bias - variance regression

Question 1

what does S,x and y mean in regression problem

Answer 1

Let S be a training dataset sampled from some distribution.
Let x ∈Xdenote an input and y ∈R the corresponding output.

Question 2

what is true relationship

Answer 2

y= f (x) + ϵ,
where ϵ is random noise with E[ϵ] = 0 and variance Var(ϵ) = σ2.
ˆA learning algorithm produces an estimator
f (x; S).