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Mathematics of Machine Learning
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Predicates are a big step toward properly formalizing mathematical thinking, but we are not quite there yet. To give you an example from machine learning, let’s talk about finding the minima of loss functions (that is, training a model).
A point x is said to be the global minimum of a function f(x) if, for all other y in its domain D, f(x) ≤f(y) holds. For instance, the point x = 0 is a minima of the function f(x) = x2.
How would you express this in our formal language? For one, we could say that
where we fix f(x) = x2 and x = 0. There are two parts of this sentence: for all y ∈D, and f(x) ≤f(y) is true. The second one is a predicate:
where y ∈ℝ.
The second part seems new, as we have never seen the words “for all” in our formal language before. They express a kind of quantification about when the predicate P(y) is true.
In mathematical logic, there are two quantifiers we need to...