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Alex's concepts deck
2.11MB. 0 audio & 26 images. Updated 2020-12-01.
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| Front | What is an algebra over sets of a space \(\Omega\)? |
| Back | A set of sets which is closed under complements, finite unions and intersections, and which contains \(\emptyset\) and \(\Omega\). \(\sigma\)-algebra if the operations can be countable. |
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| Front | When is a metric space complete / Banach? |
| Back | When every Cauchy sequence converges to something in the space |
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| Front | In the context of statistical learning theory, what is a teaching set? Motivate why this is useful, and give an example of concepts which have a large minimum teaching set. |
| Back | Consider a set of concepts \(C\) (like, "dog", "cat", "1", etc). Let \(X\) be a finite set of examples. \(S \subseteq X\) is a teaching set for concept \(c\in C\) when \(\forall c' \neq c: \exists s \in S: c'(s)\neq c(s)\). In other words, the teaching set gives us a chance of distinguishing between concepts. However, the rule may not help us generalize: imagine a teaching set for the concepts which classify the RGB value of the top-left pixel of an image. We might have a lot of trouble finding the right hypothesis, even given a teaching set. The teaching set concept seems good as a necessary condition: if you want to train a loss-minimizing discrete classifier, you need to at least give it a teaching set for the label space!\(S\) is a minimal teaching set for \(c\) when it's as small as possible. A large minimal teaching set would be, say, concepts corresponding to animal types, and a set of example photos where each had to contain at least 50 different kinds of animals. |
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