\u53c2\u8003<\/a>\u3002\u8fd9\u91cc\u5f20\u8d34\u5f53\u524d\u7248\u672c\u5982\u4e0b\uff1a<\/p>\n\n\n\n\"\"\"\nsuper simple neural networks (bear version)\n * input x is matrix 784x1 (where 784 = 28*28 which is MNIST image data)\n * 30 neurons for the only hidden layer, as n^{[1]} = 30\n * output layer: one neuron for classification(logistic)\n * using relu for activation function in hidden layer\n\ninput layer:\n x: (shape 784x1)\n X: m samples where m = 12665 , X: 784 x 12665\n as : a^{[0]}: 784 x 12665 n^{[0]} = 784\nhidden layer:\n n^{[1]}: 30\n W^{[1]}: (30,784) as (n^{[1]},n^{[0]})\n Z^{[1]}: (30,12665) as (n^{[1]},m)\n A^{[1]}: (30,12665) as (n^{[1]},m)\noutput layer:\n n^{[2]}: 1\n W^{[2]}: (1,30) as (n^{[2]},n^{[1]})\n Z^{[2]}: (1,12665) as (n^{[2]},m)\n A^{[2]}: (1,12665) as (n^{[2]},m)\n\noutput:\n y \\in [0,1] or p \\in {0,1}\n Y: (1 x m) ndarray\nstructure for only one sample:\n x_1 -> W*X + B -> relu ->\n x_2 -> W*X + B -> relu -> \\\n ... -> ... -> .. -> -> w*x+b -> logistic\n x_784 -> W*X + B -> relu -> \/\n ------ -------------------- ------------------\n | | |\n V V V\n input 30 neurons one neuron\n feature relu activation output layer\n\n By numpy with m samples:\n np.logistic(W2@g(W1@X+b1)+b2) as \\hat{Y}: (1 x m) ndarray\n\n dimension analysis:\n W2 : (n2,n1)\n g(W1@X+b1): (n1,m)\n W1 : (n1,n0)\n X : (n0,m)\n b1 : (n1,1) with broadcasting to (n1,m)\n b2: (n2,1) with broadcasting to (n2,m)\n\ngrad and notaion:\n forward propagation : A1 A2 Z1 Z2\n backward propagation: dW1 dW2 db1 db2\n\n more details:\n Z1 = W1@X + b1\n Z2 = W2@A1 + b2\n A1 = g(Z1) -- g for relu\n A2 = \\sigma(Z2) -- sigma for logistic\n\n dW2 = ((1\/m)*(A2-Y))@A1.T\n dW2 = dZ2@A1.T where dZ2 = (1\/m)*(A2-Y)\n A2.shape:(1,m) Y.shape:(1,m) A1.T.shape:(n1,m)\n so: dW2.shape: (1,n1)\n\n dW1 = (W2.T@((1\/m)*(A2-Y))*g_prime(Z1))@A0.T\n dW1 = dZ1@A1.T\n where\n dZ1 = W2.T@dZ2 * g_prime(Z1)\n g_prime is derivative of relu\n dW2.shape: (n1,n0)\n note: @ for matrix multiply; * for dot product\/element-wise\n\nChallenges\n 1. Understanding the MNIST dataset and labels\n 2. Understanding gradient caculate and the gradient descent\n 3. Understanding logistic regression loss function and the caculation\n 3. Knowing feature normalization\n\nabout it:\n it's a simple project for human learning how machine learning\n version ant : scalar input\/one neuron\/one layer\/binary classification\n version bear: vector input\/30+1 neurons \/two layer\/binary classification\n by orczhou.com\n\"\"\"\n\nfrom keras.datasets import mnist\nimport numpy as np\n\n# return only data lable 0 or 1 from MNIST for the binary classification\ndef filter_mnist_data(data_X, data_y):\n data_filter = np.where((data_y == 0) | (data_y == 1))\n filtered_data_X, filtered_data_y = data_X[data_filter], data_y[data_filter]\n r_data_X = filtered_data_X.reshape(filtered_data_X.shape[0],filtered_data_X.shape[1]*filtered_data_X.shape[2])\n return (r_data_X, filtered_data_y)\n\n(train_all_X, train_all_y), (test_all_X, test_all_y) = mnist.load_data()\n(train_X,train_y) = filter_mnist_data(train_all_X, train_all_y)\n(test_X ,test_y ) = filter_mnist_data(test_all_X, test_all_y)\n\nX = train_X.T\nY = train_y.reshape(1,train_y.shape[0])\n\nm = X.shape[1] # number of samples\n\n# hyper-parameter; read the comments above for structure of the NN\nn0 = X.shape[0] # number of input features\nn1 = 10 # nerons of the hidden layer\nn2 = 1 # nerons of the output layer\niteration_count = 500\nlearning_rate = 0.5\n\n# feature scaling \/ Normalization\nmean = np.mean(X,axis = 1,keepdims = True)\nstd = np.std( X,axis = 1,keepdims = True)+0.000000001\nX = (X-mean)\/std\n\n# initial parameters: W1 W2 b1 b2 size\nnp.random.seed(561)\nW1 = np.random.randn(n1,n0)*0.01\nW2 = np.random.randn(n2,n1)*0.01\nb1 = np.zeros([n1,1])\nb2 = np.zeros([n2,1])\n\n# logistic function\ndef logistic_function(x):\n return 1\/(1+np.exp(-x))\n\nabout_the_train = '''\\\ntry to train the model with:\n learning rate: {:f}\n iteration : {:d}\n neurons in hidden layer: {:d}\n\\\n'''\nprint(about_the_train.format(learning_rate,iteration_count,n1))\n\n# forward\/backward propagation (read the comment above:\"grad and notaion\")\ncost_last = np.inf # very large data,maybe better , what about np.inf\nfor i in range(iteration_count):\n # forward propagation\n A0 = X\n\n Z1 = W1@X + b1 # W1 (n1,n0) X: (n0,m)\n A1 = np.maximum(Z1,0) # relu\n Z2 = W2@A1 + b2\n A2 = logistic_function(Z2)\n\n dZ2 = (A2-Y)\/m\n dW2 = dZ2@A1.T\n db2 = np.sum(dZ2,axis=1,keepdims = True)\n\n dZ1 = W2.T@dZ2*(np.where(Z1 > 0, 1, 0)) # np.where(Z1 > 0, 1, 0) is derivative of relu function\n dW1 = dZ1@A0.T\n db1 = np.sum(dZ1,axis=1,keepdims = True)\n\n cost_current = np.sum(-(Y*(np.log(A2))) - ((1-Y)*(np.log(1-A2))))\/m\n if (i+1)%(iteration_count\/20) == 0:\n print(\"iteration: {:5d},cost_current:{:f},cost_last:{:f},cost reduce:{:f}\".format( i+1,cost_current,cost_last,cost_last-cost_current))\n\n cost_last = cost_current\n W1 = W1 - learning_rate*dW1\n W2 = W2 - learning_rate*dW2\n b1 = b1 - learning_rate*db1\n b2 = b2 - learning_rate*db2\n\nprint(\"Label:\")\nprint(np.round( Y[0][:20]+0.,0))\nprint(\"Predict:\")\nprint(np.round(A2[0][:20],0))\n\n# Normalization for test dataset\nX = (test_X.T - mean)\/std\nY = test_y.reshape(1,test_y.shape[0])\n\nY_predict = (logistic_function(W2@np.maximum((W1@X+b1),0)+b2) > 0.5).astype(int)\n\nfor index in (np.where(Y != Y_predict)[1]):\n print(f\"failed to recognize: {index}\")\n # np.set_printoptions(threshold=np.inf)\n # np.set_printoptions(linewidth=np.inf)\n # print(test_X[index].reshape(28,28))\n\nprint(\"total test set:\" + str(Y.shape[1]) + \",and err rate:\"+str((np.sum(np.square(Y-Y_predict)))\/Y.shape[1]))<\/code><\/pre>\n\n\n\n\u7b80\u5355\u7edf\u8ba1\uff0c\u4ee3\u7801\u603b\u8ba1180\u884c\uff0c\u5176\u4e2d\u6ce8\u91ca\u7ea690\u884c\u3002<\/p>\n\n\n\n
\u8fd0\u884c\u7ed3\u679c\u8bf4\u660e<\/h4>\n\n\n\n
\u8fd0\u884c\u8be5\u7a0b\u5e8f\u4f1a\u6709\u5982\u4e0b\u8f93\u51fa\uff1a<\/p>\n\n\n\n
try to train the model with:\n learning rate: 0.500000\n iteration : 500\n neurons in hidden layer: 10\niteration: 25,cost_current:0.009138,cost_last:0.009497,cost reduce:0.000358\n...\niteration: 500,cost_current:0.000849,cost_last:0.000850,cost reduce:0.000001\nfailed to recognize: 1749\nfailed to recognize: 2031\ntotal test set:2115,and err rate:0.0009456264775413711<\/code><\/pre>\n\n\n\n\u5728\u672c\u6b21\u8bad\u7ec3\u540e\uff0c\u5bf9\u4e8e\u6d4b\u8bd5\u96c6\u4e2d\u7684 2115 \u4e2d\u56fe\u7247\uff0c\u8bc6\u522b\u7684\u9519\u8bef\u7387\u4e3a 0.094%\uff0c\u5373\u6709\u4e24\u5f20\u56fe\u7247\u672a\u80fd\u591f\u6b63\u786e\u8bc6\u522b\u3002\u8fd9\u4e24\u5f20\u56fe\u7247\u7684\u7f16\u53f7\u5206\u522b\u662f1749\u548c2031\uff0c\u5bf9\u5e94\u7684\u56fe\u50cf\u5982\u4e0b\uff1a<\/p>\n\n\n\n
\n
\n
![\"\"](\"https:\/\/www.orczhou.com\/wp-content\/uploads\/2024\/11\/image-15.png\")
<\/figure>\n<\/div>\n\n\n\n\n
![\"\"](\"https:\/\/www.orczhou.com\/wp-content\/uploads\/2024\/11\/image-16.png\")
<\/figure>\n<\/div>\n<\/div>\n\n\n\n\u6700\u540e<\/h4>\n\n\n\n
\u8be5\u795e\u7ecf\u7f51\u7edc\u5728\u8bad\u7ec3\u540e\uff0c\u5bf9\u4e8e2115\u5f20\u6d4b\u8bd5\u96c6\u4e2d\u7684\u56fe\u7247\uff0c\u4ec5\u67092\u5f20\u8bc6\u522b\u5931\u8d25\u3002\u901a\u8fc7\u8be5\u7a0b\u5e8f\uff0c\u53ef\u4ee5\u770b\u5230\uff0c\u795e\u7ecf\u7f51\u7edc\u7684\u5f3a\u5927\u4e0e\u795e\u5947\u3002<\/p>\n\n\n\n
\u56e0\u4e3a\u795e\u7ecf\u7f51\u7edc\u7684 Backward Propagation \u7684\u63a8\u5bfc\u975e\u5e38\u590d\u6742\uff0c\u672c\u6587\u4e2d\u76f4\u63a5\u7ed9\u51fa\u4e86\u76f8\u5173\u516c\u5f0f\uff0c\u540e\u7eed\u5c06\u72ec\u7acb\u4ecb\u7ecd\u8be5\u90e8\u5206\u5185\u5bb9\u3002\u5982\u679c\u6709\u4efb\u4f55\u95ee\u9898\u53ef\u4ee5\u7559\u8a00\u8ba8\u8bba\u6216\u516c\u4f17\u53f7\u540e\u53f0\u7ed9\u4f5c\u8005\u7559\u8a00\u3002<\/p>\n\n\n\n
\u8865\u5145\u8bb0\u5f55<\/h4>\n\n\n\n
\u5728\u8fd9\u4e2a\u8be5\u7a0b\u5e8f\u5b9e\u73b0\u4e2d\uff0c\u8fd8\u9047\u5230\u4e86\u4e00\u4e9b\u5176\u4ed6\u7684\u95ee\u9898\uff0c\u8bb0\u5f55\u5982\u4e0b\u3002<\/p>\n\n\n\n
\u4e00\u4e9b\u62a5\u9519<\/h5>\n\n\n\n
\u201cRuntimeWarning: overflow encountered in exp: return 1\/(1+np.exp(-x))\u201d <\/p>\n\n\n\n
\u5f53\u8bad\u7ec3\u6216\u8005\u9884\u6d4b\u8ba1\u7b97\u8fc7\u7a0b\u4e2d\uff0c\u5982\u679c\u51fa\u73b0\u90e8\u5206-x<\/code>\u503c\u6bd4\u8f83\u5927\uff0c\u90a3\u4e48\u5c31\u4f1a\u51fa\u73b0np.exp(-x)<\/code>\u6ea2\u51fa\u7684\u95ee\u9898\u3002\u4e0d\u8fc7\uff0c\u8fd8\u597d\u8be5\u95ee\u9898\u5728\u8be5\u573a\u666f\u4e0b\u5e76\u4e0d\u4f1a\u5f71\u54cd\u8ba1\u7b97\u7ed3\u679c\uff0c\u56e0\u4e3a\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c1\/(1+np.exp(-x))<\/code>\u4f9d\u65e7\u4f1a\u8fd4\u56de0\uff0c\u800c\u8fd9\u4e5f\u662f\u9884\u671f\u4e2d\u7684\u3002<\/p>\n\n\n\n\u96be\u4ee5\u5206\u6790<\/h5>\n\n\n\n
\u5728\u8fd9\u4e2a\u8bc6\u522b\u7a0b\u5e8f\u4e2d\uff0c\u6700\u540e\u53d1\u73b0\u6709\u90e8\u5206\u56fe\u7247\u662f\u96be\u4ee5\u8bc6\u522b\u7684\uff0c\u800c\u4e14\u65e0\u8bba\u5982\u4f55\u8c03\u6574\u53c2\u6570\uff0c\u90e8\u5206\u56fe\u7247\u8fd8\u662f\u96be\u4ee5\u8bc6\u522b\u3002\u6700\u597d\u7684\u5904\u7406\u65b9\u6cd5\u5f53\u7136\u662f\u589e\u52a0\u8bad\u7ec3\u6570\u636e\u96c6\uff0c\u4f46\u73b0\u5b9e\u4e2d\u53ef\u80fd\u5e76\u4e0d\u603b\u662f\u80fd\u591f\u505a\u5230\u3002<\/p>\n\n\n\n
\u800c\u5173\u4e8e\u6a21\u578b\u5e94\u8be5\u5982\u4f55\u4f18\u5316\uff0c\u8fd9\u4f3c\u4e4e\u662f\u96be\u4ee5\u5206\u6790\uff0c\u65e0\u4ece\u5165\u624b\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"
\u66f4\u8fdb\u4e00\u6b65\uff0c\u672c\u6587\u5c06\u4ece\u96f6\u6784\u5efa\u4e00\u4e2a\u6d45\u5c42\u7684\u524d\u9988\u795e\u7ecf\u7f51\u7edc\uff0c\u5b9e\u73b0\u624b\u5199\u6570\u5b570\u548c1\u7684\u8bc6\u522b\u3002\u793a\u4f8b\u624b\u5199\u6570\u5b57\u5982\u4e0b\u56fe\u6240\u793a\u3002\u6574\u4f53\u5b9e\u73b0\u5305\u62ec\u6570\u636e\u9884\u5904\u7406\u3001\u53c2\u6570\u521d\u59cb\u5316\u3001forward\/backword propagation\u3001\u8bad\u7ec3\u4e0e\u9884\u6d4b\u3002<\/p>\n","protected":false},"author":1,"featured_media":21430,"comment_status":"open","ping_status":"closed","sticky":false,"template":"wp-custom-template-a-1440-px-width-template","format":"standard","meta":{"_eb_attr":"","inline_featured_image":false,"_tocer_settings":[],"footnotes":""},"categories":[137],"tags":[],"class_list":["post-15501","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-learning-more"],"_links":{"self":[{"href":"https:\/\/www.orczhou.com\/index.php\/wp-json\/wp\/v2\/posts\/15501","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.orczhou.com\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.orczhou.com\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.orczhou.com\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.orczhou.com\/index.php\/wp-json\/wp\/v2\/comments?post=15501"}],"version-history":[{"count":152,"href":"https:\/\/www.orczhou.com\/index.php\/wp-json\/wp\/v2\/posts\/15501\/revisions"}],"predecessor-version":[{"id":21985,"href":"https:\/\/www.orczhou.com\/index.php\/wp-json\/wp\/v2\/posts\/15501\/revisions\/21985"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.orczhou.com\/index.php\/wp-json\/wp\/v2\/media\/21430"}],"wp:attachment":[{"href":"https:\/\/www.orczhou.com\/index.php\/wp-json\/wp\/v2\/media?parent=15501"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.orczhou.com\/index.php\/wp-json\/wp\/v2\/categories?post=15501"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.orczhou.com\/index.php\/wp-json\/wp\/v2\/tags?post=15501"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
![\"\"](\"https:\/\/www.orczhou.com\/wp-content\/uploads\/2024\/11\/image-27.png\")
<\/figure>\n<\/div>\n\n\n\n![\"\"](\"https:\/\/www.orczhou.com\/wp-content\/uploads\/2024\/11\/image-28.png\")
<\/figure>\n<\/div>\n\n\n\n![\"\"](\"https:\/\/www.orczhou.com\/wp-content\/uploads\/2024\/11\/image-31.png\")
<\/figure>\n<\/div>\n\n\n\n![\"\"](\"https:\/\/www.orczhou.com\/wp-content\/uploads\/2024\/11\/image-30.png\")
<\/figure>\n<\/div>\n\n\n\n![\"\"](\"https:\/\/www.orczhou.com\/wp-content\/uploads\/2024\/11\/image-32.png\")
<\/figure>\n<\/div>\n<\/div>\n\n\n\n![\"\"](\"https:\/\/www.orczhou.com\/wp-content\/uploads\/2024\/11\/image-24-1024x498.png\")
<\/figure>\n<\/div>\n\n\n\n![\"\"](\"https:\/\/www.orczhou.com\/wp-content\/uploads\/2024\/11\/image-25.png\")
<\/figure>\n<\/div>\n<\/div>\n\n\n\n