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

Andrej Karpathy
16 Aug 2022145:52

TLDRIn this lecture, Andre, an experienced deep neural network trainer, introduces the concept of neural networks and backpropagation through the construction of 'micrograd', a lightweight autograd engine. He demonstrates how to manually implement backpropagation to understand the core of neural network training, covering mathematical expressions, loss functions, and optimization techniques. Andre also compares micrograd's functionality with PyTorch, showcasing the simplicity of creating custom neural network modules.


  • 🧠 The presenter, Andre, has over a decade of experience training deep neural networks and introduces the concept of neural network training 'under the hood'.
  • 📘 The lecture demonstrates building 'micrograd', a library released by Andre on GitHub, which implements backpropagation for neural network training.
  • 🔍 Micrograd is an autograd engine, short for automatic gradient, which is essential for evaluating the gradient of a loss function with respect to the weights of a neural network.
  • 🌟 Backpropagation is the mathematical core of modern deep neural network libraries and allows for the iterative tuning of neural network weights to minimize loss and improve accuracy.
  • 📚 The functionality of micrograd is shown through examples, building mathematical expressions and demonstrating how to calculate derivatives, which are crucial for understanding how inputs affect the output.
  • 🤖 Neural networks are a specific class of mathematical expressions, and backpropagation is a general mechanism that can be applied to any mathematical expression, not just neural networks.
  • 📉 The lecture includes a step-by-step implementation of micrograd, starting with understanding derivatives and moving towards building the value objects and expression graphs necessary for neural network training.
  • 🔢 Micrograd operates on scalar values, breaking down neural networks to their fundamental components, which simplifies understanding but is not used in production due to efficiency concerns.
  • 💡 The importance of visualizing expression graphs is highlighted for better understanding of the flow of calculations and the application of backpropagation.
  • 🔧 The process of manually implementing backpropagation is demonstrated, showing how gradients are calculated and propagated backward through the expression graph.
  • 🚀 The lecture concludes with the claim that micrograd contains all that's needed to train neural networks, with everything else being about efficiency, and that the autograd engine is only 100 lines of simple Python code.

Q & A

  • What is the main focus of the lecture given by Andre?

    -The lecture focuses on explaining neural network training, particularly the process of building and training a neural network from scratch using a library called Micrograd, which implements backpropagation for efficient gradient evaluation.

  • What is Micrograd and why is it significant in the context of this lecture?

    -Micrograd is a library released by Andre on GitHub that serves as an autograd engine, short for automatic gradient. It is significant because the lecture walks through the process of building Micrograd step by step, explaining how it facilitates the implementation of backpropagation for neural networks.

  • Can you explain the role of backpropagation in training neural networks?

    -Backpropagation is an algorithm that efficiently evaluates the gradient of a loss function with respect to the weights of a neural network. It allows for the iterative tuning of the network's weights to minimize the loss function, thereby improving the network's accuracy.

  • What is the purpose of the 'Value' object in Micrograd?

    -The 'Value' object in Micrograd wraps individual scalar numbers and is used to build mathematical expressions. It maintains pointers to its child nodes, allowing Micrograd to track the entire expression graph and perform backpropagation efficiently.

  • How does the chain rule from calculus play a role in backpropagation?

    -The chain rule is used in backpropagation to recursively apply the derivative of a loss function through the expression graph, starting from the output and moving backward to the inputs. This process evaluates the gradient of the loss function with respect to all internal nodes and inputs.

  • What is the mathematical core of modern deep neural network libraries like PyTorch or JAX?

    -The mathematical core of modern deep neural network libraries is backpropagation, which is used for efficient gradient computation in the training process.

  • Why does Andre claim that Micrograd is all you need to train networks, and what does he mean by 'everything else is just efficiency'?

    -Andre claims that Micrograd encompasses the fundamental concepts required for training neural networks. By saying 'everything else is just efficiency', he means that while additional features and optimizations in other libraries can improve performance and speed, they do not change the underlying mathematical principles that Micrograd already implements.

  • What is the significance of the 'dot' attribute in the context of Micrograd?

    -In Micrograd, the 'dot' attribute is used to access the data of a 'Value' object. For instance, after performing a forward pass, the output value of 'g' can be accessed using the 'dot data' attribute.

  • How does the lecture demonstrate the concept of derivatives in the context of neural networks?

    -The lecture demonstrates the concept of derivatives by first defining a scalar-valued function and then numerically approximating its derivative at various points. This helps in understanding how small changes in the input (weights or data) affect the output (loss function), which is crucial for gradient-based optimization in neural networks.

  • What is the role of the 'backward' function in Micrograd?

    -The 'backward' function in Micrograd is used to initiate the backpropagation process. When called on a 'Value' object, it starts the backward pass through the expression graph, applying the chain rule to compute the gradients of the loss function with respect to all the nodes in the graph.



🧠 Introduction to Neural Network Training

Andre introduces himself as an experienced trainer of deep neural networks and outlines his goal for the lecture: to demystify the training process of neural networks by building and training a neural network from scratch in a Jupyter notebook, using a library called Micrograd. Micrograd is an autograd engine that facilitates backpropagation, a fundamental algorithm for training neural networks by efficiently calculating the gradient of a loss function with respect to the weights of the network.


🛠️ Building Micrograd: An Autograd Engine

Andre explains the concept of Micrograd, an autograd engine he released on GitHub, which implements backpropagation for neural networks. He guides through the step-by-step process of building Micrograd, commenting on its components and demonstrating its capabilities through mathematical expressions. The functionality of Micrograd is illustrated by creating a simple expression graph with two inputs and showing how to perform forward and backward passes to calculate gradients.


🔍 Understanding Derivatives and Chain Rule

The lecture delves into the importance of understanding derivatives and the chain rule in the context of neural network training. Andre defines a scalar-valued function and explores the concept of derivatives at various points, demonstrating numerically how to approximate the derivative of a function. He emphasizes the significance of derivatives in indicating the sensitivity and slope of a function's response to changes in input.


🌐 Visualizing and Navigating Mathematical Expressions

Andre discusses the necessity of visualizing complex mathematical expressions and introduces a method to draw expression graphs using Graphviz, an open-source graph visualization software. He explains how to create nodes and edges for the graph and labels them for clarity. The process of visualizing the forward and backward passes of a mathematical expression is demonstrated, highlighting the computation of gradients for each node in the graph.


🤖 Implementing Backpropagation Manually

In this section, Andre manually implements backpropagation to illustrate the process of calculating gradients for each node in a computation graph. He emphasizes the importance of understanding the chain rule in calculus for combining local derivatives to compute the overall gradient with respect to the output. The process involves setting initial gradients and recursively applying the chain rule to propagate gradients backwards through the graph.


🔢 Debugging and Fixing Gradient Accumulation

Andre identifies a bug related to gradient accumulation in the backpropagation process, where gradients are not reset after each update, leading to incorrect gradient values. He explains the importance of zeroing the gradients before each backward pass to ensure accurate gradient descent. The fix involves adding a method to reset gradients to zero for all parameters before the backward pass.


📈 Training Neural Networks with Gradient Descent

The lecture demonstrates how to use the calculated gradients to update the weights and biases of a neural network in a process known as gradient descent. Andre shows how to iteratively perform forward and backward passes, followed by updates to the network's parameters to minimize the loss function. He also discusses the importance of choosing an appropriate learning rate to avoid overshooting or slow convergence.


🔧 Refining Neural Network Training with PyTorch

Andre compares the manual process of training a neural network with the functionalities provided by PyTorch, a production-grade deep learning library. He shows how to implement a neuron, layer, and multi-layer perceptron (MLP) in PyTorch, emphasizing the efficiency and simplicity of using tensors and built-in functions for operations like addition, multiplication, and activation.


🏗️ Constructing a Complete Neural Network Model

The focus shifts to constructing a complete neural network model, starting from individual neurons to layers of neurons and finally to a full multi-layer perceptron (MLP). Andre outlines the process of defining the architecture of an MLP, including the input layer, hidden layers, and output layer, and demonstrates how to perform a forward pass through the network to obtain predictions.


📉 Defining and Minimizing Loss Functions

Andre introduces the concept of loss functions as a measure of the neural network's performance, explaining how the mean squared error loss function works. He discusses the process of calculating individual loss components for each prediction and the ground truth, and how to aggregate these components into a total loss value that the network aims to minimize.


🔄 Iterative Optimization and Convergence

The lecture concludes with an iterative optimization process, where Andre demonstrates how to repeatedly perform forward and backward passes, followed by parameter updates to minimize the loss function. He highlights the importance of monitoring the loss and the predictions to ensure the network is converging towards accurate results. The process illustrates the fundamental workflow of training a neural network using gradient descent.



💡Neural Networks

Neural Networks are a set of algorithms designed to recognize patterns. They are inspired by the human brain's neural network and are capable of learning from data. In the video, Andre demonstrates the construction and training of neural networks, showcasing how they can be built from scratch using the micrograd library to perform tasks like binary classification.


Backpropagation is the cornerstone of training neural networks. It refers to the process of calculating the gradient of the loss function concerning the weights of the network, which is essential for optimizing these weights. The script explains backpropagation as the algorithm that allows the evaluation of these gradients efficiently, using the chain rule from calculus.


Micrograd is a lightweight automatic differentiation engine created by Andre. It is used to implement backpropagation and is the focus of the lecture. The script walks through building Micrograd step by step, explaining how it facilitates the creation of mathematical expressions and the computation of derivatives, which are vital for training neural networks.


Autograd is short for automatic differentiation, which is a system used to compute derivatives of mathematical functions with respect to their inputs. In the script, Andre mentions that micrograd is essentially an autograd engine, highlighting its role in implementing backpropagation for neural network training.


In the context of the video, a gradient is the derivative of a loss function with respect to the weights of a neural network. It indicates the direction and magnitude by which the weights should be adjusted to minimize the loss. The script discusses how gradients are computed and used to iteratively tune the weights of the network.

💡Jupyter Notebook

A Jupyter Notebook is an open-source web application that allows users to create and share documents containing live code, equations, visualizations, and narrative text. Andre starts the lecture with a blank Jupyter Notebook and builds up the micrograd library within it, demonstrating the process of defining and training a neural network.

💡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 script explains that the purpose of training a neural network is to minimize this loss function, which in turn improves the accuracy of the network's predictions.

💡Deep Neural Networks

Deep Neural Networks refer to neural networks with multiple layers between the input and output layers, allowing them to model complex patterns and relationships. Andre has been training deep neural networks for over a decade and shares insights into their training process through the construction of micrograd.

💡Forward Pass

The forward pass in neural networks is the process of feeding input data through the network to obtain an output or prediction. In the script, Andre explains and demonstrates the forward pass in the context of building mathematical expressions with micrograd and how it is used to calculate the value of the output.

💡Chain Rule

The chain rule is a fundamental principle in calculus used to compute the derivative of a composite function. In the context of the video, the chain rule is applied during backpropagation to calculate the gradients of the loss function with respect to the network's weights, allowing for the iterative improvement of the network's performance.


A tensor is a generalization of vectors and matrices to potentially higher dimensions. In deep learning, tensors are used to represent the data structures that store the inputs, weights, and outputs of neural networks. Andre mentions that while micrograd operates on scalar values for educational purposes, modern deep learning libraries like PyTorch use tensors to leverage computational efficiency.


Introduction to the process of building Micrograd, a library that simplifies the understanding of neural network training.

Micrograd is an autograd engine that implements backpropagation for efficient gradient evaluation in neural networks.

The importance of backpropagation in tuning neural network weights to minimize loss functions and improve accuracy.

A step-by-step tutorial on building mathematical expressions using Micrograd's Value objects.

Explanation of how Micrograd maintains pointers to Value objects to track the creation of mathematical expressions.

Visualization of expression graphs in Micrograd to understand the flow of data and operations within the network.

The significance of the derivative in understanding the sensitivity of a function and its applications in neural network training.

Demonstration of numerical approximation of derivatives and its comparison with analytical solutions.

Building and visualizing more complex expressions involving multiple inputs and operations in Micrograd.

Understanding the role of scalar-valued autograd engines in breaking down neural networks to individual scalars for educational purposes.

The simplicity of Micrograd's codebase, consisting of only two files, showcasing the core of neural network training.

Introduction to the concept of neural networks as mathematical expressions and the generality of backpropagation.

Explanation of how neural networks are trained using the machinery of backpropagation, regardless of the network's complexity.

The pedagogical approach of using scalar values in Micrograd to understand neural network training before optimizing for efficiency.

Implementation of the backward pass in Micrograd to perform backpropagation and update the gradients of the network parameters.

The process of topological sorting to ensure the correct order of backpropagation through the computational graph.

Identification and resolution of a bug related to the accumulation of gradients when a variable is used more than once.

The ability to break down complex functions like tanh into more atomic operations for a deeper understanding of neural network operations.

Integration of Micrograd with PyTorch to demonstrate the alignment of its autograd functionality with a production-grade library.

Building a multi-layer perceptron (MLP) using Micrograd and understanding its role in binary classification tasks.

The iterative process of forward pass, backward pass, and parameter updates in training a neural network to minimize loss.

Common pitfalls in neural network training, such as forgetting to zero gradients before backward propagation.

Final thoughts on the simplicity and power of neural networks as mathematical expressions and the importance of understanding their underlying mechanisms.