We address a problem of estimation of an unknown regression function f at a given point x₀ from noisy observations y_i = f(x_i)+ e_i, ;i =1, ..., n. Here x_i in ε R^k are observable regressors and (e_i) are normal i.i.d. (unobservable) disturbances. The problem is analyzed in the minimax framework, namely, we suppose that f belongs to some functional class F, such that its finite-dimensional cut F_n = {f(x_i), f ε F, i =0, ..., n, } is a convex compact set. For an arbitrary fixed regression plan X_n =(x₁;...;x_n) we study minimax on F_n confidence intervals of affine estimators and construct an estimator which attains the minimax performance on the class of arbitrary estimators when the confidence level approaches 1.