ทำการแพร่ย้อนกลับของโครงข่ายเล็ก ๆ ด้วยมือ

Let's hand-trace the smallest net: one input x, one weight w, no bias. The output is simply y = w times x. Tiny enough to do on paper.

We compare the output to a target t with squared error. This loss punishes being far from the target and is easy to differentiate by hand.

Plug in x = 2, w = 3, t = 10. Forward gives y = 6 and loss = (6 - 10) squared = 16. Now we walk backward to find dloss/dw.

First link: how does loss change with y? The derivative of (y - t) squared is 2(y - t), which here is 2 times (6 - 10) = -8.

Second link: how does y change with w? Since y = w times x, dy/dw is just x, which is 2 in our example.

The chain rule multiplies the links: dloss/dw = dloss/dy times dy/dw = -8 times 2 = -16. That single number is our gradient.

A negative gradient means increasing w would lower the loss. So we should nudge w upward to do better next time.

With a learning rate of 0.1, the update is w = w - 0.1 times (-16) = 3 + 1.6 = 4.6. The weight moved toward a better value.

Redo the forward pass with w = 4.6: y = 9.2 and loss = (9.2 - 10) squared = 0.64. Far below 16, so the step truly helped.

PyTorch gets the same gradient automatically. Set requires_grad on w, run forward, call backward, and read w.grad to see -16.

A million-weight net does exactly this, just with more links chained together. The math you did by hand scales straight up.

Let's check your hand trace.

You traced a tiny net: forward for values, then backward multiplying local derivatives to get the gradient, then one step that lowered the loss. ✏️