My formulation is broad enough that it doesn’t have to be a dedicated piece of logic, there just has to be some way of looking at the reset of the network that depends on X and Y being the same.
But X and Y are not the same! For example, if the model is intended to classify images of animals, the computation X may correspond to [how many legs does the animal have?] and Y may correspond to [how large is the animal?]
This is what I take issue with—if there is a way to change both components simultaneously to have an effect on the loss, SGD will happily do that.
This seems to me wrong. SGD updates the weights in the direction of the gradient, and if changing a given weight alone does not affect the loss then the gradient component that is associated with that weight will be 0 and thus SGD will not change that weight.
SGD updates the weights in the direction of the gradient, and if changing a given weight alone does not affect the loss then the gradient component that is associated with that weight will be 0 and thus SGD will not change that weight.
If the partial derivative wrt two different parameters is zero, i.e ∂f∂θ1=∂f∂θ2=0, then it must be that changing both simultaneously does not change the loss either (to be precise, limh→0f(x+h(θ1+θ2))h=0).
I don’t see how this is relevant here. If it is the case that changing only w1 does not affect the loss, and changing only w2 does not affect the loss, then SGD would not change them (their gradient components will be zero), even if changing them both can affect the loss.
It’s relevant because it demonstrates that in differentiable functions, if it is the case that changing only w1 does not affect the loss, and changing only w2 does not affect the loss, then it is not possible that changing them both can affect the loss either.
But X and Y are not the same! For example, if the model is intended to classify images of animals, the computation X may correspond to [how many legs does the animal have?] and Y may correspond to [how large is the animal?]
This seems to me wrong. SGD updates the weights in the direction of the gradient, and if changing a given weight alone does not affect the loss then the gradient component that is associated with that weight will be 0 and thus SGD will not change that weight.
If the partial derivative wrt two different parameters is zero, i.e ∂f∂θ1=∂f∂θ2=0, then it must be that changing both simultaneously does not change the loss either (to be precise, limh→0f(x+h(θ1+θ2))h=0).
I don’t see how this is relevant here. If it is the case that changing only w1 does not affect the loss, and changing only w2 does not affect the loss, then SGD would not change them (their gradient components will be zero), even if changing them both can affect the loss.
It’s relevant because it demonstrates that in differentiable functions, if it is the case that changing only w1 does not affect the loss, and changing only w2 does not affect the loss, then it is not possible that changing them both can affect the loss either.