Manuscript Psy03-B@1 · Psy03-B-v1.tex · theorem · 8ca3caf9

Theorem: Nested constraints order optimal achievable risk

Optimal capacity and actual performance are different · lines 865–885 · proof included or linked

Psy03-B@1#thm:optimalrisk, Psy03-B@1#eq:optimalrisk

If $S \subseteq S^{'}$ , then

F (S^{'}) \subseteq F (S) .

For a common nonnegative loss,

f \in F (S^{'}) in f E ℓ (f (X), Y) \geq f \in F (S) in f E ℓ (f (X), Y) .

An empty feasible class has infimum $+ \infty$ .

Proof. An implementation satisfying the additional zero constraints also satisfies the original constraints. An infimum over a smaller set cannot be smaller.

Displayed formulas in this passage

unlabeled · line 868
$F (S^{'}) \subseteq F (S) .$
Psy03-B@1#eq:optimalrisk · line 872
$f \in F (S^{'}) in f E ℓ (f (X), Y) \geq f \in F (S) in f E ℓ (f (X), Y) .$

Nearby in this section

definition: Definition: Common feasible class
section: Optimal capacity and actual performance are different
proposition: Proposition: Deletion can improve, worsen, or preserve a checkpoint
section: Optimal capacity and actual performance are different
proposition: Proposition: Permanence is initially a latent restriction
section: Optimal capacity and actual performance are different