[Verse 1]
Neural network spits predictions, but the truth ain't matching up
Error signals screaming loud, gotta trace where things corrupt
Forward pass delivered lies, now we calculate the cost
Gradient descent needs direction, or our training will be lost
Chain rule mathematics, derivatives cascade down
From output layer backwards, error's throne gets dethroned
[Chorus]
Back-prop, back-prop, errors flowing upstream
Gradient times the delta, fulfilling the dream
Weight updates, weight updates, learning from mistakes
Chain rule multiplication, whatever it takes
Back-prop, back-prop, neurons getting wise
Partial derivatives dancing, minimizing lies
[Verse 2]
Hidden layers hold the secrets, but they're buried deep inside
Activation functions squashing, where the gradients reside
Delta equals error times derivative of z
Propagate through every synapse, set the parameters free
Matrix multiplication backwards, transpose the weight array
Each neuron gets its punishment for leading us astray
[Chorus]
Back-prop, back-prop, errors flowing upstream
Gradient times the delta, fulfilling the dream
Weight updates, weight updates, learning from mistakes
Chain rule multiplication, whatever it takes
Back-prop, back-prop, neurons getting wise
Partial derivatives dancing, minimizing lies
[Bridge]
Vanishing gradients haunt the deeper layers
Exploding gradients make the training prayers
Learning rate's the throttle, momentum smooths the ride
Batch normalization keeps the signals amplified
[Verse 3]
Computational graph unravels, every operation tracked
Automatic differentiation, keeping gradients intact
Stochastic mini-batches shuffle, epochs cycling through
Backpropagation algorithm makes the networks true
From perceptrons to transformers, same principle applies
Error signals teach the system, optimization never dies
[Outro]
Mathematics of the mind, encoded in each weight
Backpropagation builds the bridge between mistake and fate