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The Xcode Subversion client includes tools to review the version history of your projects and the ability to compare two versions. Having that gold-standard copy on the server means that (provided the server is accessible) you are able to share a project with others. Each of you follows the work cycle: update the working copy, code, save, commit the new revision. As a result each of you will get all of the changes made by each of you when you update. If you and a colleague have both been working on the same piece of code, then you will get a conflict alert when you do a commit or an update. Subversion has processes to help you resolve conflicts, but undoubtedly this all works better if the various team members divide up the coding so that they don't tread on each other's toes. We're assuming for the sake of the exercises in this book that you are working alone on your projects, but the repository approach is useful for you too, because it means that you can access your code from more than one computer and make sure that you are always working on the right version. Suppose you work on your iMac at home and your MacBook at the office; by having a code repository on a server, you can use the update, code, commit approach to make sure both computers are kept in sync.

This is the generalization of Equation (72) to the case in which the learner still picks the best hypothesis h E H, but where the best hypothesis may have nonzero training error Notice that m depends logarithmically on H and on 116, as it did in the more restrictive case of Equation (72) However, in this less restrictive situation m now grows as the square of 116, rather than linearly with 116

Now that we have a bound indicating the number of training examples sufficient to probably approximately learn the target concept, we can use it to determine the sample complexity and PAC-learnability of some specific concept classes Consider the class C of target concepts described by conjunctions of boolean literals A boolean literal is any boolean variable (eg, Old), or its negation (eg, -Old) Thus, conjunctions of boolean literals include target concepts such as "Old A -Tallv Is C PAC-learnable We can show that the answer is yes by first showing that any consistent learner will require only a polynomial number of training examples to learn any c in C, and then suggesting a specific algorithm that uses polynomial time per training example Consider any consistent learner L using a hypothesis space H identical to C We can use Equation (72) to compute the number m of random training examples sufficient to ensure that L will, with probability (1 - S), output a hypothesis with maximum error E To accomplish this, we need only determine the size IHI of the hypothesis space Now consider the hypothesis space H defined by conjunctions of literals based on n boolean variables The size 1HI of this hypothesis space is 3" To see this, consider the fact that there are only three possibilities for each variable in

When you are planning your Subversion setup you need to make a strategic decision whether to have one repository to manage all of your projects or one repository for each project. Both approaches have their benefits and drawbacks, so let s take a look at each in turn.

any given hypothesis: Include the variable as a literal in the hypothesis, include its negation as a literal, or ignore it Given n such variables, there are 3" distinct hypotheses Substituting IH I = 3" into Equation (72) gives the following bound for the sample complexity of learning conjunctions of up to n boolean literals

For example, if a consistent learner attempts to learn a target concept described by conjunctions of up to 10 boolean literals, and we desire a 95% probability that it will learn a hypothesis with error less than l , then it suffices to present m randomly drawn training examples, where rn = -$(10 1n 3 ln(1105)) = 140 Notice that m grows linearly in the number of literals n, linearly in 116, and logarithmically in 116 What about the overall computational effort That will depend, of course, on the specific learning algorithm However, as long as our learning algorithm requires no more than polynomial computation per training example, and no more than a polynomial number of training examples, then the total computation required will be polynomial as well In the case of learning conjunctions of boolean literals, one algorithm that meets this requirement has already been presented in 2 It is the FIND-S algorithm, which incrementally computes the most specific hypothesis consistent with the training examples For each new positive training example, this algorithm computes the intersection of the literals shared by the current hypothesis and the new training example, using time linear in n Therefore, the FIND-S algorithm PAC-learns the concept class of conjunctions of n boolean literals with negations