采用附加动量法和自适应学习率设计来改进bp神经网络的迭代速度，如果不迭代学习率会提高精度；迭代学习率（自适应）会加快收敛，但精度降低（Matlab代码实现）

clear all;
close all;
clc;
er = [];
load mnist_uint8;  %用自己的数据
for idj = 1:10
train_x = double(train_x) / 255;
test_x  = double(test_x)  / 255;
train_y = double(train_y);
test_y  = double(test_y);
mu=mean(train_x);    
sigma=max(std(train_x),eps);
train_x=bsxfun(@minus,train_x,mu);   
train_x=bsxfun(@rdivide,train_x,sigma);   
test_x=bsxfun(@minus,test_x,mu);
test_x=bsxfun(@rdivide,test_x,sigma);
arc = [784 300 10]; 
n=numel(arc);
W = cell(1,n-1); 
for i=2:n
    W{i-1} = (rand(arc(i),arc(i-1)+1)-0.5) * 8 *sqrt(6 / (arc(i)+arc(i-1)));
end
learningRate = 2; 
numepochs = 1;   
batchsize = 200; 
m = size(train_x, 1);
numbatches = m / batchsize;
%% 训练
L = zeros(numepochs*numbatches,1);
ll=1;
for i = 1 : numepochs
    kk = randperm(m);
    for l = 1 : numbatches
        batch_x = train_x(kk((l - 1) * batchsize + 1 : l * batchsize), :);
        batch_y = train_y(kk((l - 1) * batchsize + 1 : l * batchsize), :);
       %% 正向传播
        mm = size(batch_x,1);
        x = [ones(mm,1) batch_x];
        a{1} = x;
        %隐藏层用tanh
        for ii = 2 : n-1
            a{ii} = 1.7159*tanh(2/3.*(a{ii - 1} * W{ii - 1}'));   
            a{ii} = [ones(mm,1) a{ii}];
        end
        %最后一层使用sigmoid
        a{n} = 1./(1+exp(-(a{n - 1} * W{n - 1}')));
        e = batch_y - a{n};
        L(ll) = 1/size(e,2) * sum(sum(e.^2)) / mm; 
       %% 反向传播
        d{n} = -e.*(a{n}.*(1 - a{n}));
        for ii = (n - 1) : -1 : 2
            d_act = 1.7159 * 2/3 * (1 - 1/(1.7159)^2 * a{ii}.^2);
            if ii+1==n    
                d{ii} = (d{ii + 1} * W{ii}) .* d_act; 
            else 
                d{ii} = (d{ii + 1}(:,2:end) * W{ii}).* d_act;
            end          
        end
        for ii = 1 : n-1
            if ii + 1 == n
                     dW{ii} = (d{ii + 1}' * a{ii}) / size(d{ii + 1}, 1);
            else
                     dW{ii} = (d{ii + 1}(:,2:end)' * a{ii}) / size(d{ii + 1}, 1);      
            end
        end
       %% 更新参数
       if ll == 1
           learningRateS = cell(1,n-1);
            for ii = 1 : n - 1       
                 W{ii} = W{ii} - learningRate.*dW{ii};
                 learningRateS{ii} = ones(size(dW{ii})) * learningRate;
            end
            pre_dW = dW;
       else
            for ii = 1 : n - 1       
                 W{ii} = W{ii} +pre_dW{ii} * 0.5 - learningRateS{ii}.*dW{ii} .* 1.1.^sign1(dW{ii}.*pre_dW{ii});
                 learningRateS{ii} = learningRateS{ii}.* 1.1.^sign1(dW{ii}.*pre_dW{ii});
            end
            pre_dW = dW;
       end
        ll=ll+1;
    end
end
%% 测试
mm = size(test_x,1);
x = [ones(mm,1) test_x];
a{1} = x;
for ii = 2 : n-1    
    a{ii} = 1.7159 * tanh( 2/3 .* (a{ii - 1} * W{ii - 1}'));  
    a{ii} = [ones(mm,1) a{ii}];
end
a{n} = 1./(1+exp(-(a{n - 1} * W{n - 1}')));
[~, i] = max(a{end},[],2);
labels = i;                  
[~, expected] = max(test_y,[],2);
bad = find(labels ~= expected); 
er = [er, numel(bad) / size(x, 1)];    
end
mean(er)
std(er,1)
plot(L);
xlabel('更新次数');
ylabel('误差');
function x = sign1(x)
x(x>=0) = 1;
x(x<0) = -1;
end