homework1b solutions

homework1b solutions - Massachusetts Institute of...

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Massachusetts Institute of Technology 6.867 Machine Learning, Fall 2006 Problem Set 1 Solutions Section B 1. (a) The functions are on the course website hw1/solutions . The major cause of variation among the solutions was the choice of the order in which points were picked for the update step. This was also the cause of a few subtle mistakes as well. The most straightforward strategy for picking points (in the order of their occurrence in the dataset), produces θ = 2 . 3799 radians or θ = 136 . 3582 . The number of updates, as per the above strategy, was 10. The analysis described in the lectures indicated that the number of perceptron updates (mistakes) was necessarily 2 bounded by R/γ where γ is the maximum geometric margin for this problem. geom geom A small number of updates therefore suggests that γ is reasonably large in comparison geom to the radius R of the enclosing sphere. In other words, the two class populations appear to be well-separated. (b) The most straightforward strategy for picking points (in the order of their occurrence in the dataset), produces θ = 2 . 3566 radians or θ = 136 . 3582 . We will accept answers in the range (2 . 3552 , 2 . 3592) radians or (134 . 9454 , 135 . 1730 ). The latter ranges corresponds to decision boundaries going through points at the margins of the max-margin classifier (see Prob 2). The number of updates, as per the above strategy, was 152. The bounding sphere is about the same for these points, however. We would therefore expect that the geometric margin γ is larger for this problem. geom (c) We can also evaluate γ geom , the margin actually achieved by the perceptron algorithm. This is not the maximum margin but may nevertheless be indicative of how hard the problem is. Given X and θ , γ geom can be calculated in MATLAB as follows: gamma_geom = min(abs(X*theta / norm(theta))) We get γ a = 1 . 6405 and γ b = 0 . 0493, again with some variation due to the order geom geom in which one selects the training examples. These margins appear to be consistent with our analysis, at least in terms of their relative magnitude. The bound on the number of updates holds for any margin, maximum or not, but gives the tightest guarantee with the maximum margin.
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