Use the Pigeonhole principle to show that there must be a pair i j ≤ d + 1 such that _i and _ j use the same xk and use that fact to derive a contradiction to the requirements from the conjunctions hi ,h j .

1. Show that |Hd con | ≤ 3+1.

2. Conclude that VCdim(H) ≤ log3.

3. Show that Hd con shatters the set of unit vectors {e≤ d}.

4. (**) Show that VCdim(Hd con) ≤ d.

Hint: Assume by contradiction that there exists a set = {c1, . . ., cd+1} that is shattered by Hd con. Let h1, . . .,hd+1 be hypotheses in Hd con that satisfy ∀∈ [+1]hi (c j ) = _ 0 j 1 otherwise For each ∈ [+1], hi (or more accurately, the conjunction that corresponds to hi ) contains some literal _which is false on ci and true on c j for each _= i. Use the Pigeonhole principle to show that there must be a pair <>≤ + 1 such that _and use the same xk and use that fact to derive a contradiction to the requirements from the conjunctions hi ,h j .

find the cost of your paper

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