Stability of optimal predictor schemes under a broader class of reductions
post by Vanessa Kosoy (vanessa-kosoy) · 2016-04-30T14:17:35.000Z · LW · GW · 0 commentsContents
Definition 1 Definition 2 Theorem Proposition 1 Proposition 2 Proof of Proposition 2 Proof of Theorem None No comments
We define a new class of reductions which preserve optimal predictor schemes, generalizing the previous notion of pseudo-invertible reductions. These are reductions that preserve the estimation problem on average but allow for large variance.
Definition 1
Fix an error space of rank 2. Consider , distributional estimation problems, a -valued -bischeme. is called a -pseudo-invertible weak reduction of to when there is a polynomial s.t. the following conditions hold:
(i)
(ii)
(iii) There is and a -valued -bischeme s.t.
(iv) There is a -valued -scheme s.t.
Such is called a -pseudo-inverse of .
a -valued -scheme is called a -pseudo-invertible weak reduction of to when it becomes such when adding trivial dependence on .
Definition 2
An error space of rank 2 is called stable when for any non-constant polynomial and , the function is in .
Theorem
Fix a stable error space of rank 2. Suppose there is a polynomial s.t. . Consider , distributional estimation problems, a -pseudo-invertible weak reduction of to and a -optimal predictor scheme for . Define by
Here, we assume the lengths of the and are compatible with and respectively. Then, is a -optimal predictor scheme for .
Proposition 1
Consider , distributional estimation problems. Suppose is a weak -pseudo-invertible reduction of to and is it's -pseudo-inverse. Then there is s.t. for any bounded function
Proposition 1 proved exactly as Proposition 2 for unidistributional estimation problems.
Proposition 2
Consider a finite set, a probability measure on , and . Then
Proof of Proposition 2
Proof of Theorem
Consider a -predictor scheme. Let be the -predictor scheme defined by
We have
where .
Applying the definitive property of to the left hand side we get
where . Using property (ii) of pseudo-invertible reductions, we get
Using the definitive property of and property (ii) of pseudo-invertible reductions on the right-hand side, we get
where . Applying Proposition 1
where . Applying the stability of to and putting everything together
for . Applying Proposition 2
Applying property (i) of pseudo-invertible reductions
for . Using the definition of we get
for . Finally we get
0 comments
Comments sorted by top scores.