如何使用1-k編碼來推導多類邏輯迴歸分類器的目標函數？

Question

我得到了這個wiki頁面所說的（http://en.wikipedia.org/wiki/Multinomial_logistic_regression），但我不知道如何獲得隨機梯度下降的更新規則。很抱歉在這裏提出這個問題（這實際上只是關於機器學習理論，而不是實際的實施）。有人可以提供解決方案嗎？提前致謝！

Answer 1

我碰巧編寫代碼來implent SOFTMAX，我指大部分頁面http://ufldl.stanford.edu/wiki/index.php/Softmax_Regression

這是我在MATLAB寫的代碼，希望這將有助於

function y = sigmoid_multi(weight,x,class_index) 
%% weight feature_dim * class_num 
%% x  feature_dim * 1 
%% class_index scalar 
    sum = eps; 
    class_num = size(weight,2); 
    for i = 1:class_num 
     sum = sum + exp(weight(:,i)'*x); 
    end 
    y = exp(weight(:,class_index)'*x)/sum; 
end 

function g = gradient(train_patterns,train_labels,weight) 
    m = size(train_patterns,2); 
    class_num = size(weight,2); 
    g = zeros(size(weight)); 
    for j = 1:class_num 
     for i = 1:m 
      if(train_labels(i) == j) 
       g(:,j) = g(:,j) + (1 - log(sigmoid_multi(weight,train_patterns(:,i),j) + eps))*train_patterns(:,i); 
      end 
     end 
    end 
    g = -(g/m); 
end 
function J = object_function(train_patterns,train_labels,weight) 
    m = size(train_patterns,2); 
    J = 0; 
    for i = 1:m 
     J = J + log(sigmoid_multi(weight,train_patterns(:,i),train_labels(i)) + eps); 
    end 
    J = -(J/m); 
end 

function weight = multi_logistic_train(train_patterns,train_labels,alpha) 
%% weight feature_dim * class_num 
%% train_patterns featur_dim * sample_num 
%% train_labels 1 * sample_num 
%% alpha scalar 
    class_num = length(unique(train_labels)); 
    m = size(train_patterns,2); %% sample_number; 
    n = size(train_patterns,1); % feature_dim; 
    weight = rand(n,class_num); 
    for i = 1:40 
     J = object_function(train_patterns,train_labels,weight); 
     fprintf('objec function value : %f\n',J); 
     weight = weight - alpha*gradient(train_patterns,train_labels,weight); 
    end  
end