Some time ago a small system for digital identification was developed based on the BP neural network algorithm and interacted with MFC. In the process of implementation, we also try to find some source code. Generally speaking, the portability of these source code is not good. Most of the interactive parts and the core algorithm code are mixed together, so that not only the code reading difficulties, but also the core algorithm is important. Not portable. The importance of design patterns and design patterns! Therefore, the core algorithm of the BP neural network is implemented in standard C++, so the portability is guaranteed, and then the interaction between different GUI libraries based on the core algorithm (MFC, QT) can quickly build a good system. The following describes the principle of the BP algorithm (see "digital image processing and machine vision" is very suitable for engineering partners), while giving the code to achieve, and finally based on the core algorithm to build interactive examples.
The theoretical basis of artificial neural network
Sensor
The sensor is an artificial neuron with two simple outputs, as shown in the figure below.
2. Linear unit
Only the perceptual outputs of the 1 and -1 outputs actually limit their ability to process and sort. The following figure shows a simple generalization, namely a perceptron without a threshold.
3. Error criteria
A commonly used error metric, the square error criterion, is used. The formula is as follows.
Where D is the training sample, td is the training output of the training observation d, and ot is the actual observation of the observation d. If it's a convex function (that's fine, it's nice to hear the convex function, huh, huh!), but you can still use the gradient descent method to find its parameter w.
4. Gradient decline derivation
In higher mathematics, the concept of a gradient is actually a direction vector, which is the direction in which the directional derivative is the greatest, that is, along this direction, the value of the function changes fastest. Here we do a gradient descent, then we update the value of the parameter w in the negative direction of the gradient to quickly reach the minimum value of the E function. The steps of this gradient descent algorithm are basically as follows:
1) Initialize the parameter w (random, or other method).
2) Find the gradient.
3) Update the parameter w along the gradient direction to add a learning rate, that is, how much the step is down.
4) Repeat 1), 2), 3) until the set condition is reached (the number of iterations, or the reduction of E is less than a certain threshold).
The gradient expression is as follows:
So how do you get gradients? This is the process of deriving a complex function as follows:
Where xid is the input component xi corresponding to the dth observation in the sample. In this way, the updated expression of the parameter w in the training process is as follows (in which a learning rate is added, that is, the decreasing step):
So the update increment of the parameter wi is:
For the problem of the selection of the learning rate, the smaller one is usually able to guarantee convergence, see the figure below.
5. Incremental gradient drop
The disadvantage of the gradient descent algorithm in 4 is that sometimes the convergence speed is slow. If there are multiple local minimum values ​​on the error surface, the algorithm cannot guarantee that the global minimum can be found. In order to improve these disadvantages, an incremental gradient descent algorithm is proposed. Incremental gradient descent, which differs from gradient descent in 4, is that the updating of parameter w in 4 is calculated based on the error of the observations in the entire sample, whereas the incremental gradient descent algorithm is based on the single observation in the sample. The value of the error to calculate the update of w.
6. Gradient test
This is a more practical content. How to determine your own code is certainly not wrong? Because it is very easy to make mistakes when we are looking for a gradient, I have committed it, er, tune it out for two days and found out that an array is wrong in the table below. If you look earlier at Stanford University's deep learning basic tutorial, Now, this is just a part of the screenshot, and have time to take a closer look.
Multilayer neural network
Well, with the previous foundation, we can now conduct actual combat and construct a multilayer neural network.
Sigmoid neurons
Sigmoid neurons can be represented by the following diagram:
2. Neural network layer
A three-layer BP neural network can be represented by the following figure:
3. Standard C++ definitions of neurons and neural network layers
As can be seen from the schematic diagram of the three-layer BP neural network in 2, the hidden layer and the output layer have a similar structure. The neuron and neural network layers are defined as follows:
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