如何使用1-of-k编码推导多类逻辑回归分类器的目标函数

Question

我明白这个wiki页面的意思了(http://en.wikipedia.org/wiki/Multinomial_logistic_regression)，但我不知道如何获得随机梯度下降的更新规则。很抱歉在这里问这个问题（这实际上只是关于机器学习理论，而不是实际实现）。有人能提供一个带解释的解决方案吗？提前感谢！

Answer 1

我碰巧为implement softmax写代码，我引用的最多的是页面http://ufldl.stanford.edu/wiki/index.php/Softmax_Regression

这是我在matlab中写的代码，希望它能帮助

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 : %fn',J);
        weight = weight - alpha*gradient(train_patterns,train_labels,weight);
     end    
end