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(Solved): Consider the function f:R^(n)->R defined by f(x)=10(x_(1)-1)^(2) \sum_(i=2)^n x_(i)^(2) The opt ...



Consider the function

f:R^(n)->R

defined by

f(x)=10(x_(1)-1)^(2) \sum_(i=2)^n x_(i)^(2)

The optimal minimer of

f

is

x^(**)=(1,0,0,dots,0)

. Show that the GD algorithm with an arbitrary starting point

x_(0)

and suitable step-size (choose the step-size based on smoothness), satisfies the following: After

k

iterations, we have

||x_(k)-x^(**)||_(2)^(2)<=||x_(0)-x^(**)||_(2)^(2)(1 c)^(-k)

for some universal constant

c>0

(it doesn't matter what constant

c

you get to receive fullcredit). In other words, the distance of the

k

'th iterate to the optimal decreases exponentially in the number of iterations. [4 points] [Hint: Show that the function is smooth and see if you can show a complementary inequality:

f(x)-f(x^(**))>=c||x-x^(**)||_(2)^(2)

for some constant

c

. Now, use this with the main inequality that we used in the analysis of GD (the one we obtained by combining smoothness upperbound and convexity). You can use these to get a multiplicative decrease in the distance to optimum at every step.]



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