Deep Neural Networks: A Practitioner’s Overview and Implementation

Introduction

• Trainer: Andre, with over a decade of experience in training deep neural networks.
• Goal: To understand the detailed workings of neural network training by building micrograd, an autograd engine that implements backpropagation, from scratch.

Micrograd Overview

• Micrograd: A library designed for educational purposes to help understand backpropagation and the training process of neural networks.
• Backpropagation: Algorithm to efficiently calculate the gradient of the loss function with respect to neural network weights, enabling weight tuning to minimize loss and improve accuracy.

Micrograd Example

1. Value Object: Wraps scalar values into a custom class that supports operations needed for forward and backward passes.

   • Operations: Addition, multiplication, exponentiation, negation, etc.
   • Maintenance: Keeps track of children nodes and the operation (op) that produced the value.

2. Expression Graph: A directed acyclic graph where nodes are values and edges represent operations.

   • Example: Construct a graph with operations on a and b leading to a final node g and observe both forward and backward passes.

Calculating Derivatives and Backpropagation

1. Numerical Gradient Check: Empirical approach using the limit definition of derivatives.

   • Derivative of f(x): (f(x + h) - f(x)) / h as h approaches 0.

2. Chain Rule: Fundamental for backpropagation, allowing the composition of derivatives through intermediate variables.

   • Example: For a composite function g(f(x)), the derivative dg/dx is dg/df * df/dx.

3. Manual Backpropagation Example: Step-by-step manual calculation of gradients for a simple expression, demonstrating the chain rule.

Building Classes for Neural Network Components

1. Neuron: Represents a single neuron with input weights and bias and performs a forward pass using a non-linear activation function like tanh.

   • Instantiation: Define neurons with specified input dimensions and weights initialized randomly.

2. Layer: Comprising multiple neurons with independent evaluations.

   • Instantiation: Define a layer with multiple neurons sharing the same input dimensions.

3. Multi-Layer Perceptron (MLP): Stack layers of neurons to form a full MLP.

   • Instantiation: Comprised of several layers feeding into one another sequentially.

Training the Neural Network

1. Forward Pass: Calculate the output of neural net given input data using defined weights.
2. Loss Function: Define a loss (e.g., Mean Squared Error) to evaluate the difference between predicted and true values.
3. Backward Pass: Compute gradients of the loss w.r.t all parameters using backpropagation.
4. Gradient Descent: Adjust parameters in the direction that minimizes the loss function.
5. Training Loop: Iteratively perform forward passes, compute loss, backward passes, and parameter updates until convergence.
   • Learning Rate: Important hyperparameter to control step size in gradient descent.
   • Zero Gradients: Ensure gradients are reset between iterations to avoid incorrect accumulation.

Practical Insights

1. PyTorch Comparison: Explains how concepts implemented in micrograd translate into production-grade libraries like PyTorch.
2. Pedagogical Tools: Simple tools like micrograd help understand fundamental concepts which are otherwise abstracted in high-level libraries.
3. Bugs and Debugging: Emphasize common pitfalls (e.g., not zeroing gradients) and the importance of debugging in developing neural networks.

Conclusion

• Key Takeaways: Understanding the theory and implementation of backpropagation, neural network components, and training routines.
• Framework Building: How simple theoretical foundations build up to more complex and efficient libraries used in practice.
• Final Thoughts: Mastering these fundamentals empowers one to explore more advanced deep learning architectures and applications.