A simple program for an ANN to cover the Mnist dataset – III – forward propagation

I continue with my efforts of writing a small Python class by which I can setup and test a Multilayer Perceptron [MLP] as a simple example for an artificial neural network [ANN]. In the last two articles of this series

A simple program for an ANN to cover the Mnist dataset – I
A simple program for an ANN to cover the Mnist dataset – II

I defined some code elements, which controlled the layers, their node numbers and built weight matrices. We succeeded in setting random initial values for the weights. This enables us to work on the forward propagation algorithm in this article.

Methods to cover training and mini-batches

As we later on need to define methods which cover "training epochs" and the handling of "mini-batches" comprising a defined number of training records we extend our set of methods already now by

An "epoch" characterizes a full training step comprising

  • propagation, cost and derivative analysis and weight correction of all data records or samples in the set of training data, i.e. a loop over all mini-batches.

Handling of a mini-batch comprises

  • (vectorized) propagation of all training records of a mini-batch,
  • cumulative cost analysis for all training records of a batch,
  • cumulative, averaged gradient evaluation of the cost function by back-propagation of errors and summation over all records of a training batch,
  • weight corrections for nodes in all layers based on averaged gradients over all records of the batch data.

Vectorized propagation means that we propagate all training records of a batch in parallel. This will be handled by Numpy matrix multiplications (see below).
We shall see in a forthcoming that we can also cover the cumulative gradient calculation over all batch samples by matrix-multiplications where we shift the central multiplication and summation operations to appropriate rows and columns.

However, we do not care for details of training epochs and complete batch-operations at the moment. We use the two methods "_fit()" and "_handle_mini_batch()" in this article only as envelopes to trigger the epoch loop and the matrix operations for propagation of a batch, respectively.

Modified "__init__"-function

We change and extend our "__init_"-function of class MyANN a bit:

    def __init__(self, 
                 my_data_set = "mnist", 
                 n_hidden_layers = 1, 
                 ay_nodes_layers = [0, 100, 0], # array which should have as much elements as n_hidden + 2
                 n_nodes_layer_out = 10,  # expected number of nodes in output layer 
                                                  
                 my_activation_function = "sigmoid", 
                 my_out_function        = "sigmoid",   
                 
                 n_size_mini_batch = 50,  # number of data elements in a mini-batch 
                 
                 n_epochs      = 1,
                 n_max_batches = -1,  # number of mini-batches to use during epochs - > 0 only for testing 
                                      # a negative value uses all mini-batches 
                 
                 vect_mode = 'cols', 
                 
                 figs_x1=12.0, figs_x2=8.0, 
                 legend_loc='upper right',
                 
                 b_print_test_data = True
                 
                 ):
        '''
        Initialization of MyANN
        Input: 
            data_set: type of dataset; so far only the "mnist", "mnist_784" datsets are known 
                      We use this information to prepare the input data and learn about the feature dimension. 
                      This info is used in preparing the size of the input layer.     
            n_hidden_layers = number of hidden layers => between input layer 0 and output layer n 

            ay_nodes_layers = [0, 100, 0 ] : We set the number of nodes in input layer_0 and the output_layer to zero 
                              Will be set to real number afterwards by infos from the input dataset. 
                              All other numbers are used for the node numbers of the hidden layers.
            n_nodes_out_layer = expected number of nodes in the output layer (is checked); 
                                this number corresponds to the number of categories NC = number of labels to be distinguished
            
            my_activation_function : name of the activation function to use 
            my_out_function : name of the "activation" function of the last layer which produces the output values 
            
            n_size_mini_batch : Number of elements/samples in a mini-batch of training data 
                                The number of mini-batches will be calculated from this
            
            n_epochs : number of epochs to calculate during training
            n_max_batches : > 0: maximum of mini-batches to use during training 
                            < 0: use all mini-batches  
            
            vect_mode: Are 1-dim data arrays (vctors) ordered by columns or rows ?

            figs_x1=12.0, figs_x2=8.0 : Standard sizing of plots , 
            legend_loc='upper right': Position of legends in the plots
            
            b_print_test_data: Boolean variable to control the print out of some tests data 
             
         '''
        
        # Array (Python list) of known input data sets 
        self._input_data_sets = ["mnist", "mnist_784", "mnist_keras"]  
        self._my_data_set = my_data_set
        
        # X, y, X_train, y_train, X_test, y_test  
            # will be set by analyze_input_data 
            # X: Input array (2D) - at present status of MNIST image data, only.    
            # y: result (=classification data) [digits represent categories in the case of Mnist]
        self._X       = None 
        self._X_train = None 
        self._X_test  = None   
        self._y       = None 
        self._y_train = None 
        self._y_test  = None
        
        # relevant dimensions 
        # from input data information;  will be set in handle_input_data()
        self._dim_sets     = 0  
        self._dim_features = 0  
        self._n_labels     = 0   # number of unique labels - will be extracted from y-data 
        
        # Img sizes 
        self._dim_img      = 0 # should be sqrt(dim_features) - we assume square like images  
        self._img_h        = 0 
        self._img_w        = 0 
        
        # Layers
        # ------
        # number of hidden layers 
        self._n_hidden_layers = n_hidden_layers
        # Number of total layers 
        self._n_total_layers = 2 + self._n_hidden_layers  
        # Nodes for hidden layers 
        self._ay_nodes_layers = np.array(ay_nodes_layers)
        # Number of nodes in output layer - will be checked against information from target arrays
        self._n_nodes_layer_out = n_nodes_layer_out
        
        
        # Weights 
        # --------
        # empty List for all weight-matrices for all layer-connections
        # Numbering : 
        # w[0] contains the weight matrix which connects layer 0 (input layer ) to hidden layer 1 
        # w[1] contains the weight matrix which connects layer 1 (input layer ) to (hidden?) layer 2 
        self._ay_w = []  
        
        # --- New -----
        # Two lists for output of propagation
        # __ay_x_in  : input data of mini-batches on the different layers; the contents is calculated by the propagation algorithm    
        # __ay_a_out : output data of the activation function; the contents is calculated by the propagation algorithm
        # Note that the elements of these lists are numpy arrays     
        self.__ay_X_in  = []  
        self.__ay_a_out = [] 
        
        
        # Known Randomizer methods ( 0: np.random.randint, 1: np.random.uniform )  
        # ------------------
        self.__ay_known_randomizers = [0, 1]

        # Types of activation functions and output functions 
        # ------------------
        self.__ay_activation_functions = ["sigmoid"] # later also relu 
        self.__ay_output_functions     = ["sigmoid"] # later also softmax 
        
        # the following dictionaries will be used for indirect function calls 
        self.__d_activation_funcs = {
            'sigmoid': self._sigmoid, 
            'relu':    self._relu
            }
        self.__d_output_funcs = { 
            'sigmoid': self._sigmoid, 
            'softmax': self._softmax
            }  
          
        # The following variables will later be set by _check_and set_activation_and_out_functions()            
        self._my_act_func = my_activation_function
        self._my_out_func = my_out_function
        self._act_func = None    
        self._out_func = None    

        # number of data samples in a mini-batch 
        self._n_size_mini_batch = n_size_mini_batch
        self._n_mini_batches = None  # will be determined by _get_number_of_mini_batches()

        # number of epochs 
        self._n_epochs = n_epochs
        # maximum number of batches to handle (<0 => all!) 
        self._n_max_batches = n_max_batches


        # print some test data 
        self._b_print_test_data = b_print_test_data

        # Plot handling 
        # --------------
        # Alternatives to resize plots 
        # 1: just resize figure  2: resize plus create subplots() [figure + axes] 
        self._plot_resize_alternative = 1 
        # Plot-sizing
        self._figs_x1 = figs_x1
        self._figs_x2 = figs_x2
        self._fig = None
        self._ax  = None 
        # alternative 2 does resizing and (!) subplots() 
        self.initiate_and_resize_plot(self._plot_resize_alternative)        
        
        
        # ***********
        # operations 
        # ***********
        
        # check and handle input data 
        self._handle_input_data()
        # set the ANN structure 
        self._set_ANN_structure()
        
        # Prepare epoch and batch-handling - sets mini-batch index array, too 
        self._prepare_epochs_and_batches()
        
        # perform training 
        start_c = time.perf_counter()
        self._fit(b_print=True, b_measure_batch_time=False)
        end_c = time.perf_counter()
        print('\n\n ------') 
        print('Total training Time_CPU: ', end_c - start_c) 
        print("\nStopping program regularily")
        sys.exit()

 
Readers who have followed me so far will recognize that I renamed the parameter "n_mini_batch" to "n_size_mini_batch" to indicate its purpose a bit more clearly. We shall derive the number of required mini-batches form the value of this parameter.
I have added two new parameters:

  • n_epochs = 1
  • n_max_batches = -1

"n_epochs" will later receive the user's setting for the number of epochs to follow during training. "n_max_Batches" allows us to limit the number of mini-batches to analyze during tests.

The kind reader will also have noticed that I encapsulated the series of operations for preparing the weight-matrices for the ANN in a new method "_set_ANN_structure()"

    
    '''-- Main method to set ANN structure --''' 
    def _set_ANN_structure(self):
        # check consistency of the node-number list with the number of hidden layers (n_hidden)
        self._check_layer_and_node_numbers()
        # set node numbers for the input layer and the output layer
        self._set_nodes_for_input_output_layers() 
        self._show_node_numbers() 

        # create the weight matrix between input and first hidden layer 
        self._create_WM_Input() 
        # create weight matrices between the hidden layers and between tha last hidden and the output layer 
        self._create_WM_Hidden() 

        # check and set activation functions 
        self._check_and_set_activation_and_out_functions()
        
        return None

 
The called functions have remained unchanged in comparison to the last article.

Preparing epochs and batches

We can safely assume that some steps must be performed to prepare epoch- and batch handling. We, therefore, introduced a new function "_prepare_epochs_and_batches()". For the time being this method only calculates the number of mini-batches from the input parameter "n_size_mini_batch". We use the Numpy-function "array_split()" to split the full range of input data into batches.

 
    ''' -- Main Method to prepare epochs -- '''
    def _prepare_epochs_and_batches(self):
        # set number of mini-batches and array with indices of input data sets belonging to a batch 
        self._set_mini_batches()
        return None
##    
    ''' -- Method to set the number of batches based on given batch size -- '''
    def _set_mini_batches(self, variant=0): 
        # number of mini-batches? 
        self._n_mini_batches = math.ceil( self._y_train.shape[0] / self._n_size_mini_batch )
        print("num of mini_batches = " + str(self._n_mini_batches))
        
        # create list of arrays with indices of batch elements 
        self._ay_mini_batches = np.array_split( range(self._y_train.shape[0]), self._n_mini_batches )
        print("\nnumber of batches : " + str(len(self._ay_mini_batches)))
        print("length of first batch : " + str(len(self._ay_mini_batches[0])))
        print("length of last batch : "  + str(len(self._ay_mini_batches[self._n_mini_batches - 1]) ))
        return None

 
Note that the approach may lead to smaller batch sizes than requested by the user.
array_split() cuts out a series of sub-arrays of indices of the training data. I.e., "_ay_mini_batches" becomes a 1-dim array, whose elements are 1-dim arrays, too. Each of the latter contains a collection of indices for selected samples of the training data - namely the indices for those samples which shall be used in the related mini-batch.

Preliminary elements of the method for training - "_fit()"

For the time being method "_fit()" is used for looping over the number of epochs and the number of batches:

 
    ''' -- Method to set the number of batches based on given batch size -- '''
    def _fit(self, b_print = False, b_measure_batch_time = False):
        # range of epochs
        ay_idx_epochs  = range(0, self._n_epochs)
        
        # limit the number of mini-batches
        n_max_batches = min(self._n_max_batches, self._n_mini_batches)
        ay_idx_batches = range(0, n_max_batches)
        if (b_print):
            print("\nnumber of epochs = " + str(len(ay_idx_epochs)))
            print("max number of batches = " + str(len(ay_idx_batches)))
        
        # looping over epochs
        for idxe in ay_idx_epochs:
            if (b_print):
                print("\n ---------")
                print("\nStarting epoch " + str(idxe+1))
            
            # loop over mini-batches
            for idxb in ay_idx_batches:
                if (b_print):
                    print("\n ---------")
                    print("\n Dealing with mini-batch " + str(idxb+1))
                if b_measure_batch_time: 
                    start_0 = time.perf_counter()
                # deal with a mini-batch
                self._handle_mini_batch(num_batch = idxb, b_print_y_vals = False, b_print = b_print)
                if b_measure_batch_time: 
                    end_0 = time.perf_counter()
                    print('Time_CPU for batch ' + str(idxb+1), end_0 - start_0) 
        
        return None
#

 
We limit the number of mini_batches. The double-loop-structure is typical. We tell function "_handle_mini_batch(num_batch = idxb,...)" which batch it should handle.

Preliminary steps for the treatment of a mini-batch

We shall build up the operations for batch handling over several articles. In this article we clarify the operations for feed forward propagation, only. Nevertheless, we have to think a step ahead: Gradient calculation will require that we keep the results of propagation layer-wise somewhere.

As the number of layers can be set by the user of the class we save the propagation results in two Python lists:

  • ay_Z_in_layer = []
  • ay_A_out_layer = []

The Z-values define a collection of input vectors which we normally get by a matrix multiplication from output data of the last layer and a suitable weight-matrix. The "collection" is our mini-batch. So, "ay_Z_in_layer" actually is a 2-dimensional array.

For the ANN's input layer "L0", however, we just fill in an excerpt of the "_X"-array-data corresponding to the present mini-batch.

Array "ay_A_out_layer[n]" contains the results of activation function applied onto the elements of "ay_Z_in_layer[n]" of Layer "Ln". (In addition we shall add a value for a bias neutron; see below).

Our method looks like:

 
    ''' -- Method to deal with a batch -- '''
    def _handle_mini_batch(self, num_batch = 0, b_print_y_vals = False, b_print = False):
        '''
        For each batch we keep the input data array Z and the output data A (output of activation function!) 
        for all layers in Python lists
        We can use this as input variables in function calls - mutable variables are handled by reference values !
        We receive the A and Z data from propagation functions and proceed them to cost and gradient calculation functions
        
        As an initial step we define the Python lists ay_Z_in_layer and ay_A_out_layer 
        and fill in the first input elements for layer L0  
        '''
        ay_Z_in_layer  = [] # Input vector in layer L0;  result of a matrix operation in L1,...
        ay_A_out_layer = [] # Result of activation function 
    
        #print("num_batch = " + str(num_batch))
        #print("len of ay_mini_batches = " + str(len(self._ay_mini_batches))) 
        #print("_ay_mini_batches[0] = ")
        #print(self._ay_mini_batches[num_batch])
    
        # Step 1: Special treatment of the ANN's input Layer L0
        # Layer L0: Fill in the input vector for the ANN's input layer L0 
        ay_Z_in_layer.append( self._X_train[(self._ay_mini_batches[num_batch])] ) # numpy arrays can be indexed by an array of integers
        #print("\nPropagation : Shape of X_in = ay_Z_in_layer = " + str(ay_Z_in_layer[0].shape))           
        if b_print_y_vals:
            print("\n idx, expected y_value of Layer L0-input :")           
            for idx in self._ay_mini_batches[num_batch]:
                print(str(idx) + ', ' + str(self._y_train[idx]) )
        
        # Step 2: Layer L0: We need to transpose the data of the input layer 
        ay_Z_in_0T       = ay_Z_in_layer[0].T
        ay_Z_in_layer[0] = ay_Z_in_0T

        # Step 3: Call the forward propagation method for the mini-batch data samples 
        self._fw_propagation(ay_Z_in = ay_Z_in_layer, ay_A_out = ay_A_out_layer, b_print = b_print) 
        
        if b_print:
            # index range of layers 
            ilayer = range(0, self._n_total_layers)
            print("\n ---- ")
            print("\nAfter propagation through all layers: ")
            for il in ilayer:
                print("Shape of Z_in of layer L" + str(il) + " = " + str(ay_Z_in_layer[il].shape))
                print("Shape of A_out of layer L" + str(il) + " = " + str(ay_A_out_layer[il].shape))

        
        # Step 4: To be done: cost calculation for the batch 
        # Step 5: To be done: gradient calculation via back propagation of errors 
        # Step 6: Adjustment of weights  
        
        # try to accelerate garbage handling
        if len(ay_Z_in_layer) > 0:
            del ay_Z_in_layer
        if len(ay_A_out_layer) > 0:
            del ay_A_out_layer
        
        return None

 
Why do we need to transpose the Z-matrix for layer L0?
This has to do with the required matrix multiplication of the forward propagation (see below).

The function "_fw_propagation()" performs the forward propagation of a mini-batch through all of the ANN's layers - and saves the results in the lists defined above.

Important note: We transfer our lists (mutable Python objects) to "_fw_propagation()"! This has the effect that the array of the corresponding values is referenced from within "_fw_propagation()"; therefore will any elements added to the lists also be available outside the called function! Therefore we can use the calculated results also in further functions for e.g. gradient calculations which will later be called from within "_handle_mini_batch()".

Note also that this function leaves room for optimization: It is e.g. unnecessary to prepare ay_Z_in_0T again and again for each epoch. We will transfer the related steps to "_prepare_epochs_and_batches()" later on.

Forward Propagation

In one of my last articles in this blog I already showed how one can use Numpy's Linear Algebra features to cover propagation calculations required for information transport between two adjacent layers of a feed forward "Artificial Neural Network" [ANN]:
Numpy matrix multiplication for layers of simple feed forward ANNs

The result was that we can cover propagation between neighboring layers by a vectorized multiplication of two 2-dim matrices - one containing the weights and the other vectors of feature data for all mini-batch samples. In the named article I discussed in detail which rows and columns are used for the central multiplication with weights and summations - and that the last dimension of the input array should account for the mini-batch samples. This requires the transpose operation on the input array of Layer L0. All other intermediate layer results (arrays) do already get the right form for vectorizing.

"_fw_propagation()" takes the following form:

 
    ''' -- Method to handle FW propagation for a mini-batch --'''
    def _fw_propagation(self, ay_Z_in, ay_A_out, b_print= False):
        
        b_internal_timing = False
        
        # index range of layers 
        ilayer = range(0, self._n_total_layers-1)

        # propagation loop
        for il in ilayer:
            if b_internal_timing: start_0 = time.perf_counter()
            
            if b_print: 
                print("\nStarting propagation between L" + str(il) + " and L" + str(il+1))
                print("Shape of Z_in of layer L" + str(il) + " (without bias) = " + str(ay_Z_in[il].shape))
            
            # Step 1: Take input of last layer and apply activation function 
            if il == 0: 
                A_out_il = ay_Z_in[il] # L0: activation function is identity 
            else: 
                A_out_il = self._act_func( ay_Z_in[il] ) # use real activation function 
            
            # Step 2: Add bias node 
            A_out_il = self._add_bias_neuron_to_layer(A_out_il, 'row')
            # save in array     
            ay_A_out.append(A_out_il)
            if b_print: 
                print("Shape of A_out of layer L" + str(il) + " (with bias) = " + str(ay_A_out[il].shape))
            
            # Step 3: Propagate by matrix operation 
            Z_in_ilp1 = np.dot(self._ay_w[il], A_out_il) 
            ay_Z_in.append(Z_in_ilp1)
            
            if b_internal_timing: 
                end_0 = time.perf_counter()
                print('Time_CPU for layer propagation L' + str(il) + ' to L' + str(il+1), end_0 - start_0) 
        
        # treatment of the last layer 
        il = il + 1
        if b_print:
            print("\nShape of Z_in of layer L" + str(il) + " = " + str(ay_Z_in[il].shape))
        A_out_il = self._out_func( ay_Z_in[il] ) # use the output function 
        ay_A_out.append(A_out_il)
        if b_print:
            print("Shape of A_out of last layer L" + str(il) + " = " + str(ay_A_out[il].shape))
        
        return None
#

 
First we set a range for a loop over the layers. Then we apply the activation function. In "step 2" we add a bias-node to the layer - compare this to the number of weights, which we used during the initialization of the weight matrices in the last article. In step 3 we apply the vectorized Numpy-matrix multiplication (np.dot-operation). Note that this is working for layer L0, too, because we already transposed the input array for this layer in "_handle_mini_batch()"!

Note that we need some special treatment for the last layer: here we call the out-function to get result values. And, of course, we do not add a bias neuron!

It remains to have a look at the function "_add_bias_neuron_to_layer(A_out_il, 'row')", which extends the A-data by a constant value of "1" for a bias neuron. The function is pretty simple:

    ''' Method to add values for a bias neuron to A_out '''
    def _add_bias_neuron_to_layer(self, A, how='column'):
        if how == 'column':
            A_new = np.ones((A.shape[0], A.shape[1]+1))
            A_new[:, 1:] = A
        elif how == 'row':
            A_new = np.ones((A.shape[0]+1, A.shape[1]))
            A_new[1:, :] = A
        return A_new    

A first test

We let the program run in a Jupyter cell with the following parameters:

This produces the following output ( I omitted the output for initialization):

 
Input data for dataset mnist_keras : 
Original shape of X_train = (60000, 28, 28)
Original Shape of y_train = (60000,)
Original shape of X_test = (10000, 28, 28)
Original Shape of y_test = (10000,)

Final input data for dataset mnist_keras : 
Shape of X_train = (60000, 784)
Shape of y_train = (60000,)
Shape of X_test = (10000, 784)
Shape of y_test = (10000,)

We have 60000 data sets for training
Feature dimension is 784 (= 28x28)
The number of labels is 10

Shape of y_train = (60000,)
Shape of ay_onehot = (10, 60000)

Values of the enumerate structure for the first 12 elements : 
(0, 6)
(1, 8)
(2, 4)
(3, 8)
(4, 6)
(5, 5)
(6, 9)
(7, 1)
(8, 3)
(9, 8)
(10, 9)
(11, 0)

Labels for the first 12 datasets:

Shape of ay_onehot = (10, 60000)
[[0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 1.]
 [0. 0. 0. 0. 0. 0. 0. 1. 0. 0. 0. 0.]
 [0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.]
 [0. 0. 0. 0. 0. 0. 0. 0. 1. 0. 0. 0.]
 [0. 0. 1. 0. 0. 0. 0. 0. 0. 0. 0. 0.]
 [0. 0. 0. 0. 0. 1. 0. 0. 0. 0. 0. 0.]
 [1. 0. 0. 0. 1. 0. 0. 0. 0. 0. 0. 0.]
 [0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.]
 [0. 1. 0. 1. 0. 0. 0. 0. 0. 1. 0. 0.]
 [0. 0. 0. 0. 0. 0. 1. 0. 0. 0. 1. 0.]]

The node numbers for the 4 layers are : 
[784 100  50  10]

Shape of weight matrix between layers 0 and 1 (100, 785)
Creating weight matrix for layer 1 to layer 2
Shape of weight matrix between layers 1 and 2 = (50, 101)
Creating weight matrix for layer 2 to layer 3
Shape of weight matrix between layers 2 and 3 = (10, 51)

The activation function of the standard neurons was defined as "sigmoid"
The activation function gives for z=2.0:  0.8807970779778823

The output function of the neurons in the output layer was defined as "sigmoid"
The output function gives for z=2.0:  0.8807970779778823
num of mini_batches = 300

number of batches : 300
length of first batch : 200
length of last batch : 200

number of epochs = 1
max number of batches = 2

 ---------

Starting epoch 1

 ---------

 Dealing with mini-batch 1

Starting propagation between L0 and L1
Shape of Z_in of layer L0 (without bias) = (784, 200)
Shape of A_out of layer L0 (with bias) = (785, 200)

Starting propagation between L1 and L2
Shape of Z_in of layer L1 (without bias) = (100, 200)
Shape of A_out of layer L1 (with bias) = (101, 200)

Starting propagation between L2 and L3
Shape of Z_in of layer L2 (without bias) = (50, 200)
Shape of A_out of layer L2 (with bias) = (51, 200)

Shape of Z_in of layer L3 = (10, 200)
Shape of A_out of last layer L3 = (10, 200)

 ---- 

After propagation through all layers: 
Shape of Z_in of layer L0 = (784, 200)
Shape of A_out of layer L0 = (785, 200)
Shape of Z_in of layer L1 = (100, 200)
Shape of A_out of layer L1 = (101, 200)
Shape of Z_in of layer L2 = (50, 200)
Shape of A_out of layer L2 = (51, 200)
Shape of Z_in of layer L3 = (10, 200)
Shape of A_out of layer L3 = (10, 200)

 ---------

 Dealing with mini-batch 2

Starting propagation between L0 and L1
Shape of Z_in of layer L0 (without bias) = (784, 200)
Shape of A_out of layer L0 (with bias) = (785, 200)

Starting propagation between L1 and L2
Shape of Z_in of layer L1 (without bias) = (100, 200)
Shape of A_out of layer L1 (with bias) = (101, 200)

Starting propagation between L2 and L3
Shape of Z_in of layer L2 (without bias) = (50, 200)
Shape of A_out of layer L2 (with bias) = (51, 200)

Shape of Z_in of layer L3 = (10, 200)
Shape of A_out of last layer L3 = (10, 200)

 ---- 

After propagation through all layers: 
Shape of Z_in of layer L0 = (784, 200)
Shape of A_out of layer L0 = (785, 200)
Shape of Z_in of layer L1 = (100, 200)
Shape of A_out of layer L1 = (101, 200)
Shape of Z_in of layer L2 = (50, 200)
Shape of A_out of layer L2 = (51, 200)
Shape of Z_in of layer L3 = (10, 200)
Shape of A_out of layer L3 = (10, 200)


 ------
Total training Time_CPU:  0.010270356000546599

Stopping program regularily
stopped

 
We see that the dimensions of the Numpy arrays fit our expectations!

If you raise the number for batches and the number for epochs you will pretty soon realize that writing continuous output to a Jupyter cell costs CPU-time. You will also notice strange things regarding performance, multithreading and the use of the Linalg library OpenBlas on Linux system. I have discussed this extensively in a previous article in this blog:
Linux, OpenBlas and Numpy matrix multiplications – avoid using all processor cores

So, for another tests we set the following environment variable for the shell in which we start our Jupyter notebook:

export OPENBLAS_NUM_THREADS=4

This is appropriate for my Quad-core CPU with hyperthreading. You may choose a different parameter on your system!

We furthermore stop printing in the epoch loop by editing the call to function "_fit()":

self._fit(b_print=False, b_measure_batch_time=False)

We change our parameter setting to:

Then the last output lines become:

The node numbers for the 4 layers are : 
[784 100  50  10]

Shape of weight matrix between layers 0 and 1 (100, 785)
Creating weight matrix for layer 1 to layer 2
Shape of weight matrix between layers 1 and 2 = (50, 101)
Creating weight matrix for layer 2 to layer 3
Shape of weight matrix between layers 2 and 3 = (10, 51)

The activation function of the standard neurons was defined as "sigmoid"
The activation function gives for z=2.0:  0.8807970779778823

The output function of the neurons in the output layer was defined as "sigmoid"
The output function gives for z=2.0:  0.8807970779778823
num of mini_batches = 150

number of batches : 150
length of first batch : 400
length of last batch : 400


 ------
Total training Time_CPU:  146.44446582399905

Stopping program regularily
stopped

Good !
The time required to repeat this kind of forward propagation for a network with only one hidden layer with 50 neurons and 1000 epochs is around 160 secs. As backward propagation is not much more complex than forward propagation this already indicates that we should be able to train such a most simple MLP with 60000 28x28 images in less than 10 minutes on a standard CPU.

Conclusion

In this article we saw that coding forward propagation is a pretty straight-forward exercise with Numpy! The tricky thing is to understand the way numpy.dot() handles vectorizing of a matrix product and which structure of the matrices is required to get the expected numbers!

In the next article we shall start working on cost and gradient calculation.

 

A simple program for an ANN to cover the Mnist dataset – I

For beginners both in Python and Machine Learning [ML] the threshold to do some real programming and create your own ANN seems to be relatively high. Well, some readers might say: Why program an ANN by yourself at a basic Python level at all when Keras and TensorFlow [TF] are available? Answer: For learning! And eventually to be able to do some things TF has not been made for. And as readers of this blog will see soon, I have some ideas along this line ...

So, I thought, just let us set up a small Python3 and Numpy based program to create a simple ANN and train it for the MNIST dataset.

I take a shortcut and assume that readers are already acquainted with the following topics:

  1. what simple neural networks with a hidden layer look like,
  2. what a cost function and the gradient descent method is,
  3. what logistic regression is and what the cost function J for it looks like,
  4. why the back propagation of deviations from known values gives you the required partial derivatives for typical cost functions,
  5. what a mini-batch approach to optimization is.

These basics are well documented in literature; e.g. in the books of Geron and Rashka (see the references at the end of this article). However, I shall briefly comment on these topics whilst building the code.

We need a relatively well defined first objective for the usage of our ANN. We shall concentrate on classification tasks. As a first example we shall use the conventional MNIST data set. The MNIST data set consists of 28x28 px images of handwritten numbers. It is a standard data set used in many elementary courses on ML. The challenge for the ANN is that it should be able to recognize hand-written digits from a digitized gray-color image after some training.

Note that this task does NOT require the use of an ANN. "Stochastic Gradient Descent"-approaches to determine (linear) separation surfaces in combination with a One-versus-All strategy for multi-category-classification may be sufficient. See chapters 3 to 5 in the book of Geron for more information.

Regarding the build up of the ANN program, I basically follow an approach described by Raschka in his book. However, at multiple points I take the freedom to organize the code differently and comment in my own way ... I am only a beginner in Python; I hope my insights are helpful for others in the same situation.

In any case you should make yourself familiar with numpy arrays and their "shapes". I assume that you understand the multidimensional structure of Numpy arrays ....

Wording

To avoid confusion, I use the following wording:

Category: Each input data element is associated with a category to which it belongs. The classification algorithm (here: the ANN) may achieve a distinction between the association of input data with multiple categories. It should - after some training - detect (non-linear) separation interfaces for categories in a multidimensional feature space. In the case of MNIST we speak about ten categories corresponding to 10 digits, including zero.

Label: A category may be described by a label. Training data may provide a so called "target label array" _y_train for all input data. We must be prepared to transform target labels for input data into a usable form for an ANN, i.e. into a vectorized form which selects a specific category out of many. This process is called "label encoding".

Input data set: A complete set of input data. Such a set consists of individual "elements" or "records". The MNIST input set of training data has 60000 records - which we provide via an array _X_train.

Output data set: A complete set of output data after propagation through the ANN for the input data set. The number of elements or records of the output data sets is equal to the number of records in the input data set. The output set will be represented by an array "_ay_a_Out".

A data record of the input data set: One distinct element of the input data set (and its array). Note that such an element itself may be a multidimensional array covering all features in a distinct form. Such arrays represents a so called "tensor".

A data record of the output data set: One distinct element of the output data set. Note that such an element itself may be an array covering all possible categories in a distinct form.

A simple neural network - layers, nodes, weights

A neural network is composed of a series of sequential layers with nodes. All nodes of a specific layer can be connected with all nodes of neighboring layers. The simplified sketch below displays an ANN with just three layers - an input layer, a "hidden" middle layer and an output layer. Note that there can be (many) more hidden layers than just one.

Input layer and its number of nodes
To feed input data into the ANN we need an input layer with input nodes. How many? Well, this depends on the number of features your data set represents. In the MNIST case an 28x28 px image with simple gray values (integer number between 0 and 256) represents 28x28 = 768 different "features" - i.e. numbers for color values. We need as many nodes to represent the full image information by the input layer.

For other input data the number and structure of features may be different; the number of input nodes must then be adjusted accordingly. The number of nodes should, therefore, be a parameter or be derived from information on the type of input data. How complicated and structured features are mapped to a one dimensional input vector is a question one should think about carefully. (Most people today deal with a time set of data as just a special form of a feature - I regard this as questionable in some cases, but this is beyond this article series ...)

Output layer and its number of nodes
We shall use our ANN for classification tasks in the beginning. We, therefore, assume that the output of the ANN should distinguish between "strong>NC" different categories an input data set can belong to. In case of the MNIST dataset we can distinguish between 10 different digits. Thus an output layer must in this case at least comprise 10 different nodes. To be able to cover other data sets with a different number of categories the number of output nodes must be a parameter, too.

How we indicate the belonging to a category numerically - by a probability number between "0" and "1" or just a "1" at the right category and zeros otherwise - can be a matter of discussion. It is also a question of the cost function we wish to use. We will come back to this point later.

The numbers of "hidden layers" and their nodes
We want these numbers to be parameters, too. For simple datasets we do not need big networks, but we want to able to play around a bit with 1 up to 3 layers. (For an ANN to recognize hand written digits an input layer "layer 1" and only one hidden layer "layer 2" before the output layer"layer 3" - are required.)

Activation and output functions
The nodes in hidden layers use a so called "activation function" to transform the aggregated input from different feeding nodes of the previous layer into one distinct value in a given interval - e.g. between 0 and 1. Again, we should be prepared to have a parameter to choose between different activation functions.

We should be aware of the fact that the nodes of the output layers need special consideration as the "activation function" there produces the final output - which must allow for a distinction of categories. This may lead to a special form - e.g. a kind of probability function. So, output functions should also be regarded as variable.

An ANN class and its interface

I develop my code as a Python module in an PyDEV/Eclipse which uses a virtual Python environment. I described the setup of such an environment in detail in XXXX. In the directory structure there I place a module "myann.py" at the location "...../ml_1/mynotebooks/mycode/myann.py". We need to import some libraries first

Modules and libraries to import

'''
Module to create a simple layered neural network for the MNIST data set
Created on 23.08.2019
@author: ramoe
'''
import numpy as np
import math 
import sys
import time
import tensorflow
from sklearn.datasets import fetch_mldata
from sklearn.datasets import fetch_openml
from keras.datasets import mnist as kmnist
from scipy.special import expit  
from matplotlib import pyplot as plt
#from matplotlib.colors import ListedColormap
#import matplotlib.patches as mpat 
#from keras.activations import relu

 

Why do I import "tensorflow" and "keras"?
Well, only for the purpose to create the input data of MNIST quickly. Sklearn's "fetchml_data" is doomed to end. "fetch_openml" does not use caching in some older versions and is also otherwise terribly slow. But, "keras", which in turn needs tensorflow as a backend, provides its own tool to provide the MNIST data.

We need "scipy" to get an optimized version of the "sigmoid"-function which is one important version of an activation function. "numpy" and "math" are required for fast array- and math-operations. "time" is required to measure the run time of program segments and "mathplotlib" will help to visualize some information during training.

The "__init__"-function of our class

We encapsulate most of the functionality in a class and its methods. Python provides the "__init__"-function, which we can use as a kind of "constructor" - although it technically is not the same as a constructor in other languages. Anyway, we can use it as an interface to provide parameters and initialize variables in a class instance.

We shall build up our "__init__"-function during the next articles step by step in form of well separated segments; in the beginning we only look a attributes and methods required to set up (MNIST) input data, to create the basic network layers and to create arrays with the (MNIST) input data.

Parameters

class MyANN:
    def __init__(self, 
                 my_data_set = "mnist", 
                 n_hidden_layers = 1, 
                 ay_nodes_layers = [0, 100, 0], # array which should have as much elements as n_hidden + 2
                 n_nodes_layer_out = 10,  # number of nodes in output layer 
                 
                 my_activation_function = "sigmoid", 
                 my_out_function        = "sigmoid",   
                 
                 vect_mode = 'cols', 
                 
                 figs_x1=12.0, figs_x2=8.0, 
                 legend_loc='upper right'
                 ):
        '''
        Initialization of MyANN
        Input: 
            data_set: type of dataset; so far only the "mnist", "mnist_784" and the "mnist_keras" datsets are known. 
                      We use this information to prepare the input data and learn about the feature dimension. 
                      This info is used in preparing the size of the input layer.     
            n_hidden_layers = number of hidden layers => between input layer 0 and output layer n 
            ay_nodes_layers = [0, 100, 0 ] : We set the number of nodes in input layer_0 and the output_layer to zero 
                                             Will be set to real number afterwards by infos from the input dataset. 
                                             All other numbers are used for the node numbers of the hidden layers.
            n_nodes_layer_out = expected number of nodes in the output layer (is checked); 
                                this number corresponds to the number of categories to be distinguished   
            
            my_activation_function : name of the activation function to use 
            my_out_function : name of the "activation" function of the last layer whcih produces the output values 
            
            vect_mode: Are 1-dim data arrays (vectors) ordered by columns or rows ?
            
            figs_x1=12.0, figs_x2=8.0 : Standard sizing of plots , 
            legend_loc='upper right': Position of legends in the plots 
            
         '''

 

You see that I defined multiple parameters, which are explained in the doc string. We use a name to choose the dataset to train our ANN on. We assume that special methods to fetch the required data are implemented in our class. This requires that the class knows exactly which data sets it is capable to handle. We provide an list with this information below. The other parameters should be clear from their explanation.

Initialization of class attributes

We first initialize a bunch of class attributes which we shall use to define the network of layers, nodes, weights, to keep our input data and functions.

        # Array (Python list) of known input data sets 
        self.__input_data_sets = ["mnist", "mnist_784", "mnist_keras"]  
        self._my_data_set = my_data_set
        
        # X, y, X_train, y_train, X_test, y_test  
            # will be set by analyze_input_data 
            # X: Input array (2D) - at present status of MNIST image data, only.    
            # y: result (=classification data) [digits represent categories in the case of Mnist]
        self._X       = None 
        self._X_train = None 
        self._X_test  = None   
        self._y       = None 
        self._y_train = None 
        self._y_test  = None
        
        # relevant dimensions 
        # from input data information;  will be set in handle_input_data()
        self._dim_sets     = 0  
        self._dim_features = 0  
        self._n_labels     = 0   # number of unique labels - will be extracted from y-data 
        
        # Img sizes 
        self._dim_img      = 0 # should be sqrt(dim_features) - we assume square like images  
        self._img_h        = 0 
        self._img_w        = 0 
        
        # Layers
        # ------
        # number of hidden layers 
        self._n_hidden_layers = n_hidden_layers
        # Number of total layers 
        self._n_total_layers = 2 + self._n_hidden_layers  
        # Nodes for hidden layers 
        self._ay_nodes_layers = np.array(ay_nodes_layers)
        # Number of nodes in output layer - will be checked against information from target arrays
        self._n_nodes_layer_out = n_nodes_layer_out
        
        
        # Weights 
        # --------
        # empty List for all weight-matrices for all layer-connections
        # Numbering : 
        # w[0] contains the weight matrix which connects layer 0 (input layer ) to hidden layer 1 
        # w[1] contains the weight matrix which connects layer 1 (input layer ) to (hidden?) layer 2 
        self._ay_w = []  
        
        # Known Randomizer methods ( 0: np.random.randint, 1: np.random.uniform )  
        # ------------------
        self.__ay_known_randomizers = [0, 1]

        # Types of activation functions and output functions 
        # ------------------
        self.__ay_activation_functions = ["sigmoid"] # later also relu 
        self.__ay_output_functions     = ["sigmoid"] # later also softmax 
        
        # the following dictionaries will be used for indirect function calls 
        self.__d_activation_funcs = {
            'sigmoid': self._sigmoid, 
            'relu':    self._relu
            
            }
        self.__d_output_funcs = { 
            'sigmoid': self._sigmoid, 
            'softmax': self._softmax
            }    

        # The following variables will later be set by _check_and set_activation_and_out_functions()            
        self._my_act_func = my_activation_function
        self._my_out_func = my_out_function
        self._act_func = None    
        self._out_func = None    

        # Plot handling 
        # --------------
        # Alternatives to resize plots 
        # 1: just resize figure  2: resize plus create subplots() [figure + axes] 
        self._plot_resize_alternative = 1 
        # Plot-sizing
        self._figs_x1 = figs_x1
        self._figs_x2 = figs_x2
        self._fig = None
        self._ax  = None 
                # alternative 2 does resizing and (!) subplots() 
        self.initiate_and_resize_plot(self._plot_resize_alternative)  

        # ***********
        # operations 
        # ***********
        # check and handle input data 
        self._handle_input_data()
        
        print("\nStopping program regularily")
        sys.exit()
       

 
To make things not more complicated as necessary in the beginning I omit the usage of properties and full encapsulation of private attributes. For convenience reasons I also use only one underscore for some attributes and functions/methods to allow for external usage. This is helpful in a testing phase. However, many items can in the end be switched to really private properties or methods.

List of known input datasets
The list of known input data sets is kept in the variable "self.__input_data_sets". The variables

self._X, self._X_train, self._X_test, self._y, self._y_train, self._y_test

will be used to keep all data of the chosen dataset, the training data, the test data for checking of the reliability of the algorithm after training, and the corresponding target data (y_...) for classification. The target data in the MNIST case contain the digit the respective image (X_..) represents.

All of the named attributes will become numpy arrays. A method called "_handle_input_data(self)" will load the (MNIST) input data and fill the numpy arrays.

The input arrays "X_..." will via their dimensions provide the information on the number of data sets (_dim_sets) and the number of features (_dim_features). The target data arrays "Y_..." provide the number of "classes" (MNIST: 10 digits) to distinguish between by the ANN algorithm. We keep this number in the attribute "_n_labels".

It is also useful to keep the pixel dimensions of input image data. At least for MNIST we assume quadratic images (_img_h = img_w = _dim_img).

Layers and weights
The number of total layers ("_n_total_layers") is by 2 bigger than the number of hidden layers (_n_hidden_layers).

We take the number of nodes in the layers from the respective list provided as an input parameter "ay_nodes_layers" to our class. We transform the list into a numpy array "_ay_nodes_layers". The expected number of nodes in the output layer is used for consistency checks and saved in "_n_nodes_layer_out".

The "weights" of a network must be given in form of matrices - as they describe connections between nodes of different adjacent layers. As the number of layers is not fixed but can be set by the user, I use a list "_ay_w" to collect such matrices in the order of layer_0 (input) to layer_n (output).

Weights, i.e. the matrix elements must initially be defined by random numbers. To provide such numbers we have to use randomizer functions. Depending on the kind (floating point numbers, integer numbers) we use at least two randomizers (randint, uniform).

Usable activation and output functions are named in Python dictionaries which point to respective methods. This allows for "indirect addressing" of these functions later on. You may recognize this by the direct refernece of the dictionary elements to defined class methods (no strings are used!).

For the time being we work with the "sigmoid" and the "relu" functions for activation and the "sigmoid" and "softmax" functions for output creation. The attributes "self._act_func" and "self._out_func" are used later on to invoke the functions requested by the respective parameters of the classes interface.

The final part of the code segment given above is used for plot sizing with the help of "matplotlib"; a method "initiate_and_resize_plot()" takes care of this. It can use 2 alternative ways of doing so.

Read and provide the input data

Now let us turn to some methods. We first need to read in and prepare the input data. We use a method "_handle_input_data()" to work on this problem.

For the time being we have only three different ways to load the MNIST dataset from different origins. We need not yet call different methods but deal with each MNIST source within the method:

    # Method to handle different types of input data sets 
    def _handle_input_data(self):    
        '''
        Method to deal with the input data: 
        - check if we have a known data set ("mnist" so far)
        - reshape as required 
        - analyze dimensions and extract the feature dimension(s) 
        '''
        # check for known dataset 
        try: 
            if (self._my_data_set not in self._input_data_sets ): 
                raise ValueError
        except ValueError:
            print("The requested input data" + self._my_data_set + " is not known!" )
            sys.exit()   

        
        # handle the mnist original dataset 
        if ( self._my_data_set == "mnist"): 
            mnist = fetch_mldata('MNIST original')
            self._X, self._y = mnist["data"], mnist["target"]
            print("Input data for dataset " + self._my_data_set + " : \n" + "Original shape of X = " + str(self._X.shape) +
              "\n" + "Original shape of y = " + str(self._y.shape))
            self._X_train, self._X_test, self._y_train, self._y_test = self._X[:60000], self._X[60000:], self._y[:60000], self._y[60000:] 
            
        # handle the mnist_784 dataset 
        if ( self._my_data_set == "mnist_784"): 
            mnist2 = fetch_openml('mnist_784', version=1, cache=True, data_home='~/scikit_learn_data') 
            self._X, self._y = mnist2["data"], mnist2["target"]
            print ("data fetched")
            # the target categories are given as strings not integers 
            self._y = np.array([int(i) for i in self._y])
            print ("data modified")
            print("Input data for dataset " + self._my_data_set + " : \n" + "Original shape of X = " + str(self._X.shape) +
              "\n" + "Original shape of y = " + str(self._y.shape))
            self._X_train, self._X_test, self._y_train, self._y_test = self._X[:60000], self._X[60000:], self._y[:60000], self._y[60000:] 

        # handle the mnist_keras dataset 
        if ( self._my_data_set == "mnist_keras"): 
            (self._X_train, self._y_train), (self._X_test, self._y_test) = kmnist.load_data()
            len_train =  self._X_train.shape[0]
            #print(len_train)
            print("Input data for dataset " + self._my_data_set + " : \n" + "Original shape of X_train = " + str(self._X_train.shape) +
              "\n" + "Original Shape of y_train = " + str(self._y_train.shape))
            len_test =  self._X_test.shape[0]
            #print(len_test)
            print("Original shape of X_test = " + str(self._X_test.shape) +
              "\n" + "Original Shape of y_test = " + str(self._y_test.shape))
            self._X_train = self._X_train.reshape(len_train, 28*28) 
            self._X_test  = self._X_test.reshape(len_test, 28*28) 

        # Common Mnist handling 
        if ( self._my_data_set == "mnist" or self._my_data_set == "mnist_784" or self._my_data_set == "mnist_keras" ): 
            self._common_handling_of_mnist()    
        
        # Other input data sets can not yet be handled 

 
We first check, whether the input parameter fits a known dataset - and raise an error if otherwise. The data come in different forms for the three sources of MNIST. For each set we want to extract arrays

self._X_train, self._X_test, self._y_train, self._y_test.

We have to do this a bit differently for the 3 cases. Note that the "mnist_784" set from "fetch_openml" gives the target
category values in form of strings and not integers. We correct this after loading.

The fastest method is the MNIST dataset based on "keras"; the keras function "kmnist.load_data()" provides already a 60000:10000 ratio for training and test data. However, we get the images in a (60000, 28, 28) array shape; we therefore reshape the "_X_train"-array to (60000, 784) and "_X_test"-array to (10000, 784).

Analyze the input data and encode the provided target labels

The further handling of the MNIST data requires some common analysis.

    # Method for common input data handling of Mnist data sets
    def _common_handling_of_mnist(self):    
        print("\nFinal input data for dataset " + self._my_data_set + 
              " : \n" + "Shape of X_train = " + str(self._X_train.shape) + 
              "\n" + "Shape of y_train = " + str(self._y_train.shape) + 
              "\n" + "Shape of X_test = " + str(self._X_test.shape) + 
              "\n" + "Shape of y_test = " + str(self._y_test.shape) 
              )
        
        # mixing the training indices 
        shuffled_index = np.random.permutation(60000)
        self._X_train, self._y_train = self._X_train[shuffled_index], self._y_train[shuffled_index]
        
        # set dimensions 
        self._dim_sets = self._y_train.shape[0]
        self._dim_features = self._X_train.shape[1] 
        self._dim_img = math.sqrt(self._dim_features)
        # we assume square images 
        self._img_h = int(self._dim_img)
        self._img_w = int(self._dim_img)
         
        # Print dimensions  
        print("\nWe have " + str(self._dim_sets) + " data sets for training") 
        print("Feature dimension is " + str(self._dim_features) + " (= " + str(self._img_w)+ "x" + str(self._img_h) + ")") 

        # we need to encode the digit labels of mnist 
        self._get_num_labels()
        self._encode_all_mnist_labels()

 
As you see we retrieve some of our class attributes which we shall use during training and do some printing. This is trivial. Not so trivial is the handling of the output data.

What shape do we expect for the "_X_train" and "_y_train"? Each element of the input data set is an array with values for all features. Thus the "_X_train.shape" should be (60000, 784). For _y_train we expect a simple integer describing the digit to which the MNIST input image corresponds. Thus we expect a one dimensional array with _y_train.shape = (60000).

However, the output data of our ANN for one input element come as an array of values for different categories - and not as a simple number. This shows that we need to encode the "_y_train"-data, i.e. the target labels, into a usable array form. We use two methods to achieve this:

    # Method to encode mnist labels 
    def _get_num_labels(self):
        self._n_labels = len(np.unique(self._y_train))
        print("The number of labels is " + str(self._n_labels))
        

    # Method to encode all mnist labels 
    def _encode_all_mnist_labels(self, b_print=True):
        '''
        We shall use vectorized input and output - i.e. we process a whole batch of input data sets in parallel
        (see article in the Linux blog) 
        The output array will then have the form OUT(i_out_node, idx) where 
            i_out_node enumerates the node of the last layer (i.e. the category)    
            idx enumerates the data set within a batch,
        After training, if y_train[idx] = 6, we would expect an output value of OUT[6,idx] = 1.0 and OUT[i_node, idx]=0.0 otherwise 
        for a categorization decision in the ideal case. Realistically, we will get a distribution of numbers over the nodes 
        with values between 0.0 and 1.0 - with hopefully the maximum value at the right node OUT[6,idx].  
        
        The following method creates an arrays OneHot[i_out_node, idx] with 
        OneHot[i_node_out, idx] = 1.0, if i_node_out =  int(y[idx])
        OneHot(i_node_out, idx] = 0.0, if i_node_out != int(y[idx])  
        
        This will allow for a vectorized comparison of calculated values and knwon values during training 
        '''
       
        self._ay_onehot = np.zeros((self._n_labels, self._y_train.shape[0]))
        # ay_oneval is just for convenience and printing purposes 
        self._ay_oneval = np.zeros((self._n_labels, self._y_train.shape[0], 2))
        
        if b_print: 
            print("\nShape of y_train = " + str(self._y_train.shape))
            print("Shape of ay_onehot = " + str(self._ay_onehot.shape))
        
        # the next block is just for illustration purposes and a better understanding
        if b_print: 
            values = enumerate(self._y_train[0:12])
            print("\nValues of the enumerate structure for the first 12 elements : = ")

            for iv in values: 
                print(iv)
        
        # here we prepare the array for vectorized comparison 
        print("\nLabels for the first 12 datasets:")
        for idx, val in enumerate(self._y_train):
            self._ay_onehot[val, idx ]   = 1.0
            self._ay_oneval[val, idx, 0] = 1.0 
            self._ay_oneval[val, idx, 1] = val 
       
        if b_print: 
            print("\nShape of ay_onehot = " + str(self._ay_onehot.shape))
            print(self._ay_onehot[:, 0:12])
            #print("Shape of ay_oneval = " + str(self._ay_oneval.shape))
            #print(self._ay_oneval[:, 0:12, :])

 
The first method only determines the number of labels (= number of categories).

We see from the code of the second method that we encode the target labels in the form of two arrays. The relevant one for our optimization algorithm will be "_ay_onehot". This array is 2-dimensional. Why?

Working with mini-batches

A big advantage of the optimization method we shall use later on during training is that we will perform weight adjustment for a whole bunch of training data in one step. Meaning:

We propagate a whole bunch of test data in parallel throughout the grid, get an array with result data (output array) for which we then calculate a value of the total cost function and an array containing the difference of the output array to an array of correct values (error) for all test data of the bunch. The "error" (i.e. the difference) will be back-propagated to determine the derivative of the cost function for corrections of the node weights.

Such a bunch is called a "batch" and if it is significantly smaller than the whole set of training data - a "mini-batch". Working with "mini-batches" during optimization is a compromise between using the full data set for gradient determination during each optimization step ("batch approach") or using just one input data element of the training set for gradient descent correction ("stochastic approach"). See chapter 4 of the book of Geron and chapter 2 in the book of Raschka for some information on this topic.

The advantage of mini-batches is that we can use vectorized linear algebra operations over all elements of the batch. Linear Algebra libraries are optimized to perform such operations on modern CPUs and GPUs.

You really should keep in mind for understanding the code for the propagation and optimization algorithms discussed in forthcoming articles that the cost function is determined over all elements of a batch and the derivative determination is a matrix like operation involving all input elements of each defined batch! This corresponds to the point that the separation interface in the feature space must be adjusted with respect to all given data points in the end. Mini-batches only help in so far as we look at selected samples of data to achieve bigger steps of our gradient guided descent into an optimum in the beginning - with the disadvantage of making some jumpy stochastic turns instead of a smoother approach.

What is the shape of the output array?
A single input element of the bunch is an array of 784 feature values. The corresponding output array is an array with values for 10 categories (here digits). But, what about a whole bunch of test data, i.e. a "batch"?

As I have explained already in my last article
Numpy matrix multiplication for layers of simple feed forward ANNs
that the output array for a bunch of test data will have the form "_ay_a_Out[i_out_node, idx]" with:

  • i_out_node enumerating the node of the last layer, i.e. the various possible category
  • idx enumerating the data set within a batch of training data

We shall construct the output function such that it provides values within the interval [0,1] for each node of the output layer. We define a perfectly working grid after training as one that would produce a "1" for the correct category node (i.e. the expected digit) and "0" otherwise.

For error-back-propagation we need to compare the real results with the correct category values for the test data. To be able to do this we must build up a 2-dim array of the same shape as "_ay_a_Out" with correct output values for all test data of the batch. E.g.: If we expect the digit 7 for an input array of index idx within the set of training data, we need a 2-dim output array with the element [[0,0,0,0,0,0,0,1,0,0], idx].

By using Numpy's zero()-function and Pythons "enumerate()"-function we achieve exactly this for all data elements of the training data set. See the method "_encode_all_mnist_labels()". Thus, _ay_onehot.shape is expected to be (10, 60000). From this 2-dim array we can later cut out bunches of consecutive test data for mini-batches.

We print out the values for the first 12 elements of the input data set.

The array "_ay_oneval" is provided for convenience and print purposes: it provides the expected digit value in addition.

Tests and output via a Jupyter notebook

Let us test the reading of the input data and the label encoding with a Jupyter notebook. In previous articles I have described already how to do so.

I build a Jupyter notebook called "myANN" (in my present working directory "/projekte/GIT/ai/ml1/mynotebooks").

I start it with

myself@mytux:/projekte/GIT/ai/ml1> source bin/activate 
(ml1) myself@mytux:/projekte/GIT/ai/ml1> jupyter notebook
[I 15:07:30.953 NotebookApp] Writing notebook server cookie secret to /run/user/21001/jupyter/notebook_cookie_secret
[I 15:07:38.754 NotebookApp] jupyter_tensorboard extension loaded.
[I 15:07:38.754 NotebookApp] Serving notebooks from local directory: /projekte/GIT/ai/ml1
[I 15:07:38.754 NotebookApp] The Jupyter Notebook is running at:
[I 15:07:38.754 NotebookApp] http://localhost:8888/?token=06c2626c8724f65d1e3c4a50457da0d6db414f88a40c7baf
[I 15:07:38.755 NotebookApp] Use Control-C to stop this server and shut down all kernels (twice to skip confirmation).
[C 15:07:38.771 NotebookApp] 

and add two cells. The first one is for imports of libraries.

By the last line I import my present class code. With the second cell I start my class; the "__init__"-function is automatically executed:

The output is:

Note that the display of "_ay_onehot" shows the categories in vertical direction (rows) and the index for the input data element in horizontal direction (columns)! You see that correspondence of the labels in the enumerate structure correspond to the "1"s in the "_ay_onehot"-array.

Conclusion

Reading the MNIST dataset into Numpy arrays via Keras is simple. We have prepared an array "_ay_onehot"- which we shall use during optimization to calculate a difference between calculated output of the ANN and the expected category for an element of our training data set.

In the next article
A simple program for an ANN to cover the Mnist dataset – II
we shall define initial weights for our ANN.

Literature and links

Referenced Books
"Python machine Learning", Seb. Raschka, 2016, Packt Publishing, Birmingham, UK
"Machine Learning mit Sckit-Learn & TensorFlow", A. Geron, 2018, O'REILLY, dpunkt.verlag GmbH, Heidelberg, Deutschland

Links regarding cost (or loss) functions and logistic regression
https://towardsdatascience.com/introduction-to-logistic-regression-66248243c148
https://cmci.colorado.edu/classes/INFO-4604/files/slides-5_logistic.pdf
Wikipedia article on Loss functions for classification
https://towardsdatascience.com/optimization-loss-function-under-the-hood-part-ii-d20a239cde11
https://stackoverflow.com/questions/32986123/why-the-cost-function-of-logistic-regression-has-a-logarithmic-expression
https://medium.com/technology-nineleaps/logistic-regression-gradient-descent-optimization-part-1-ed320325a67e
https://blog.algorithmia.com/introduction-to-loss-functions/
uni leipzig on logistic regression