Wednesday, November 11, 2009

Lecture 19 notes

I think I was over-simplifying the proof of Fact 2 (the expected number of violated constraints by a random r-set is at most md/r, where m is the total number of constraints, d the dimension and r the sample size): however, here is a just-as-simple proof of that fact.

Note that given a set R, a constraint h (not in R) is violated by the optimal point for R only if h is in the optimal basis for (R union h). Take a random h outside a random r-set R---now h is a random element of (R union h)---and hence h is one of the d basis constraints with probability d/(r+1). Hence the expected number of violated constraints is (m-r)d/(r+1) <= md/r. QED.

This is a "backwards-analysis" style proof: instead of considering what happens when we add a random constraint h to a random r-set R, we consider what happens when we remove a random constraint from a random (r+1)-set R'.

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