Stories with Tag backpropagation

• Differentiable Programming from Scratch

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  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.

• NoProp: Training neural networks without back-propagation or forward-propagation

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  Posted: 2025-04-14 00:03:51

  Verbose tl;dr

  NoProp introduces a novel method for training neural networks that eliminates both backpropagation and forward propagation. Instead of relying on gradient-based updates, it uses a direct feedback mechanism based on a layer's contribution to the network's output error. This contribution is estimated by randomly perturbing the layer's output and observing the resulting change in the loss function. These perturbations and loss changes are used to directly adjust the layer's weights without explicitly calculating gradients. This approach simplifies the training process and potentially opens up new possibilities for hardware acceleration and network architectures.

  The paper "NoProp: Training Neural Networks without Back-Propagation or Forward-Propagation" introduces a novel approach to training neural networks that eliminates the need for both backpropagation and even the explicit calculation of forward activations. This contrasts sharply with traditional training methods, which rely heavily on these two processes. Backpropagation is typically used to calculate gradients of the loss function with respect to the network's weights, guiding updates that minimize the loss. Forward propagation, of course, is the fundamental process of passing input data through the network to generate predictions.

  NoProp, short for No Propagation, achieves this radical departure by utilizing a direct relationship between the weights of the network and the output loss. The core idea is to consider the output of the neural network as a function of its weights. This allows for a direct approximation of the gradient of the loss with respect to the weights without needing to explicitly calculate the activations at each layer during a forward pass or the gradients through backpropagation.

  Instead of the traditional iterative process of forward and backward passes, NoProp employs a Monte Carlo estimation of the gradient. For each weight, the algorithm samples random perturbations around the current weight value. The loss is then evaluated for each perturbed weight, and this information is used to estimate the gradient of the loss with respect to that specific weight. This process is performed for each weight in the network independently, eliminating the dependency chain between layers inherent in backpropagation.

  The authors achieve this Monte Carlo estimation by employing what they term a signed output sum. This method involves calculating the difference between the loss evaluated at a positively perturbed weight and the loss evaluated at a negatively perturbed weight. This difference, scaled appropriately, serves as an unbiased estimator of the gradient. Furthermore, the authors explore different variance reduction techniques, such as antithetic sampling, to improve the efficiency and accuracy of the gradient estimation.

  The paper also investigates alternative optimization methods, specifically evolutionary strategies, to update the weights using the estimated gradients. These methods, which are inherently parallelizable, further enhance the potential computational advantages of NoProp.

  The performance of NoProp is evaluated on several benchmark datasets, including MNIST and CIFAR-10. While the results don't yet surpass the state-of-the-art achieved by traditional backpropagation-based methods, they demonstrate the feasibility of this fundamentally different approach to neural network training. The authors highlight the potential of NoProp, particularly for extremely deep or recurrent networks, where backpropagation can face challenges related to vanishing or exploding gradients. Furthermore, the inherent parallelism of NoProp opens doors for novel hardware implementations and potentially significant computational advantages in the future. The authors suggest that further research could unlock the full potential of NoProp and potentially lead to significant advancements in the field of deep learning.

  • neural networks
  • deep learning
  • backpropagation
  • forward propagation
  • training algorithms
  • optimization
  • NoProp
  • machine learning
  • artificial intelligence
  • gradient-free optimization
  • alternative training methods
  • arXiv

  Summary of Comments ( 43 )
  https://news.ycombinator.com/item?id=43676837

  Verbose tl;dr

  Hacker News users discuss the implications of NoProp, questioning its practicality and scalability. Several commenters express skepticism about its performance on complex tasks compared to backpropagation, particularly regarding computational cost and the "hyperparameter hell" it might introduce. Some highlight the potential for NoProp to enable training on analog hardware and its theoretical interest, while others point to similarities with other direct feedback alignment methods. The biological plausibility of NoProp also sparks debate, with some arguing that it offers a more realistic model of learning in biological systems than backpropagation. Overall, there's cautious optimism tempered by concerns about the method's actual effectiveness and the need for further research.

  The Hacker News post titled "NoProp: Training neural networks without back-propagation or forward-propagation" (https://news.ycombinator.com/item?id=43676837) discusses the pre-print paper proposing a novel neural network training method called NoProp. The comments section contains a mix of intrigue, skepticism, and requests for clarification.

  Several commenters express fascination with the potential implications of eliminating backpropagation, a computationally expensive process. They highlight the potential for energy efficiency and speed improvements if NoProp proves viable. Some wonder about its applicability to different network architectures and problem domains beyond the simple tasks explored in the paper.

  A recurring theme is the desire for more experimental validation. Commenters acknowledge the novelty of the approach but emphasize the need for further testing on more complex datasets and architectures to truly assess NoProp's capabilities and limitations. Some express skepticism about its scalability and generalizability.

  Some users delve into the technical details, questioning the random weight initialization and local optimization aspects of NoProp. They discuss the potential for suboptimal solutions and the role of the selection algorithm in finding suitable weights. One commenter draws parallels to genetic algorithms, given the evolutionary nature of NoProp's weight selection process.

  Another point of discussion revolves around the paper's clarity. Some commenters find the explanation of the algorithm difficult to follow, requesting more detailed descriptions and pseudocode. They also question the paper's claim of "no forward propagation," arguing that the evaluation process inherently involves some form of forward pass, albeit a potentially simplified one.

  Finally, there's a discussion around the practical significance of NoProp. While acknowledging the theoretical interest, some commenters question whether the proposed method offers substantial advantages over existing techniques in real-world scenarios. They suggest that the computational cost of the selection process might offset the gains from eliminating backpropagation, especially for large networks.

  Overall, the comments section reflects a cautious optimism tempered by a healthy dose of scientific skepticism. There's a clear interest in exploring this new direction in neural network training, but also a recognition that further research and experimentation are necessary to determine its true potential.

• Matrix Calculus (For Machine Learning and Beyond)

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  Posted: 2025-03-29 20:00:33

  Verbose tl;dr

  "Matrix Calculus (For Machine Learning and Beyond)" offers a comprehensive guide to matrix calculus, specifically tailored for its applications in machine learning. It covers foundational concepts like derivatives, gradients, Jacobians, Hessians, and their properties, emphasizing practical computation and usage over rigorous proofs. The resource presents various techniques for matrix differentiation, including the numerator-layout and denominator-layout conventions, and connects these theoretical underpinnings to real-world machine learning scenarios like backpropagation and optimization algorithms. It also delves into more advanced topics such as vectorization, chain rule applications, and handling higher-order derivatives, providing numerous examples and clear explanations throughout to facilitate understanding and application.

  The arXiv preprint "Matrix Calculus (For Machine Learning and Beyond)" by Erik Learned-Miller presents a comprehensive and meticulously detailed guide to matrix calculus, specifically tailored for its applications in machine learning but extending its relevance to other fields as well. The author argues that existing treatments of matrix calculus are often fragmented, inconsistent in notation, or lacking the pedagogical depth required for a robust understanding. This work aims to rectify these issues by offering a unified and rigorous framework.

  The paper meticulously develops the foundational concepts of matrix calculus, starting with a thorough review of essential prerequisites such as linear algebra and multivariate calculus. It emphasizes the importance of understanding differentials as infinitesimal changes, drawing a clear distinction between differentials and derivatives. This groundwork is crucial for correctly interpreting and applying the chain rule in matrix calculus, a frequent source of confusion.

  The core of the paper revolves around the concept of the differential form of derivatives. This form, expressed as df = Tr(A dX), offers a flexible and consistent way to represent derivatives involving matrices and vectors. The trace operator plays a key role in simplifying expressions and facilitating manipulations. The authors meticulously derive the differential forms for various common matrix operations, including matrix multiplication, inverse, determinant, and eigenvalue decomposition.

  A significant portion of the paper is dedicated to elaborating on the chain rule in the context of matrix calculus. The authors introduce a step-by-step procedure for applying the chain rule, emphasizing the importance of identifying intermediate quantities and their respective differentials. They demonstrate the application of this procedure through several worked examples, highlighting the nuances and potential pitfalls. This systematic approach helps demystify the chain rule and makes it more accessible for practical computations.

  The paper also addresses the issue of converting between the differential form of derivatives and the more conventional gradient or Jacobian forms. It provides explicit formulas and procedures for these conversions, acknowledging the prevailing notational ambiguity in the field and offering clarity. This allows practitioners to connect the differential form, which is advantageous for derivations, with the more familiar gradient or Jacobian representations.

  Furthermore, the paper delves into advanced topics such as Hessian matrices, which describe the second-order derivatives of functions involving matrices and vectors. It explores the calculation of Hessians using the differential form, illustrating the power and elegance of this approach. The treatment of Hessians provides further insight into the optimization problems frequently encountered in machine learning.

  Throughout the paper, the author emphasizes practical applications in machine learning. Examples are drawn from various machine learning domains, including linear regression, neural networks, and Gaussian processes. These examples demonstrate how the developed framework can be applied to derive gradients and Hessians for common loss functions and model parameters, enabling efficient optimization algorithms.

  Finally, the paper concludes by summarizing the key concepts and providing a comprehensive table of derivatives in both differential and gradient/Jacobian forms. This serves as a valuable quick reference for practitioners and reinforces the unified approach presented throughout the work. The overall goal is to empower readers with a robust understanding of matrix calculus, equipping them to tackle complex derivations and contribute to the advancement of machine learning and other related disciplines.

  • matrix calculus
  • machine learning
  • linear algebra
  • mathematics
  • calculus
  • Derivatives
  • Vectors
  • Matrices
  • Multivariable Calculus
  • optimization
  • deep learning
  • artificial intelligence
  • Computer Science
  • Tensor Calculus
  • Gradient Descent
  • backpropagation
  • Numerical Computation
  • Preprint
  • arXiv

  Summary of Comments ( 27 )
  https://news.ycombinator.com/item?id=43518220

  Verbose tl;dr

  Hacker News users discussed the accessibility and practicality of the linked matrix calculus resource. Several commenters appreciated its clear explanations and examples, particularly for those without a strong math background. Some found the focus on differentials beneficial for understanding backpropagation and optimization algorithms. However, others argued that automatic differentiation makes manual matrix calculus less crucial in modern machine learning, questioning the resource's overall relevance. A few users also pointed out the existence of other similar resources, suggesting alternative learning paths. The overall sentiment leaned towards cautious praise, acknowledging the resource's quality while debating its necessity in the current machine learning landscape.

  The Hacker News post titled "Matrix Calculus (For Machine Learning and Beyond)" linking to an arXiv paper on the same topic generated a modest number of comments, primarily focused on the utility and accessibility of resources for learning matrix calculus.

  Several commenters discussed their preferred resources, often contrasting them with the perceived dryness or complexity of typical mathematical texts. One commenter recommended the book "Matrix Differential Calculus with Applications in Statistics and Econometrics" by Magnus and Neudecker, praising its focus on practical applications and relative clarity compared to other dense mathematical treatments. Another commenter concurred with the challenges of learning matrix calculus, recounting their struggles with a dense textbook and expressing appreciation for resources that prioritize clarity and intuitive understanding.

  The discussion also touched upon the balance between theoretical depth and practical application in learning matrix calculus. One commenter argued for the importance of understanding the underlying theory, suggesting that a strong foundation facilitates more effective application and debugging. Another commenter countered this perspective, suggesting that for many machine learning practitioners, a more pragmatic approach focusing on readily applicable formulas and identities might be more efficient. They specifically pointed out the usefulness of the "Matrix Cookbook" as a quick reference for common operations.

  A separate thread emerged discussing the merits of using index notation versus matrix notation. While acknowledging the elegance and conciseness of matrix notation, one commenter highlighted the potential for ambiguity and errors when dealing with complex expressions. They argued that index notation, while less visually appealing, can provide greater clarity and precision. Another commenter agreed, adding that index notation can be particularly helpful for deriving and verifying complex matrix identities.

  Finally, one commenter mentioned the relevance of automatic differentiation in modern machine learning, suggesting that it might alleviate the need for deep dives into manual matrix calculus for many practitioners. However, they also acknowledged that understanding the underlying principles could still be valuable for advanced applications and debugging.

  In summary, the comments on the Hacker News post reflect a common sentiment among practitioners: matrix calculus can be a challenging but essential tool for machine learning. The discussion revolves around the search for accessible and practical resources, the balance between theoretical understanding and practical application, and the relative merits of different notational approaches.

• The Matrix Calculus You Need for Deep Learning

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  Posted: 2025-03-29 16:01:22

  Verbose tl;dr

  "The Matrix Calculus You Need for Deep Learning" provides a practical guide to the core matrix calculus concepts essential for understanding and working with neural networks. It focuses on developing an intuitive understanding of derivatives of scalar-by-vector, vector-by-scalar, vector-by-vector, and scalar-by-matrix functions, emphasizing the denominator layout convention. The post covers key topics like the Jacobian, gradient, Hessian, and chain rule, illustrating them with clear examples and visualizations related to common deep learning scenarios. It avoids delving into complex proofs and instead prioritizes practical application, equipping readers with the tools to derive gradients for various neural network components and optimize their models effectively.

  The online article "The Matrix Calculus You Need for Deep Learning," hosted on explained.ai, provides a comprehensive yet accessible introduction to the fundamental concepts of matrix calculus essential for understanding and working with deep learning algorithms. It meticulously explains the mathematical tools required to derive gradients and perform optimization in neural networks.

  The article commences by establishing the importance of matrix calculus in deep learning, highlighting its role in gradient-based optimization methods. It then proceeds to define key concepts like derivatives and gradients in the context of scalar-valued functions, laying a solid foundation for later discussions on higher-dimensional operations. The article carefully distinguishes between derivatives, which represent the rate of change of a function with respect to a single variable, and gradients, which encompass the rates of change with respect to multiple variables, forming a vector.

  Building upon these foundational concepts, the article delves into the intricacies of matrix calculus, focusing on the differentiation of various function types. It starts with simple scalar-by-vector derivatives, elaborately explaining the process of differentiating a scalar function with respect to a vector input. This is followed by a detailed exploration of vector-by-vector derivatives, where both the function output and input are vectors. Critically, the article emphasizes the Jacobian matrix, which captures all the partial derivatives of a vector-valued function. The treatment of Jacobian matrices includes a discussion of its dimensions and how these relate to the input and output vectors.

  The exposition continues with vector-by-matrix and matrix-by-vector derivatives, providing clear explanations and illustrative examples for each case. The authors meticulously describe how these derivatives are calculated and represented, emphasizing the proper arrangement of partial derivatives within resulting matrices or higher-order tensors. These sections delve into the nuances of dimensionality and the practical implications of these derivative computations for gradient calculations in neural networks.

  A central focus of the article is the chain rule and its application in deep learning. It explains how the chain rule allows for the computation of complex derivatives by breaking them down into simpler, manageable steps. This concept is crucial for calculating gradients in deep neural networks with multiple layers, where the output of one layer serves as the input for the subsequent layer. The authors provide detailed examples of applying the chain rule in various scenarios, demonstrating its versatility and power.

  The article concludes by bringing together these concepts to demonstrate how they are applied in the context of training neural networks. It explains how backpropagation, a core algorithm in deep learning, leverages the chain rule and matrix calculus to efficiently compute the gradients of the loss function with respect to the network's parameters. This enables the iterative adjustment of these parameters to minimize the loss and improve the network's performance. The final sections reiterate the significance of understanding matrix calculus for anyone seeking a deeper understanding of the inner workings and optimization processes of deep learning models. The article emphasizes that a solid grasp of these mathematical principles is essential for effectively designing, implementing, and debugging complex neural network architectures.

  • matrix calculus
  • deep learning
  • linear algebra
  • mathematics
  • neural networks
  • backpropagation
  • Derivatives
  • gradients
  • multivariate calculus
  • machine learning
  • optimization
  • vector calculus
  • chain rule
  • Jacobians
  • Hessians

  Summary of Comments ( 1 )
  https://news.ycombinator.com/item?id=43516506

  Verbose tl;dr

  Hacker News users generally praised the article for its clarity and accessibility in explaining matrix calculus for deep learning. Several commenters appreciated the visual explanations and step-by-step approach, finding it more intuitive than other resources. Some pointed out the importance of denominator layout notation and its relevance to backpropagation. A few users suggested additional resources or alternative notations, while others discussed the practical applications of matrix calculus in machine learning and the challenges of teaching these concepts effectively. One commenter highlighted the article's helpfulness in understanding the chain rule in a multi-dimensional context. The overall sentiment was positive, with many considering the article a valuable resource for those learning deep learning.

  The Hacker News post titled "The Matrix Calculus You Need for Deep Learning" (linking to explained.ai/matrix-calculus/) generated several comments discussing the resource and its relevance to deep learning.

  Several commenters praised the clarity and comprehensiveness of the explained.ai resource. One user described it as a "great resource," highlighting its ability to break down complex concepts into understandable chunks. Another commenter appreciated the detailed explanations and practical examples provided, stating it filled gaps in their understanding. The site's focus on providing intuition and geometrical interpretations, rather than just rote formulas, was also lauded by multiple users. One individual specifically mentioned how helpful the explanations of the chain rule and backpropagation were, emphasizing the importance of these concepts in deep learning.

  Some commenters offered alternative resources and learning approaches. One suggested a different website and book that they found useful for learning matrix calculus. Another emphasized the value of deriving formulas oneself for deeper understanding, even if pre-derived versions are readily available. Someone else pointed out that, in practice, automatic differentiation libraries like those found in TensorFlow and PyTorch handle the complexities of matrix calculus, minimizing the need for manual calculations. However, they acknowledged that understanding the underlying principles is still beneficial.

  A few commenters discussed the practical application of matrix calculus in deep learning. While acknowledging its theoretical importance, some argued that a deep understanding isn't always essential for practitioners. They suggested focusing on the high-level concepts and letting the software handle the details. Others countered this viewpoint, arguing that a strong foundation in matrix calculus is crucial for debugging, optimizing models, and pushing the boundaries of the field.

  There was a brief exchange regarding the notation used in the article. One commenter expressed a preference for denominator layout notation, while another explained why numerator layout is generally preferred in the context of deep learning.

  Finally, there were a couple of meta-comments. One user asked about the background of the author of the explained.ai resource. Another commenter mentioned encountering broken links within the website.