深度学习笔记一：稀疏自编码器

开始学习深度学习了，既然确定目标就要努力前行！为自己加油！——2015.6.11

Sparse Encoder

1.神经网络
概念：假设我们有训练样本集 (x(^ i),y(^ i)) ，那么神经网络算法能够提供一种复杂且非线性的假设模型 h_{W,b}(x) ，它具有参数 W, b ，可以以此参数来拟合我们的数据。
激活函数：
f(z)=sigmoid(z)=1/(1+exp(-z))
导数：f’(z)=f(z)(1-f(z)) 很重要，求代价函数极值的时候要用到
模型：
一个简单的神经网络，只有输入层，一个隐藏层和输出层组成。每加一层就相当于对输入多进行一次非线性处理，进而形成复杂的目标函数hw,b(x).
(http://img.blog.csdn.net/20150611084555088)
目标值从前往后计算：
Z2=W1*data+b1
a2=f(Z2)
Z3=W2*a2+b2
a3=f(Z3)
目标函数的代价函数：
第一部分是：直接误差——m个输入的平均误差
第二部分是：权值惩罚——所有W元素的平方和，目的是为了减少权重的幅度，防止过度拟合
(http://img.blog.csdn.net/20150611085816673)
为使代价函数最小，可以使用批梯度下降法，从而确定参数W1，W2，b1,b2。
步骤：（1）给W1,W2,b1,b2初始值：初始值设计很关键，否则将会得到不好的结果，练习里面是这样设计的：

r  = sqrt(6) / sqrt(hiddenSize+visibleSize+1);   % we'll choose weights uniformly from the interval [-r, r]W1 = rand(hiddenSize, visibleSize) * 2 * r - r;W2 = rand(visibleSize, hiddenSize) * 2 * r - r;b1 = zeros(hiddenSize, 1);b2 = zeros(visibleSize, 1);

（2）需要给出代价函数:即上述代码公式
（3）需要给出代价函数对W和b的偏导

说明：这里需要的是后面有的1/m括号里的部分。括号里的第一部分其实就是每个输入对W,b求导值的平均。
2、反向传导
这部分很关键，用于计算每个输入对的求导。为了对W求导，先对Z求导，对Z求导的结果就是残差。从后向前计算，有点类似于计算图的关键路径的计算方法。

3、稀疏自编码器
代价函数和W偏导与普通神经网络有一点区别，代价函数需要加入稀疏代价。

心得：感觉稀疏自编码器就是对输入进行压缩表示，前面编码，最后一层解码。做个实验试了一下两个隐藏层的情况，想第一次发现边，第二次发现拐角，然而效果好差！翻了一下教程，发现后面有专门的栈式自编码器，汗~~~不过至少说明自己思考的方向是对滴~

代码完成中出现的问题：
错误总结：

Jcost=(0.5/m)*sum(sum((a3-data).^2));   %%正确sum((a3-data).^2)，写成sum(a3-data).^2导致错误，找了好久的原因啊，原来是因为一对括号！

Jweight=0.5*(sum(sum(W1.^2))+sum(sum(W2.^2)));%%写成了Jweight=0.5*sum(sum(W1.^2))+sum(sum(W2.^2));还是少了括号!!

经验：
minFun的用法

addpath minFunc/options.Method = 'lbfgs'; % Here, we use L-BFGS to optimize our cost                          % function. Generally, for minFunc to work, you                          % need a function pointer with two outputs: the                          % function value and the gradient. In our problem,                          % sparseAutoencoderCost.m satisfies this.options.maxIter = 400;    % Maximum number of iterations of L-BFGS to run options.display = 'on';[opttheta, cost] = minFunc( @(p) sparseAutoencoderCost(p, ...                                   visibleSize, hiddenSize, ...                                   lambda, sparsityParam, ...                                   beta, patches), ...                              theta, options);                              %需要提供一个返回代价函数和偏导的函数sparseAutoencoderCost

sparseAutoencoderCost.h

function [cost,grad] = sparseAutoencoderCost(theta, visibleSize, hiddenSize, ...                                             lambda, sparsityParam, beta, data)% visibleSize: the number of input units (probably 64) % hiddenSize: the number of hidden units (probably 25) % lambda: weight decay parameter% sparsityParam: The desired average activation for the hidden units (denoted in the lecture%                           notes by the greek alphabet rho, which looks like a lower-case "p").% beta: weight of sparsity penalty term% data: Our 64x10000 matrix containing the training data.  So, data(:,i) is the i-th training example. % The input theta is a vector (because minFunc expects the parameters to be a vector). % We first convert theta to the (W1, W2, b1, b2) matrix/vector format, so that this % follows the notation convention of the lecture notes. W1 = reshape(theta(1:hiddenSize*visibleSize), hiddenSize, visibleSize);W2 = reshape(theta(hiddenSize*visibleSize+1:2*hiddenSize*visibleSize), visibleSize, hiddenSize);b1 = theta(2*hiddenSize*visibleSize+1:2*hiddenSize*visibleSize+hiddenSize);b2 = theta(2*hiddenSize*visibleSize+hiddenSize+1:end);% Cost and gradient variables (your code needs to compute these values). % Here, we initialize them to zeros. cost = 0;W1grad = zeros(size(W1)); W2grad = zeros(size(W2));b1grad = zeros(size(b1)); b2grad = zeros(size(b2));%% ---------- YOUR CODE HERE --------------------------------------%  Instructions: Compute the cost/optimization objective J_sparse(W,b) for the Sparse Autoencoder,%                and the corresponding gradients W1grad, W2grad, b1grad, b2grad.%% W1grad, W2grad, b1grad and b2grad should be computed using backpropagation.% Note that W1grad has the same dimensions as W1, b1grad has the same dimensions% as b1, etc.  Your code should set W1grad to be the partial derivative of J_sparse(W,b) with% respect to W1.  I.e., W1grad(i,j) should be the partial derivative of J_sparse(W,b) % with respect to the input parameter W1(i,j).  Thus, W1grad should be equal to the term % [(1/m) \Delta W^{(1)} + \lambda W^{(1)}] in the last block of pseudo-code in Section 2.2 % of the lecture notes (and similarly for W2grad, b1grad, b2grad).% % Stated differently, if we were using batch gradient descent to optimize the parameters,% the gradient descent update to W1 would be W1 := W1 - alpha * W1grad, and similarly for W2, b1, b2. % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%ych代码Jcost = 0;%直接误差Jweight = 0;%权值惩罚Jsparse = 0;%稀疏性惩罚[n m] = size(data);%m为样本的个数，n为样本的特征数% % %前向算法计算各神经网络节点的线性组合值和active值z2=W1*data+repmat(b1,1,m);a2=sigmoid(z2);z3=W2*a2+repmat(b2,1,m);a3=sigmoid(z3); Jcost=(0.5/m)*sum(sum((a3-data).^2));   %%正确sum((a3-data).^2)，写成sum(a3-data).^2导致错误，找了好久的原因啊，原来是因为一对括号！ Jweight=0.5*(sum(sum(W1.^2))+sum(sum(W2.^2)));rho=(1/m).*sum(a2,2);Jsparse=sum(sparsityParam.*log(sparsityParam./rho)+(1-sparsityParam).*log((1-sparsityParam)./(1-rho)));cost=Jcost+lambda*Jweight+beta*Jsparse;d3=-(data-a3).*(sigmoid(z3).*(1-sigmoid(z3)));sterm = beta*(-sparsityParam./rho+(1-sparsityParam)./(1-rho));d2=(W2'*d3+repmat(sterm,1,m)).*(sigmoid(z2).*(1-sigmoid(z2)));W1grad=(1/m).*(d2*data')+lambda.*W1;W2grad=(1/m).*(d3*a2')+lambda.*W2;b1grad=(1/m).*sum(d2,2);b2grad=(1/m).*sum(d3,2);%-------------------------------------------------------------------% After computing the cost and gradient, we will convert the gradients back% to a vector format (suitable for minFunc).  Specifically, we will unroll% your gradient matrices into a vector.grad = [W1grad(:) ; W2grad(:) ; b1grad(:) ; b2grad(:)];endfunction sigm = sigmoid(x)    sigm = 1 ./ (1 + exp(-x));end

本文参考：http://ufldl.stanford.edu/wiki/index.php/UFLDL%E6%95%99%E7%A8%8B
http://www.cnblogs.com/tornadomeet/tag/Deep%20Learning/

第一次写博客，内容有些凌乱，格式也不规范，当做自己的学习笔记，不当之处敬请指正