We address a problem of estimation of an unknown regression function f at a given point x0 from noisy observations yi=f(xi)+ei, i=1,...,n. Here xi in Rk are observable regressors and ei 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 Fn={f(xi), f in F, i=0,...,n} is a convex compact set. For an arbitrary fixed regression plan Xn=(x1...xn) we study minimax on Fn 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.