# Backpropagation Algorithm Implementation: A Deep Dive into AI Engineering From Scratch

> Learn the backpropagation algorithm implementation from scratch using reverse-mode automatic differentiation. Discover how computational graphs and the chain rule drive AI engineering.

- Repository: [Rohit Ghumare/ai-engineering-from-scratch](https://github.com/rohitg00/ai-engineering-from-scratch)
- Tags: deep-dive
- Published: 2026-07-19

---

**The repository implements reverse-mode automatic differentiation from scratch using only standard library components, wrapping scalar values in a `Value` class that tracks computational graphs and applies the chain rule through topological sorting.**

The **backpropagation algorithm implementation** in the `rohitg00/ai-engineering-from-scratch` repository provides a transparent, educational view of how neural networks learn. Located in the Deep Learning Core phase (Lesson 03), this framework-free code demonstrates the mechanics of gradient computation without hidden abstractions.

## Core Architecture of the Implementation

The implementation centers on a **computational graph** built dynamically during the forward pass. Each operation creates nodes that remember their parents and how to compute local gradients.

### The Value Wrapper and Automatic Differentiation

In [`phases/03-deep-learning-core/03-backpropagation/code/main.py`](https://github.com/rohitg00/ai-engineering-from-scratch/blob/main/phases/03-deep-learning-core/03-backpropagation/code/main.py), the `Value` class serves as the fundamental unit of computation. Each instance stores:
- `data`: The forward-pass scalar value
- `grad`: The accumulated gradient initialized to 0.0
- `_prev`: A set of parent nodes (children in the dependency graph)
- `_backward`: A closure implementing the local gradient computation

When the final loss node calls its `backward()` method, it triggers a reverse topological traversal that applies the chain rule to every preceding operation.

### Operator Overloads and Chain Rule Application

Arithmetic operations override standard Python operators to build the graph automatically. For multiplication, the `_backward` closure implements the product rule:

```python
def _backward():
    self.grad += other.data * out.grad
    other.grad += self.data * out.grad

```

This pattern extends to addition, power operations, and activation functions. Each operator defines how to distribute the incoming gradient (`out.grad`) to its inputs according to the derivative of that specific operation.

## Topological Sorting and Gradient Flow

Before gradients flow backward, the `build_topo` function performs a depth-first search starting from the loss node. This creates a topological ordering ensuring that a node never receives gradient updates before all its dependents have been processed.

The backward pass executes in three stages:

1. **Gradient seeding**: The loss node's gradient is set to 1.0 (`self.grad = 1.0`)
2. **Reverse traversal**: Nodes are visited in reverse topological order
3. **Accumulation**: Each node's `_backward` function adds its contribution to parent gradients

This approach guarantees that gradients propagate correctly from outputs back to inputs, handling complex computational graphs with shared parameters.

## Neural Network Structure and Training

The repository implements a complete **multilayer perceptron (MLP)** using the `Value` infrastructure, demonstrating end-to-end training on classic problems like XOR classification and circle classification.

### Multilayer Perceptron Components

The Python implementation defines three hierarchical classes in [`main.py`](https://github.com/rohitg00/ai-engineering-from-scratch/blob/main/main.py):
- `Neuron`: Maintains weights as `Value` objects and applies **sigmoid** activation
- `Layer`: Aggregates multiple neurons and manages forward passes
- `Network`: Composes layers and computes the **squared-error loss** (`mse_loss`)

The Julia implementation in `main.jl` uses a mutable `MLP` struct with explicit weight matrices, implementing forward passes through the `forward!` function and manual gradient calculations for each parameter.

### Training Loop Mechanics

Both implementations follow identical training protocols. The `train_xor` and `train_circle` functions demonstrate the standard optimization loop:

```python
from phases.03_deep_learning_core.03_backpropagation.code.main import train_xor

if __name__ == "__main__":
    train_xor()

```

The training procedure executes:

1. **Forward propagation**: Computing predictions and loss
2. **Gradient reset**: Calling `zero_grad()` to clear previous gradients
3. **Backward pass**: Invoking `backward()` on the total loss to populate gradients
4. **Parameter update**: Applying gradient descent (`p.data -= lr * p.grad`)

## Julia Implementation and Gradient Verification

The Julia code provides additional educational value through explicit gradient checking. After computing analytical gradients via backpropagation, the implementation verifies correctness against numerical finite-difference estimates:

```julia
using .phases.03_deep_learning_core.03_backpropagation.code.main: main

main()  # Runs XOR, circle classification, and gradient verification

```

This verification ensures that the manually-derived backward functions in `main.jl` correctly implement the chain rule for the sigmoid activation and squared-error loss functions.

## Summary

The **backpropagation algorithm implementation** in this repository demonstrates:

- **Reverse-mode automatic differentiation** through a dynamically built computational graph
- **Topological sorting** to ensure correct gradient propagation order in `build_topo`
- **Framework-free operation** using only Python's `math`/`random` modules and Julia's standard library
- **Educational transparency** with explicit `Value` wrappers and visible chain-rule applications
- **Verification methods** including gradient checking against numerical derivatives in the Julia version

## Frequently Asked Questions

### How does the Value class enable automatic differentiation?

The `Value` class in [`phases/03-deep-learning-core/03-backpropagation/code/main.py`](https://github.com/rohitg00/ai-engineering-from-scratch/blob/main/phases/03-deep-learning-core/03-backpropagation/code/main.py) enables automatic differentiation by recording every mathematical operation as a node in a computational graph. Each instance stores references to its parent nodes and a `_backward` closure that knows how to compute local gradients using the chain rule. When `backward()` is called on the loss node, it traverses this graph in reverse topological order, accumulating gradients at each step without requiring manual derivative calculations from the user.

### What is the role of topological sorting in backpropagation?

Topological sorting ensures that gradients flow correctly from the output back to inputs by visiting nodes only after all their children have been processed. The `build_topo` function in the Python implementation performs a depth-first search to create this ordering, preventing situations where a parent node might receive partial gradient updates before all its dependents have contributed their gradients. This guarantees that the chain rule is applied comprehensively across the entire computational graph.

### How are gradients verified in the Julia implementation?

The Julia implementation in `phases/03-deep-learning-core/03-backpropagation/code/main.jl` includes a gradient-checking mechanism that compares analytical gradients computed via backpropagation against numerical approximations using finite differences. This verification computes the loss at slightly perturbed parameter values and estimates the derivative numerically, confirming that the manual chain-rule implementations in the `backward` functions produce mathematically correct results within a small epsilon tolerance.

### Why is this implementation considered educational?

This implementation is considered educational because it exposes every component of the backpropagation process that frameworks like PyTorch or TensorFlow hide behind optimized C++ backends. By using only standard library functions and explicitly defining how each operation computes its backward pass, the code in `rohitg00/ai-engineering-from-scratch` makes the chain rule, topological sorting, and gradient accumulation visible and modifiable. Students can inspect the `_backward` closures, modify the `Value` class behavior, and trace exactly how gradients flow through a multilayer perceptron during training.