Stories with Tag Automatic Differentiation

• Differentiable Programming from Scratch

  permalink

  Posted: 2025-04-17 04:30:47

  Verbose tl;dr

  This blog post provides a gentle introduction to automatic differentiation (AD), explaining how it computes derivatives of functions efficiently. It focuses on the forward mode of AD, building the concept from basic calculus and dual numbers. The post illustrates the process with clear, step-by-step examples, calculating derivatives of simple functions like f(x) = x² + 2x + 1 and more complex composite functions. It demonstrates how to implement forward mode AD in Python, emphasizing the recursive nature of the computation and how dual numbers facilitate tracking both function values and derivatives. The post concludes by hinting at the reverse mode of AD, a more efficient approach for functions with many inputs.

  This blog post, "Differentiable Programming from Scratch," provides a comprehensive yet accessible introduction to the core concepts of automatic differentiation (AD), specifically focusing on the forward mode. It meticulously deconstructs the process of calculating derivatives computationally, eschewing reliance on symbolic differentiation or numerical approximations like finite differences. Instead, it leverages the principle of dual numbers, extending real numbers with an infinitesimal component, ε, which obeys the rule ε² = 0.

  The post begins by establishing the foundational mathematical concepts. It explains how dual numbers, represented as a + bε, can be used to calculate derivatives by exploiting the Taylor expansion of a function. Evaluating a function with a dual number argument yields the function's value and its derivative as the real and infinitesimal components, respectively. This eliminates the need for symbolic manipulation of equations.

  The core of the implementation revolves around overloading arithmetic operations for dual numbers. The post meticulously details how these operations are defined, showcasing how addition, subtraction, multiplication, and division work with the inclusion of the infinitesimal component. This allows existing functions that operate on real numbers to seamlessly compute derivatives when provided with dual number inputs.

  Furthermore, the post extends the concept to encompass elementary functions like exponentiation, logarithm, sine, and cosine. It provides clear, step-by-step derivations of the dual number equivalents of these functions, demonstrating how the Taylor series expansion and the properties of ε facilitate the computation of their derivatives. This effectively builds a comprehensive toolkit for automatic differentiation of a wide range of mathematical expressions.

  The culmination of the post is a practical demonstration of the implemented AD system. It presents a simple example of calculating the derivative of a polynomial function. By inputting a dual number with the desired input value and an infinitesimal component of 1, the code returns both the function's value and its derivative at that point. This concrete example solidifies the practical application of the theoretical concepts discussed earlier.

  The post concludes by highlighting the elegance and efficiency of this approach. It emphasizes how automatic differentiation, implemented using dual numbers and operator overloading, provides a robust and precise method for computing derivatives, avoiding the pitfalls of symbolic manipulation complexity and the inaccuracy of numerical approximations. This method provides a foundation for more sophisticated applications in fields like machine learning and optimization, where accurate gradient calculations are paramount. The overall presentation emphasizes clarity and pedagogical value, breaking down a complex concept into digestible steps with illustrative examples and code snippets.

  • Differentiable Programming
  • Automatic Differentiation
  • Autodiff
  • backpropagation
  • Computational Graphs
  • machine learning
  • deep learning
  • neural networks
  • calculus
  • optimization
  • programming
  • Tutorial
  • from scratch
  • Implementation

  Summary of Comments ( 13 )
  https://news.ycombinator.com/item?id=43713140

  Verbose tl;dr

  HN users generally praised the article for its clear explanation of automatic differentiation (AD), particularly its focus on building intuition and avoiding unnecessary jargon. Several commenters appreciated the author's approach of starting with simple examples and progressively building up to more complex concepts. Some highlighted the article's effectiveness in explaining the difference between forward and reverse mode AD. A few users with experience in machine learning frameworks like TensorFlow and PyTorch pointed out that understanding AD's underlying principles is crucial for effective use of these tools. One commenter noted the article's relevance to fields beyond machine learning, such as scientific computing and optimization. A minor point of discussion revolved around the nuances of terminology, specifically the distinction between "dual numbers" and other approaches to representing derivatives.

  The Hacker News post "Differentiable Programming from Scratch" (linking to an article explaining automatic differentiation) sparked a moderately active discussion with 16 comments. Several commenters focused on the practical applications and limitations of automatic differentiation (AD), particularly in the context of machine learning.

  One commenter highlighted the difference between symbolic differentiation (which can lead to expression swell) and AD, pointing out that while AD avoids expression swell, it can still be computationally intensive, especially with higher-order derivatives. They mentioned the use of dual numbers and hyper-dual numbers for calculating first and second derivatives respectively, emphasizing the increasing complexity with higher orders. This commenter also touched upon the challenges of implementing AD efficiently, suggesting that achieving optimal performance often requires specialized hardware and software.

  Another commenter emphasized the benefits of JAX, a Python library specifically designed for high-performance numerical computation, including AD. They praised JAX's ability to handle complex derivatives efficiently, making it a valuable tool for researchers and practitioners working with large-scale machine learning models.

  A different thread of discussion revolved around the practical limitations of AD in real-world applications. One commenter expressed skepticism about the widespread applicability of AD, noting that many functions encountered in practice are not differentiable. They argued that while AD is undoubtedly useful in certain domains like machine learning, its limitations should be acknowledged. This prompted a counter-argument suggesting that even with non-differentiable functions, approximations and relaxations can often be employed to make AD applicable. The discussion touched upon the concept of subgradients and their use in optimizing non-differentiable functions.

  Some commenters also discussed alternative approaches to differentiation, such as numerical differentiation. While acknowledging its simplicity, they pointed out its limitations in terms of accuracy and computational cost, especially when dealing with higher-dimensional functions.

  Finally, a few comments focused on the pedagogical aspects of the linked article, praising its clarity and accessibility. One commenter appreciated the article's intuitive explanation of AD, making it easier for readers without a strong mathematical background to grasp the underlying concepts.

  In summary, the comments on Hacker News reflect a nuanced understanding of automatic differentiation, covering its strengths, limitations, and practical implications. The discussion highlights the importance of AD in machine learning while acknowledging the challenges associated with its implementation and application to real-world problems. The commenters also touch upon alternative differentiation techniques and appreciate the pedagogical value of the linked article.

• PyTorch Internals: Ezyang's Blog

  permalink

  Posted: 2025-03-22 14:39:04

  Verbose tl;dr

  Edward Yang's blog post delves into the internal architecture of PyTorch, a popular deep learning framework. It explains how PyTorch achieves dynamic computation graphs through operator overloading and a tape-based autograd system. Essentially, PyTorch builds a computational graph on-the-fly as operations are performed, recording each step for automatic differentiation. This dynamic approach contrasts with static graph frameworks like TensorFlow v1 and offers greater flexibility for debugging and control flow. The post further details key components such as tensors, variables (deprecated in later versions), functions, and modules, illuminating how they interact to enable efficient deep learning computations. It highlights the importance of torch.autograd.Function as the building block for custom operations and automatic differentiation.

  Edward Z. Yang's blog post, "PyTorch Internals," offers a comprehensive dive into the underlying architecture of the PyTorch deep learning framework, aiming to demystify its operation for advanced users and developers. He begins by outlining the core principles that guide PyTorch's design, emphasizing its focus on flexibility and enabling cutting-edge research. This includes a "user-first" approach prioritizing ease of use and debugging, and a dynamic computation graph that constructs the computational graph as the operations are executed, as opposed to statically defining it beforehand. This dynamic nature allows for greater flexibility in model construction and control flow, especially beneficial for research involving complex or varying network architectures.

  The blog post then delves into the technical details of how PyTorch achieves this dynamic computation. Central to this is the Tensor object, which not only holds the numerical data but also, crucially, a grad_fn attribute. This grad_fn acts as a pointer to the function that created the tensor, forming the backward links in the dynamic computation graph. This allows PyTorch to automatically compute gradients for backpropagation during training by traversing this dynamically built graph. Yang elaborates on the Function class, which represents these operations within the graph. Each Function object contains a forward method, which performs the actual computation, and a backward method, which computes the gradients with respect to its inputs.

  The post then elucidates the automatic differentiation (autograd) engine in PyTorch. It explains how the autograd engine recursively applies the chain rule using the grad_fn pointers and the backward methods of the Function objects to compute gradients of a scalar loss with respect to all tensors involved in its computation. This automated gradient computation is a cornerstone of PyTorch's ability to train deep learning models efficiently.

  Yang proceeds to discuss the interaction between the autograd engine and the tensor data itself. He clarifies the distinction between the .data attribute, which provides access to the raw tensor values, and the tensor object itself, which is involved in tracking the computation history for autograd. Modifying the .data attribute directly bypasses the autograd engine and allows for manipulation of tensor values without affecting the gradient computation.

  The blog post also touches on the role of the dispatcher in PyTorch. The dispatcher is responsible for directing operations to the correct backend implementations, allowing PyTorch to support various hardware acceleration options like CPUs, GPUs, and TPUs. This component enables the framework to perform computations efficiently on diverse hardware without requiring users to write hardware-specific code.

  Finally, Yang concludes with a brief overview of how custom operators can be implemented in PyTorch. This extensibility allows researchers and developers to incorporate specialized operations or integrate with other libraries seamlessly. The ability to define custom Function objects and register them with the dispatcher provides a powerful mechanism for extending the capabilities of the framework. This post thus provides a valuable resource for anyone seeking a deeper understanding of the internal mechanics that power PyTorch's flexibility and efficiency in the dynamic landscape of deep learning research.

  • PyTorch
  • deep learning
  • machine learning
  • Internals
  • Implementation
  • Tensors
  • Automatic Differentiation
  • Computational Graphs
  • performance
  • optimization
  • C++
  • Python
  • ezyang
  • blog

  Summary of Comments ( 22 )
  https://news.ycombinator.com/item?id=43445931

  Verbose tl;dr

  Hacker News users discuss Edward Yang's blog post on PyTorch internals, praising its clarity and depth. Several commenters highlight the value of understanding how automatic differentiation works, with one calling it "critical for anyone working in the field." The post's explanation of the interaction between Python and C++ is also commended. Some users discuss their personal experiences using and learning PyTorch, while others suggest related resources like the "Tinygrad" project for a simpler perspective on automatic differentiation. A few commenters delve into specific aspects of the post, like the use of Variable and its eventual deprecation, and the differences between tracing and scripting methods for graph creation. Overall, the comments reflect an appreciation for the post's contribution to understanding PyTorch's inner workings.

  The Hacker News post titled "PyTorch Internals: Ezyang's Blog," linking to an article on the same topic, has generated a significant number of comments discussing various aspects of PyTorch's internal workings and comparing it to other frameworks like TensorFlow and JAX.

  Several commenters praise the clarity and depth of the original blog post, finding it a valuable resource for understanding PyTorch's architecture. One commenter specifically appreciates the explanation of how PyTorch's define-by-run approach simplifies the creation of dynamic computation graphs, contrasting it with the more static graph construction required by TensorFlow 1.x. This dynamic nature is highlighted as a key advantage for research and experimentation.

  The discussion also delves into the performance implications of PyTorch's design. While some acknowledge that define-by-run can introduce overhead, others argue that its flexibility outweighs this drawback, particularly in research settings where rapid prototyping and experimentation are paramount. The evolution of PyTorch's tracing capabilities and the introduction of TorchScript are mentioned as mechanisms for bridging the performance gap with static graph approaches. A commenter notes that for production environments, tracing or scripting dynamic models can achieve performance comparable to static graph frameworks.

  Comparisons with JAX are also prevalent, with some commenters highlighting JAX's functional approach and its potential for optimization through techniques like automatic differentiation and just-in-time compilation. However, others note that PyTorch's imperative style might be more intuitive for some users and allows for easier debugging. The trade-offs between the two frameworks are discussed in terms of performance, ease of use, and debugging experience.

  One commenter raises the point that PyTorch's design has influenced other machine learning frameworks, citing TensorFlow 2.x's eager execution mode as an example of this convergence. Another discussion thread revolves around the challenges of scaling PyTorch to distributed computing environments and managing the complexity of distributed training.

  Several commenters share their personal experiences and anecdotes about using PyTorch, offering practical insights into its strengths and weaknesses. These anecdotes provide real-world context to the technical discussion, illustrating how PyTorch is used in practice across various domains. One such commenter mentions the benefits of PyTorch's extensibility, highlighting how custom operators and extensions can be easily integrated into the framework. The overall sentiment towards PyTorch appears to be positive, with many commenters expressing appreciation for its design, flexibility, and growing ecosystem.

• Auto-Differentiating Any LLM Workflow: A Farewell to Manual Prompting

  permalink

  Posted: 2025-01-29 05:15:45

  Verbose tl;dr

  The paper "Auto-Differentiating Any LLM Workflow: A Farewell to Manual Prompting" introduces a method to automatically optimize LLM workflows. By representing prompts and other workflow components as differentiable functions, the authors enable gradient-based optimization of arbitrary metrics like accuracy or cost. This eliminates the need for manual prompt engineering, allowing users to simply specify their desired outcome and let the system learn the best prompts and parameters automatically. The approach, called DiffPrompt, uses a continuous relaxation of discrete text and employs efficient approximate backpropagation through the LLM. Experiments demonstrate the effectiveness of DiffPrompt across diverse tasks, showcasing improved performance compared to manual prompting and other automated methods.

  The arXiv preprint "Auto-Differentiating Any LLM Workflow: A Farewell to Manual Prompting" introduces a novel methodology for optimizing Large Language Model (LLM) workflows by leveraging automatic differentiation. Traditionally, refining LLM prompts and parameters has been a laborious manual process, requiring iterative experimentation and intuition-driven adjustments. This paper proposes a radical departure from this manual approach by framing the entire LLM workflow as a differentiable function, thus enabling the application of gradient-based optimization techniques.

  The core innovation lies in the development of a continuous relaxation of discrete LLM operations. Since LLMs operate on discrete text tokens, their outputs are not inherently differentiable. To overcome this challenge, the authors introduce a method for approximating the discrete token probabilities with continuous representations. This relaxation allows for the calculation of gradients, which indicate the direction and magnitude of changes in the input that would lead to desired changes in the output. By iteratively adjusting the input parameters – including prompt text, temperature settings, and other workflow parameters – based on these gradients, the system automatically optimizes the LLM workflow toward a specified objective.

  The paper details the mathematical underpinnings of this differentiable LLM framework, explaining how the continuous relaxation is achieved and how gradients are computed. It also demonstrates the practical applicability of the method across various LLM tasks, including text summarization, question answering, and code generation. In these experiments, the automatically optimized workflows achieved significant performance improvements compared to manually tuned baselines.

  Furthermore, the paper explores the potential for this approach to automate the design of complex LLM workflows. Instead of relying on human expertise to assemble and configure different LLM components, the differentiable framework can automatically learn optimal workflow structures and parameter settings. This opens up the possibility of creating highly sophisticated and efficient LLM applications without the need for extensive manual engineering.

  The authors conclude that their proposed method represents a significant step towards fully automated LLM workflow optimization, potentially eliminating the need for tedious manual prompt engineering. This automated approach promises to democratize access to powerful LLM capabilities, enabling users with limited technical expertise to leverage the full potential of these advanced language models. The paper also suggests several avenues for future research, including exploring different continuous relaxation techniques and developing more sophisticated optimization algorithms.

  • Large Language Models
  • LLMs
  • Prompt Engineering
  • Automatic Differentiation
  • Autodiff
  • Workflow Automation
  • deep learning
  • machine learning
  • natural language processing
  • NLP
  • optimization
  • Gradient Descent
  • Differentiable Programming
  • AI

  Summary of Comments ( 15 )
  https://news.ycombinator.com/item?id=42861815

  Verbose tl;dr

  Hacker News users discuss the potential of automatic differentiation for LLM workflows, expressing excitement but also raising concerns. Several commenters highlight the potential for overfitting and the need for careful consideration of the objective function being optimized. Some question the practical applicability given the computational cost and complexity of differentiating through large LLMs. Others express skepticism about abandoning manual prompting entirely, suggesting it remains valuable for high-level control and creativity. The idea of applying gradient descent to prompt engineering is generally seen as innovative and potentially powerful, but the long-term implications and practical limitations require further exploration. Some users also point out potential misuse cases, such as generating more effective spam or propaganda. Overall, the sentiment is cautiously optimistic, acknowledging the theoretical appeal while recognizing the significant challenges ahead.

  The Hacker News post titled "Auto-Differentiating Any LLM Workflow: A Farewell to Manual Prompting" (linking to the arXiv paper at https://arxiv.org/abs/2501.16673) generated a moderate discussion with a mix of excitement and skepticism.

  Several commenters expressed interest in the potential of automatically optimizing LLM workflows through differentiation. They saw it as a significant step towards making prompt engineering more systematic and less reliant on trial and error. The idea of treating prompts as parameters that can be learned resonated with many, as manual prompt engineering is often perceived as a tedious and time-consuming process. Some envisioned applications beyond simple prompt optimization, such as fine-tuning entire workflows involving multiple LLMs or other components.

  However, skepticism was also present. Some questioned the practicality of the approach, particularly regarding the computational cost of differentiating through complex LLM pipelines. The concern was raised that the resources required for such optimization might outweigh the benefits, especially for smaller projects or individuals with limited access to computational power. The reliance on differentiable functions within the workflow was also pointed out as a potential limitation, restricting the types of operations that could be included in the optimized pipeline.

  Another point of discussion revolved around the black-box nature of LLMs. Even with automated optimization, understanding why a particular prompt or workflow performs well remains challenging. Some commenters argued that this lack of interpretability could hinder debugging and further development. The potential for overfitting to specific datasets or benchmarks was also mentioned as a concern, emphasizing the need for careful evaluation and generalization testing.

  Finally, some commenters drew parallels to existing techniques in machine learning, such as hyperparameter optimization and neural architecture search. They questioned whether the proposed approach offered significant advantages over these established methods, suggesting that it might simply be a rebranding of familiar concepts within the context of LLMs. Despite the potential benefits, some believed that manual prompt engineering would still play a crucial role, especially in defining the initial structure and objectives of the LLM workflow.