Updated equations of gradients and y_pred

[im-shriram]

Refactor Backpropagation Equations for Readability and Mathematical Clarity

Summary

This PR updates the manual backpropagation logic in polynomial_tensor.py to use dot products (torch.matmul) for gradient calculations, making the implementation more mathematically intuitive and readable.

Changes

• Backpropagation: Replaced element-wise multiplication with torch.matmul for grad_b, grad_c, and grad_d. This better represents the dot product operation in the gradient equations.
• Forward Pass: Minor refactoring of the y_pred equation for improved clarity.

Rationale

Using matmul for these gradients aligns the code closer to standard linear algebra notation, making the tutorial easier to interpret for users learning manual gradient computation.

[pytorch-bot]

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PR Review: Updated equations of gradients and y_pred

• Gather context and read the changed file
• Analyze the mathematical correctness of the changes
• Provide review feedback

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Review Summary

This PR has correctness concerns and pedagogical trade-offs that should be addressed before merging.

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Issues Found

1. Redundant .sum() after torch.matmul on 1D tensors (Bug/Misleading)

File: beginner_source/examples_tensor/polynomial_tensor.py:52-54

grad_b = torch.matmul(input=grad_y_pred, other=x).sum()
grad_c = torch.matmul(input=grad_y_pred, other=x.pow(2)).sum()
grad_d = torch.matmul(input=grad_y_pred, other=x.pow(3)).sum()

Both grad_y_pred and x are 1D tensors of shape (2000,). When torch.matmul is called with two 1D tensors, it computes the dot product and returns a scalar tensor. The subsequent .sum() is therefore a no-op — it sums a scalar, returning the same scalar.

This isn't technically wrong (it produces the correct result), but it's misleading. It suggests the author believed matmul returns a vector here, which it does not. For a beginner tutorial, this is confusing.

2. Pedagogical regression for a beginner tutorial

The original code:

grad_b = (grad_y_pred * x).sum()

This explicitly shows the two-step process: element-wise multiply, then sum — which is how you manually compute a dot product. For learners encountering gradient computation for the first time, this transparency is valuable. Replacing it with torch.matmul hides the mechanics behind an API call, which works against the tutorial's purpose of teaching manual backpropagation.

3. grad_y_pred rewrite is a no-op change

# Before:
grad_y_pred = 2.0 * (y_pred - y)
# After:
grad_y_pred = -2.0 * (y - y_pred)

These are algebraically identical: -2(y - ŷ) = 2(ŷ - y). This doesn't improve clarity or correctness — it just introduces a double negative that's arguably harder to read.

4. Unnecessary parentheses in y_pred

# Before:
y_pred = a + b * x + c * x ** 2 + d * x ** 3
# After:
y_pred = a + (b * x) + (c * x.pow(2)) + (d * x.pow(3))

Python's operator precedence already guarantees the correct evaluation order (** > * > +). The added parentheses are redundant. Replacing ** with .pow() is a style preference, not a clarity improvement.

5. Import reorder (positive)

Moving import math before import torch correctly follows PEP 8 (stdlib before third-party). This is the one unambiguously good change. ✓

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Recommendation

I would not recommend merging this PR as-is. The changes don't fix a bug, and they make a beginner tutorial less transparent by hiding computation behind torch.matmul while introducing a misleading redundant .sum(). The tutorial intentionally uses manual, explicit operations to teach how gradient computation works under the hood.

If the goal is to use matmul for dot products, the .sum() calls should be removed. But even then, the pedagogical trade-off should be considered — explicit element-wise operations are arguably better for a beginner tutorial about manual backpropagation.

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