Gradient descent with vector-valued loss
I see clearly that this works for $l(w) \in \mathbb{R}$, but am wondering how it generalizes to vector-valued loss functions, i.e. $l(w) \in \mathbb{R}^n$ for $n &gt; 1$.


Generally in neural network optimisers it does not*, because it is not possible to define what optimising a multi-value function means whilst keeping the values separate. If you have a multi-valued loss function, you will need to reduce it to a single value in order to optimise.

When a neural network has multiple outputs, then typically the loss function that is optimised is a (possibly weighted) sum of the individual loss functions calculated from each prediction/ground truth pair in the output vector.

If your loss function is naturally a vector, then you must choose some reduction of it to scalar value e.g. you can minimise the magnitude or maximise some dot-product of a vector, but you cannot "minimise a vector".



* There is a useful definition of multi-objective optimisation, which effectively finds multiple sets of parameters that cannot be improved upon (for a very specific definition of optimality called Pareto optimality). I do not think it is commonly used in neural network frameworks such as TensorFlow. Instead I suspect that passing a vector loss function into TensorFlow optimiser will cause it to optimise a simple sum of vector components.