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L.A. Wasserman 529●●●● ●●● ●●●● ●● ●●● ●●●●●●● ● ●●●●● ● ●●● ● ●●●●●● ●● ●●●●●● ●●●●●●● ●● ●● ●● ●●●●●●●●●● ●●● ●●●●●●●● ● ●●● ●●●●●●●●●●● ●● ●●●● ● ●● ●●● ●●● ●●●●●●● ●●●●●●●●●●● ●● ●●●●●●●●●● ● ●●●● ● ●● ●●●●●●●●●●FIGURE 44.2Labeled and unlabeled data.1. There is a semi-supervised estimator ̂f such that( ) 2sup R P ( ̂f) C2+ξ≤ , (44.1)P ∈P nnwhere R P ( ̂f) =E{ ̂f(X) − f(X)} 2 is the risk of the estimator ̂f underdistribution P .2. For supervised estimators S n ,wehaveinf̂f∈S n(sup R P ( ̂f) C≥P ∈P nn3. Combining these two results, we conclude thatinf ̂f∈SSNsup P ∈Pn R P ( ̂f) ( Cinf ̂f∈Snsup P ∈Pn R P ( ̂f)≤ n) 2d−1. (44.2)) 2(d−3−ξ)(2+ξ)(d−1)−→ 0 (44.3)and hence, semi-supervised estimation dominates supervised estimation.The class P n consists of distributions such that the marginal for X is highlyconcentrated near some lower dimensional set and such that the regressionfunction is smooth on this set. We have not proved that the class must beof this form for semi-supervised inference to improve on supervised inferencebut we suspect that is indeed the case. Our framework includes a parameter αthat characterizes the strength of the semi-supervised assumption. We showedthat, in fact, one can use the data to adapt to the correct value of α.