# Gradient calculation from scalar Loss?

Greetings everyone!

I’m trying to understand the implementation of a custom Model in keras, and particularly with respect to the calculation of the gradients from the loss.

From what I understand, when implementing a custon train_step, loss should return a scalar value, instead of a loss value for each batch sample (batch_size, ). This is the case in the implementation of the following VAE (e.g., Variational AutoEncoder ), and is stated more generally here: Losses , where it suggests that in a custom training loop, a Class instance loss should be used, retuning a scalar.

My question is, how does the tape.gradient calculates the average gradient? If we average the losses of the batch, don’t we lose the per sample gradient information? Theoretically, I understand that we need to calculate the average gradient of the per sample gradients in the batch, and not the gradient of the average loss. Here is a very nice explanation why the two are not equivalent: backpropagation - What exactly is averaged when doing batch gradient descent? - Artificial Intelligence Stack Exchange

I would appreciate any clarifications and help!

Certainly! When training a neural network with mini-batches in TensorFlow, the loss computed for a batch is typically a scalar value representing the average of individual losses for each sample in the batch. This scalar loss is used to calculate gradients with respect to the model’s parameters using TensorFlow’s `tf.GradientTape`.

Here’s why averaging works and is commonly used:

1. Stability: Averaging loss over a batch stabilizes training by reducing noise from individual sample losses.
2. Scale Consistency: It ensures the magnitude of loss (and thus gradients) doesn’t depend on batch size, making training behavior more consistent.

The gradient of this average loss effectively approximates the average of the individual gradients for each sample. This is computationally efficient and is standard practice due to the linearity of differentiation: the gradient of the sum (or average) of losses is equal to the sum (or average) of individual gradients.

Therefore, even though the loss is a scalar average, the gradient calculation does not lose per-sample gradient information; rather, it computes an efficient estimate of the average gradient across the batch. This is how TensorFlow’s automatic differentiation handles gradient calculations in mini-batch training.