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PytorchTutorialCodes/5_define_autograd_function_explanation.md

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+# Custom autograd function in PyTorch: Step-by-step explanation
+
+Let's break down what happens in your code, especially how you get the gradient numbers, what they mean, and how PyTorch's autograd system works when you define your own function.
+
+***
+
+## 1. What is a custom autograd function?
+
+- In PyTorch, you can create your own mathematical operation and tell PyTorch **how to compute its gradient** (how it changes with respect to its input).
+- You do this by subclassing `torch.autograd.Function` and implementing two methods:
+    - `forward`: computes the output from the input (normal math)
+    - `backward`: computes the gradient of the output with respect to the input (how the output changes if you nudge the input)
+
+***
+
+## 2. How do you get the gradient numbers?
+
+- **Forward pass:** You calculate the output for your function. Here, it's the Legendre polynomial:
+
+$$
+P_3(x) = \frac{1}{2}(5x^3 - 3x)
+$$
+- **Backward pass:** You tell PyTorch the formula for the derivative of your function with respect to its input. For Legendre polynomial:
+
+$$
+\frac{dP_3}{dx} = \frac{1}{2}(15x^2 - 3) = 1.5(5x^2 - 1)
+$$
+- When you call `loss.backward()`, PyTorch uses your `backward` method to calculate how much the loss would change if you changed the input a little bit. This is the **gradient**.
+- PyTorch automatically chains these gradients through all operations in your model, so you get the gradient of the loss with respect to every parameter (a, b, c, d).
+
+***
+
+## 3. What do these numbers mean?
+
+- The gradient for each parameter (e.g., `a.grad`, `b.grad`, etc.) tells you **how much the loss would change if you nudged that parameter up or down**.
+- If the gradient is large and positive, increasing the parameter will increase the loss (bad), so you want to decrease it.
+- If the gradient is large and negative, increasing the parameter will decrease the loss (good), so you want to increase it.
+- You use these numbers to update your parameters in the direction that makes the loss smaller (gradient descent).
+
+***
+
+## 4. How does PyTorch get these numbers?
+
+- During the forward pass, PyTorch builds a **computation graph** of all operations.
+- When you call `loss.backward()`, PyTorch walks backward through this graph, using the `backward` methods you defined (and built-in ones for standard operations), applying the chain rule to compute all gradients.
+- The numbers you see in `.grad` are the result of this process: they are the derivatives of the loss with respect to each parameter.
+
+***
+
+## 5. Step-by-step in your code
+
+1. **Forward pass:**
+    - You compute `y_pred = a + b * P3(c + d * x)` using your custom Legendre polynomial function.
+2. **Loss calculation:**
+    - You compute the loss as the sum of squared errors between `y_pred` and the true `y`.
+3. **Backward pass:**
+    - You call `loss.backward()`. PyTorch uses your custom `backward` method to compute the gradients for all parameters.
+4. **Update parameters:**
+    - You use the gradients to update `a`, `b`, `c`, and `d` so the loss gets smaller.
+
+***
+
+## 6. Summary
+
+- **Custom autograd functions** let you define both the math and the gradient for new operations.
+- **Gradients** are numbers that tell you how to change each parameter to make the loss smaller.
+- **PyTorch computes these automatically** using your `backward` method and the chain rule.
+- You use these gradients to update your parameters and train your model.
+
+If you want, I can walk through a concrete example with numbers, or show how the gradients change during training!
+<span style="display:none">[^1][^2][^3][^4][^5][^6][^7][^8][^9]</span>
+
+<div align="center">⁂</div>
+
+[^1]: https://docs.pytorch.org/tutorials/beginner/examples_autograd/polynomial_custom_function.html
+
+[^2]: https://apxml.com/courses/advanced-pytorch/chapter-1-pytorch-internals-autograd/custom-autograd-functions
+
+[^3]: https://docs.pytorch.org/tutorials/beginner/blitz/autograd_tutorial.html
+
+[^4]: https://brsoff.github.io/tutorials/beginner/examples_autograd/two_layer_net_custom_function.html
+
+[^5]: https://ghamrouni.github.io/stn-tuto/beginner/examples_autograd/two_layer_net_custom_function.html
+
+[^6]: https://docs.pytorch.org/tutorials/beginner/pytorch_with_examples.html
+
+[^7]: https://docs.pytorch.org/tutorials/intermediate/custom_function_double_backward_tutorial.html
+
+[^8]: https://www.kaggle.com/code/peggy1502/learning-pytorch-2-new-autograd-functions
+
+[^9]: https://stackoverflow.com/questions/54586938/how-to-wrap-pytorch-functions-and-implement-autograd
+