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Problem 2 : Sub - gradient Method Consider a least squares problem with l 1 regularization: m i n x [ f ( x )
Problem : Subgradient Method
Consider a least squares problem with regularization:
This problem is often called LASSO least absolute shrinkage and selection operator and is known to induce lem sparse solutions with few
nonzero elements in which can have advantages in terms of computation and interpretability. This problem is nonsmooth due to the
regularization term. It is also not strongly convex when A has more columns than rows. We ie you will solve this problem using several
different algorithms in this class. We start with what we have seen thus far: the subgradient method.
The dataset represented in the matrices provided in the numpy binary files Anpy and bnpy are from a diabetes dataset scikit
learn.orgstablemodulesgeneratedsklearndatasets.loaddiabetes.html with features that has been corrupted with an additional noisy
features. Thus a sparse solution should be very effective. Below you will find some skeleton code to help with loading the data, running the
algorithm and plotting the results. Don't use stock optimization code, you should develop the core part of this assignment yourself.
Minimize using iterations of the subgradient method starting with and
Part A
Use a decreasing step size of with values for that roughly optimize the empirical performance. Separately record the unsquared
error and the regularization term
Part B
Now use a more slowly decreasing step size of with values for that roughly optimize the empirical performance. Separately
record the unsquared error and the regularization term
Part C
Now try to find the best fixed step size. Plot the results and compare to the decreasing step size you see above.
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