Softmax takes in an n-dimensional vector x = R and outputs another n-dimensional vector y R...

  

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Softmax takes in an n-dimensional vector x = R" and outputs another n-dimensional vector y R" with the ith element y(i) = n k=1 expx(i) expx(k) Vi E {1, 2,...,n}. In this problem, please calculate the gradient of y with respect to x. Let di.j 1. (5 Points): Please derive an expression for dii, where i = {1, 2, ..., n}. Hint: Recall that for a general function in the form of h(x): respect to the variable x is (3) y (i) - ax(j) f(x) = 9(x)' its derivative h(x) af (x) g(x) ag(x) : f(x) n (9(x)) 2(x(i)) 2. (5 Points): In a similar way, please derive the expression for di,j, where i j and i, j = {1, 2,...,n}. Softmax takes in an n-dimensional vector x = R" and outputs another n-dimensional vector y R" with the ith element y(i) = n k=1 expx(i) expx(k) Vi E {1, 2,...,n}. In this problem, please calculate the gradient of y with respect to x. Let di.j 1. (5 Points): Please derive an expression for dii, where i = {1, 2, ..., n}. Hint: Recall that for a general function in the form of h(x): respect to the variable x is (3) y (i) - ax(j) f(x) = 9(x)' its derivative h(x) af (x) g(x) ag(x) : f(x) n (9(x)) 2(x(i)) 2. (5 Points): In a similar way, please derive the expression for di,j, where i j and i, j = {1, 2,...,n}.