared error as the loss function to train a simple binary classifier.

💡Weights

Weights in a neural network are the parameters that are adjusted during training to improve the performance of the network. They are typically represented as numerical values associated with the connections between neurons. Andre shows how the weights are initialized and updated using gradient descent based on the calculated gradients.

💡Activation Function

Activation functions are mathematical equations that determine the output of a neural network's neurons. They introduce non-linearity into the network, allowing it to learn more complex patterns. Andre uses the hyperbolic tangent (tanh) function as an example of an activation function in the neural network model.

💡Forward Pass

The forward pass is the process of passing input data through a neural network to obtain an output. It involves the calculation of the output for each neuron in the network based on the weighted sum of its inputs and the application of an activation function. Andre demonstrates the forward pass in the context of building a mathematical expression graph in micrograd.

💡Chain Rule

The chain rule is a fundamental principle in calculus used to compute the derivative of a composed function. In the context of neural networks, it is used during backpropagation to calculate the gradients of the loss function with respect to the network's weights. Andre explains the application of the chain rule in backpropagation through the expression graph.

💡Gradient Descent

Gradient descent is an optimization algorithm used to minimize the loss function by updating the weights of the neural network in the direction opposite to the gradient. Andre illustrates how to perform gradient descent by adjusting the weights based on the calculated gradients to improve the neural network's predictions.

💡Tensors

Tensors are multi-dimensional arrays of numbers used in deep learning to represent complex data structures, such as images or sequences. Andre mentions tensors in the context of PyTorch, a popular deep learning library, to contrast the scalar-focused approach of micrograd with the more general approach used in production-level libraries.

Highlights

Andre, an experienced deep neural network trainer, guides through the construction of neural networks from scratch.

Introduction to Micrograd, a library for building mathematical expressions and performing backpropagation.

Micrograd's autograd engine efficiently evaluates the gradient of a loss function with respect to neural network weights.

The process of iteratively tuning neural network weights to minimize the loss function is explained.

Building a mathematical expression graph with Micrograd and performing a forward pass to calculate the output value.

Backpropagation is demonstrated by calculating the derivative of the output with respect to all internal nodes and inputs.

The importance of the derivative in understanding how inputs affect the output of a neural network is discussed.

Andre illustrates how neural networks are a specific class of mathematical expressions that can be optimized using backpropagation.

Micrograd's simplicity is highlighted, with the core autograd engine consisting of just 100 lines of Python code.

The concept of scalar-valued autograd engines and their use in pedagogical explanations of neural network training is covered.

Efficiency in neural network libraries is shown to be about tensor operations and parallelism, not changing the underlying math.

A step-by-step implementation of Micrograd is provided, starting with understanding derivatives and their significance.

The numerical approximation of derivatives using small increments is demonstrated for better understanding.

Andre explains the chain rule in calculus as a fundamental part of backpropagation for neural networks.

The application of the chain rule to compute gradients through a neural network is shown through manual backpropagation.

The use of topological sorting to ensure the correct order of backpropagation through a computation graph is discussed.

Micrograd's ability to handle complex expressions and operations, such as exponentiation and division, is demonstrated.

The implementation of a non-linearity function, such as tanh, in Micrograd and its importance in neural networks is shown.

Andre provides a comparison of Micrograd with PyTorch, a modern deep neural network library, to show similarities in API and functionality.

The construction of a multi-layer perceptron (MLP) using Micrograd and the execution of a forward pass through the network.

Introduction of a mean squared error loss function to measure the performance of the neural network.

The process of updating neural network weights based on gradient descent to minimize the loss function is explained.

The importance of resetting gradients to zero before the backward pass to prevent accumulation of gradients is highlighted.

A more complex binary classification problem is solved using Micrograd, demonstrating the library's capabilities.