💡 [REQUEST] - Tutorial: Visualizing the Mathematics Behind Neural Network Training

从 beginner_source/ 开始，比较 autogradqs_tutorial.py、autograd_tutorial.py 和 buildmodel_tutorial.py，以确定剩余的教学空缺。构建一个可在 CPU 上运行的教程，涵盖提议的六个部分，包括可观察的矩阵运算、梯度、grad_fn 以及优化器的变化。当数学步骤可以在代码中追踪并验证时，即表示完成。

🚀 Describe the improvement or the new tutorial

A beginner-friendly tutorial that traces the mathematical operations inside a PyTorch training loop — connecting linear algebra to torch.nn.Linear, the chain rule to torch.autograd.backward(), and gradient descent to torch.optim — with runnable code that makes each step observable and verifiable.

Why this is needed:
Many learners use autograd as a black box. Existing tutorials explain the API (how to call .backward(), how to access .grad) but do not trace what is happening mathematically at each step. This tutorial bridges that gap by making the math visible through code:

1. Print a weight matrix, verify it matches x @ W.T + b
2. Call .backward(), then manually verify the gradient using the chain rule
3. Walk the grad_fn graph to see the operations autograd recorded
4. Compare parameters before and after optimizer.step() to see gradient descent in action

Proposed structure:

1. Linear Algebra — what nn.Linear computes, verified with manual matrix multiplication
2. Activation Functions — why nonlinearity matters, visualizing ReLU gradients
3. The Chain Rule — what backward() computes, manually verifying gradients
4. The Computational Graph — inspecting grad_fn to trace operations
5. Gradient Descent — what optimizer.step() does, before/after comparison
6. Putting It Together — forward, backward, step mapped to three branches of mathematics

This approach has been validated in practice: I presented this material at Grace Hopper Celebration 2025 ("The Math Behind the Magic: Understanding the Role of Mathematics in Deep Learning"). Talk materials: https://github.com/AddyM/Math_behind_ML

Difficulty level: Beginner
Target location: beginner_source/

I am happy to write and submit this tutorial as a PR.

Existing tutorials on this topic

1. autogradqs_tutorial.py — covers autograd API usage (requires_grad, backward, grad). Focuses on how to use autograd, not what it computes mathematically.
   https://pytorch.org/tutorials/beginner/basics/autogradqs_tutorial.html

2. autograd_tutorial.py — conceptual overview of automatic differentiation and computational graphs. Does not include manual verification of gradients or tracing the chain rule step by step.
   https://pytorch.org/tutorials/beginner/blitz/autograd_tutorial.html

3. buildmodel_tutorial.py — covers nn.Module and model building. Does not connect layer operations to their underlying linear algebra.
   https://pytorch.org/tutorials/beginner/basics/buildmodel_tutorial.html

Additional context

PyTorch's eager execution model makes the mathematics of deep learning observable in a way static graph frameworks do not allow. This tutorial leans into that strength — every intermediate result is printable, every gradient is inspectable, every operation in the computational graph is traceable.
No GPU required — the entire tutorial runs on CPU, making it accessible to all learners.
References:

GHC 2025 talk materials: https://github.com/AddyM/Math_behind_ML
PyTorch Discuss thread on this approach: https://discuss.pytorch.org/ (search: "Building a PyTorch curriculum that teaches math through code")