深度神经网络

本文的布局如下：首先是本文的基本思想，然后是结合实例的输入和输出，然后是每个函数所起的作用，之后是函数的具体实现，最后是备注
一、基本思想
1、输入样本数据，然后进行训练，然后进行测试
2、深度神经网络训练过程中：首先是进行初始化，根据需求设置神经网络的基本结构；然后进行前向传递（feedforward），层与层之间进行传递，求得误差；然后进行反向传播（back propogation），根据误差最小化原则，使用随机梯度下降法，对各个参数进行求导，确定下降方向，对各个参数进行更新（权重和偏置，该方法类似于单隐层前馈神经网络中的BP神经网络求解算法），在使用样本对神经网络进行训练的过程中，有一个小的case，即单个样本可以多次使用，原因在神经网络发生变化后，那么对该样本的学习能力就会不一样（有点类似于嚼甘蔗，或者说读书，一本书，比如平凡的世界这本书，在自己初中的时候看，在高三看，在复读的那一年看，在本科时候看，在工作的时候看，在研究生阶段看，不同的生命阶段看，总会有不同的体验的，惊喜地发现了这一点，一个深度神经网络也类似于一个正在成长的人，具有成长属性，像一个生命）
3、前向传递阶段：上一层的隐层输出做为本层的输入，具体的原理可参照BP神经网络的原理，如果有由于多层而造成的不同，则会另外进行补充
4、反向传播亦是如此
二、输入和输出
本文以MNIST手写数字识别为研究对象，输入的是10000幅像素为28*28的手写图片，输出的是图片所属的类别（1-10，这10个数字）
对于其他例子，分类问题，亦是如此
三、相关函数
1、function nn = nnsetup(architecture)：神经网络的初始化，可以是一层，也可以是多层；返回一个神经网络结构
2、function [nn, L] = nntrain(nn, train_x, train_y, opts, val_x, val_y)：神经网络的训练；返回一个神经网络，它更新了激励函数，误差，权重和偏置
3、function nn = nnff(nn, x, y)：神经网络的前向传递；返回更新了层激活函数，误差和损失的神经网络结构
4、function nn = nnbp(nn)：神经网络的反向传播；返回更新过权重的神经网络结构
5、function nn = nnapplygrads(nn)：根据计算出来的参数的梯度对参数（权重和偏置）进行更新；返回更新过权重和偏置的神经网络结构
6、function [loss] = nneval(nn, loss, train_x, train_y, val_x, val_y)：评估神经网络的性能；返回更新之后的损失结构体
四、函数具体实现
1、主方法

function test_example_NNload mnist_uint8;train_x = double(train_x) / 255;test_x  = double(test_x)  / 255;train_y = double(train_y);test_y  = double(test_y);% normalize[train_x, mu, sigma] = zscore(train_x);test_x = normalize(test_x, mu, sigma);%% ex1 vanilla neural netrand('state',0)nn = nnsetup([784 100 10]);opts.numepochs =  1;   %  Number of full sweeps through dataopts.batchsize = 100;  %  Take a mean gradient step over this many samples[nn, L] = nntrain(nn, train_x, train_y, opts);[er, bad] = nntest(nn, test_x, test_y);assert(er < 0.08, 'Too big error');%% ex2 neural net with L2 weight decayrand('state',0)nn = nnsetup([784 100 10]);nn.weightPenaltyL2 = 1e-4;  %  L2 weight decayopts.numepochs =  1;        %  Number of full sweeps through dataopts.batchsize = 100;       %  Take a mean gradient step over this many samplesnn = nntrain(nn, train_x, train_y, opts);[er, bad] = nntest(nn, test_x, test_y);assert(er < 0.1, 'Too big error');%% ex3 neural net with dropoutrand('state',0)nn = nnsetup([784 100 10]);nn.dropoutFraction = 0.5;   %  Dropout fraction opts.numepochs =  1;        %  Number of full sweeps through dataopts.batchsize = 100;       %  Take a mean gradient step over this many samplesnn = nntrain(nn, train_x, train_y, opts);[er, bad] = nntest(nn, test_x, test_y);assert(er < 0.1, 'Too big error');%% ex4 neural net with sigmoid activation functionrand('state',0)nn = nnsetup([784 100 10]);nn.activation_function = 'sigm';    %  Sigmoid activation functionnn.learningRate = 1;                %  Sigm require a lower learning rateopts.numepochs =  1;                %  Number of full sweeps through dataopts.batchsize = 100;               %  Take a mean gradient step over this many samplesnn = nntrain(nn, train_x, train_y, opts);[er, bad] = nntest(nn, test_x, test_y);assert(er < 0.1, 'Too big error');%% ex5 plotting functionalityrand('state',0)nn = nnsetup([784 20 10]);opts.numepochs         = 5;            %  Number of full sweeps through datann.output              = 'softmax';    %  use softmax outputopts.batchsize         = 1000;         %  Take a mean gradient step over this many samplesopts.plot              = 1;            %  enable plottingnn = nntrain(nn, train_x, train_y, opts);[er, bad] = nntest(nn, test_x, test_y);assert(er < 0.1, 'Too big error');%% ex6 neural net with sigmoid activation and plotting of validation and training error% split training data into training and validation datavx   = train_x(1:10000,:);tx = train_x(10001:end,:);vy   = train_y(1:10000,:);ty = train_y(10001:end,:);rand('state',0)nn                      = nnsetup([784 20 10]);     nn.output               = 'softmax';                   %  use softmax outputopts.numepochs          = 5;                           %  Number of full sweeps through dataopts.batchsize          = 1000;                        %  Take a mean gradient step over this many samplesopts.plot               = 1;                           %  enable plottingnn = nntrain(nn, tx, ty, opts, vx, vy);                %  nntrain takes validation set as last two arguments (optionally)[er, bad] = nntest(nn, test_x, test_y);assert(er < 0.1, 'Too big error');

2、function nn = nnsetup(architecture)

%NNSETUP creates a Feedforward Backpropagate Neural Network% nn = nnsetup(architecture) returns an neural network structure with n=numel(architecture)% layers, architecture being a n x 1 vector of layer sizes e.g. [784 100 10]    nn.size   = architecture;    nn.n      = numel(nn.size);    nn.activation_function              = 'tanh_opt';   %  Activation functions of hidden layers: 'sigm' (sigmoid) or 'tanh_opt' (optimal tanh).    nn.learningRate                     = 2;            %  learning rate Note: typically needs to be lower when using 'sigm' activation function and non-normalized inputs.    nn.momentum                         = 0.5;          %  Momentum    nn.scaling_learningRate             = 1;            %  Scaling factor for the learning rate (each epoch)    nn.weightPenaltyL2                  = 0;            %  L2 regularization    nn.nonSparsityPenalty               = 0;            %  Non sparsity penalty    nn.sparsityTarget                   = 0.05;         %  Sparsity target    nn.inputZeroMaskedFraction          = 0;            %  Used for Denoising AutoEncoders    nn.dropoutFraction                  = 0;            %  Dropout level (http://www.cs.toronto.edu/~hinton/absps/dropout.pdf)    nn.testing                          = 0;            %  Internal variable. nntest sets this to one.    nn.output                           = 'sigm';       %  output unit 'sigm' (=logistic), 'softmax' and 'linear'    for i = 2 : nn.n           % weights and weight momentum        nn.W{i - 1} = (rand(nn.size(i), nn.size(i - 1)+1) - 0.5) * 2 * 4 * sqrt(6 / (nn.size(i) + nn.size(i - 1)));        nn.vW{i - 1} = zeros(size(nn.W{i - 1}));        % average activations (for use with sparsity)        nn.p{i}     = zeros(1, nn.size(i));       endend

3、function [nn, L] = nntrain(nn, train_x, train_y, opts, val_x, val_y)

%NNTRAIN trains a neural net% [nn, L] = nnff(nn, x, y, opts) trains the neural network nn with input x and% output y for opts.numepochs epochs, with minibatches of size% opts.batchsize. Returns a neural network nn with updated activations,% errors, weights and biases, (nn.a, nn.e, nn.W, nn.b) and L, the sum% squared error for each training minibatch.assert(isfloat(train_x), 'train_x must be a float');assert(nargin == 4 || nargin == 6,'number ofinput arguments must be 4 or 6')loss.train.e               = [];loss.train.e_frac          = [];loss.val.e                 = [];loss.val.e_frac            = [];opts.validation = 0;if nargin == 6    opts.validation = 1;endfhandle = [];if isfield(opts,'plot') && opts.plot == 1    fhandle = figure();endm = size(train_x, 1);batchsize = opts.batchsize;numepochs = opts.numepochs;numbatches = m / batchsize;assert(rem(numbatches, 1) == 0, 'numbatches must be a integer');L = zeros(numepochs*numbatches,1);n = 1;for i = 1 : numepochs    tic;    kk = randperm(m);    for l = 1 : numbatches        batch_x = train_x(kk((l - 1) * batchsize + 1 : l * batchsize), :);        %Add noise to input (for use in denoising autoencoder)        if(nn.inputZeroMaskedFraction ~= 0)            batch_x = batch_x.*(rand(size(batch_x))>nn.inputZeroMaskedFraction);        end        batch_y = train_y(kk((l - 1) * batchsize + 1 : l * batchsize), :);        nn = nnff(nn, batch_x, batch_y);        nn = nnbp(nn);        nn = nnapplygrads(nn);        L(n) = nn.L;        n = n + 1;    end    t = toc;    if opts.validation == 1        loss = nneval(nn, loss, train_x, train_y, val_x, val_y);        str_perf = sprintf('; Full-batch train mse = %f, val mse = %f', loss.train.e(end), loss.val.e(end));    else        loss = nneval(nn, loss, train_x, train_y);        str_perf = sprintf('; Full-batch train err = %f', loss.train.e(end));    end    if ishandle(fhandle)        nnupdatefigures(nn, fhandle, loss, opts, i);    end    disp(['epoch ' num2str(i) '/' num2str(opts.numepochs) '. Took ' num2str(t) ' seconds' '. Mini-batch mean squared error on training set is ' num2str(mean(L((n-numbatches):(n-1)))) str_perf]);    nn.learningRate = nn.learningRate * nn.scaling_learningRate;endend

4、function nn = nnff(nn, x, y)

%NNFF performs a feedforward pass% nn = nnff(nn, x, y) returns an neural network structure with updated% layer activations, error and loss (nn.a, nn.e and nn.L)    n = nn.n;    m = size(x, 1);    x = [ones(m,1) x];    nn.a{1} = x;    %feedforward pass    for i = 2 : n-1        switch nn.activation_function             case 'sigm'                % Calculate the unit's outputs (including the bias term)                nn.a{i} = sigm(nn.a{i - 1} * nn.W{i - 1}');            case 'tanh_opt'                nn.a{i} = tanh_opt(nn.a{i - 1} * nn.W{i - 1}');        end        %dropout        if(nn.dropoutFraction > 0)            if(nn.testing)                nn.a{i} = nn.a{i}.*(1 - nn.dropoutFraction);            else                nn.dropOutMask{i} = (rand(size(nn.a{i}))>nn.dropoutFraction);                nn.a{i} = nn.a{i}.*nn.dropOutMask{i};            end        end        %calculate running exponential activations for use with sparsity        if(nn.nonSparsityPenalty>0)            nn.p{i} = 0.99 * nn.p{i} + 0.01 * mean(nn.a{i}, 1);        end        %Add the bias term        nn.a{i} = [ones(m,1) nn.a{i}];    end    switch nn.output         case 'sigm'            nn.a{n} = sigm(nn.a{n - 1} * nn.W{n - 1}');        case 'linear'            nn.a{n} = nn.a{n - 1} * nn.W{n - 1}';        case 'softmax'            nn.a{n} = nn.a{n - 1} * nn.W{n - 1}';            nn.a{n} = exp(bsxfun(@minus, nn.a{n}, max(nn.a{n},[],2)));            nn.a{n} = bsxfun(@rdivide, nn.a{n}, sum(nn.a{n}, 2));     end    %error and loss    nn.e = y - nn.a{n};    switch nn.output        case {'sigm', 'linear'}            nn.L = 1/2 * sum(sum(nn.e .^ 2)) / m;         case 'softmax'            nn.L = -sum(sum(y .* log(nn.a{n}))) / m;    endend

5、function nn = nnbp(nn)

%NNBP performs backpropagation% nn = nnbp(nn) returns an neural network structure with updated weights     n = nn.n;    sparsityError = 0;    switch nn.output        case 'sigm'            d{n} = - nn.e .* (nn.a{n} .* (1 - nn.a{n}));        case {'softmax','linear'}            d{n} = - nn.e;    end    for i = (n - 1) : -1 : 2        % Derivative of the activation function        switch nn.activation_function             case 'sigm'                d_act = nn.a{i} .* (1 - nn.a{i});            case 'tanh_opt'                d_act = 1.7159 * 2/3 * (1 - 1/(1.7159)^2 * nn.a{i}.^2);        end        if(nn.nonSparsityPenalty>0)            pi = repmat(nn.p{i}, size(nn.a{i}, 1), 1);            sparsityError = [zeros(size(nn.a{i},1),1) nn.nonSparsityPenalty * (-nn.sparsityTarget ./ pi + (1 - nn.sparsityTarget) ./ (1 - pi))];        end        % Backpropagate first derivatives        if i+1==n % in this case in d{n} there is not the bias term to be removed                         d{i} = (d{i + 1} * nn.W{i} + sparsityError) .* d_act; % Bishop (5.56)        else % in this case in d{i} the bias term has to be removed            d{i} = (d{i + 1}(:,2:end) * nn.W{i} + sparsityError) .* d_act;        end        if(nn.dropoutFraction>0)            d{i} = d{i} .* [ones(size(d{i},1),1) nn.dropOutMask{i}];        end    end    for i = 1 : (n - 1)        if i+1==n            nn.dW{i} = (d{i + 1}' * nn.a{i}) / size(d{i + 1}, 1);        else            nn.dW{i} = (d{i + 1}(:,2:end)' * nn.a{i}) / size(d{i + 1}, 1);              end    endend

6、function nn = nnapplygrads(nn)

%NNAPPLYGRADS updates weights and biases with calculated gradients% nn = nnapplygrads(nn) returns an neural network structure with updated% weights and biases    for i = 1 : (nn.n - 1)        if(nn.weightPenaltyL2>0)            dW = nn.dW{i} + nn.weightPenaltyL2 * [zeros(size(nn.W{i},1),1) nn.W{i}(:,2:end)];        else            dW = nn.dW{i};        end        dW = nn.learningRate * dW;        if(nn.momentum>0)            nn.vW{i} = nn.momentum*nn.vW{i} + dW;            dW = nn.vW{i};        end        nn.W{i} = nn.W{i} - dW;    endend

7、function [loss] = nneval(nn, loss, train_x, train_y, val_x, val_y)

%NNEVAL evaluates performance of neural network% Returns a updated loss structassert(nargin == 4 || nargin == 6, 'Wrong number of arguments');nn.testing = 1;% training performancenn                    = nnff(nn, train_x, train_y);loss.train.e(end + 1) = nn.L;% validation performanceif nargin == 6    nn                    = nnff(nn, val_x, val_y);    loss.val.e(end + 1)   = nn.L;endnn.testing = 0;%calc misclassification rate if softmaxif strcmp(nn.output,'softmax')    [er_train, dummy]               = nntest(nn, train_x, train_y);    loss.train.e_frac(end+1)    = er_train;    if nargin == 6        [er_val, dummy]             = nntest(nn, val_x, val_y);        loss.val.e_frac(end+1)  = er_val;    endendend

8、function nnupdatefigures(nn,fhandle,L,opts,i)

%NNUPDATEFIGURES updates figures during trainingif i > 1 %dont plot first point, its only a point       x_ax = 1:i;    % create legend    if opts.validation == 1        M            = {'Training','Validation'};    else        M            = {'Training'};    end    %create data for plots    if strcmp(nn.output,'softmax')        plot_x       = x_ax';        plot_ye      = L.train.e';        plot_yfrac   = L.train.e_frac';    else        plot_x       = x_ax';        plot_ye      = L.train.e';    end    %add error on validation data if present    if opts.validation == 1        plot_x       = [plot_x, x_ax'];        plot_ye      = [plot_ye,L.val.e'];    end    %add classification error on validation data if present    if opts.validation == 1 && strcmp(nn.output,'softmax')        plot_yfrac   = [plot_yfrac, L.val.e_frac'];            end%    plotting    figure(fhandle);       if strcmp(nn.output,'softmax')  %also plot classification error        p1 = subplot(1,2,1);        plot(plot_x,plot_ye);        xlabel('Number of epochs'); ylabel('Error');title('Error');        title('Error')        legend(p1, M,'Location','NorthEast');        set(p1, 'Xlim',[0,opts.numepochs + 1])        p2 = subplot(1,2,2);        plot(plot_x,plot_yfrac);        xlabel('Number of epochs'); ylabel('Misclassification rate');        title('Misclassification rate')        legend(p2, M,'Location','NorthEast');        set(p2, 'Xlim',[0,opts.numepochs + 1])    else        p = plot(plot_x,plot_ye);        xlabel('Number of epochs'); ylabel('Error');title('Error');        legend(p, M,'Location','NorthEast');        set(gca, 'Xlim',[0,opts.numepochs + 1])    end    drawnow;endend

9、function [er, bad] = nntest(nn, x, y)

 labels = nnpredict(nn, x);    [dummy, expected] = max(y,[],2);    bad = find(labels ~= expected);        er = numel(bad) / size(x, 1);end

10、function nnchecknumgrad(nn, x, y)

epsilon = 1e-6;    er = 1e-7;    n = nn.n;    for l = 1 : (n - 1)        for i = 1 : size(nn.W{l}, 1)            for j = 1 : size(nn.W{l}, 2)                nn_m = nn; nn_p = nn;                nn_m.W{l}(i, j) = nn.W{l}(i, j) - epsilon;                nn_p.W{l}(i, j) = nn.W{l}(i, j) + epsilon;                rand('state',0)                nn_m = nnff(nn_m, x, y);                rand('state',0)                nn_p = nnff(nn_p, x, y);                dW = (nn_p.L - nn_m.L) / (2 * epsilon);                e = abs(dW - nn.dW{l}(i, j));                assert(e < er, 'numerical gradient checking failed');            end        end    endend

五、参考文献
https://github.com/rasmusbergpalm/DeepLearnToolbox
注：该深度学习工具箱主要是针对于matlab而言的，属于源代码级别的，相对于研究生而言，逻辑清晰比较易懂；但是在实际的工程应用中，多使用Python编程语言，并且也有许多大公司出产的相对更加健全的平台，如tensorflow，theano等；因此本文只是用来结合论文，来理解其基本思想，基础入门研究之用