Question: We say ???? : ???? ???? f:RR has bounded gradients if for every ???? , ???? ???? x,yR, ???? ( ???? ) ???? ( ????

We say ???? : ???? ???? f:RR has bounded gradients if for every ???? , ???? ???? x,yR, ???? ( ???? ) ???? ( ???? ) ???? ???? ???? f (x)f (y)Lxy for some absolute constant ???? L that does not depend on ???? x or ???? y. (The expression ???? f is the derivative of ???? f.) Now consider the function ???? ( ???? ) = ???? 2 . f(x)=x2. Is this function convex? Explain why or why not. Does this function have bounded gradients (assume the gradient at the point ???? = 2 x=2 is defined to be 0)? If so, for what value of ???? L? Consider starting at the point ???? = 1.05 x=1.05 and running gradient descent to find the minimum of ???? ( ???? ) f(x). Assume you use a learning rate equal to 0.1. Will you converge to the global minimum? Explain

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