Week 5 Check-in Quiz

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Please complete the following quiz by class on Wed. 10/14/2020.

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Which of the following functions are convex? Check all that apply.

2 points

f(x) = |x| for a scalar x.

f(x) = sqrt(|x|) for a scalar x.

f(x) = min_i(x_i) for a vector x with entries x_1, ..., x_n.

f(x) = max_i(x_i) for a vector x with entries x_1, ..., x_n.

f(x) = x^2 for a scalar x.

Which of the following sets are convex? Check all that apply.

2 points

All vectors x with max_i(x_i) < 1.

All vectors x with min_i(x_i) > 1.

All vectors x with ||Ax||_2 < 1 for some fixed matrix A.

All vectors x with ||x||_1 < 1, where ||x||_1 denotes the L1 norm, ||x||_1 = sum_i=1^n |x_i|.

Give an example of a non-convex scalar function with multiple local minima, and give an example of a non-convex scalar function with only one local minimum.

2 points

When minimizing a function with gradient descent, knowing that function has a large Lipschitz constant G might lead you to:

1 point

Increase the learning rate parameter eta.

Take a larger number of iterations T.

Use a projection oracle.

Return the last iterate x^(T).

None of the above.

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