【面向代码】学习 Deep Learning（一）Neural Network ._scaling factor for the learning rate (each epoch)

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最近一直在看Deep Learning，各类博客、论文看得不少

但是说实话，这样做有些疏于实现，一来呢自己的电脑也不是很好，二来呢我目前也没能力自己去写一个toolbox

只是跟着Andrew Ng的UFLDL tutorial 写了些已有框架的代码(这部分的代码见github)

后来发现了一个matlab的Deep Learning的toolbox，发现其代码很简单，感觉比较适合用来学习算法

再一个就是matlab的实现可以省略掉很多数据结构的代码，使算法思路非常清晰

所以我想在解读这个toolbox的代码的同时来巩固自己学到的，同时也为下一步的实践打好基础

(本文只是从代码的角度解读算法，具体的算法理论步骤还是需要去看paper的

我会在文中给出一些相关的paper的名字，本文旨在梳理一下算法过程，不会深究算法原理和公式)

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使用的代码：DeepLearnToolbox  ，下载地址：点击打开，感谢该toolbox的作者

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第一章从分析NN(neural network)开始，因为这是整个deep learning的大框架，参见UFLDL

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首先看一下\tests\test_example_NN.m ，跳过对数据进行normalize的部分，最关键的就是：

（为了注释显示有颜色，我把matlab代码中的%都改成了//）

nn = nnsetup([784 100 10]);
opts.numepochs =  1;   //  Number of full sweeps through data
opts.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);

很简单的几步就训练了一个NN，我们发现其中最重要的几个函数就是nnsetup,nntrain和nntest了

那么我们分别来分析着几个函数，\NN\nnsetup.m

nnsetup

function nn = nnsetup(architecture)
//首先从传入的architecture中获得这个网络的整体结构，nn.n表示这个网络有多少层，可以参照上面的样例调用nnsetup([784 100 10])加以理解
 
    nn.size   = architecture;
    nn.n      = numel(nn.size);
    //接下来是一大堆的参数，这个我们到具体用的时候再加以说明
    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'
    //对每一层的网络结构进行初始化，一共三个参数W,vW，p，其中W是主要的参数  
    //vW是更新参数时的临时参数，p是所谓的sparsity，(等看到代码了再细讲)    
   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));   
    end
end

nntrain

setup大概就这样一个过程，下面就到了train了，打开\NN\nntrain.m

我们跳过那些检验传入数据是否正确的代码，直接到关键的部分

denoising 的部分请参考论文：Extracting and Composing Robust Features with Denoising Autoencoders

m = size(train_x, 1);
//m是训练样本的数量
//注意在调用的时候我们设置了opt，batchsize是做batch gradient时候的大小
batchsize = opts.batchsize; numepochs = opts.numepochs;
numbatches = m / batchsize;  //计算batch的数量
assert(rem(numbatches, 1) == 0, 'numbatches must be a integer');
L = zeros(numepochs*numbatches,1);
n = 1;
//numepochs是循环的次数
for i = 1 : numepochs
    tic;
    kk = randperm(m);
    //把batches打乱顺序进行训练，randperm(m)生成一个乱序的1到m的数组
    for l = 1 : numbatches
        batch_x = train_x(kk((l - 1) * batchsize + 1 : l * batchsize), :);
        //Add noise to input (for use in denoising autoencoder)
        //加入noise，这是denoising autoencoder需要使用到的部分
        //这部分请参见《Extracting and Composing Robust Features with Denoising Autoencoders》这篇论文
        //具体加入的方法就是把训练样例中的一些数据调整变为0，inputZeroMaskedFraction表示了调整的比例
        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), :);
        //这三个函数
        //nnff是进行前向传播，nnbp是后向传播，nnapplygrads是进行梯度下降
        //我们在下面分析这些函数的代码
        nn = nnff(nn, batch_x, batch_y);
        nn = nnbp(nn);
        nn = nnapplygrads(nn);
        L(n) = nn.L;
        n = n + 1;
    end
    
    t = toc;
    if ishandle(fhandle)
        if opts.validation == 1
            loss = nneval(nn, loss, train_x, train_y, val_x, val_y);
        else
            loss = nneval(nn, loss, train_x, train_y);
        end
        nnupdatefigures(nn, fhandle, loss, opts, i);
    end
        
    disp(['epoch ' num2str(i) '/' num2str(opts.numepochs) '. Took ' num2str(t) ' seconds' '. Mean squared error on training set is ' num2str(mean(L((n-numbatches):(n-1))))]);
    nn.learningRate = nn.learningRate * nn.scaling_learningRate;
end

下面分析三个函数nnff,nnbp和nnapplygrads

nnff

nnff就是进行feedforward pass，其实非常简单，就是整个网络正向跑一次就可以了

当然其中有dropout和sparsity的计算

具体的参见论文“Improving Neural Networks with Dropout“和Autoencoders and Sparsity

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
        //根据选择的激活函数不同进行正向传播计算
        //你可以回过头去看nnsetup里面的第一个参数activation_function
        //sigm就是sigmoid函数,tanh_opt就是tanh的函数，这个toolbox好像有一点改变
        //tanh_opt是1.7159*tanh(2/3.*A)
        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的计算部分部分 dropoutFraction 是nnsetup中可以设置的一个参数
        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
        //计算sparsity，nonSparsityPenalty 是对没达到sparsitytarget的参数的惩罚系数
        //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
	//计算error
    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;
    end
end

nnbp

代码：\NN\nnbp.m

nnbp呢是进行back propagation的过程，过程还是比较中规中矩，和ufldl中的Neural Network讲的基本一致

值得注意的还是dropout和sparsity的部分

        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

这只是实现的内容，代码中的d{i}就是这一层的delta值，在ufldl中有讲的

dW{i}基本就是计算的gradient了，只是后面还要加入一些东西，进行一些修改

具体原理参见论文“Improving Neural Networks with Dropout“ 以及 Autoencoders and Sparsity的内容

nnapplygrads

代码文件：\NN\nnapplygrads.m

    for i = 1 : (nn.n - 1)
        if(nn.weightPenaltyL2>0)
            dW = nn.dW{i} + nn.weightPenaltyL2 * nn.W{i};
        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;
    end

这个内容就简单了，nn.weightPenaltyL2 是weight decay的部分，也是nnsetup时可以设置的一个参数

有的话就加入weight Penalty，防止过拟合，然后再根据momentum的大小调整一下，最后改变nn.W{i}即可

nntest

nntest再简单不过了，就是调用一下nnpredict，在和test的集合进行比较

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

nnpredict

代码文件：\NN\nnpredict.m

function labels = nnpredict(nn, x)
    nn.testing = 1;
    nn = nnff(nn, x, zeros(size(x,1), nn.size(end)));
    nn.testing = 0;
    
    [~, i] = max(nn.a{end},[],2);
    labels = i;
end

继续非常简单，predict不过是nnff一次，得到最后的output~~

max(nn.a{end},[],2); 是返回每一行的最大值以及所在的列数，所以labels返回的就是标号啦

(这个test好像是专门用来test 分类问题的，我们知道nnff得到最后的值即可)

总结