Is the target assumed to be a noisy version of the output of the model in machine learning? I wonder if the following equation (you can find it in almost every ML book) refers to a general assumption that we make when using machine learning:
$$y = f(x)+\epsilon,$$
where $y$ is our output, $f$ is e.g. a neural network and $\epsilon$ is an independent noise term.
Does this mean that we assume the $y$'s contained in our training data set come from a noised version of our network output?
Not necessarily. The neural network (or whatever else you use) is a model of what you are trying to do, and usually models are not able to perfectly model reality, as it is too complex. A noise term is generally used to represent that, ie the imperfection of the model's relationship with the actual world.