前言

吳恩達的課程堪稱經典，有必要總結一下。
學以致用，以學促用，通過筆記總結，鞏固學習成果，復習新學的概念。

目錄

文章目錄

• 前言
• 目錄
• 正文
• • 反向傳播算法
  • 前向傳播過程
  • 反向傳播算法圖解
  • 實現技巧參數合一
  • 梯度檢查
  • 參數隨機初始化
  • 總結
• 作業答案

正文

本節主要討論，神經網絡的誤差函數。
神經網絡圖示
邏輯回歸和神經網絡的誤差函數。

反向傳播算法

計算誤差和誤差的導數（梯度）

前向傳播過程

梯度計算的前向傳播過程
誤差反向傳遞圖示
反向傳播算法的流程

反向傳播算法圖解

前向傳播流程
前向傳播示例
反向傳播算法的功能
前向傳播計算誤差，反向傳播修正梯度

實現技巧參數合一

參數合成一列

梯度檢查

為了確保實現時候的正確性，可以手工檢查一下梯度的正確與否。
實現細節。

參數隨機初始化

初始參數需要我們設置，如果全部設置為0的話又會陷入停滯。
隨機初始化，打破對稱。

總結

神經網絡模型流程

神經網絡實現流程。
誤差函數隨參數變化的圖形化展示

作業答案

ex4.m

%% Machine Learning Online Class - Exercise 4 Neural Network Learning% Instructions % ------------ % % This file contains code that helps you get started on the % linear exercise. You will need to complete the following functions % in this exericse: % % sigmoidGradient.m % randInitializeWeights.m % nnCostFunction.m % % For this exercise, you will not need to change any code in this file, % or any other files other than those mentioned above. %%% Initialization clear ; close all; clc%% 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)%% =========== Part 1: Loading and Visualizing Data ============= % We start the exercise by first loading and visualizing the dataset. % You will be working with a dataset that contains handwritten digits. %% 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;%% ================ Part 2: Loading Parameters ================ % In this part of the exercise, we load some pre-initialized % neural network parameters.fprintf('\nLoading Saved Neural Network Parameters ...\n')% Load the weights into variables Theta1 and Theta2 load('ex4weights.mat');% Unroll parameters nn_params = [Theta1(:) ; Theta2(:)];%% ================ Part 3: Compute Cost (Feedforward) ================ % To the neural network, you should first start by implementing the % feedforward part of the neural network that returns the cost only. You % should complete the code in nnCostFunction.m to return cost. After % implementing the feedforward to compute the cost, you can verify that % your implementation is correct by verifying that you get the same cost % as us for the fixed debugging parameters. % % We suggest implementing the feedforward cost *without* regularization % first so that it will be easier for you to debug. Later, in part 4, you % will get to implement the regularized cost. % fprintf('\nFeedforward Using Neural Network ...\n')% Weight regularization parameter (we set this to 0 here). lambda = 0;J = nnCostFunction(nn_params, input_layer_size, hidden_layer_size, ...num_labels, X, y, lambda);fprintf(['Cost at parameters (loaded from ex4weights): %f '...'\n(this value should be about 0.287629)\n'], J);fprintf('\nProgram paused. Press enter to continue.\n'); pause;%% =============== Part 4: Implement Regularization =============== % Once your cost function implementation is correct, you should now % continue to implement the regularization with the cost. %fprintf('\nChecking Cost Function (w/ Regularization) ... \n')% Weight regularization parameter (we set this to 1 here). lambda = 1;J = nnCostFunction(nn_params, input_layer_size, hidden_layer_size, ...num_labels, X, y, lambda);fprintf(['Cost at parameters (loaded from ex4weights): %f '...'\n(this value should be about 0.383770)\n'], J);fprintf('Program paused. Press enter to continue.\n'); pause;%% ================ Part 5: Sigmoid Gradient ================ % Before you start implementing the neural network, you will first % implement the gradient for the sigmoid function. You should complete the % code in the sigmoidGradient.m file. %fprintf('\nEvaluating sigmoid gradient...\n')g = sigmoidGradient([-1 -0.5 0 0.5 1]); fprintf('Sigmoid gradient evaluated at [-1 -0.5 0 0.5 1]:\n '); fprintf('%f ', g); fprintf('\n\n');fprintf('Program paused. Press enter to continue.\n'); pause;%% ================ Part 6: Initializing Pameters ================ % In this part of the exercise, you will be starting to implment a two % layer neural network that classifies digits. You will start by % implementing a function to initialize the weights of the neural network % (randInitializeWeights.m)fprintf('\nInitializing Neural Network Parameters ...\n')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(:)];%% =============== Part 7: Implement Backpropagation =============== % Once your cost matches up with ours, you should proceed to implement the % backpropagation algorithm for the neural network. You should add to the % code you've written in nnCostFunction.m to return the partial % derivatives of the parameters. % fprintf('\nChecking Backpropagation... \n');% Check gradients by running checkNNGradients checkNNGradients;fprintf('\nProgram paused. Press enter to continue.\n'); pause;%% =============== Part 8: Implement Regularization =============== % Once your backpropagation implementation is correct, you should now % continue to implement the regularization with the cost and gradient. %fprintf('\nChecking Backpropagation (w/ Regularization) ... \n')% Check gradients by running checkNNGradients lambda = 3; checkNNGradients(lambda);% Also output the costFunction debugging values debug_J = nnCostFunction(nn_params, input_layer_size, ...hidden_layer_size, num_labels, X, y, lambda);fprintf(['\n\nCost at (fixed) debugging parameters (w/ lambda = %f): %f ' ...'\n(for lambda = 3, this value should be about 0.576051)\n\n'], lambda, debug_J);fprintf('Program paused. Press enter to continue.\n'); pause;%% =================== Part 8: Training NN =================== % You have now implemented all the code necessary to train a neural % network. To train your neural network, we will now use "fmincg", which % is a function which works similarly to "fminunc". Recall that these % advanced optimizers are able to train our cost functions efficiently as % long as we provide them with the gradient computations. % fprintf('\nTraining Neural Network... \n')% 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));fprintf('Program paused. Press enter to continue.\n'); pause;%% ================= Part 9: Visualize Weights ================= % You can now "visualize" what the neural network is learning by % displaying the hidden units to see what features they are capturing in % the data.fprintf('\nVisualizing Neural Network... \n')displayData(Theta1(:, 2:end));fprintf('\nProgram paused. Press enter to continue.\n'); pause;%% ================= Part 10: Implement Predict ================= % After training the neural network, we would like to use it to predict % the labels. You will now implement the "predict" function to use the % neural network to predict the labels of the training set. This lets % you compute the training set accuracy.pred = predict(Theta1, Theta2, X);fprintf('\nTraining Set Accuracy: %f\n', mean(double(pred == y)) * 100);

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

nnCostFunction.m

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)); Y=zeros(m,num_labels); for i=1:m Y(i,y(i))=1; end % ====================== 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 X2=[ones(m,1) X]; z_2=X2*Theta1'; z_2=[ones(m,1) z_2]; a_2=sigmoid(X2*Theta1'); a_2=[ones(m,1) a_2]; z_3=a_2*Theta2'; a_3=sigmoid(z_3); for i=1:mfor k=1:num_labelsJ=J-Y(i,k)*log(a_3(i,k))-(1-Y(i,k))*log(1-a_3(i,k));end end J=J/m; J = J + lambda/2/m * (sum(sum(Theta1(:, 2:end).^2)) + sum(sum(Theta2(:, 2:end).^2)));% 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. % Y=Y'; D1=zeros(size(Theta1)); D2=zeros(size(Theta2));for t=1:mdel_3=a_3(t,:)'-Y(:,t);del_2=Theta2'*del_3.*sigmoidGradient(z_2(t,:)');D1=D1+del_2(2:end)*X2(t,:);D2=D2+del_3*a_2(t,:); end Theta1_grad=D1/m; Theta1_grad(:,2:end)=Theta1_grad(:,2:end)+lambda/m*Theta1(:,2:end); Theta2_grad=D2/m; Theta2_grad(:,2:end)=Theta2_grad(:,2:end)+lambda/m*Theta2(:,2:end);% 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. % % ------------------------------------------------------------- % ========================================================================= % Unroll gradients grad = [Theta1_grad(:) ; Theta2_grad(:)]; end

randinitializeWeights.m

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; W = rand(L_out, 1 + L_in) * 2 * epsilon_init-epsilon_init; % =========================================================================end

總結

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