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That is as you d expect, since the two values are different. To get a test that passes, you would ideally need to assert that the values are different, but sadly there is no STAssertsNotEqual macro. However, you can test this by using a different macro, STAssertFalse. Change the method to that shown in Listing 11 4.

In the above section we showed that sample complexity for PAC learning grows as the logarithm of the size of the hypothesis space While Equation (72) is quite useful, there are two drawbacks to characterizing sample complexity in terms of IHI First, it can lead to quite weak bounds (recall that the bound on 6 can be significantly greater than 1 for large I H I) Second, in the case of infinite hypothesis spaces we cannot apply Equation (72) at all! Here we consider a second measure of the complexity of H, called the Vapnik-Chervonenkis dimension of H (VC dimension, or VC(H), for short) As we shall see, we can state bounds on sample complexity that use VC(H) rather than IHI In many cases, the sample complexity bounds based on VC(H) will be tighter than those from Equation (72) In addition, these bounds allow us to characterize the sample complexity of many infinite hypothesis spaces, and can be shown to be fairly tight

The VC dimension measures the complexity of the hypothesis space H, not by the number of distinct hypotheses 1 H 1, but instead by the number of distinct instances from X that can be completely discriminated using H To make this notion more precise, we first define the notion of shattering a set of instances Consider some subset of instances S E X For example, Figure 73 shows a subset of three instances from X Each hypothesis h from H imposes some dichotomy on S; that is, h partitions S into the two subsets {x E Slh(x) = 1) and {x E Slh(x) = 0) Given some instance set S, there are 2ISI possible dichotomies, though H may be unable to represent some of these We say that H shatters S if every possible dichotomy of S can be represented by some hypothesis from H

Listing 11 4. A Better Way to Test a Not Equal Assertion - (void)test100CShouldNotConvertTo100F { [testTempConverter setValue:[NSNumber numberWithFloat:100.0] forKey:@"originalTemp"]; NSNumber *newTemperatureInF = [NSNumber numberWithFloat:(float)[testTempConverter convertCToF]]; STAssertFalse(100.0f==[newTemperatureInF floatValue], @"Result should be %f, not 100.0F", [newTemperatureInF floatValue]); }

Definition: A set of instances S is shattered by hypothesis space H if and only if for every dichotomyof S there exists some hypothesis in H consistent with this dichotomy

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Figure 73 illustrates a set S of three instances that is shattered by the hypothesis space Notice that each of the 23 dichotomies of these three instances is covered by some hypothesis Note that if a set of instances is not shattered by a hypothesis space, then there must be some concept (dichotomy) that can be defined over the instances, but that cannot be represented by the hypothesis space The ability of H to shatter

In this case, you are asserting that an expression saying the returned value is 100.0 is false (as it is, since the returned value is 212.0). Run the test, and it should pass. To confirm that this is a valid test, change the assertion line to read as follows:

FIGURE 73 A set of three instances shattered by eight hypotheses For every possible dichotomy of the instances, there exists a corresponding hypothesis

a set of instances is thus a measure of its capacity to represent target concepts defined over these instances

STAssertFalse(212.0f==[newTemperatureInF floatValue],

The ability to shatter a set of instances is closely related to the inductive bias of a hypothesis space Recall from 2 that an unbiased hypothesis space is one capable of representing every possible concept (dichotomy) definable over the instance space X Put briefly, an unbiased hypothesis space H is one that shatters the instance space X What if H cannot shatter X, but can shatter some large subset S of X Intuitively, it seems reasonable to say that the larger the subset of X that can be shattered, the more expressive H The VC dimension of H is precisely this measure

Definition: The Vapnik-Chervonenkis dimension, V C ( H ) , of hypothesis space H defined over instance space X is the size of the largest finite subset of X shattered by H If arbitrarily large finite sets of X can be shattered by H, then V C ( H ) = oo

@"Result should be %f, not 100.0F", [newTemperatureInF floatValue]);

Note that for any finite H, VC(H) 5 log2 IHI To see this, suppose that VC(H) = d Then H will require 2d distinct hypotheses to shatter d instances Hence, 2d 5 IHI, andd = VC(H) s l o g 2 ( H (