Backpropagation in Neural Networks: A Mathematical, Computational, and Practical Treatise - From the Chain Rule to the Transformer Era (Foundations of Modern Machine Learning)
English | May 13, 2026 | ASIN: B0H1N9T42R | 106 pages | Epub | 513.63 KB
Backpropagation is the algorithm that quietly powers every modern neural network - from the two-layer model on a laptop to the trillion-parameter language model on a supercomputer. This treatise is a complete, mathematically rigorous, and engineering-aware treatment of how it works, why it works, and how to make it work in practice.Backpropagation in Neural Networks begins with the historical credit-assignment problem and the calculus that solves it, then derives every gradient - for dense, convolutional, recurrent, normalization, and attention-based layers - from first principles. Closed-form derivations sit alongside a vectorized NumPy reference implementation, a numerical worked example, and an extended case study of training a modern transformer at scale.What you will learnThe calculus and statistics needed to express the algorithm cleanly: the chain rule, partial derivatives, Jacobians, the Hessian, and reverse-mode automatic differentiation.The full backpropagation derivation, presented three ways: as scalar mathematics, as matrix calculus, and as a vectorized implementation.The forward pass for every canonical layer type and the matching backward pass - including the elegant cross-entropy-with-softmax simplification.Gradient descent and its modern descendants: SGD, momentum, RMSProp, Adam, AdamW, AdaFactor, second-order methods, and natural gradient.Regularization that actually works in practice: weight decay, dropout, batch and layer normalization, early stopping, and data augmentation.The engineering of distributed training: data parallelism, tensor parallelism, pipeline parallelism, ZeRO, mixed-precision, and gradient compression.A complete case study of backpropagation through the transformer - scaled dot-product attention, multi-head attention, residuals, embeddings, and the full training lifecycle. Designed for serious studyThe book is structured for both linear reading and reference use. Each chapter ends with star-rated exercises ranging from routine to research level, with selected solutions in the appendix. A complete glossary, an index, a notation table, a topic-organized reading list, and 110+ references support deeper exploration.Who this book is forAdvanced undergraduates, graduate students, and practicing researchers in machine learning, applied mathematics, computational neuroscience, and statistics. Familiarity with multivariable calculus, linear algebra, and probability is assumed at the level of any rigorous undergraduate sequence.Why it mattersModern artificial intelligence is, at its heart, the chain rule scaled to industrial proportions. Mastery of backpropagation is the most durable investment a student of the field can make - every architecture that comes next will still rely on it.