吴恩达机器学习 - 神经网络的反向传播算法

题目链接:点击打开链接


笔记:

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因为这一部分的内容确实难度比较大,所以我准备按最后一页笔记的思路一点一点的写出实现的思路和我的想法。

首先让数据可视化

执行代码

load('ex4data1.mat');
m = size(X, 1);

sel = randperm(size(X, 1));     %乱序后随机选择100组数据进行展示
sel = sel(1:100);

displayData(X(sel, :));

结果是:

这里写图片描述

用到是函数是displayData.m:

function [h, display_array] = displayData(X, example_width)
%DISPLAYDATA Display 2D data in a nice grid
% [h, display_array] = DISPLAYDATA(X, example_width) displays 2D data
% stored in X in a nice grid. It returns the figure handle h and the 
% displayed array if requested.

% Set example_width automatically if not passed in
if ~exist('example_width', 'var') || isempty(example_width) 
    example_width = round(sqrt(size(X, 2)));
end

% Gray Image
colormap(gray);

% Compute rows, cols
[m n] = size(X);
example_height = (n / example_width);

% Compute number of items to display
display_rows = floor(sqrt(m));
display_cols = ceil(m / display_rows);

% Between images padding
pad = 1;

% Setup blank display
display_array = - ones(pad + display_rows * (example_height + pad), ...
                       pad + display_cols * (example_width + pad));

% Copy each example into a patch on the display array
curr_ex = 1;
for j = 1:display_rows
    for i = 1:display_cols
        if curr_ex > m, 
            break; 
        end
        % Copy the patch

        % Get the max value of the patch
        max_val = max(abs(X(curr_ex, :)));
        display_array(pad + (j - 1) * (example_height + pad) + (1:example_height), ...
                      pad + (i - 1) * (example_width + pad) + (1:example_width)) = ...
                        reshape(X(curr_ex, :), example_height, example_width) / max_val;
        curr_ex = curr_ex + 1;
    end
    if curr_ex > m, 
        break; 
    end
end

% Display Image
h = imagesc(display_array, [-1 1]);

% Do not show axis
axis image off

drawnow;

end

表示一下神经网络模型:

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我们可以得到以下信息:

  • 3层网络
  • 输入层有400(20*20的图像样本)个单元(这里不包括偏置单元)
  • 输出层有10个(表示0,1,2,…,9)单元
  • 隐藏层有25个单元

代价函数

公式

这里写图片描述

sigmoid.m代码(这个已经没有难度,只是下面要调用,先粘出来):

function g = sigmoid(z)
%SIGMOID Compute sigmoid functoon
% J = SIGMOID(z) computes the sigmoid of z.

g = 1.0 ./ (1.0 + exp(-z));
end

代价函数的计算:nnCostFunction.m中填充的代码(暂时没加正则化)(这里要求有任意维度的输出层都通用):

%计算各层的z(x)
a1 = [ones(m,1), X];        %input
z2 = a1*Theta1';       %hidden
a2 = [ones(m,1), sigmoid(z2)];
z3 = a2*Theta2';       %output
a3 = sigmoid(z3);

%转换y向量
Y = zeros(m, size(Theta2, 1));        %适应不同维度的输出层
for i = 1:size(Theta2, 1)
    Y(find(y==i), i) = 1;
end

%然后计算J
J = sum(sum(-(Y.*log(a3)+(1-Y).*log(1-a3))))/m;

代价函数正则化(在上面的代码下添加):

%对J进行正则化
J = J + lambda/(2.0*m)* ...
    (sum(sum(Theta1(:,2:size(Theta1,2)).^2))+ ...
    sum(sum(Theta2(:,2:size(Theta2,2)).^2)));

反向传播

Sigmoid导数的实现(sigmoidGradient.m):

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

随机初始化(randInitializeWeights.m)(因为权重不能全为0嘛,笔记上解释了为什么):

function W = randInitializeWeights(L_in, L_out)
%RANDINITIALIZEWEIGHTS Randomly initialize the weights of a layer with L_in
%incoming connections and L_out outgoing connections
% W = RANDINITIALIZEWEIGHTS(L_in, L_out) randomly initializes the weights 
% of a layer with L_in incoming connections and L_out outgoing 
% connections. 
%
% Note that W should be set to a matrix of size(L_out, 1 + L_in) as
% the first column of W handles the "bias" terms
%

% You need to return the following variables correctly 
W = zeros(L_out, 1 + L_in);

% ====================== YOUR CODE HERE ======================
% Instructions: Initialize W randomly so that we break the symmetry while
% training the neural network.
%
% Note: The first column of W corresponds to the parameters for the bias unit
%

epsilon_init = 0.12;        %这个数字要小一点从而保证较高的学习效率
W = rand(L_out, 1+L_in)*2*epsilon-epsilon_init;

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

end

另外这里还有一个比较好的选取epsilon的方法:
这里写图片描述

反向传播(填充在nnCostFunction.m中,代价函数计算代码之下,注意这里没有正则化):

公式和图示:
这里写图片描述

%开始反向传播,分5部计算梯度
%1.对于输入层,计算每个样本的激活值(上面已经实现)

%2.计算输出层的误差值
delta3 = a3 - Y;

%3.计算l=2层的误差值(这里由于矩阵的方向的不同,和文档中式子不太一样)
delta2 = delta3*Theta2(:,2:end).*sigmoidGradient(z2);

%4.用公式计算DELTA(delta的大写形式)
DELTA1 = delta2'*a1;
DELTA2 = delta3'*a2;

%5.除以样本数得到梯度
Theta1_grad = DELTA1./m;
Theta2_grad = DELTA2./m;

梯度检验(checkNNGradients.m):

function checkNNGradients(lambda)
%CHECKNNGRADIENTS Creates a small neural network to check the
%backpropagation gradients
% CHECKNNGRADIENTS(lambda) Creates a small neural network to check the
% backpropagation gradients, it will output the analytical gradients
% produced by your backprop code and the numerical gradients (computed
% using computeNumericalGradient). These two gradient computations should
% result in very similar values.
%

if ~exist('lambda', 'var') || isempty(lambda)
    lambda = 0;
end

input_layer_size = 3;
hidden_layer_size = 5;
num_labels = 3;
m = 5;

% We generate some 'random' test data
Theta1 = debugInitializeWeights(hidden_layer_size, input_layer_size);
Theta2 = debugInitializeWeights(num_labels, hidden_layer_size);
% Reusing debugInitializeWeights to generate X
X  = debugInitializeWeights(m, input_layer_size - 1);
y  = 1 + mod(1:m, num_labels)'; % Unroll parameters nn_params = [Theta1(:) ; Theta2(:)]; % Short hand for cost function costFunc = @(p) nnCostFunction(p, input_layer_size, hidden_layer_size, ... num_labels, X, y, lambda); [cost, grad] = costFunc(nn_params); numgrad = computeNumericalGradient(costFunc, nn_params); % Visually examine the two gradient computations. The two columns % you get should be very similar. disp([numgrad grad]); fprintf(['The above two columns you get should be very similar.\n' ...
         '(Left-Your Numerical Gradient, Right-Analytical Gradient)\n\n']);

% Evaluate the norm of the difference between two solutions. 
% If you have a correct implementation, and assuming you used EPSILON = 0.0001 
% in computeNumericalGradient.m, then diff below should be less than 1e-9
diff = norm(numgrad-grad)/norm(numgrad+grad);

fprintf(['If your backpropagation implementation is correct, then \n' ... 'the relative difference will be small (less than 1e-9). \n' ... '\nRelative Difference: %g\n'], diff);

end

好了,检验通过我们进行下一步:正则化梯度(nnCostFunction.m)(在上面计算梯度的代码下填充):

%正则化梯度
Theta1_grad(:,2:end) = Theta1_grad(:,2:end) + lambda/m*Theta1(:,2:end);
Theta2_grad(:,2:end) = Theta2_grad(:,2:end) + lambda/m*Theta2(:,2:end);

到此为止,计算代价以及计算梯度的过程我们已经完成了,下面贴出nnCostFunction.m的完整代码

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.
%

%本地数据的矩阵大小
%Theta1:25*401
%Theta2:10*26
%X:5000*400
%z1:5000*401
%z2:5000*25
%z3:5000*10
%y:5000*1
%Y:5000*10
%a1:5000*401
%a2:5000*26
%a3:5000*10
%delta3:5000*10
%delta2:5000*25

%计算各层的z(x)
a1 = [ones(m,1), X];        %input
z2 = a1*Theta1';       %hidden
a2 = [ones(m,1), sigmoid(z2)];
z3 = a2*Theta2';       %output
a3 = sigmoid(z3);

%转换y向量
Y = zeros(m, size(Theta2, 1));        %适应不同维度的输出层
for i = 1:size(Theta2, 1)
    Y(find(y==i), i) = 1;
end

%然后计算J
J = sum(sum(-(Y.*log(a3)+(1-Y).*log(1-a3))))/m;

%对J进行正则化
J = J + lambda/(2.0*m)* ...
    (sum(sum(Theta1(:,2:size(Theta1,2)).^2))+ ...
    sum(sum(Theta2(:,2:size(Theta2,2)).^2)));


%开始反向传播,分5部计算梯度
%1.对于输入层,计算每个样本的激活值(上面已经实现)

%2.计算输出层的误差值
delta3 = a3 - Y;

%3.计算l=2层的误差值(这里由于矩阵的方向的不同,和文档中式子不太一样)
delta2 = delta3*Theta2(:,2:end).*sigmoidGradient(z2);

%4.用公式计算DELTA(delta的大写形式)
DELTA1 = delta2'*a1;
DELTA2 = delta3'*a2;

%5.除以样本数得到梯度
Theta1_grad = DELTA1./m;
Theta2_grad = DELTA2./m;

%正则化梯度
Theta1_grad(:,2:end) = Theta1_grad(:,2:end) + lambda/m*Theta1(:,2:end);
Theta2_grad(:,2:end) = Theta2_grad(:,2:end) + lambda/m*Theta2(:,2:end);


% -------------------------------------------------------------

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

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


end

学习使用高级优化来求解(如果对写法有疑惑,可以参考这篇文章:点击打开链接):

% Create "short hand" for the cost function to be minimized
costFunction = @(p) nnCostFunction(p, ...
                                   input_layer_size, ...
                                   hidden_layer_size, ...
                                   num_labels, X, y, lambda);

% Now, costFunction is a function that takes in only one argument (the
% neural network parameters)
[nn_params, cost] = fmincg(costFunction, initial_nn_params, options);

可视化隐藏层(使用displayData(Theta1(:, 2:end));):

displayData.m:
function [h, display_array] = displayData(X, example_width)
%DISPLAYDATA Display 2D data in a nice grid
% [h, display_array] = DISPLAYDATA(X, example_width) displays 2D data
% stored in X in a nice grid. It returns the figure handle h and the 
% displayed array if requested.

% Set example_width automatically if not passed in
if ~exist('example_width', 'var') || isempty(example_width) 
    example_width = round(sqrt(size(X, 2)));
end

% Gray Image
colormap(gray);

% Compute rows, cols
[m n] = size(X);
example_height = (n / example_width);

% Compute number of items to display
display_rows = floor(sqrt(m));
display_cols = ceil(m / display_rows);

% Between images padding
pad = 1;

% Setup blank display
display_array = - ones(pad + display_rows * (example_height + pad), ...
                       pad + display_cols * (example_width + pad));

% Copy each example into a patch on the display array
curr_ex = 1;
for j = 1:display_rows
    for i = 1:display_cols
        if curr_ex > m, 
            break; 
        end
        % Copy the patch

        % Get the max value of the patch
        max_val = max(abs(X(curr_ex, :)));
        display_array(pad + (j - 1) * (example_height + pad) + (1:example_height), ...
                      pad + (i - 1) * (example_width + pad) + (1:example_width)) = ...
                        reshape(X(curr_ex, :), example_height, example_width) / max_val;
        curr_ex = curr_ex + 1;
    end
    if curr_ex > m, 
        break; 
    end
end

% Display Image
h = imagesc(display_array, [-1 1]);

% Do not show axis
axis image off

drawnow;

end

来看看隐藏层有什么秘密:

这里写图片描述


最后,我们可以用不同的λ来获得一个更精确的参数,这个以后再实验

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