# If a neural network is a universal function approximator, is the irreducible error 0?

How to marry the fact that (most) neural networks with a single hidden layer are universal function approximators with the fact that in the bias-variance decomposition we consider there to be an irreducible error?

Can the irreducible error be 0? (Then it would be inappropriate named.)

The bias-variance tradeoff holds in the context that $$y=f(X)+\varepsilon$$, with $$\varepsilon$$ having mean 0 and variance $$\sigma^2$$; and $$\sigma^2$$ turns out to be the irreducible error, so named because it is a lower bound on the expected squared-error. So to answer your last question, the irreducible error can be zero, if there is no noise in the data.