Mathematical Reinforcement to the Minibatch of Deep Learning

We elucidate a practical method in Deep Learning called the minibatch which is very useful to avoid local minima. The mathematical structure of this method is, however, a bit obscure. We emphasize that a certain condition, which is not explicitly stated in ordinary expositions, is essential for the minibatch method. We present a comprehensive description Deep Learning for non-experts with the mathematical reinforcement.


Introduction
Deep Learning is one of Machine Learning methods and it has attracted much attention recently.Leaning in general is divided into the supervised learning and unsupervised learning.In this paper we discuss only supervised learning.For general introduction to Deep Learning, see for example [1].The monographs [2] [3] [4] are standard textbooks.
Deep Learning is based on big data, so the supervised learning will give a heavy load to the computer.In order to alleviate the burden we usually use a practical method called the minibatch (a collection of randomly selected small data from big data, see Figure 5).Although the method is commonly used it is not widely understood why it is so effective from the mathematical viewpoint.
We point out in Theorem II that a certain condition, which is practically satisfied in ordinary applications, is essential for the effectiveness of the minibatch method.
In this paper we present a short and concise review of Deep Learning for non-experts and provide a mathematical reinforcement to the minibatch from the viewpoint of Linear Algebra.Theorem I and II in the text are our main results and some related problems are presented.
After reading this paper, readers are encouraged to tackle a remarkable paper [5] which is definitely a monumental achievement.

Simple Neural Network
As a general introduction to Neuron Dynamics see for example [6] [7].
For non-experts let us draw a neuron model based on the step function ( ) S x .
In Figure 1 the set { } , , , n w w w  is the weights of synaptic connections.The parameter θ is the threshold of the neuron. Here,

( )
z S x = is a simple step function defined by ( ) Since an algorithm called the backpropagation in Neural Network System uses derivatives of some activation functions, the simple step function ( ) S x is not suitable.Therefore, we usually use a function called the sigmoid function instead of the step function (see Figure 2).This function is used for a nonlinear compression of data.

K. Fujii
Definition The sigmoid function is defined by ( ) ( ) It is easy to see that the function satisfies and its derivatives are expressed as polynomials of the original function The graph is as follows.
Comment In Mathematics this is a famous function called the standard logistic function, while in Deep Learning this is called the sigmoid function.
A general logistic function ( ) , the overall height L, the parameter λ specifying the inverse width of the transient region and the location of the center 0 x : ( ) The standard logistic function is characterized by lecture note [8] as to why the function is so important.
Let us draw a neuron model based on the sigmoid function, See Figure 3.
The most important thing is to improve the set of weights of the synoptic connections { } for simplicity.This causes no problem for the present explanation.There is a suitable value of θ for each specific application.
For 1,2, , Input data : , , , , Teacher signal : where T denotes the transpose of a vector (or a matrix).
Figure 3. Simple neuron model modified.
Our method is to determine the weights of synaptic connections which match the output data with given teacher signals as much as possible.For the purpose we consider a square error like ( ) and determine ( ) We want to realize this purpose by repeated learning.For that we consider a discrete dynamical system.Namely, we set for ( ) The initial value ( ) is given appropriately (not so important).

Gradient Descent
For a general introduction to Gradient Descent see for example [9].
The gradient descent is performed by changing ( ) and we evaluate the error with the renewed weights ( ) ( ) 1 .

E E t = + w
This renewal of the weights is the learning process.This is very popular in Mathematics 1 .We repeat this procedure T steps which is large enough like = w for simplicity (t omitted).From (5) simple calculation gives and from (6) we obtain By substituting this into (5) we obtain and the square error becomes which is complicated enough.The renewal of the weights terminates when all ( ) , which is a critical point as shown in the next subsection.

Number of Critical Points
Here we evaluate the number of critical points to be encountered in the process.
First of all, let us show the definition of a critical point with a simple example.
Definition Let

( )
, , f f x y z = be a smooth function.Then a critical (or stationary) point ( ) ) x A critical point is not necessarily a point giving local minimum or local maximum.
We make an important assumption with respect to the input data.
are linearly independent.
The assumption tells that there is a set of integers Let us recall and try to find if E has a critical point.Suppose E has a critical point at the weights ( ) Since the determinant of coefficient matrix is non-zero from (9) we have ( ) 0 for 1, 2, , The result shows that there is no critical point in this process except for 0 E = .Therefore, when modifying ( ) k w t by the gradient descent (6) successively there is no danger of being trapped by points taking local minima.

Let us summarize the result :
Theorem I Under the assumption (8) there is only one global minimum Next, let us explain how to choose linearly independent data.We assume that n is huge and m is sufficiently smaller than n.
For m input data If m is not large, evaluation of the Gram's determinant is not so difficult.It is well-known that the input data are linearly independent if the Gram's determinant is non-zero (see for example [10]).
Let us give a short explanation for non-experts.For simplicity we consider the case of 3 m = and set , , , is linearly independent .
Therefore, if we choose m vectors from R n randomly they are almost all linearly independent.

Deep Neural Network
Deep Learning is the heart of Artificial Intelligence (AI) and we provide a concise review for non-experts.Deep Learning is a deep neural network consisting of the input layer (K components), the (multiplex) intermediate layers , , , M M N  components and the output layer (N components).See Figure 4 (in which ).Let us explain some notations.

Backpropagation
( ) is the input data.In this case, the weights of synaptic connections are represented by matrices , , U V W given by and our aim is educating these matrices.
The synaptic connections are expressed as A comment is in order.The flow of data is as follows: .
Now, if Let us consider (discrete) time evolution of the model.For 0,1,2, t = , we must determine and we expect if t is large enough (the end of Learning !). Assume now that the system is in the t-th Then the time evolution of the weights of synaptic connections is determined by the gradient descent: DOI: 10.4236/apm.2018.83016 which is a natural generalization of that of Section 2.1.
Before the calculation let us make a comment on the derivatives of a composite function.
Comment Let us explain with a simple example.For a composite function the derivative is given by ( A composite function is in general complicated, while its derivative is not so complicated.

Calculation
Let us perform the calculation of (14).
Since the determinant of coefficient matrix is non-zero from (16) we have Moreover, these equations are independent of 1,2, , i N =  , so that we finally obtain Let us emphasize that the assumption (16) is "local" because it refers to the linear independence of the data of the layer one step before the last of the flow: .
A question arises naturally.
Problem I What is the meaning of the assumption in the total flow of data ?
Last in this section let us comment on the minibatch of Deep Learning.When input data A is huge the calculation of time evolution of the weights of synaptic connections will give a heavy load to the computer.In order to alleviate it the minibatch is practical and very useful, see Figure 5. Then a question also arises naturally.
Problem II Is there a relation between A and T?
One may conjecture [ ] where [k] is the Gauss symbol, which is the greatest integer less than or equal to k.For example, [3.14] = 3.However, I do not believe this is correct.

Concluding Remarks
Nowadays, studying Deep Learning is standard for university students in the world.However, it is not so easy for them because Deep Learning requires a lot of knowledge.
In this paper, we treated the minibatch of Deep Learning and gave the mathematical reinforcement to it from the viewpoint of Linear Algebra and presented some related problems.We expect that young researchers will solve the problems in the near future.
As a related topic of Artificial Intelligence, there is Information Retrieval.
hard learning.For the purpose we prepare m input data and teacher signals.In the following we assume m n for simplicity.The equation can be written as DOI: 10.4236/apm.
are linearly dependent.For example, we can set Short calculation after substituting this into the Gram's determinant above gives R a set of all n m × matrices over R and denote ( )

First, let us
show a figure of deep neural network 2 (the figure is short in intermediate layers).

2 Figure 4
Figure 4 is of course not perfect because drawing a figure of a big size is not easy.