题目下载【传送门】
题目简述:识别图片中的数字,训练该模型,求参数θ。
第1步:读取数据文件:
%% Setup the parameters you will use for this exercise input_layer_size = 400; % 20x20 Input Images of Digits hidden_layer_size = 25; % 25 hidden units num_labels = 10; % 10 labels, from 1 to 10 % (note that we have mapped "0" to label 10) % Load Training Data fprintf('Loading and Visualizing Data ...\n') load('ex4data1.mat'); m = size(X, 1); % Randomly select 100 data points to display sel = randperm(size(X, 1)); sel = sel(1:100); displayData(X(sel, :)); fprintf('Program paused. Press enter to continue.\n'); pause; fprintf('\nLoading Saved Neural Network Parameters ...\n') % Load the weights into variables Theta1 and Theta2 load('ex4weights.mat'); % Unroll parameters nn_params = [Theta1(:) ; Theta2(:)];
第2步:初始化参数:
initial_Theta1 = randInitializeWeights(input_layer_size, hidden_layer_size); initial_Theta2 = randInitializeWeights(hidden_layer_size, num_labels); % Unroll parameters initial_nn_params = [initial_Theta1(:) ; initial_Theta2(:)];
其中randInitializeWeights函数实现初始化θ:
function W = randInitializeWeights(L_in, L_out) % You need to return the following variables correctly W = zeros(L_out, 1 + L_in); epsilon_init = 0.12; W = rand(L_out, 1 + L_in) * 2 * epsilon_init - epsilon_init; end
第3步:实现nnCostFunction函数,计算 J 和 D:
function [J grad] = nnCostFunction(nn_params, ... input_layer_size, ... hidden_layer_size, ... num_labels, ... X, y, lambda) % 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)); % X:5000*400 % Y:5000*10 % a1:5000*401(后5000*400) % z2:5000*25 % a2:5000*26(后5000*25) % z3:5000*10 % a3:5000*10 % Theta1:25*401 % Theta2:10*26 % delta3:5000*10 % delta2:5000*25 % bigDelta1:25*401 % bigDelta2:10*26 % Theta1_grad:25*401 % Theta2_grad:10*26 Y = zeros(size(X, 1), num_labels); for i = 1: size(X, 1), Y(i, y(i, 1)) = 1; end a1 = [ones(m, 1) X]; z2 = a1*Theta1'; a2 = sigmoid(z2); a2 = [ones(size(a2, 1), 1) a2]; z3 = a2*Theta2'; a3 = sigmoid(z3); J = 1 / m * sum(sum(-Y .* log(a3) - (1 - Y) .* log(1 - a3))); Theta1_copy = Theta1(:, 2: end); Theta2_copy = Theta2(:, 2: end); J = J + lambda * (sum(sum(Theta1_copy.^2)) + sum(sum(Theta2_copy.^2))) / (2*m); delta3 = a3 - Y; delta2 = delta3 * Theta2_copy .* sigmoidGradient(z2); bigDelta1 = zeros(size(Theta1)); bigDelta2 = zeros(size(Theta2)); bigDelta1 = delta2' * a1; bigDelta2 = delta3' * a2; Theta1_grad = bigDelta1 / m + lambda / m * Theta1; Theta2_grad = bigDelta2 / m + lambda / m * Theta2; Theta1_grad(:, 1) = bigDelta1(:, 1) / m; Theta2_grad(:, 1) = bigDelta2(:, 1) / m; % Unroll gradients grad = [Theta1_grad(:) ; Theta2_grad(:)]; end
其中sigmoid函数:
function g = sigmoid(z) g = 1.0 ./ (1.0 + exp(-z)); end
其中sigmoidGradient函数:
function g = sigmoidGradient(z) g = zeros(size(z)); g = sigmoid(z) .* (1 - sigmoid(z)) end
第4步:梯度检测:
% Check gradients by running checkNNGradients lambda = 3; checkNNGradients(lambda);
其中checkNNGradients函数实现梯度检测:
function checkNNGradients(lambda) 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
其中数值方法计算函数computeNumericalGradient实现:
function numgrad = computeNumericalGradient(J, theta) numgrad = zeros(size(theta)); perturb = zeros(size(theta)); e = 1e-4; for p = 1:numel(theta) % Set perturbation vector perturb(p) = e; loss1 = J(theta - perturb); loss2 = J(theta + perturb); % Compute Numerical Gradient numgrad(p) = (loss2 - loss1) / (2*e); perturb(p) = 0; end end
其中测试数据初始化函数debugInitializeWeights函数:
function W = debugInitializeWeights(fan_out, fan_in) % Set W to zeros W = zeros(fan_out, 1 + fan_in); % Initialize W using "sin", this ensures that W is always of the same % values and will be useful for debugging W = reshape(sin(1:numel(W)), size(W)) / 10; end
第5步:训练模型,计算最优解:
% After you have completed the assignment, change the MaxIter to a larger % value to see how more training helps. options = optimset('MaxIter', 50); % You should also try different values of lambda lambda = 1; % 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); % Obtain Theta1 and Theta2 back from nn_params 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));
第6步:可视化隐藏层:
displayData(Theta1(:, 2:end));
运行结果:
第7步:计算准确率:
pred = predict(Theta1, Theta2, X); fprintf('\nTraining Set Accuracy: %f\n', mean(double(pred == y)) * 100);
其中predict函数:
function p = predict(Theta1, Theta2, X) % Useful values m = size(X, 1); num_labels = size(Theta2, 1); % You need to return the following variables correctly p = zeros(size(X, 1), 1); h1 = sigmoid([ones(m, 1) X] * Theta1'); h2 = sigmoid([ones(m, 1) h1] * Theta2'); [dummy, p] = max(h2, [], 2); end
运行结果:
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