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A diagram of the signals in the network when learning the back propagation algorithm is shown in the following figure:

All training images are alternately presented to the input of the network, so that it does not, figuratively speaking, forget some as it remembers others. Usually this is done in random order, but because our data will be placed in matrices and computed as a whole as a single set, in our implementation we will introduce another element of randomness.

The use of matrices means that the weights of all layers, as well as the input and target training data, will be represented by matrices, and the above formulas, and thus the algorithms, will take a matrix form. In other words, we will not be able to operate with separate vectors of input and training data, and the entire cycle from steps 2 to 7 will be calculated for the entire data set at once. One such cycle is called a learning epoch.

Implementing a neural network in a MatrixNet class:

The number n does not include the input layer because it does not require weights.

Training of NS is performed on the training data available at the moment. Based on this data the network configuration (number of layers, number of neurons per layers, etc.), learning rate and other characteristics are chosen. Therefore, in principle, it is always possible to design a network powerful enough to give any sufficiently small error on training data. However, the main strength of NS and the purpose of using NS is that it works well on future unknown data (with the same implicit dependencies as the training set). The effect when a trained NS adjusts too well to the training data, but fails the "forward test" is called overtraining, and should be eradicated in every possible way. For this purpose, the so-called regularization is used - adding some additional conditions to the model or training method that evaluate the generalization ability of the network. There are many different ways of regularization, in particular: analyzing the performance of the trained network on an additional validation dataset (different from the training one); randomly discarding of part of neurons or connections during training; pruning the network after the training; introducing noise into the input data; Artificial data multiplication; Weak constant reduction of the amplitude of weights during training; experimental selection of the volume and configuration of the network on a fine-grained edge, when the network is still capable of learning but no longer retrains on the available data; We implement some of these in our class. To begin with, we will provide for passing not only input and output training data (data and target parameters, respectively), but also a validation set (it also consists of input and paired output vectors: validation and check) to the training method. As the training progresses, network error on training data, as a rule, decreases monotonically enough (the phrase "as a rule" is used, because if the wrong choice of training speed or network capacity is made, the process can go unstable). However, if we count the network error on the validation set in parallel, it will also decrease at first (while the network identifies the most important regularities in the data), and then it will start to grow as it overtrains (when the network adapts already to the specific features of the training sample, but not the validation sample). Thus, the learning process should be stopped when the validation error begins to grow. This approach is called "early stopping". In addition to the two data sets, the train method allows us to set the maximum number of epochs (epochs), the desired accuracy (i.e., the average minimum error, which is enough for us: in this case, training also ends with a success sign), and the method of error calculation (lf). The training speed is set equal to the accuracy, but in order to increase the flexibility of the settings, they can be separated if necessary. This is done because further on we will support automatic speed adjustment, and the initial approximate value is not so important.

First of all, we check that the error value is a valid number. Otherwise there is an overflow in the network or invalid data is fed to the input.

If the error continues to decrease or, at least, does not increase, remember the new error values for comparison in the next epoch. If the error reaches the required accuracy, we consider the training completed and also exit the cycle.

In addition, the loop calls a virtual progress method, which can be overridden in derived network classes, and with it interrupt learning in response to some user action. I will show the standard implementation of progress below.

Finally, if the loop was not interrupted by any of the above conditions, run a backProp of the network error.

The default progress method logs the training values once per second.

The return value true continues learning, while false will cause the loop to terminate. In addition to "early stop", the MatrixNet class knows how to randomly "dropout" some connections a la "dropout". The canonical method of "dropout" involves temporarily excluding randomly selected neurons from the network. However, we cannot implement this without large overhead, because the algorithm uses matrix operations. To exclude neurons from the layer, we would need to reformat the weight matrices and partially copy them at each iteration. It is much easier and more efficient to set random weights to 0, i.e. to break connections. Of course, at the beginning of each epoch, the program should restore temporarily disconnected weights to their previous state, and then select some new ones to disconnect in the next epoch. The number of ties that are temporarily zeroed is set using the enableDropOut method as a percentage of the total number of weights in the network. By default, the dropOutRate variable is 0, and the mode is disabled.

The principle of "dropout" consists in saving the current state of weight matrices in some additional storage (the DropOutState class implements it) and zeroing out randomly chosen connections of the network. After training the network in the obtained modified form for one epoch, the zeroed matrix elements are restored from the storage, and the procedure is repeated: other random weights are selected and zeroed, the network is trained with them, and so on.

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