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

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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 – II – initial random weight values
A simple program for an ANN to cover the Mnist dataset – I – a starting point

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',
                 
n                 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 28×28 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

A simple program for an ANN to cover the Mnist dataset – IV – the concept of a cost or loss function

we shall start working on cost and gradient calculation.

 

Linux, OpenBlas and Numpy matrix multiplications – avoid using all processor cores

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Recently, I tested the propagation methods of a small Python3/Numpy class for a multilayer perceptron [MLP]. I unexpectedly ran into a performance problem with OpenBlas.

The problem had to do with the required vectorized matrix operations for forward propagation – in my case through an artificial neural network [ANN] with 4 layers. In a first approach I used 784, 100, 50, 10 neurons in 4 consecutive layers of the MLP. The weight matrices had corresponding dimensions.

The performance problem was caused by extensive multi-threading; it showed a strong dependency on mini-batch sizes and on basic matrix dimensions related to the neuron numbers per layer:

  • For the given relatively small number of neurons per layer and for mini-batch sizes above a critical value (N > 255) OpenBlas suddenly occupied all processor cores with a 100% work load. This had a disastrous impact on performance.
  • For neuron numbers as 784, 300, 140, 10 OpenBlas used all processor cores with a 100% work load right from the beginning, i.e. even for small batch sizes. With a seemingly bad performance over all batch sizes – but decreasing somewhat with large batch sizes.

This problem has been discussed elsewhere with respect to the matrix dimensions relevant for the core multiplication and summation operations – i.e. the neuron numbers per layer. However, the vectorizing aspect of matrix multiplications is interesting, too:

One can imagine that splitting the operations for multiple independent test samples is in principle ideal for multi-threading. So, using as many processor cores as possible (in my case 8) does not look like a wrong decision of OpenBlas at first.

Then I noticed that for mini-batch sizes “N” below a certain number (N < 250) the system only seemed to use up to 3-4 cores; so there remained plenty of CPU capacity left for other tasks. Performance for N < 250 was better by at least a factor of 2 compared to a situation with an only slightly bigger batch size (N ≥ 260). I got the impression that OpenBLAS under certain conditions just decides to use as many threads as possible – which no good outcome.

In the last years I sometimes had to work with optimizing multi-threaded database operations on Linux systems. I often got the impression that you have to be careful and leave some CPU resources free for other tasks and to avoid heavy context switching. In addition bottlenecks appeared due to the concurrent access of may processes to the CPU cache. (RAM limitations were an additional factor; but this should not be the case for my Python program.) Furthermore, one should not forget that Python/Numpy experiments on Jupyter notebooks require additional resources to handle the web page output and page update on the browser. And Linux itself also requires some free resources.

So, I wanted to find out whether reducing the number of threads – or available cores – for Numpy and OpenBlas would be helpful in the sense of an overall gain in performance.

All data shown below were gathered on a desktop system with some background activity due to several open browsers, clementine and pulse-audio as active audio components, an open mail client (kontact), an open LXC container, open Eclipse with Pydev and open ssh connections. Program tests were performed with the help of Jupyter notebooks. Typical background CPU consumption looks like this on Ksysguard:

Most of the consumption is due to audio. Small spikes on one CPU core due to the investigation of incoming mails were possible – but always below 20%.

Basics

One of
the core ingredients to get an ANN running are matrix operations. More precisely: multiplications of 2-dim Numpy matrices (weight matrices) with input vectors. The dimensions of the weight matrices reflect the node-numbers of consecutive ANN-layers. The dimension of the input vector depends on the node number of the lower of two neighbor layers.

However, we perform such matrix operations NOT sequentially sample for sample of a collection of training data – we do it vectorized for so called mini-batches consisting of between 50 and 600000 individual samples of training data. Instead of operating with a matrix on just one feature vector of one training sample we use matrix multiplications whereby the second matrix often comprises many vectors of data samples.

I have described such multiplications already in a previous blog article; see Numpy matrix multiplication for layers of simple feed forward ANNs.

In the most simple case of an MLP with e.g.

  • an input layer of 784 nodes (suitable for the MNIST dataset),
  • one hidden layer with 100 nodes,
  • another hidden layer with 50 nodes
  • and an output layer of 10 nodes (fitting again the MNIST dataset)

and “mini”-batches of different sizes (between 20 and 20000). An input vector to the first hidden layer has a dimension of 100, so the weight matrix creating this input vector from the “output” of the MLP’s input layer has a shape of 784×100. Multiplication and summation in this case is done over the dimension covering 784 features. When we work with mini-batches we want to do these operations in parallel for as many elements of a mini-batch as possible.

All in all we have to perform 3 matrix operations

(784×100) matrix on (784)-vector, (100×50) matrix on (100)-vector, (50×10) matrix on (50) vector

on our example ANN with 4 layers. However, we collect the data for N mini-batch samples in an array. This leads to Numpy matrix multiplications of the kind

(784×100) matrix on an (784, N)-array, (100×50) matrix on an (100, N)-array, (50×10) matrix on an (50, N)-array.

Thus, we deal with matrix multiplications of two 2-dim matrices. Linear algebra libraries should optimize such operations for different kinds of processors.

The reaction of OpenBlas to an MLP with 4 layers comprising 784, 100, 50, 10 nodes

On my Linux system Python/Numpy use the openblas-library. This is confirmed by the output of command “np.__config__.show()”:

openblas_info:
    libraries = ['openblas', 'openblas']
    library_dirs = ['/usr/local/lib']
    language = c
    define_macros = [('HAVE_CBLAS', None)]
blas_opt_info:
    libraries = ['openblas', 'openblas']
    library_dirs = ['/usr/local/lib']
    language = c
    define_macros = [('HAVE_CBLAS', None)]
openblas_lapack_info:
    libraries = ['openblas', 'openblas']
    library_dirs = ['/usr/local/lib']
    language = c
    define_macros = [('HAVE_CBLAS', None)]
lapack_opt_info:
    libraries = ['openblas', 'openblas']
    library_dirs = ['/usr/local/lib']
    language = c
    define_macros = [('HAVE_CBLAS', None)]

and by 

(ml1) myself@mytux:/projekte/GIT/ai/ml1/lib64/python3.6/site-packages/numpy/core> ldd  _multiarray_umath.cpython-36m-x86_64-linux-gnu.so
        linux-vdso.so.1 (0x00007ffe8bddf000)
        libopenblasp-r0-2ecf47d5.3.7.dev.so => /projekte/GIT/ai/ml1/lib/python3.6/site-packages/numpy/core/./../.libs/libopenblasp-r0-2ecf47d5.3.7.dev.so (0x00007fdd9d15f000)
        libm.so.6 => /lib64/libm.so.6 (0x00007fdd9ce27000)
        libpthread.so.0 => /lib64/libpthread.so.0 (0x00007fdd9cc09000)
        libc.so.6 => /lib64/libc.so.6 (0x00007fdd9c84f000)
        /lib64/ld-
linux-x86-64.so.2 (0x00007fdd9f4e8000)
        libgfortran-ed201abd.so.3.0.0 => /projekte/GIT/ai/ml1/lib/python3.6/site-packages/numpy/core/./../.libs/libgfortran-ed201abd.so.3.0.0 (0x00007fdd9c555000)

In all tests discussed below I performed a series of calculations for different batch sizes

N = 50, 100, 200, 250, 260, 500, 2000, 10000, 20000

and repeated the full forward propagation 30 times (corresponding to 30 epochs in a full training series – but here without cost calculation and weight adjustment. I just did forward propagation.)

In a first experiment, I did not artificially limit the number of cores to be used. Measured response times in seconds are indicated in the following plot:

Runtime for a free number of cores to use and different batch-sizes N

We see that something dramatic happens between a batch size of 250 and 260. Below you see the plots for CPU core consumption for N=50, N=200, N=250, N=260 and N=2000.

N=50:

N=200:

N=250:

N=260:

N=2000:

The plots indicate that everything goes well up to N=250. Up to this point around 4 cores are used – leaving 4 cores relatively free. After N=260 OpenBlas decides to use all 8 cores with a load of 100% – and performance suffers by more than a factor of 2.

This result support the idea to look for an optimum of the number of cores “C” to use.

The reaction of OpenBlas to an MLP with layers comprising 784, 300, 140, 10 nodes

For a MLP with neuron numbers (784, 300, 140, 10) I got the red curve for response time in the plot below. The second curve shows what performance is possible with just using 4 cores:

Note the significantly higher response times. We also see again that something strange happens at the change of the batch-size from 250 to 260.

The 100% CPU
consumption even for a batch-size of only 50 is shown below:

Though different from the first test case also these plots indicate that – somewhat paradoxically – reducing the number of CPU cores available to OpenBlas could have a performance enhancing effect.

Limiting the number of available cores to OpenBlas

A bit of Internet research shows that one can limit the number of cores to use by OpenBlas e.g. via an environment variable for the shell, in which we start a Jupyter notebook. The relevant command to limit the number of cores “C” to 3 is :

export OPENBLAS_NUM_THREADS=3

Below you find plots for the response times required for the batch sizes N listed above and core numbers of

C=1, C=2, C=3, C=4, C=5, C=6, C=7, C=8 :

For C=5 I did 2 different runs; the different results for C=5 show that the system reacts rather sensitively. It changes its behavior for larger core number drastically.

We also find an overall minimum of the response time:
The overall optimum occurs for 400 < N < 500 for C=1, 2, 3, 4 – with the minimum region being broadest for C=3. The absolute minimum is reached on my CPU for C=4.

We understand from the plots above that the number of cores to use become hyper-parameters for the tuning of the performance of ANNs – at least as long as a standard multicore-CPU is used.

CPU-consumption

CPU consumption for N=50 and C=2 looks like:

For comparison see the CPU consumption for N=20000 and C=4:

CPU consumption for N=20000 and C=6:

We see that between C=5 and C=6 CPU resources get heavily consumed; there are almost no reserves left in the Linux system for C ≥ 6.

Dependency on the size of the weight-matrices and the node numbers

For a full view on the situation I also looked at the response time variation with node numbers for a given number of CPU cores.

For C=4 and node number cases

  • 784, 300, 140, 10
  • 784, 200, 100, 10
  • 784, 100, 50, 10
  • 784, 50, 20, 10

I got the following results:

There is some broad variation with the weight-matrix size; the bigger the weight-matrix the longer the calculation time. This is, of course, to be expected. Note that the variation with the batch-size number is relatively smooth – with an optimum around 400.

Now, look at the same plot for C=6:

Note that the response time is significantly bigger in all cases compared to the previous situation with C=4. In cases of a large matrix by around 36% for N=2000. Also the variation with batch-size is more pronounced.

Still, even with 6 cores you do not get factors between 1.4 and 2.0 as compared to the case of C=8 (see above)!

Conclusion

As I do not know what the authors of OpenBlas are doing exactly, I refrain from technically understanding and interpreting the causes of the data shown above.

However, some consequences seem to be clear:

  • It is a bad idea to provide all CPU cores to OpenBlas – which unfortunately is the default.
  • The data above indicate that using only 4 out of 8 core threads is reasonable to get an optimum performance for vectorized matrix multiplications on my CPU.
  • Not leaving at least 2 CPU cores free for other tasks is punished by significant performance losses.
  • When leaving the decision of how many cores to use to OpenBlas a critical batch-size may exist for which the performance suddenly breaks down due to heavy multi-threading.

Whenever you deal with ANN or MLP simulations on a standard CPU (not GPU!) you should absolutely care about how many cores and related threads you want to offer to OpenBlas. As far as I understood from some Internet articles the number of cores to be used can be not only be controlled by Linux (shell) environment variables but also by os-commands in a Python program. You should perform tests to find optimum values for your CPU.

Links

stackoverflow: numpy-suddenly-uses-all-cpus

stackoverflow: run-openblas-on-multicore

stackoverflow: multiprocessing-pool-makes-numpy-matrix-multiplication-slower

scicomp: why-isnt-my-matrix-vector-multiplication-scaling/1729

Setting the number of threads via Python
stackoverflow:
set-max-number-of-threads-at-runtime-on-numpy-openblas
codereview.stackexchange: better-way-to-set-number-of-threads-used-by-numpy

 

Numpy matrix multiplication for layers of simple feed forward ANNs

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I move a small step forward on my trials as a beginner in machine learning. This is once again going to be an article from which experts on “Artificial Neural networks” [ANNs] will learn nothing new. But some of my readers may also be newbies both to Python, to numpy and to ANNs. For them this article may be helpful.

I shall have a brief look at some matrix (or array) operations which constitute important parts of the information propagation on ANNs. How would such operations look like in Python with Numpy? One should have a clear understanding of the so called “dot”-operation between multidimensional arrays in this context. Such operations can be performed highly optimized on GPUs – and whole sets of data samples can be handled in form of vectorized code instructions. I will try to explain this in form of matrix terms required to simulate information transport between some layers of a very simple network.

Propagation operations on ANNs

Basic Artificial Neural Networks consist of layers with neurons. Each layer has a defined number of neurons. Each neuron is connected with all neurons of the next layer (receiving neurons). The output “O(k, L)” of a neuron N(k,L) of Layer “L” to a neuron N(m,L+1) is multiplied by some “weight” W[(m, L+1),(k, L),].

The contributions of all neurons on layer “L” to N(m, L+1) are summed up. The result is then transformed by some local “activation” function and afterwards presented by N(m, L+1) as output which shall be transported to the neurons of layer L+2. The succession of such operations is called “propagation“.

At the core of propagation we, therefore, find operations which we can express in the following form :

Input vector at N(m, L+1) : I(m, L+1) = SUM(over_all_k) [ W[(m, L+1), (k, L)] * O(k, L) ]
Output vector at N(m, L+1) : O(m, L+1) = f_act( I(m, L+1) )

To make things simpler: Let us set f_act = 1. Then we get

O(m, L+1) = SUM(over all k) [ W[(m, L+1), (k,N)] * O(k, L) ]

The SUM is basically a matrix operations on vector O(k, L). If output vectors were always given by singular values in “k” rows (!) then we could write

O(L+1) = W(L, L+1) * O(L) )

Batches and numerical optimization of the matrix operations for propagation

In the past I have programmed gas flow simulations in exploding stars in Fortran/C and simulated cascaded production networks for Supply Chains in PHP. There, you work with deep and complicated layer structures, too. Therefore I felt well prepared for writing a small program to set up a simple artificial neural network with some hidden layers.

Something, you will find in literature (e.g. in the book of S. Rashka, “Python machine Learning”, 2016, PACKT Publishing] are remarks on batch based training. Rashka e.g. rightfully claims in his discussion of an example code that the use of mini-batches of input data sets is advantageous (see chapter 12, page 379 in the book mentioned above):

“Mini-batch learning has the advantage over online learning that we can make use of our vectorized implementations to improve computational efficiency.”

I swallowed this remark first without much thinking. As a physicist I had used highly optimized libraries for Linear Algebra operation on supercomputers with vector registers already 35 years ago. Ok, numpy can perform a matrix operation on some vector very fast, especially on GPUs – so what?

But when I tried to follow the basic line of programming of
Rashka for an ANN in Python, I suddenly found that I did not fully understand what the code was doing whilst handling batches of input data sets. The reason was that I had not at all grasped the full implications of the quoted remark. It indicated not just the acceleration of a matrix operation on one input vector; no, instead, it indicated that we could and should perform matrix operations in parallel on all input vectors given by a full series of many different data sets in a mini-batch collection.

So, we then want to perform an optimized matrix operation on another matrix of at least 2 dimensions. When you think about it, you will quickly understand that any kind of operation of one n-dim matrix on another matrix with more and different dimensions must be defined very precisely to avoid misunderstandings and miscalculations.

To make things simple:
Let us take a (3,5)-matrix A and a (10,3)-matrix B. What should a “dot“-like operation A * B mean in this case? Would it be suitable at all, regarding the matrix element arrangements in rows and columns?

I studied the numpy documentation on “numpy.dot(A, B)”. See e.g. here. Ooops, not fully trivial: “If A is an N-D array and B is an M-D array (where M>=2), it is a sum product over the last axis of A and the second-to-last axis of B”. The matrices must obviously be arranged in a suitable way ….

So, I decided to performing some simple matrix experiments to get a clear understanding. I assumed a fictitious ANN of three layers and some direct propagation between the layers. What does propagation mean in such a case, how is it expressed in matrix operation terms in Python and what does batch-handling mean in this context?

Three simple layers and a batch of input data sets

The following graphics shows my simple network:

Our batch has 10 data sets of test data for our ANN. Each data set describes 3 properties; thus the input layer “L1” has 3 nodes. Our “hidden” layer “L2” shall have 5 nodes and our output layer “L3” only 2 nodes.

Weight matrices and propagation operations

Now, let us define some simple matrices with input vectors for transportation and weight-matrices for the layer-connections with Python. All activation functions shall in our simple example just perform a multiplication by 1.

The input to layer “L1” – i.e. a batch of 10 data sets – shall be given by a matrix of 10 lines (dimension 1) and 3 columns (dimension 2) 

import numpy as np
import scipy
import time 

A1 = np.ones((10,3))
A1[:,1] *= 2
A1[:,2] *= 4
print("\nMatrix of input vectors to layer L1:\n")
print(A1)


print("\nMatrix of input vectors to layer 1:\n")
print(A1)

Matrix of input vectors to layer 1:

Matrix of input vectors to layer L1:

[[1. 2. 4.]
 [1. 2. 4.]
 [1. 2. 4.]
 [1. 2. 4.]
 [1. 2. 4.]
 [1. 2. 4.]
 [1. 2. 4.]
 [1. 2. 4.]
 [1. 2. 4.]
 [1. 2. 4.]]

We now define the weight matrix “W1” for transport between layer L1 and layer L2 as: 

W1= np.random.randint(1, 10, 5*3)
W1 = w1.reshape(5,3)
print("\nMatrix of weights W1(L1, L2) :\n")
print(W1)

Matrix of weights W1(L1, L2) :

[[3 8 5]
 [4 9 7]
 [4 1 3]
 [8 8 4]
 [3 1 9]]

We set the first dimension to the number of nodes in the target layer (L2) and the second dimension to
the number of nodes in the lower preceding layer (here: L1). Expressed in the shape notation for an array “(dimension 1, dimension 2, …) “:

Shape of W : (dim 1, dim 2) = (node number of target layer, node number of preceding layer)

If we now tried a numpy “dot”-operation for our matrices 

A2 = np.dot(W1, A1)

we would get an error:

ValueError: shapes (5,3) and (10,3) not aligned: 3 (dim 1) != 10 (dim 0)

Well, this is actually consistent with the numpy documentation: The last dimension of “W1” should be consistent with the second to last (=first) dimension of “strong>A1“.

What we need to do here is to transpose the matrix of our input vectors for our multiple data sets: 

A1 = np.transpose(A1)
print("\nMatrix of output vectors of layer 1 to layer 2:\n")
print(A1)

Matrix of output vectors of layer 1 to layer 2:

[[1. 1. 1. 1. 1. 1. 1. 1. 1. 1.]
 [2. 2. 2. 2. 2. 2. 2. 2. 2. 2.]
 [4. 4. 4. 4. 4. 4. 4. 4. 4. 4.]]

Then 

A2 = np.dot(W1, A1)
print("\n\nA2:\n")
print(A2)

A2:

[[39. 39. 39. 39. 39. 39. 39. 39. 39. 39.]
 [50. 50. 50. 50. 50. 50. 50. 50. 50. 50.]
 [18. 18. 18. 18. 18. 18. 18. 18. 18. 18.]
 [40. 40. 40. 40. 40. 40. 40. 40. 40. 40.]
 [41. 41. 41. 41. 41. 41. 41. 41. 41. 41.]]

Now, we need a weight matrix W2 between layer L1 and layer L2: 

W2= np.random.randint(1, 3, 2*5)
W2 = W2.reshape(2,5)
print("\n\nW2:\n")
print(W2)

W2:
[[2 2 1 2 2]
 [2 2 2 2 1]]

We followed the same policy – and used the number of nodes in target layer L2 as the first dimension of the weight array.

Now, do we have to to anything with array A2, if we want to use it as an input for the next matrix operation ? 

A3 = np.dot(W2, A2)

No, we do not need to change anything! “A2” has the required form, already. The second to last dimension in “A2” is 5 in our example – as is the last dimension in “W2“!

So, the following will work directly : 

A3 = np.dot(W2, A2)
print("\n\nA3:\n")
print(A3)

A3:

[[358. 358. 358. 358. 358. 358. 358. 358. 358. 358.]
 [335. 335. 335. 335. 335. 335. 335. 335. 335. 335.]]

You can try out our complete “propagation” code in a Jupyter notebook cell. 

import numpy as np
import scipy
import time 

A1 = np.ones((10,3))
A1[:,1] *= 2
A1[:,2] *= 4
print("\nMatrix of input vectors to layer L1:\n")
print(A1)


W1= np.random.randint(1, 10, 5*3)
W1 = W1.reshape(5,3)
print("\nMatrix of weights W1(L1, L2) :\n")
print(W1)

A1 = np.transpose(A1)
print("\nMatrix of output vectors of layer L1 to layer L2:\n")
print(A1)

A2 = np.dot(W1, A1)
print("\n\nA2:\n")
print(A2)

W2= np.random.randint(1, 3, 2*5)
W2 = W2.reshape(2,5)
print("\n\nW2:\n")
print(W2)

A3 = np.dot(W2, A2)
print("\n\nA3:\n")
print(A3)

Conclusion

The first thing we learned is that matrix operations on simple vectors can be extended to a set of vectors, i.e. a matrix. In the case of ANNs one dimension of such an (numpy) array would cover the number of data sets of a typical input mini-batch. The other dimension would cover the “properties” of the input data.

The second thing we saw is that the input matrix to an ANN often must be transposed to work together with weight matrices, if such matrices follow the policy that the first dimension is given by the number of cells in the target layer and the second dimension by the number of nodes of the
preceding layer.

The weighted input to the first target layer is then given by a matrix “dot”-operation on the transposed input matrix. The outcome is a matrix where the first dimension is defined by the number of nodes of the target layer and the second dimension the number of data sets. This has already the correct form for a further propagation to the next layer. Note that the application of an activation function to each of the matrix elements would not change the required arrangement of matrix data!

After the input layer we, therefore, can just continue to directly apply weight matrices to output matrices by a numpy-“dot”-operation. The weight matrix structure just should reflect our shape policy of

(dim1, dim2) = (node number of target layer, node number of preceding layer)

.

In the next articles we shall use these insights to build a Python class for the setup and training of a simple ANN with one or two hidden layers for the analysis of the famous MNIST dataset.