The steps for gradient descent with respect to some variable is:
1. Compute gradient of loss with respect to the variable
2. Update the variable with step in direction of the negative gradient.

The statement claims that the following are different:
1. Gradient descent on the data with frozen weights
2. Computing the gradient of the loss with respect to the input data using the frozen weights, and then update the data by taking a small step in the direction of the negative gradient

However, (1), expands to:
1. Compute the gradient of the loss with respect to [the input data with frozen weights]
2. Update the input data with a small step in the direction of the negative gradient.

Therefore, (1) is equivalent to (2).

Is this argument airtight? If not, explain exactly why it's wrong using clear logic.
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