# Recognize Digits Source code of this tutorial is under [book/recognize_digits](https://github.com/PaddlePaddle/book/tree/develop/recognize_digits). For the first-time use, please refer to PaddlePaddle [installation instructions](http://www.paddlepaddle.org/doc_cn/build_and_install/index.html). ## Introduction When we learn programming, the first program is usually printing “Hello World.” In Machine Learning, or Deep Learning, this is handwritten digit recognition with [MNIST](http://yann.lecun.com/exdb/mnist/) dataset. Handwriting recognition is a typical image classification problem. The problem is relatively easy, and MNIST is a complete dataset. As a simple Computer Vision dataset, MNIST contains images of handwritten digits and corresponding labels (Fig. 1). An image is a 28x28 matrix, and a label is to one of the 10 digits from 0 to 9. Each image is normalized in size and centered.

Fig. 1. Examples of MNIST images

MNIST dataset is made from [NIST](https://www.nist.gov/srd/nist-special-database-19) Special Database 3 (SD-3) and Special Database 1 (SD-1). Since SD-3 is labeled by staffs in U.S. Census Bureau, while SD-1 is labeled by high school students in U.S., SD-3 is cleaner and easier to recognize than SD-1 is. Yann LeCun et al. used half of samples from each of SD-1 and SD-3 to make MNIST training set (60,000 samples) and test set (10,000 samples), where training set was labeled by 250 different annotators, and it was guaranteed that annotators of training set and test set are not completely overlapped. Yann LeCun, one of the founders of Deep Learning, had huge contribution on handwritten character recognition in early dates, and proposed CNN (Convolutional Neural Network), which drastically improved recognition capability for handwritten characters. CNN is now a critical key for Deep Learning. From Yann LeCun’s first proposal of LeNet, to those winning models in ImageNet, such as VGGNet, GoogLeNet, ResNet, etc. (Please refer to [Image Classification](https://github.com/PaddlePaddle/book/tree/develop/image_classification) tutorial), CNN achieved a series of impressive results in Image Classification tasks. Many algorithms are tested on MNIST. In 1998, LeCun experimented single layer linear classifier, MLP (Multilayer Perceptron) and Multilayer CNN LeNet. These algorithms constantly reduced test error from 12% to 0.7% \[[1](#References)\]. Since then, researchers worked on many algorithms such as k-NN (K-Nearest Neighbors) \[[2](#References)\], Support Vector Machine (SVM) \[[3](#References)\], Neural Networks \[[4-7](#References)\] and Boosting \[[8](#References)\], and applied various preprocessing methods, such as distortion removal, noise removal and blurring, to increase recognition accuracy. In this tutorial, we start from simple softmax regression model, and guide readers to introduction of handwritten character recognition, and step-by-step improvement of models. ## Model Overview Before introducing classification algorithms and training procedure, we provide some definitions: - $X$ is input: Input is $28\times28$ MNIST image. It is flattened to $784$ dimensional vector. $X=\left ( x_0, x_1, \dots, x_{783} \right )$. - $Y$ is output: Output of classifier is 10 class digits from 0 to 9. $Y=\left ( y_0, y_1, \dots, y_9 \right )$. Each dimension $y_i$ represents a probability that the image belongs to $i$. - $L$ is ground truth label: $L=\left ( l_0, l_1, \dots, l_9 \right )$ It is also 10 dimensional, but only one dimension is 1 and others are all 0. ### Softmax Regression The simplest softmax regression model is to feed input into fully connected layers, and directly use softmax for multiclass classification \[[9](#References)\]. Input $X$ is multiplied with weights $W$, added by bias $b$, and activated. $$ y_i = softmax(\sum_j W_{i,j}x_j + b_i) $$ where $ softmax(x_i) = \frac{e^{x_i}}{\sum_j e^{x_j}} $ For a $N$ class classification problem with $N$ output nodes, a $N$ dimensional vector is normalized to $N$ real values in [0, 1], each representing the probability of the sample to belong to the class. Here $y_i$ is the prediction probability that an image is digit $i$. In classification problem, we usually use cross entropy loss function: $$ crossentropy(label, y) = -\sum_i label_ilog(y_i) $$ Fig. 2 is softmax regression network, with weights in black, and bias in red. +1 indicates bias is 1.

Fig. 2. Softmax regression network architecture

### Multilayer Perceptron Softmax regression model uses the simplest two layer neural network, i.e. it only contains input layer and output layer, so that it's regression ability is limited. To achieve better recognition effect, we consider adding several hidden layers \[[10](#References)\] between the input layer and the output layer. 1. After the first hidden layer, we get $ H_1 = \phi(W_1X + b_1) $, where $\phi$ is activation function. Some common ones are sigmoid, tanh and ReLU. 2. After the second hidden layer, we get $ H_2 = \phi(W_2H_1 + b_2) $. 3. Finally, after output layer, we get $Y=softmax(W_3H_2 + b_3)$, the last classification result vector. Fig. 3. is Multilayer Perceptron network, with weights in black, and bias in red. +1 indicates bias is 1.

Fig. 3. Multilayer Perceptron network architecture

### Convolutional Neural Network #### Convolutional Layer

Fig. 4. Convolutional layer

Convolutional layer is the core of Convolutional Neural Network. The parameters in this layer are composed of a set of filters, or kernels. In forward step, each kernel moves horizontally and vertically, and compute dot product of the kernel and the input on corresponding positions, then add bias and apply activation function. The result is two dimensional activation map. For example, some kernel may recognize corners, and some may recognize circles. These convolution kernels may respond strongly to the corresponding features. Fig. 4 is a dynamic graph of convolutional layer, where depths are not shown for simplicity. Input is $W_1=5,H_1=5,D_1=3$. In fact, this is a common representation for colored images. The width and the height of a colored image correspond to $W_1$ and $H_1$, respectively, and the 3 color channels for RGB correspond to $D_1$. The parameters of the convolutional layer are $K=2,F=3,S=2,P=1$. $K$ is the number of kernels. Here, $Filter W_0$ and $Filter W_1$ are two kernels. $F$ is kernel size. $W0$ and $W1$ are both $3\times3$ matrix in all depths. $S$ is stride. Kernels moves leftwards or downwards by 2 units each time. $P$ is padding, an extension of the input. The gray area in the figure shows zero padding with size 1. #### Pooling Layer

Fig. 5 Pooling layer

Pooling layer performs downsampling. The main functionality is to reduce computation by reducing network parameters. It also prevents overfitting to some extent. Usually, a pooling layer is added after a convolutional layer. Pooling layer includes max pooling, average pooling, etc. Max pooling uses rectangles to segment input layer into several parts, and compute maximum value in each part as output (Fig. 5.) #### LeNet-5 Network

Fig. 6. LeNet-5 Convolutional Neural Network architecture

[LeNet-5](http://yann.lecun.com/exdb/lenet/) is one of the simplest Convolutional Neural Networks. Fig. 6. shows its architecture: 2 dimensional image input is fed into two sets of convolutional layer and pooling layer, then it is fed into fully connected layer and softmax classifier. The following three properties of convolution enable LeNet-5 to better recognize images than Multilayer fully-connected perceptrons: - 3D properties of neurons: a convolutional layer is organized by width, height and depth. Neurons in each layer are connected to only a small region in previous layer. This region is called receptive field. - Local connection: CNN utilizes local space correlation by connecting local neurons. This design guarantees learned filter has strong response to local input features. Stacking many such layers leads non-linear filter becomes more and more global. This allows the network to first obtain good representation for a small parts of input, then combine them to represent larger region. - Sharing weights: In CNN, computation is iterated with shared parameters (weights and bias) to form a feature map. This means all neurons in the same depth of output respond to the same feature. This allows detecting a feature regardless of its position in the input, and enables a property of translation equivariance. For more details of Convolutional Neural Network, please refer to [Stanford open course]( http://cs231n.github.io/convolutional-networks/ ) and [Image Classification](https://github.com/PaddlePaddle/book/blob/develop/image_classification/README.md) tutorial. ### List of Common Activation Functions - Sigmoid activation function： $ f(x) = sigmoid(x) = \frac{1}{1+e^{-x}} $ - Tanh activation function： $ f(x) = tanh(x) = \frac{e^x-e^{-x}}{e^x+e^{-x}} $ In fact, tanh function is just a rescaled version of sigmoid function. It is obtained by magnifying the value of sigmoid function and moving it downwards by 1. - ReLU activation function： $ f(x) = max(0, x) $ For more information, please refer to [Activation functions in Wikipedia](https://en.wikipedia.org/wiki/Activation_function). ## Data Preparation ### Data and Download Execute the following command to download [MNIST](http://yann.lecun.com/exdb/mnist/) dataset and unzip, then put paths of training set and test set to train.list and test.list respectively for PaddlePaddle to read. ```bash ./data/get_mnist_data.sh ``` `gzip` downloaded data. The following files can be found in `data/raw_data`: | File name | Description | |----------------------|-------------------------| |train-images-idx3-ubyte| Training images, 60,000 | |train-labels-idx1-ubyte| Training labels, 60,000 | |t10k-images-idx3-ubyte | Evaluation images, 10,000 | |t10k-labels-idx1-ubyte | Evaluation labels, 10,000 | Users can randomly generate 10 images with the following script (Refer to Fig. 1.) ```bash ./load_data.py ``` ### Provide Data for PaddlePaddle We use python interface to provide data to system. `mnist_provider.py` shows a complete example for MNIST data. ```python # Define a py data provider @provider( input_types={'pixel': dense_vector(28 * 28), 'label': integer_value(10)}) def process(settings, filename): # settings is not used currently. # Open image file with open( filename + "-images-idx3-ubyte", "rb") as f: # Read first 4 parameters. magic is data format. n is number of data. rows and cols are number of rows and columns, respectively magic, n, rows, cols = struct.upack(">IIII", f.read(16)) # With empty string as a unit, read data one by one images = np.fromfile( f, 'ubyte', count=n * rows * cols).reshape(n, rows, cols).astype('float32') # Normalize data of [0, 255] to [-1,1] images = images / 255.0 * 2.0 - 1.0 # Open label file with open( filename + "-labels-idx1-ubyte", "rb") as l: # Read first two parameters magic, n = struct.upack(">II", l.read(8)) # With empty string as a unit, read data one by one labels = np.fromfile(l, 'ubyte', count=n).astype("int") for i in xrange(n): yield {"pixel": images[i, :], 'label': labels[i]} ``` ## Model Configurations ### Data Definition In model configuration, use `define_py_data_sources2` to define reading of data from `dataprovider`. If this configuration is used for prediction, data definition is not necessary. ```python if not is_predict: data_dir = './data/' define_py_data_sources2( train_list=data_dir + 'train.list', test_list=data_dir + 'test.list', module='mnist_provider', obj='process') ``` ### Algorithm Configuration Set training related parameters. - batch_size: use 128 samples in each training step. - learning_rate: rating of iteration, related to the rate of convergence. - learning_method: use optimizer `MomentumOptimizer` for training. The parameter 0.9 indicates momentum keeps 0.9 of previous speed. - regularization: A method to prevent overfitting. Here L2 regularization is used. ```python settings( batch_size=128, learning_rate=0.1 / 128.0, learning_method=MomentumOptimizer(0.9), regularization=L2Regularization(0.0005 * 128)) ``` ### Model Architecture #### Overview First get data by `data_layer`, and get classification result by classifier. Here we provided three different classifiers. In training, we compute loss function, which is usually cross entropy for classification problem. In prediction, we can directly output results. ``` python data_size = 1 * 28 * 28 label_size = 10 img = data_layer(name='pixel', size=data_size) predict = softmax_regression(img) # Softmax Regression #predict = multilayer_perceptron(img) # Multilayer Perceptron #predict = convolutional_neural_network(img) #LeNet5 Convolutional Neural Network if not is_predict: lbl = data_layer(name="label", size=label_size) inputs(img, lbl) outputs(classification_cost(input=predict, label=lbl)) else: outputs(predict) ``` #### Softmax Regression One simple fully connected layer with softmax activation function outputs classification result. ```python def softmax_regression(img): predict = fc_layer(input=img, size=10, act=SoftmaxActivation()) return predict ``` #### MultiLayer Perceptron The following code implements a Multilayer Perceptron with two fully connected hidden layers and ReLU activation function. Output layer has Softmax activation function. ```python def multilayer_perceptron(img): # First fully connected layer with ReLU hidden1 = fc_layer(input=img, size=128, act=ReluActivation()) # Second fully connected layer with ReLU hidden2 = fc_layer(input=hidden1, size=64, act=ReluActivation()) # Output layer as fully connected layer and softmax activation. The size must be 10. predict = fc_layer(input=hidden2, size=10, act=SoftmaxActivation()) return predict ``` #### Convolutional Neural Network LeNet-5 The following is the LeNet-5 network architecture. 2D input image is first fed into two sets of convolutional layer and pooling layer, and it is fed into fully connected layer, and another fully connected layer with softmax activation. ```python def convolutional_neural_network(img): # First convolutional layer - pooling layer conv_pool_1 = simple_img_conv_pool( input=img, filter_size=5, num_filters=20, num_channel=1, pool_size=2, pool_stride=2, act=TanhActivation()) # Second convolutional layer - pooling layer conv_pool_2 = simple_img_conv_pool( input=conv_pool_1, filter_size=5, num_filters=50, num_channel=20, pool_size=2, pool_stride=2, act=TanhActivation()) # Fully connected layer fc1 = fc_layer(input=conv_pool_2, size=128, act=TanhActivation()) # Output layer as fully connected layer and softmax activation. The size must be 10. predict = fc_layer(input=fc1, size=10, act=SoftmaxActivation()) return predict ``` ## Training Model ### Training Commands and Logs 1.Configure `train.sh` to execute training: ```bash config=mnist_model.py # Select network in mnist_model.py output=./softmax_mnist_model log=softmax_train.log paddle train \ --config=$config \ # Scripts for network configuration. --dot_period=10 \ # After `dot_period` steps, print one `.` --log_period=100 \ # Print a log every batchs --test_all_data_in_one_period=1 \ # Whether to use all data in every test --use_gpu=0 \ # Whether to use GPU --trainer_count=1 \ # Number of CPU or GPU --num_passes=100 \ # Passes for training (One pass uses all data.) --save_dir=$output \ # Path to saved model 2>&1 | tee $log python -m paddle.utils.plotcurve -i $log > plot.png ``` After configuring parameters, execute `./train.sh`. Training log is as follows. ``` I0117 12:52:29.628617 4538 TrainerInternal.cpp:165] Batch=100 samples=12800 AvgCost=2.63996 CurrentCost=2.63996 Eval: classification_error_evaluator=0.241172 CurrentEval: classification_error_evaluator=0.241172 ......... I0117 12:52:29.768741 4538 TrainerInternal.cpp:165] Batch=200 samples=25600 AvgCost=1.74027 CurrentCost=0.840582 Eval: classification_error_evaluator=0.185234 CurrentEval: classification_error_evaluator=0.129297 ......... I0117 12:52:29.916970 4538 TrainerInternal.cpp:165] Batch=300 samples=38400 AvgCost=1.42119 CurrentCost=0.783026 Eval: classification_error_evaluator=0.167786 CurrentEval: classification_error_evaluator=0.132891 ......... I0117 12:52:30.061213 4538 TrainerInternal.cpp:165] Batch=400 samples=51200 AvgCost=1.23965 CurrentCost=0.695054 Eval: classification_error_evaluator=0.160039 CurrentEval: classification_error_evaluator=0.136797 ......I0117 12:52:30.223270 4538 TrainerInternal.cpp:181] Pass=0 Batch=469 samples=60000 AvgCost=1.1628 Eval: classification_error_evaluator=0.156233 I0117 12:52:30.366894 4538 Tester.cpp:109] Test samples=10000 cost=0.50777 Eval: classification_error_evaluator=0.0978 ``` 2.Use `plot_cost.py` to plot error curve during training. ```bash python plot_cost.py softmax_train.log ``` 3.Use `evaluate.py ` to select the best trained model. ```bash python evaluate.py softmax_train.log ``` ### Training Results for Softmax Regression

Fig. 7 Softmax regression error curve

Evaluation results of the models: ```text Best pass is 00013, testing Avgcost is 0.484447 The classification accuracy is 90.01% ``` From the evaluation results, the best pass for softmax regression model is pass-00013, where classification accuracy is 90.01%, and the last pass-00099 has accuracy of 89.3%. From Fig. 7, we also see that the best accuracy may not appear in the last pass. An explanation is that during training, the model may already arrive at local optimum, and it just swings around nearby in the following passes, or it gets lower local optimum. ### Results of Multilayer Perceptron

Fig. 8. Multilayer Perceptron error curve

Evaluation results of the models： ```text Best pass is 00085, testing Avgcost is 0.164746 The classification accuracy is 94.95% ``` From the evaluation results, the final training accuracy is 94.95%. It has significant improvement comparing with softmax regression model. The reason is that softmax regression is simple, and it cannot fit complex data, but Multilayer Perceptron with hidden layers has stronger fitting capacity. ### Training results for Convolutional Neural Network

图9. Convolutional Neural Network error curve

Results of model evaluation： ```text Best pass is 00076, testing Avgcost is 0.0244684 The classification accuracy is 99.20% ``` From the evaluation result, the best accuracy of Convolutional Neural Network is 99.20%. This means, for image problem, Convolutional Neural Network has better recognition effect than fully connected network. This should be related to the local connection and parameter sharing of convolutional layers. Also, in Fig. 9, Convolutional Neural Network achieves good effect in early steps, which indicates that it is fast to converge. ## Application Model ### Prediction Commands and Results Script `predict.py` can make prediction for trained models. For example, in softmax regression: ```bash python predict.py -c mnist_model.py -d data/raw_data/ -m softmax_mnist_model/pass-00047 ``` - -c sets model architecture - -d sets data for prediction - -m sets model parameters, here the best trained model is used for prediction Follow to instruction to input image ID for prediction. The classifier can output probabilities for each digit, predicted results with the highest probability, and ground truth label. ``` Input image_id [0~9999]: 3 Predicted probability of each digit: [[ 1.00000000e+00 1.60381094e-28 1.60381094e-28 1.60381094e-28 1.60381094e-28 1.60381094e-28 1.60381094e-28 1.60381094e-28 1.60381094e-28 1.60381094e-28]] Predict Number: 0 Actual Number: 0 ``` From the result, this classifier recognizes the digit on the third image as digit 0 with near to 100% probability, and the ground truth is actually consistent. ## Conclusion Softmax regression, Multilayer Perceptron and Convolutional Neural Network in this tutorial are the most basic Deep Learning models. More sophisticated models in the following tutorials are derived from them. Therefore, these models are very helpful for the future learning. At the same time, we observed that when evolving from the simplest softmax regression to slightly complex Convolutional Neural Network, recognition accuracy on MNIST data set has large improvement, due to Convolutional layers' local connections and parameter sharing. When learning new models in the future, we hope readers to understand the key ideas for a new model to improve over an old one. Moreover, this tutorial introduced basic flow of PaddlePaddle model design, starting from dataprovider, model layer construction, to final training and prediction. By becoming familiar with this flow, readers can use specific data, define specific network models, and complete training and prediction for their tasks. ## References 1. LeCun, Yann, Léon Bottou, Yoshua Bengio, and Patrick Haffner. ["Gradient-based learning applied to document recognition."](http://ieeexplore.ieee.org/abstract/document/726791/) Proceedings of the IEEE 86, no. 11 (1998): 2278-2324. 2. Wejéus, Samuel. 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