In standard backpropagation for a feedforward neural network, what quantity is propagated backward from the output layer to hidden layers?

The gradient of the loss with respect to each layer's pre-activation

In gradient checking for backpropagation, a parameter theta_i is perturbed by a small epsilon. Which finite-difference approximation is generally preferred for numerically estimating dL/dtheta_i?

[L(theta_i + epsilon) - L(theta_i - epsilon)] / (2 epsilon)

Consider a hidden layer l with pre-activation z^(l), activation a^(l), and next-layer weights W^(l+1). If delta^(l+1) = dL/dz^(l+1), which expression correctly gives delta^(l)?

delta^(l) = ((W^(l+1))^T delta^(l+1)) ⊙ g'(z^(l))

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