Assume we have K different classes in a multi-class Softmax Regression model. The for k=1,2,..., K,...

 

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Assume we have K different classes in a multi-class Softmax Regression model. The for k=1,2,..., K, where Sk (x) = Ok².x, T - posterior probability is pk = 8(Sk(x))k exp (Sk(x)) Σ1 exp (sj(x)) j=1 input x is an n-dimension vector, and K the total number of classes. 1) To learn this Softmax Regression model, how many parameters we need to estimate? What are these parameters? 2) Consider the cross-entropy cost function J(0) of m training samples {(xi, Yi)}i=1,2,...,m as below. Derive the gradient of J(0) regarding to 0k. m K ΕΣΣ yklog (p) i=1 k=1 where y(¹) = 1 if the ith instance belongs to class k; 0 otherwise. J(0) = = Assume we have K different classes in a multi-class Softmax Regression model. The for k=1,2,..., K, where Sk (x) = Ok².x, T - posterior probability is pk = 8(Sk(x))k exp (Sk(x)) Σ1 exp (sj(x)) j=1 input x is an n-dimension vector, and K the total number of classes. 1) To learn this Softmax Regression model, how many parameters we need to estimate? What are these parameters? 2) Consider the cross-entropy cost function J(0) of m training samples {(xi, Yi)}i=1,2,...,m as below. Derive the gradient of J(0) regarding to 0k. m K ΕΣΣ yklog (p) i=1 k=1 where y(¹) = 1 if the ith instance belongs to class k; 0 otherwise. J(0) = = Assume we have K different classes in a multi-class Softmax Regression model. The for k=1,2,..., K, where Sk (x) = Ok².x, T - posterior probability is pk = 8(Sk(x))k exp (Sk(x)) Σ1 exp (sj(x)) j=1 input x is an n-dimension vector, and K the total number of classes. 1) To learn this Softmax Regression model, how many parameters we need to estimate? What are these parameters? 2) Consider the cross-entropy cost function J(0) of m training samples {(xi, Yi)}i=1,2,...,m as below. Derive the gradient of J(0) regarding to 0k. m K ΕΣΣ yklog (p) i=1 k=1 where y(¹) = 1 if the ith instance belongs to class k; 0 otherwise. J(0) = = Assume we have K different classes in a multi-class Softmax Regression model. The for k=1,2,..., K, where Sk (x) = Ok².x, T - posterior probability is pk = 8(Sk(x))k exp (Sk(x)) Σ1 exp (sj(x)) j=1 input x is an n-dimension vector, and K the total number of classes. 1) To learn this Softmax Regression model, how many parameters we need to estimate? What are these parameters? 2) Consider the cross-entropy cost function J(0) of m training samples {(xi, Yi)}i=1,2,...,m as below. Derive the gradient of J(0) regarding to 0k. m K ΕΣΣ yklog (p) i=1 k=1 where y(¹) = 1 if the ith instance belongs to class k; 0 otherwise. J(0) = =

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