Deep Learning by Andrew Ng --- Softmax regression

这是UFLDL的编程练习。

Weight decay（Softmax 回归有一个不寻常的特点：它有一个“冗余”的参数集）后的cost function和梯度函数：

• cost function： $$\begin{align} J(\theta) = - \frac{1}{m} \left[ \sum_{i=1}^{m} \sum_{j=1}^{k} 1\left\{y^{(i)} = j\right\} \log \frac{e^{\theta_j^T x^{(i)}}}{\sum_{l=1}^k e^{ \theta_l^T x^{(i)} }} \right] + \frac{\lambda}{2} \sum_{i=1}^k \sum_{j=0}^n \theta_{ij}^2 \end{align}$$
• 梯度函数：

bsxfun函数的使用：

• to prevent overflow, simply subtract some large constant value from each of the $$\begin{align} \theta_j^T x^{(i)}\end{align}$$ terms before computing the exponential：
  % M is the matrix as described in the text
  M = bsxfun(@minus, M, max(M, [], 1));
• use the following code to compute the hypothesis：
  % M is the matrix as described in the text
  M = bsxfun(@rdivide, M, sum(M）

练习题答案（建议自己完成，后参考）：

• softmaxCost.m:

M = theta*data; %exp(theta(l)‘ * x(i))
M = bsxfun(@minus, M, max(M, [], 1));  
h = exp(M);
h =  bsxfun(@rdivide, h, sum(h));  
size(groundTruth);
cost = -1/numCases*sum(sum(groundTruth.*log(h)))+lambda/2*sum(sum(theta.^2));  
thetagrad = -1/numCases*((groundTruth-h)*data‘)+lambda*theta; 

• softPredict.m:

[index ,  pred]= max(theta * data,[],1);