Decision trees, boosted decision trees, branching programs, neural nets intro
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چکیده
Administrative: • Enrollment should be good now. • Next homework will go out next Wednesday or Friday. • Recap of where we are in course, and interplay between representation, optimization, and generalization; usual story is that as representation power improves, optimization and generalization become more painful. • Many basic latex errors in assignments; please consult lshort for an easy reference/tutorial. • Please keep giving feedback. Here’s the lemma from last time which we’ll need today: Lemma 1. Let continuous g : R → R and any > 0 be given. a. There exists a partition of [0, 1] into rectangles (R1, . . . , RN ) and a function h(x) := ∑N i=1 gi1[x ∈ Ri] where gi := g(x) for some x ∈ Ri such that ‖g − h‖1 ≤ . (Without loss of generality, this partition can be the uniform gridding of [0, 1] into cubes of equal volume.) b. Let a basis B given so that for any τ > 0 and rectangle R, there exists fR ∈ span(B) with ‖fR−1R‖1 ≤ τ . Then there exists f ∈ span(B) with ‖g − f‖1 ≤ .
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