Andrew Ng coursera上的《機器學習》ex4

按照課程所給的ex4的文檔要求，ex4要求完成以下幾個計算過程的代碼編寫：

exerciseName	description
sigmoidGradient.m	compute the grident of the sigmoid function
randInitializedWeights.m	randomly initialize weights
nnCostFunction.m	Neutral network function

1.sigmoidGradient.m

要求如下：

根據作業文檔給出的要求以及邏輯迴歸的梯度下降的表達式。得出如下的Octave代碼：

function g = sigmoidGradient(z)
%SIGMOIDGRADIENT returns the gradient of the sigmoid function
%evaluated at z
%   g = SIGMOIDGRADIENT(z) computes the gradient of the sigmoid function
%   evaluated at z. This should work regardless if z is a matrix or a
%   vector. In particular, if z is a vector or matrix, you should return
%   the gradient for each element.

g = zeros(size(z));

% ====================== YOUR CODE HERE ======================
% Instructions: Compute the gradient of the sigmoid function evaluated at
%               each value of z (z can be a matrix, vector or scalar).
g = sigmoid(z) .* (1 - sigmoid(z));
% =============================================================
end

需要注意的是文檔中給出的表達式只是針對一個訓練數據集的，需要對所有的訓練數據集進行操作的話，需要用到Octave的一個運算符.，小數點表示對所有數據都進行操作。

2. randInitializedWeights.m

該塊代碼不需要自己寫，所有忽略。

3.nnCostFunction.m

要求如下：

Octave代碼如下：

function [J grad] = nnCostFunction(nn_params, ...
                                   input_layer_size, ...
                                   hidden_layer_size, ...
                                   num_labels, ...
                                   X, y, lambda)
%NNCOSTFUNCTION Implements the neural network cost function for a two layer
%neural network which performs classification
%   [J grad] = NNCOSTFUNCTON(nn_params, hidden_layer_size, num_labels, ...
%   X, y, lambda) computes the cost and gradient of the neural network. The
%   parameters for the neural network are "unrolled" into the vector
%   nn_params and need to be converted back into the weight matrices. 
% 
%   The returned parameter grad should be a "unrolled" vector of the
%   partial derivatives of the neural network.
%

% Reshape nn_params back into the parameters Theta1 and Theta2, the weight matrices
% for our 2 layer neural network
Theta1 = reshape(nn_params(1:hidden_layer_size * (input_layer_size + 1)), ...
                 hidden_layer_size, (input_layer_size + 1));

Theta2 = reshape(nn_params((1 + (hidden_layer_size * (input_layer_size + 1))):end), ...
                 num_labels, (hidden_layer_size + 1));

% Setup some useful variables
m = size(X, 1);

% You need to return the following variables correctly 
J = 0;
Theta1_grad = zeros(size(Theta1));
Theta2_grad = zeros(size(Theta2));

% ====================== YOUR CODE HERE ======================
% Instructions: You should complete the code by working through the
%               following parts.
%
% Part 1: Feedforward the neural network and return the cost in the
%         variable J. After implementing Part 1, you can verify that your
%         cost function computation is correct by verifying the cost
%         computed in ex4.m
%
% Part 2: Implement the backpropagation algorithm to compute the gradients
%         Theta1_grad and Theta2_grad. You should return the partial derivatives of
%         the cost function with respect to Theta1 and Theta2 in Theta1_grad and
%         Theta2_grad, respectively. After implementing Part 2, you can check
%         that your implementation is correct by running checkNNGradients
%
%         Note: The vector y passed into the function is a vector of labels
%               containing values from 1..K. You need to map this vector into a 
%               binary vector of 1's and 0's to be used with the neural network
%               cost function.
%
%         Hint: We recommend implementing backpropagation using a for-loop
%               over the training examples if you are implementing it for the 
%               first time.
%
% Part 3: Implement regularization with the cost function and gradients.
%
%         Hint: You can implement this around the code for
%               backpropagation. That is, you can compute the gradients for
%               the regularization separately and then add them to Theta1_grad
%               and Theta2_grad from Part 2.
%

a1 = [ones(m, 1) X];
z2 = a1 * Theta1';
a2 = sigmoid(z2);
a2 = [ones(m, 1) a2];
z3 = a2 * Theta2';
h = sigmoid(z3);

yk = zeros(m, num_labels);
for i = 1:m
    yk(i, y(i)) = 1;
end
J = (1/m)* sum(sum(((-yk) .* log(h) - (1 - yk) .* log(1 - h))));


r = (lambda / (2 * m)) * (sum(sum(Theta1(:, 2:end) .^ 2))
    + sum(sum(Theta2(:, 2:end) .^ 2)));
J = J + r;

for row = 1:m
    a1 = [1 X(row,:)]';
    z2 = Theta1 * a1;
    a2 = sigmoid(z2);
    a2 = [1; a2];
    z3 = Theta2 * a2;
    a3 = sigmoid(z3);

    z2 = [1; z2];
    delta3 = a3 - yk'(:, row);
    delta2 = (Theta2' * delta3) .* sigmoidGradient(z2);
    delta2 = delta2(2:end);

    Theta1_grad = Theta1_grad + delta2 * a1';
    Theta2_grad = Theta2_grad + delta3 * a2';

end

Theta1_grad = Theta1_grad ./ m;
Theta1_grad(:, 2:end) = Theta1_grad(:, 2:end) ...
        + (lambda/m) * Theta1(:, 2:end);
Theta2_grad = Theta2_grad ./ m;
Theta2_grad(:, 2:end) = Theta2_grad(:, 2:end) + ...
        + (lambda/m) * Theta2(:, 2:end);
% -------------------------------------------------------------

% =========================================================================

% Unroll gradients
grad = [Theta1_grad(:) ; Theta2_grad(:)];
end

以上的代碼分成了三個部分：
Part 1: Feedforward the neural network and return the cost in the variable J。

Part 2: Implement the backpropagation algorithm to compute the gradients Theta1_grad and Theta2_grad. You should return the partial derivatives of the cost function with respect to Theta1 and Theta2 in Theta1_grad and Theta2_grad, respectively.
關於反向傳播的思想如下：