CHAPTER 6. RELATIONSHIP BETWEEN LEARNING CLASSIFIERS
6.2 Modeling Learning Tasks
6.2.2 k Delegating oracle
A kDelegating oracle is a procedure which when invoked by a learner in turn invokes subordi- nate oracles EX1hf1, Xs1, D1i . . . EXkhfk, Xsk, Dki with probabilities p1. . . pkrespectively. The
kDelegating oracle has access to a mapping set M = {m1, m2. . . mk} where mi = {mxi, mci};
mxi : Xsi −→ X is an attribute mapping function; and mci : Ci −→ C is a class mapping func-
tion where Ci = Range(fi) and C = Range(f ). It uses the mapping mi to convert an instance
hxsi, fi(xsi)i received form the ithsubordinate oracle to hmix(xsi), mci(fi(xsi))i before passing it
to the learner. A mapping aware k-delegating oracle is represented by EXhφ, X+, D+, M, X i, where X+ = {X1, X2. . . Xk} is the set of instance spaces, D+ = {D1, D2. . . Dk} is the set
of distributions, M is a mapping set and φ is the labeling function with which instances are returned to the learner. A schematic representation of learning using a kDelegating oracle is shown in Figure 6.1.
∀xsi ∈ Xsi, mxi(xsi) ∈ X ; ∀l ∈ Range(fi), mci(l) ∈ Range(f ) and whenever x ∈ Xsi, Xsj,
mxi(x) = mxj(x). A mapping set M is said to be admissible if ∀mi∈ M, mi is admissible. An
admissible mapping set M ensures that the examples returned by the k -delegating oracle are of the form hx, l(x)i where x ∈ X and l(x) ∈ C. We denote the set of all possible admissible mappings by M∗ and assume in the rest of the chapter that a given mapping is admissible. Consider the case of learning a function f using a k Delegating oracle kEXhφ, X+, D+, M, X i. A straight-forward analysis shows that the total number of admissible mappings is |M ∗ | = Qk
i |X ||Xsi|(we assume two mappings are different if they differ on mapping atleast one instance
from the instance space of any one of the subordinate oracles).
Let the ideal mapping set Mtrue= {m1,true, m2,true. . . Mk,true,} where mi,true=
{mx
i,true, mci,true}. The ideal mapping set is assumed to satisfy the following conditions: (a)
Mtrue ∈ M∗ and (b) hx, l(x)i the labeled instance returned to L (obtained after applying
relevant mappings in Mtrue to the labeled instance sampled from the subordinate oracle) is
the same as hx, f (x)i. The condition (b) requires that instances returned to the learner are labeled according to the target function f .
Definition 2 Semantics Preserving Class Mapping : A class mapping mci is said to be semantics preserving if ∀l ∈ Ci mci(l) = mci,true(l).
Definition 3 Semantics Preserving Attribute Mapping: An attribute mapping mxi is said to be semantics preserving whenever ∀x ∈ Xsi, fi(x) = l and mci,true(l) = l1 implies
f (mx
i(x)) = l1
Definition 4 Semantics Preserving Mapping Set : A mapping set M = {m1, m2. . . mk}
is said to be correct if ∀i ∈ {1, 2, . . . , k} mxi and mci are semantics preserving.
Definition 5 Identity Mapping A mapping set M is said to be an identity mapping if ∀mi = {mxi, mci} ∈ M, mxi(x) = x and mci(l) = l.
A semantics preserving mapping guarantees that the instances returned to the learner are labeled with the target function f . Hence, given a k Delegating oracle kEXhφ, X+, D+, M, X i
Instances with Label 0 Instances with Label 1 Instances with Label 0 Instances with Label 1 Instances with Label 0 Instances with Label 1 Instances with Label 0 Instances with Label 1
Figure 6.2 An example of a mapping with errors and a semantics preserv- ing mapping.
with semantics preserving mapping M , it is the case that φ = f where f is the target function. It is the case that Mtrue is semantics preserving. A mapping set M is said to have errors if it
is not semantics preserving. Errors in a mapping set can either be due to errors in attribute mappings or errors in class mappings. Since the class mappings are fairly straightforward (being between the binary class labels 0 and 1), we will assume throughout the chapter that only possible errors are in attribute mappings. Figure 6.2 shows an example mapping with errors and a semantics preserving mapping.
Remark A straight-forward analysis shows that the total number of semantics preserving mappings isQk i |X+| |Xsi+| × |X−||Xsi −| .
The remark above shows that, in general, there are multiple available semantics preserving mappings each of which returns examples to the learner that are labeled with the target func- tion. We now show that difference between using different semantics preserving mappings man- ifests itself in terms of difference in distribution with which examples are returned to the learner. Given a mapping mxi, let [mxi(y 7→ z)] be a indicator function that returns 1 when mxi(y) = z (for y ∈ Xsi) and returns 0 otherwise. Given a k Delegating oracle kEXhφ, X+, D+, M, X i the
distribution over X with which instances are returned to the learner L is P r[z] = k X i=1 pi X y∈Xsi [mxi(y 7→ z)]pry∈Di[y]
stances are returned to the learner L depends on the set P = {p1. . . pk} (the probabilities of
selecting the subordinate oracles), the distribution set D+ and the mapping set M (through dependence on indicator function [mxi(y 7→ z]) and is denoted by MP(D+). For easing clutter
of notation we drop the parameter P in the notation and denote it by M (D+).
While a learner L with access to a kDelegating oracle can easily model learning from semantically disparate data (including modeling errors in mappings), it is the case that a learner with access to a kDelegating oracle cannot model domain adaptation. We now introduce, in the next section, an approach to model domain adaptation.