A Step-by-step Introduction to the Implementation of Automatic Differentiation (2402.16020v2)

Abstract: Automatic differentiation is a key component in deep learning. This topic is well studied and excellent surveys such as Baydin et al. (2018) have been available to clearly describe the basic concepts. Further, sophisticated implementations of automatic differentiation are now an important part of popular deep learning frameworks. However, it is difficult, if not impossible, to directly teach students the implementation of existing systems due to the complexity. On the other hand, if the teaching stops at the basic concept, students fail to sense the realization of an implementation. For example, we often mention the computational graph in teaching automatic differentiation, but students wonder how to implement and use it. In this document, we partially fill the gap by giving a step by step introduction of implementing a simple automatic differentiation system. We streamline the mathematical concepts and the implementation. Further, we give the motivation behind each implementation detail, so the whole setting becomes very natural.

• The paper presents a comprehensive guide to implementing automatic differentiation, clarifying both forward and reverse modes for derivative computation.
• It details the construction of computational graphs by wrapping functions to maintain operation dependencies and enable accurate function evaluation.
• It highlights the role of topological ordering for efficient partial derivative calculation, connecting theoretical insights with practical machine learning applications.

Implementing Automatic Differentiation: A Step-by-Step Guide

Introduction to Automatic Differentiation

Despite its critical role in deep learning, the complexity of automatic differentiation (AD) often poses a significant barrier to learning and understanding. This paper presents a detailed, step-by-step tutorial on implementing a basic automatic differentiation system that bridges the gap between theoretical concepts and their implementation. It simplifies the complex mathematical underpinnings and implementation details, making the entire process accessible and comprehensible, especially for beginners. The essence of this documentation is to ease the learning curve associated with automatic differentiation, providing a straightforward approach to implementing it from scratch.

Automatic Differentiation Basics

Automatic differentiation, indispensable in computing derivatives in machine learning algorithms, particularly in neural networks, operates on two primary modes: forward and reverse. The paper revisits these modes, highlighting their algorithmic structures and underlying principles. It uses a simple example function to elucidate these concepts, efficiently demonstrating how these modes differ in their approach to derivative computation and their practical implications in computational graphs, showcasing automatic differentiation’s versatility and power in dealing with complex derivative computations.

Implementing Function Evaluation and Computational Graph

A significant portion of the paper is devoted to explicating the construction of the computational graph, a pivotal component in automatic differentiation. This involves detailing the creation of nodes and edges, essentially embodying the operations and variables in a function. The authors propose an innovative approach to building this graph through wrapping functions that not only perform the required operations but also maintain the relationships between nodes, thereby preserving the structural and functional integrity of the graph. This approach simplifies the graph construction process, making it more intuitive and manageable.

Topological Ordering and Partial Derivatives

One of the noteworthy contributions of this paper is its discussion on the importance of topological ordering in the computational graph for efficient derivative computation. It presents a clear, methodical strategy for obtaining a topological order and explains its necessity in ensuring a correct and efficient computation of partial derivatives along the graph. By introducing a concise yet comprehensive method to calculate these derivatives, the paper further demystifies the implementation of automatic differentiation, making it more accessible to students and practitioners alike.

Practical Implications and Theoretical Considerations

From a practical standpoint, the paper's detailed exposition on implementing automatic differentiation from the ground up has profound implications for educational purposes. It provides a solid foundation for students and new learners to understand and appreciate the complexities and capabilities of automatic differentiation without being overwhelmed. Theoretically, it reinforces the significance of computational graphs in AD and underscores the efficiency of the forward mode in specific contexts. Additionally, the discussion on topological ordering provides valuable insights into the optimization of AD processes, highlighting potential areas for future research and development.

Concluding Remarks

This step-by-step guide to implementing automatic differentiation significantly contributes to the democratization of understanding in the field of machine learning. By breaking down complex concepts into manageable implementations, it not only facilitates learning but also opens up avenues for further exploration and innovation. The practical approach adopted in the paper, supplemented by theoretical insights, provides a comprehensive understanding of automatic differentiation, encouraging a deeper investigation into its applications and potential improvements.