## 1 Introduction

Traditionally, an artificial neural network (ANN) is trained very slowly by a gradient descent algorithm such as the backpropagation algorithm [1-3] since a large number of hyperparameters of the ANN need to be fine-tuned with a larger number of training epochs. In particular, a deep neural network [4-9], such as a convolutional neural network (CNN), typically takes a long time to be trained well. Other intelligent training algorithms use various advanced optimization methods such as genetic algorithms [10-17], particle swarm optimization methods [18], and annealing algorithms [19] to try to find optimal hyperparameters of an ANN. However, these commonly used training algorithms take very long training time. An important research goal is to develop a new ANN with high computation speed and high performance, such as low validation errors, for various machine learning applications especially involving big data mining and real-time computation.

Neural network structure optimization algorithms also take a lot of time to try to find optimal or near-optimal numbers of different layers and numbers of neurons on different layers for big data mining problems. Especially, deep neural networks need much longer time. Thus, it is useful to develop fast shallow neural networks with relatively small numbers of neurons on different layers. We created a novel shallow 4-layer ANN with high-speed hyperparameter optimization. Training data are partitioned into local

-dimensional subspaces. Local shallow 4-layer ANNs are trained by using the partitioned data sets in the local -dimensional subspaces. This divide-and-conquer approach can optimize the local ANNs with simpler nonlinear functions more easily using smaller data sets in the local subspaces. Based on positive preliminary simulation results, we will continue to develop more advanced optimization algorithms to optimize both partitioned subspaces and hyperparameters to build fast and effective ANNs.## 2 Pairwise Neural Network (PairNet)

We propose a novel shallow ANN called “Pairwise Neural Network” (PairNet) that consists of only four layers of neurons to map inputs on the first layer to one output on the fourth layer.

Layer 1: Layer 1 has neuron pairs to map inputs to

outputs. Each pair has two neurons where one neuron has an increasing activation function

that generates a positive normalized value, and the other neuron has a decreasing activation function that generates a negative normalized value for .Layer 2: Layer 2 consists of neurons, where each neuron has an activation function to map inputs to an output as a complementary decision fusion. Each of the inputs is an output of one of the two neurons of each neuron pair on Layer 1. Let denote , and denote for . The activation functions of neurons on Layer 2 are given as , …, , , where are hyperparameters to be optimized for , , and . . For a special case, the weights () are equal, so for .

Layer 3: Layer 3 also consists of neurons but transforms the outputs of the second layer to individual output decisions. for , for , where . (activation functions) are defined as , where .

Layer 4: Layer 4 generates a final nonlinear output .

## 3 Fast Training Algorithm with Hyperparameter Optimization on Partitioned Subspaces

A data set has data, where each data consists of inputs for , and one output . An input has intervals in such that , , …, , and for , and . Then there are () -dimensional subspaces for . data are distributed in the -dimensional subspaces. A -dimensional subspace has data with outputs for , , and . For each -dimensional subspace such as (, , …, , and ), a PairNet can map inputs for to one output for . Thus, a local PairNet is built using all of the data points in a local -dimensional subspace. This divide-and-conquer approach can train the local PairNet using specific local data features to improve model performance.

For a regression problem, Layer 4 of a PairNet calculates a final output decision by computing a weighted average of the individual output decisions of Layer 3. The final ouptut is generated by a nonlinear function , , where . Finally, , where for .

The objective optimization function for a PairNet for is given below,

(1) |

After setting and , we have linear equations with hyperparameters ( and ) for as follows:

(2) |

The above system of linear equations (2) can be quickly solved to find optimal hyperparameters ( and ) for . Each subspace must have at least data points.
A new PairNet model selection algorithm is given in Algorithm 1.

## 4 Simulation Results

To compare an ANN and the PairNet, three different simulations using three different functions are done. The first 3-input-1-output benchmark function [20-23] is given below:

(3) |

The second 3-input-1-output function is given below:

(4) |

The third 3-input-1-output function is given below:

(5) |

Three training data sets (each with training data) are generated by the three functions shown in equations (3), (4), and (5) such that , , , where the operator is used, , , , and . Three testing data sets (each with testing data) are generated by the three functions shown in equations (3), (4), and (5) such that , , , where the operator is used, .

For simulations, the best 20-layer ANN was selected from five random 20-layer ANNs using ReLU (500 epochs), and the best PairNet was selected from five random PairNets with random

for and random 3-dimensional subspaces. Results shown in Table 1 indicated that the PairNets outperformed traditional ANNs in terms of speed and testing mean squared errors (MSEs).Method | Function | (sec) | ||
---|---|---|---|---|

PairNet | 3.06 | 0.191 | 0.225 | |

ANN | 199.3 | 0.022 | 0.249 | |

PairNet | 3.01 | 0.00075 | 0.00227 | |

ANN | 192.6 | 0.04214 | 0.02510 | |

PairNet | 2.84 | 10.930 | 7.7513 | |

ANN | 277.6 | 86.861 | 66.798 |

Simulation results for , , and shown in Table 2 indicated that the more number of different partitioned subspaces, the better a PairNet tended to perform in terms of training MSE () and testing MSE () in most cases. In addition, a PairNet with more subspaces is not always better than that with fewer subspaces. An important future work is to develop a new high-speed optimization algorithm to find both best partitioned subspaces and optimal hyperparameters for building the best PairNet.

Partitions (--) | Subspaces | ||||||
---|---|---|---|---|---|---|---|

2-2-2 | 8 | 1.926 | 0.940 | 0.1713 | 0.1325 | 258.0 | 148.4 |

2-3-4 | 24 | 0.857 | 0.673 | 0.1091 | 0.0903 | 224.3 | 132.4 |

3-3-3 | 27 | 0.939 | 0.606 | 0.0348 | 0.0302 | 78.30 | 46.39 |

3-4-5 | 60 | 0.444 | 0.624 | 0.0253 | 0.0242 | 82.60 | 44.47 |

4-4-4 | 64 | 0.534 | 0.702 | 0.0111 | 0.0160 | 37.60 | 37.60 |

4-5-6 | 120 | 0.245 | 0.426 | 0.0065 | 0.0122 | 25.94 | 35.82 |

5-5-5 | 125 | 0.291 | 0.563 | 0.0041 | 0.0085 | 14.39 | 23.27 |

6-6-6 | 216 | 0.168 | 0.245 | 0.0018 | 0.0030 | 7.966 | 7.300 |

## 5 Conclusions

The new shallow 4-layer PairNet can be trained very quickly with only one epoch since its hyperparameters are directly optimized one-time via simply solving a system of linear equations by using the multivariate least squares fitting method. Different from gradient descent training algorithms and other training algorithms such as genetic algorithms, the new training algorithm with direct hyperparameter computation can quickly train the PairNet because it does not need slow training with a large number of epochs. Initial simulation results show that the shallow PairNet is much faster than traditional ANNs. For accuracy, the PairNet may not always achieve the lowest training MSE but can achieve lower testing MSEs than traditional ANNs.

In addition, the divide-and-conquer approach used by Algorithm 1 is effective and efficient to build local PairNet models on local -dimensional subspaces. For big data mining applications, partitioning a big data space into many small data subspaces is useful since each local PairNet covering a small data subspace is built more quickly using fewer data points than a global PairNet covering the whole big data space.

## 6 Future Works

More robust simulations with much more complex data sets with more inputs will be done to further evaluate the PairNet and to further compare the PairNet and traditional ANNs. Additionally, a new PairNet with a new activation function of the neuron on Layer 4 will be created for classification applications. The new PairNet will be further evaluated by commonly used benchmark classification problems. The PairNet can be optimized to reduce the training MSE and testing MSE by model selection via optimizing partitioned local -dimensional subspaces.

Although the PairNet is a shallow neural network, it is actually a wide neural network if is large because both the second layer and the third layer have neurons with the first layer having

neurons. Thus, the PairNet has the curse of dimensionality. However, we will develop advanced divide-and-conquer methods to solve it. The preliminary simulations applied a random data partitioning method to divide a whole

-dimensional space into many -dimensional subspaces. More intelligent data partitioning methods will be created to build more effective local PairNets on optimized -dimensional subspaces.A significant future work is to develop more effective and faster hyperparameter optimization algorithms using parallel computing methods to find the best high-speed PairNet model with ideal activation functions on optimized -dimensional subspaces for various applications in real-time machine learning and big data mining.

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