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

Overview

Andrej Karpathy builds the micrograd autograd engine from scratch to demystify neural network training, demonstrating backpropagation through scalar value objects and expression graphs. He then extends this to implement neurons, layers, and multi-layer perceptrons (MLPs), showing how to train a simple MLP using gradient descent and highlighting the importance of zeroing gradients.

Key takeaways

Micrograd provides a transparent, scalar-based implementation of autograd and backpropagation, demystifying neural network training.
The core of backpropagation involves recursively applying the chain rule to compute gradients through the computation graph.
Neural networks are essentially complex mathematical expressions; training involves minimizing a loss function using gradient descent.
Key components like neurons, layers, and MLPs can be built by composing basic operations and tracking computation history.
A critical bug in gradient accumulation (forgetting to zero gradients) can lead to incorrect training, even if the network appears to work on simple problems.
Production libraries like PyTorch use optimized tensor operations and complex kernels, but the fundamental autograd principles demonstrated in micrograd remain the same.

Chapters

0:00 Introduction to Micrograd and Autograd

Micrograd is an autograd engine implementing backpropagation for efficient gradient calculation.
It allows building mathematical expressions and visualizing them as computation graphs.
The core idea is to break down complex functions into scalar operations.

13:32 Understanding Derivatives and Numerical Approximation

Explains derivatives as the sensitivity of a function's output to input changes.
Demonstrates numerical approximation of derivatives using a small 'h' value.
Illustrates how derivatives indicate the slope and direction of change at a point.

23:34 Derivatives with Multiple Inputs

Extends derivative concept to functions with multiple scalar inputs (a, b, c).
Numerically calculates partial derivatives (∂d/∂a, ∂d/∂b, ∂d/∂c).
Shows how each input's change affects the output 'd'.

32:00 Building the Value Object for Expression Graphs

Introduces the `Value` class to wrap scalar data and store computation history.
Implements `__add__` and `__mul__` to enable arithmetic operations on `Value` objects.
Adds `_prev` (children) and `_op` (operation name) attributes to track the expression graph.

41:42 Visualizing Expression Graphs with Graphviz

Introduces a `draw_dot` function to visualize the computation graph using Graphviz.
Graphs show nodes representing `Value` objects and edges representing operations.
Adds labels to nodes for clarity, showing variable names (a, b, c, d, e, f, l).

49:16 Introducing the Gradient (`grad`) Attribute

Adds a `grad` attribute to the `Value` class, initialized to zero.
This attribute will store the derivative of the final output (loss) with respect to the `Value` object.
Initializes `grad` to 1.0 for the final output node 'l' to start backpropagation.

53:33 Manual Backpropagation: Base Case and Summation

Manually sets `l.grad = 1.0` as the starting point for backpropagation.
Demonstrates backpropagating through a summation (d = c + e) by distributing the gradient.
Shows that for a sum, `∂l/∂c = ∂l/∂d * ∂d/∂c` and `∂l/∂e = ∂l/∂d * ∂d/∂e`, where `∂d/∂c = 1` and `∂d/∂e = 1`.

1:19:10 Manual Backpropagation: Multiplication

Demonstrates backpropagating through multiplication (e = a * b).
Applies the chain rule: `∂l/∂a = ∂l/∂e * ∂e/∂a` and `∂l/∂b = ∂l/∂e * ∂e/∂b`.
Shows that `∂e/∂a = b` and `∂e/∂b = a`, leading to `a.grad = l.grad * b.data` and `b.grad = l.grad * a.data`.

1:28:22 Backpropagating Through a Neuron with Tanh Activation

Models a neuron with inputs (x), weights (w), bias (b), and a Tanh activation function.
Defines the forward pass: `n = w*x + b`, `o = tanh(n)`.
Implements the `tanh` operation and its backward pass using the derivative `1 - tanh(n)^2`.

1:55:29 Automating Backpropagation with `_backward` Functions

Adds a `_backward` function attribute to the `Value` class for each operation.
Implements `_backward` for addition (distributes gradient), multiplication (applies chain rule with other operand), and Tanh (uses derivative `1 - output^2`).
The main `backward()` method performs a topological sort and calls `_backward` on nodes in reverse order.

2:17:19 Fixing the Gradient Accumulation Bug

Identifies a bug where gradients are overwritten instead of accumulated when a variable is used multiple times.
The fix is to change `grad` assignments to `+=` (plus equals) in the backward functions.
This ensures gradients from different paths correctly sum up.

2:25:48 Implementing More Operations: Power and Division

Adds `__pow__` for raising values to a constant power, using the derivative `n * x^(n-1)`.
Implements division by rewriting `a / b` as `a * b**(-1)`.
Adds `__neg__` and `__sub__` by leveraging existing operations (multiplication by -1, addition of negation).

Summary, takeaways, and chapters were generated by AI from the video's transcript and may contain errors. The video belongs to its creator, Andrej Karpathy.