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In the following two problems, we consider variants of the gradient descent algorithm for solving min x?Rn f(x), (1) where f : R n ? R is a smooth function. Given x (k) , the gradient descent algorithm finds x (k+1) by x (k+1) = arg min x?Rn f(x (k) ) + h?f(x (k) ), x ? x (k) i + 1 2?k kx ? x (k) k 2 2 ,

Gradient depends on inner product. Therefore, one variant of gradient descent is obtained by choosing a weighted inner product of R n rather than the standard one. In particular, given a symmetric positive definite matrix W ? R nn, we know that hx, yiW := x TW y defines an inner product of R n. (a) Express ?W f (the gradient of f in R n with the weighted inner product h, iW ) in terms of W and ?f (the gradient of f in R n with the standard inner product). (b) Give the gradient descent algorithm for solving (1) under the weighted inner product h, iW (i.e., find an explicit formula of x (k+1) when we replace h, i and k k2 2 in (2) by h, iW and k k2 W respectively). (By choosing a suitable W, we may obtain faster algorithms than the standard gradient descent. This technique is known as preconditioning.)

In the following two problems, we consider variants of the gradient descent algorithm for solving min f(x), (1) CERn where f : R" - R is a smooth function. Given a *), the gradient descent algorithm finds a (*+1) by (k+1) = arg min f(x(")) + (Vf(x(*)), x - 2(*)) 4 1 llac - 32 (4) 113, DERn 20k (2) where ox > 0 is a step size.3. Gradient depends on inner product. Therefore, one variant of gradient descent is obtained by choosing a weighted inner product of R" rather than the standard one. In particular, given a symmetric positive definite matrix W e Rexn, we know that (x, y)w :=a Wy defines an inner product of R". (a) Express Vwf (the gradient of f in R" with the weighted inner product (, .)w) in terms of W and Vf (the gradient of f in R" with the standard inner product). (b) Give the gradient descent algorithm for solving (1) under the weighted inner product (, .)w (i.e., find an explicit formula of a (*+1) when we replace (., .) and || . ? in (2) by (, .)w and || . lli respectively). (By choosing a suitable W, we may obtain faster algorithms than the standard gradient descent. This technique is known as preconditioning.)