Conceptual alignment deep neural networks

Abstract

Deep Neural Networks (DNNs) have powerful recognition abilities to classify different objects. Although the models of DNNs can reach very high accuracy even beyond human level, they are regarded as black boxes that are absent of interpretability. In the training process of DNNs, abstract features can be automatically extracted from high-dimensional data, such as images. However, the extracted features are usually mapped into a representation space that is not aligned with human knowledge. In some cases, the interpretability is necessary, e.g. medical diagnoses. For the purpose of aligning the representation space with human knowledge, this paper proposes a kind of DNNs, termed as Conceptual Alignment Deep Neural Networks (CADNNs), which can produce interpretable representations in the hidden layers. In CADNNs, some hidden neurons are selected as conceptual neurons to extract the human-formed concepts, while other hidden neurons, called free neurons, can be trained freely. All hidden neurons will contribute to the final classification results. Experiments demonstrate that the CADNNs can keep up with the accuracy of DNNs, even though CADNNs have extra constraints of conceptual neurons. Experiments also reveal that the free neurons could learn some concepts aligned with human knowledge in some cases.

Keywords

Deep neural networks conceptual alignment interpretability supervised learning representation learning

1 Introduction

Deep Neural Networks (DNNs) can be referred to large Artificial Neural Networks (ANNs) stacked with many layers. DNNs has very good performance in reducing high-dimensional data into a compact representation space [13]. In various recognition tasks, such as computer vision [12 , 25] and speech recognition [9, 14], DNNs show powerful recognition ability to fit the high-dimensional data. Given enough training data, DNNs trained by an optimization algorithm can automatically adjust their parameters and reach to a very high classification accuracy. Generally, it is very difficult to explain the meanings of the numerous parameters and the outputs of the hidden neurons under the cognitive competence of human beings. Therefore, the models of DNNs are usually regarded as black boxes that cannot interpret how they classify the objects according to some effective features. Similarly, human beings usually cannot explain how they can identify some objects. For example, Traditional Chinese Medicine (TCM) practitioners are difficult to explain how they diagnose the patients based on subtle observations. However, the model interpretability is necessary for many applications, especially in clinical diagnoses. People will feel an interpretable predictive model is more dependable than an unknown model that delivers results without supportive reasoning. Therefore, the model interpretability is crucial for the wide adoption in medical research and clinical decision-making.

In order to specify the recognition process, human beings form many effective abstract concepts to distinguish the different classes of objects. For example, a patient who has pale facial, skinny body, pale tongue and thin tongue coating might be classified into Qi-deficiency body constitution type in TCM [8, 26]. Before the advances in deep learning, a majority of machine learning methods are heavily dependent on feature engineering, which is a way to take advantage of the human knowledge. Human beings usually can discover the underlying explanatory factors hidden in the observed sensory data, and form the abstract concepts. In other words, human beings can map the observed sensory data into a representation space described by the abstract concepts, and reach a consensus on the recognition tasks. Compared with a black-box model that provides results without any reasoning, an accurate and interpretable model will be more dependable and attractive. This motivates many deep learning researchers to explore the representations in deep models [3]. But why can not the deep models learn some common sense along with human beings? Maybe a direct and effective way is to make the deep models learn the human-formed concepts so as to form similar cognitions.

This paper proposes a training method that can make the DNNs map the sensory data into a representation space aligned with human concepts. The advantages to make the DNNs learn the human-formed concepts are twofold. The former one is to let the DNNs become interpretable with human knowledge, and the latter one is to let the DNNs become easy to be trained by a small amount of training data with prior knowledge. Moreover, the DNNs will become easy to be debugged because the classification results can be traced back to the conceptual neurons. In the proposed Conceptual Alignment Deep Neural Networks (CADNNs), some neurons in the hidden layers are chosen as the conceptual neurons. These neurons are used to extract designated features related to the corresponding human-formed concepts. By introducing the conceptual neurons, the classification results of CADNNs become interpretable. Furthermore, the explanatory factors that contribute to the classification results can be checked. The main contributions of this paper are highlighted as follows:

A kind of new DNNs, called CADNNs, is proposed. CADNNs have the attractive property of interpretability, as well as keep the high accuracy like the DNNs.

The framework and objective of CADNNs are formalized and analyzed based on DNNs. The training procedure is also provided.

Experiments are designed to test the different architectures of CADNNs, which results demonstrate that the conceptual neurons can learn the effective representations of abstract concepts. Furthermore, some experiments show that the free neurons could also learn the representations corresponding to human-formed concepts, in some cases.

The remainder of this paper is structured as follows. Section 2 provides an overview of the related work. Section 3 describes the methods including the general framework, objective and training method. Section 4 depicts experimental results, whilst discussions and comments are presented in Section 5. Finally, conclusions and directions for future work are presented in Section 6.

2 Related work

With the increasing popularity of Deep Neural Networks (DNNs), a number of researches have attempted to explore the representations space of hidden layers of DNNs. Zeiler et al. [29, 30] use deconvolutional networks to explore the low-level and mid-level image representations. Yu et al. [28] have shown successful studies at learning hierarchical image representations from the pixel level via hierarchical sparse coding. Zhu et al. [31] propose a deep model, termed multi-view perceptron, for learning face identity and view representations. For learning robust representations of human physiology, Che et al. [5] propose to use prior knowledge to regularize parameters in the topmost layers. Recently, Che et al. [6] propose an interpretable mimic learning method to distill knowledge from DNNs via Gradient Boosting Trees (GBT) to learn interpretable models and strong prediction rules. Bengio et al. [4] provide a comprehensive review of representation learning about learning good representations. Despite more researchers realize the relationship between the representations of DNNs and human-formed concepts, there is not a clear method to incorporate the prior knowledge of human-formed concepts into the DNNs.

Transfer learning, which is motivated by the fact that human beings can apply previously learned knowledge to solve new problems faster even with better solutions [18], also confirms the DNNs can learn effective representations to distinguish the different objects, even in different classification tasks [3]. Ahmed et al. [2] propose using transfer learning to improve the training of hierarchical DNNs, and experiments show that transfer learning substantially improves the quality of Convolutional Neural Networks (CNNs) by incorporating useful prior knowledge. Long et al. [16] propose a Transfer Sparse Coding (TSC) approach to construct robust sparse representations for images. Oquab et al. [17] design a method to reuse the layers of DNNs trained on the ImageNet dataset to compute mid-level image representations for images in the PASCAL VOC dataset. They demonstrated that the transferable representations are useful in various kinds of tasks. However, the main difference of this work lies in that our aim is to make the DNNs generate not only effective but also interpretable representations.

Generative Adversarial Networks (GANs) [10] are used to learn to generate the observations from a compact low-dimensional representation space. Recently, GANs have shown promising results in learning hierarchical representations [19]. InfoGAN [7], which is a variant GAN, shows it can learn interpretable representations. In the experiments, it discovers visual concepts, such as hair styles, presence of eyeglasses and emotions [7]. Reed et al. [20, 21] demonstrate that GANs can generate images from human visual concepts. Reed et al. [22] also propose models combining CNNs and Long Short-Term Memories (LSTMs) to relate the images with fine-grained and category-specific language concepts. However, the interpretable representations produced by GANs are still very limited.

3 Methods

The information process of Deep Neural Networks (DNNs) is transmitting key information layer by layer from the input layer to the output layer. It can also be regarded as a representation space transformation when the activation codes of the neurons in a layer activate the neurons in the next layer. For the recognition tasks, the layers of DNNs trend to have smaller neurons from the input layer to the output layer. Therefore, these DNNs are to make the input high-dimensional data reduce to a compact low-dimensional representation space. Generally, only the representation space of the output layer is meaningful to human beings. The output neurons are trained to be aligned with the category labels, which are usually high-level abstract concepts. It is worth noting that the categories are also human-formed concepts. Generally, human-formed concepts can be organized hierarchically. The fine-grained concepts form the high-level abstract concepts. In this way, the human-formed concepts should be hierarchically distributed in a DNN. Based on this conception, a framework is devised as follows.

3.1 General framework

The general DNNs are illustrated as Fig. 1, where they are trained to fit the input and output variables, whilst the variables represented by the neurons in hidden layers are not concerned. This paper proposes to train the DNNs having hierarchical interpretable hidden neurons aligned with human-formed concepts, termed Conceptual Alignment Deep Neural Networks (CADNNs), as illustrated in Fig. 2. The method is to assign some effective meanings to some selected hidden neurons, called conceptual neurons. In addition, there are still hidden neurons, called free neurons, used to be trained in their own way. The number of conceptual neurons in each hidden layer depends on specific applications. It can be zero conceptual neuron or full of conceptual neurons in a hidden layer. The conceptual neurons need to be manually assigned according to human knowledge. In shallow hidden layers, conceptual neurons should be assigned to represent low-level abstract concepts. And high-level abstract concepts should be represented by conceptual neurons in deep hidden layers.

Fig.1

The illustration of a general DNN. The circles represent the neurons of DNN. Only the neurons of input layer and output layer are assigned meanings, and they are drawn as shaded circles.

Fig.2

The illustration of a CADNN. The circles represent the neurons. The shaded circles in hidden layers are conceptual neurons, and the hollow circles are free neurons, which can be trained freely.

A m-layer CADNN can be defined as Net ( W , b ), where W and b denote the model parameters. Assume the neurons of the input layer are represented by a vector x . The non-linear transformation in the first hidden layer is $h_{1} = (h_{1 f}, h_{1 c}) = f (x),$ (1) where h ₁ is a vector that represents the free neurons and conceptual neurons in the first hidden layer, h _1f denotes the free neurons, h _1c denotes the conceptual neurons, and f (·) is the non-linear transformation function. The typical function is a linear weighted sum with a non-linear activation function as $f (x) = σ (W_{1} x + b_{1}),$ (2) where W ₁ is the weight matrix for the first hidden layer, b₁ is the bias scalar for the first hidden layer, and σ (·) is the activation function, which can be sigmoid function, tanh function, softmax, Rectified Linear Unit (ReLU), or any other variants [11]. In a similar way, the non-linear transformation in the ith hidden layer can be formulated as $h_{(i)} = (h_{(i) f}, h_{(i) c}) = f (h_{(i - 1)}) .$ (3)

The parameters of f ( h _(i-1)) are denoted as W _i and b _i. Finally, the result of the output layer is $o = f (h_{(m - 2)}),$ (4) where o is a vector that represents the neurons in the output layer, the parameters of f ( h _(m-2)) are W _(m-1) and b_(m-1).

3.2 Cost function

Given a training sample x , and the corresponding conceptual label y = { y ₁, . . . , y _i, . . . , y _(m-1)} where y ₁ is a vector that is used to label the conceptual neurons in the first hidden layer and has the same size with them, y _i is used to label the conceptual neurons in the ith hidden layer and has the same size with them, and y _(m-1) is the label for the output layer. The cost of first hidden layer can be measured by ℓ₂-norm, written as $J_{1} (x) = \frac{1}{2} ∥ y_{1} - h_{1 c} ∥_{2}^{2},$ (5) where the h _1c is calculated according to Equation (1). In a similar way, the cost of ith hidden layer can be written as $J_{i} (x) = \frac{1}{2} ∥ y_{i} - h_{(i) c} ∥_{2}^{2},$ (6) where the h _(i)c is calculated using the forward propagation method according to Equation (3). The cost of the output layer can be written as $J_{(m - 1)} (x) = \frac{1}{2} ∥ y_{(m - 1)} - o ∥_{2}^{2} .$ (7) where the o is calculated using the forward propagation method according to Equation (4). And the total cost is the accumulated cost of each layer adding a regularization term, written as $J (x) = \sum_{i = 1}^{m - 1} (β_{i} J_{i} (x) + \frac{λ}{2} ∥ W_{i} ∥_{2}^{2}),$ (8) where λ is a hyper-parameter, which is used to avoid over-fitting by controlling scale of the weights. The hyper-parameters β = {β₁, …, β_m-1} are used to trade off the objectives of each one of layers.

For a training data set ( X , Y ) containing n samples { x ^(j), y ^(j)}, (1 ⩽ j ⩽ n), the batch cost can be evaluated using the average cost, as $J (X) = \frac{1}{n} \sum_{j = 1}^{n} \sum_{i = 1}^{m - 1} (β_{i} J_{i} (x^{(j)}) + \frac{λ}{2} ∥ W_{i} ∥_{2}^{2}) .$ (9)

3.3 Training procedure

Once an objective function as Equation (9) is chosen, the parameters of a CADNN model can be trained by optimization methods, such as Stochastic Gradient Descend (SGD). In order to use the optimization methods, the partial derivatives of the objective function with respect to parameters need to be calculated. Back-Propagation (BP) algorithm [23], which is a common method to train the multi-layer artificial neural networks, also can be used to train the CADNNs. BP algorithm uses a chain rule to calculate the partial derivatives. At the start, a feedforward pass computation runs from the input layer to the output layer, and the loss values of the output neurons can be calculated. Then, the loss values can be propagated backwards and be used to calculate the partial derivatives with respect to the parameters of each layer. The partial derivatives of the weights W _(m-1) and the biases b_(m-1) with respect to the output layer of CADNNs can be obtained as follows, $\frac{\partial J (x)}{\partial W_{(m - 1)}} = \frac{\partial J_{(m - 1)} (x)}{\partial W_{(m - 1)}} + λ ∥ W_{(m - 1)} ∥_{2},$ (10) $\frac{\partial J (x)}{\partial b_{(m - 1)}} = \frac{\partial J_{(m - 1)} (x)}{\partial b_{(m - 1)}} .$ (11)

And they can directly compute the results according to Equation (7). The partial derivatives of the weights W _i and the biases b _i with respect to the ith layer of CADNNs can be obtained as follows, $\frac{\partial J (x)}{\partial W_{i}} = \sum_{t = i}^{m - 1} \frac{\partial J_{t} (x)}{\partial W_{i}} + λ ∥ W_{i} ∥_{2},$ (12) $\frac{\partial J (x)}{\partial b_{i}} = \sum_{t = i}^{m - 1} \frac{\partial J_{t} (x)}{\partial b_{i}} .$ (13)

Each $\frac{\partial J_{t} (x)}{\partial W_{i}}$ and $\frac{\partial J_{t} (x)}{\partial b_{i}}$ in Equations (12) and (13) can be calculated according to the chain rule of BP algorithm. The partial derivatives of all the layers can be calculated from the output layer to the input layer. The training procedure for CADNNs is described as Algorithm 1. First of all, the specific architecture of a CADNN Net ( W , b ) needs to be defined. At the start of training, the parameters ( W , b ) of the defined CADNN are set randomly. For each training epoch, a mini-batch of training data ( X _m, Y _m) is sampled from the entire training dataset ( X , Y ). Based on BP algorithm and Equations (12) and (13), the mini-batch can be used to calculate the partial derivatives (g_w, g_b) with respect to the weights and biases, which are then used for updating the parameters according to a learning rate α. A training epoch contains a number of iterations of the parameter updates so as to traverse the entire training dataset. And a number of training epoches are needed to optimize the parameters to fit the data. When the training loss J ( X ) is converged, the parameters ( W , b ) of the network Net ( W , b ) are obtained.

In addition, the conceptual neurons of CADNNs can be pre-trained from the input layer to the output layer. Once the training for a layer is finished, the learning rate for the parameters related to the aligned conceptual neurons need to be decreased in the following training process, and the complete CADNNs should be fine-tuned in the end.

Algorithm 1 CADNN training algorithm, the training procedure of Conceptual Alignment Deep Neural Networks (CADNNs)

Input:

Training dataset ( X , Y ), network definition Net ( W , b ), objective function J ( X ), learning rate α, regularization hype-parameter λ, and layer-wise hype-parameters β .

Output:

Parameters ( W , b ), loss value J ( X ).

1: Randomly set the initial parameters ( W , b ) of the networks.

2: fort = 1, …, Tdo

3: fork = 1, …, Kdo

4: Sample a mini-batch ( X _m, Y _m) from training data set ( X , Y )

5: Use BP algorithm to compute the partial derivatives according to Equations (12) and (13):

g_w ← ∇ _wJ ( X_mitiinbi> ),

g_b ← ∇ _bJ ( X_mitiinbi> ).

6: Update the corresponding parameters:

w ← w - α · g_w, b ← b - α · g_b.

7: end for

8: end for

9: return ( W , b ) and J ( X ).

4 Experiments

4.1 Dataset

Fashion-MNIST dataset [27] is used in the experiments. It is a dataset that has the same format as the popular and overused MNIST dataset consisting of a training set of 60,000 examples and a test set of 10,000 examples. Each example is a 28 × 28 grayscale image, associated with a label from 10 classes. In contrast with the simple handwritten digits in MNIST dataset, the examples in Fashion-MNIST dataset are images about different clothes, trousers, shoes and bags, as shown in Table 1.

Table 1
The labels and the examples of Fashion-MNIST dataset

4.2 Experimental design

4.2.1 Hierarchical label generation

In order to obtain the hierarchical labels, we further reduce ten classes of examples into four classes. They are top, bottom, shoe and bag, as shown in Table 2. The top class includes T-shirt, pullover, dress, coat and shirt corresponding to the original label 0, 2, 4 and 6 respectively. The bottom class includes the trouser corresponding to the original label 1. The shoe class includes sandal, sneaker and ankle boot corresponding to the original label 5, 7 and 9 respectively. The bag class includes bag corresponding to the original label 8. Therefore, a training dataset of two-layer labels is obtained. The original labels can be used for training the conceptual neurons in the hidden layer, while the four new labels are used for training the neurons in the output layer.

Table 2
The high-level concepts and the label definition

Label Concept Original labels

0 Top 0 (T-shirt), 2 (Pullover), 3 (Dress), 4 (Coat), 6 (Shirt)

1 Bottom 1 (Trouser)

2 Shoe 5 (Sandal), 7 (Sneaker), 9 (Ankle boot)

3 Bag 8 (Bag)

Label	Concept	Original labels
0	Top	0 (T-shirt), 2 (Pullover), 3 (Dress), 4 (Coat), 6 (Shirt)
1	Bottom	1 (Trouser)
2	Shoe	5 (Sandal), 7 (Sneaker), 9 (Ankle boot)
3	Bag	8 (Bag)

4.2.2 Experimental architectures

The neurons in the input layer and the output layer are prescribed in the experiments. The size of the input layer neurons is 784, which is the number of pixels of an input image (28 × 28). The output layer has 4 neurons for four generated labels as shown in Table 2. And the activation function for output layer is softmax. A simple CADNN architecture $𝔸_{1}$ , which has three fully-connected layers, is defined for exploration at start. As illustrated in Fig. 3, there are a total of 100 hidden neurons, 10 conceptual neurons and other 90 free neurons in the hidden layer. The activation function is sigmoid function. In order to check the performance of the conceptual neurons, there is also an extreme architecture $𝔸_{2}$ . It has only ten conceptual neurons and no free neuron, as illustrated in Fig. 4. The activation functions and the connection types are the same with architecture $𝔸_{1}$ .

Fig.3

The illustration of architecture $𝔸_{1}$ . The input layer has 784 neurons, the output layer has 4 neurons, and its activation function is softmax. The hidden layer has 100 neurons including 10 conceptual neurons and 90 free neurons. The activation function of hidden layer is sigmoid function. The two connection types between the three layers are fully-connected.

Fig.4

The illustration of architecture $𝔸_{2}$ . It is an extreme case that all 10 conceptual neurons and no free neuron are found in the hidden layer. The 10 conceptual neurons are used to be aligned with the ten original labels. The activation functions and connection types are the same with $𝔸_{1}$ .

As shown in Table 2, if there are ten conceptual neurons corresponding to ten original labels, the new defined labels are completely represented by ten conceptual neurons. In order to test the incomplete representation situation, another architecture $𝔸_{3}$ , which cuts down the conceptual neurons, is also explored. As illustrated in Fig. 5, architecture $𝔸_{3}$ also deploy 10 neurons in the hidden layer, but there are just 9 conceptual neurons aligned with nine original labels, which do not contain the last conceptual label “ankle boot”. In addition, there is one free neuron that can be trained freely. The activation function for the hidden layer is also sigmoid function, and the layers are fully connected. In case of incomplete representations of 9 conceptual neurons, an interesting question is that whether the free neuron can learn an effective representation aligned with the veiled label “ankle boot”.

Fig.5

The illustration of architecture $𝔸_{3}$ . There are 9 conceptual neurons and one free neuron in the hidden layer. The 9 conceptual neurons are aligned with nine labels in the ten original labels, and the free neurons can be trained freely. The activation functions and connection types are the same with $𝔸_{1}$ .

The deeper architectures $𝔸_{4}$ and $𝔸_{5}$ are designed. Architecture $𝔸_{4}$ is an extension of $𝔸_{2}$ . Four layers are interpolated between the input layer and the hidden layer of $𝔸_{2}$ . They are two convolutional layers followed two max-pooling layers respectively. In the same way, architecture $𝔸_{5}$ extends the architecture $𝔸_{3}$ by adding more hidden layers, as illustrated in Fig. 6. Architecture $𝔸_{4}$ has almost the same structure with $𝔸_{5}$ except that the last hidden layer are filled with 10 conceptual neurons, so the illustration of architecture $𝔸_{4}$ is omitted. The kernel sizes of convolutions are all 5 × 5 using the SAME mode, and the kernel sizes of the max-poolings are all 2 × 2. There are 8 kernels for first convolutional layer, and 16 kernels for second convolutional layer. ReLU is specified as activation function for two convolutional layers. The activation functions for last hidden layer and output layer are still sigmoid and softmax respectively. The input layer is resized to a 28 × 28 map. Through first convolutional computations, the input data will map to 8 feature maps. The size of each feature map is 28 × 28. Therefore, there are 6272 neurons in the first hidden layer. After max-pooling computations, the sizes of feature maps are reduced to 8 × 14 × 14. The number of neurons in second hidden layer is reduced to 1568. Second convolutional computations map the outputs of first max-pooling layer into 16 feature maps of size 14 × 14. Then second max-pooling computations reduce them into 16 feature maps of size 7 × 7. So there are 3136 neurons in the third hidden layer and 784 neurons in the fourth hidden layer. The 784 neurons are fully connected to the 10 neurons in last hidden layer. Finally, the neurons of last hidden layer are fully connected to the 4 neurons in output layer.

Fig.6

The illustration of the architecture $𝔸_{5}$ . Architecture $𝔸_{4}$ has almost the same structure except that there are full of 10 conceptual neurons in the last hidden layer. The convolutions use the SAME mode. The activation functions for the convolutional layers, the last hidden layer, and the output layer are ReLU, sigmoid and softmax respectively. The numbers of neurons from the first hidden layer to the fifth hidden layer are 6272, 1568, 3136, 784 and 10 respectively.

4.3 Experimental results

The designed architectures are implemented and tested using TensorFlow [1]. The TensorFlow (version 1.0) is installed on Python 3.5 (64-bit), and the computing environment is based on Windows 10 (64-bit), 2.30 GHz CPU (Intel i3-2350M) and 4 GB memory. A training process of $𝔸_{1}$ compared with a same structure with all free neurons is illustrated in Fig. 7. The gradient descent optimizer is used, and the learning rate is set to 1. The hyper-parameters β₁ and β₂ are all set to 1. Architecture $𝔸_{1}$ quickly reaches a high accuracy after about 10 training epoches and acquires an accuracy near 99% at the end of 100 training epoches. The performance of architecture $𝔸_{1}$ is almost the same with the corresponding DNN architecture. Then, several comparative architectures, which have almost the same structure with $𝔸_{1}$ except that the number of its free neurons were reduced to 10, 5 and 2, have been tested. The experimental results are almost the same with $𝔸_{1}$

Fig.7

The illustration of the training processes of architecture $𝔸_{1}$ (CADNN) and architecture $𝔸_{1}$ with all free neurons (DNN). The horizontal axis indicates the training epoch and the vertical axis indicates the accuracy. It shows that $𝔸_{1}$ has almost the same performance with the corresponding DNN architecture.

To the extreme case, another architecture $𝔸_{2}$ with only 10 conceptual neurons in the hidden layer also has been tested. The gradient descent optimizer is used again, and the learning rate is set to 1 too. The hyper-parameters β₁ and β₂ are also set to 1. As expected, the experimental results are almost the same with $𝔸_{1}$ except for a tiny accuracy reduction to about 98.5%, because the ten conceptual neurons construct a complete representation space for the four generated labels. On the other hand, the accuracy of the conceptual neurons used in identifying the ten original labels is checked in order to observe the performance of conceptual neurons. Figure 8 illustrates the performance of conceptual neurons in the architecture $𝔸_{2}$ . In the training framework of CADNNs, the conceptual neurons demonstrate that they have learnt the representations aligned with the conceptual labels. The free neurons in DNN framework can learn some representations that contribute to a high accuracy in the end, but they have just learnt some inexplicable representations, which make no sense for human beings.

Fig.8

The illustration of the performance of conceptual neurons in the architecture $𝔸_{2}$ . This figure shows the accuracies of conceptual neurons in identifying ten original labels in 100 training epoches. As a comparison, the accuracies of ten hidden neurons trained freely in a DNN framework are also plotted. It shows that conceptual neurons are trained to be aligned with ten original labels rather than unaccountable representations.

In order to answer that whether a free neuron can learn the representation aligned with the veiled concept of “ankle boot”, some experiments on architecture $𝔸_{3}$ are explored. Figure 9 shows a training result using the hyper-parameters β₁ = 1 and β₂ = 1. Although the accuracy of the output layer remains high, the accuracy of the hidden layer drops sharply. This result shows that the hidden layer is adjusted to another representation space. There is a trade off that the training algorithm has to keep the accuracy of output layer according to the objective function, refer to Equation (9). Generally, a complex objective function has numerous local minimums, and optimization algorithms usually fall into a local optimal solution. Therefore, the result of Fig. 9 is just an undesired local optimal solution. It is possible to increase the value of parameter β₁ to promote the conceptual neurons be aligned with conceptual labels. Figure 10 shows a training result using the hyper-parameters β₁ = 10 and β₂ = 1. In this case, Stochastic Gradient Descent (SGD) algorithm adjusts the parameters of $𝔸_{3}$ to converge into a local optimal solution that the hidden layer achieves an accuracy of about 82%, which is close to the results of architecture $𝔸_{2}$ . It demonstrates that the representation space of hidden layer has aligned with ten original labels. That is to say, the free neuron has learnt the representation of label “ankle boot”.

Fig.9

The illustration of the training result of the architecture $𝔸_{3}$ , β₁ = 1 and β₂ = 1. It is a trade off that sacrifices the accuracy of the hidden layer for a high accuracy of output layer. The output layer remains a high accuracy in recognizing four newly generated labels. However, the hidden layer achieves a low accuracy in identifying ten original labels.

Fig.10

The illustration of the training result of the architecture $𝔸_{3}$ , β₁ = 10 and β₂ = 1. The output layer remains a high accuracy in recognizing the four generated labels. Moreover, the hidden layer achieves almost the same accuracy as architecture $𝔸_{2}$ in identifying ten original labels.

As a comparison with architecture $𝔸_{2}$ , a result of the deeper architecture $𝔸_{4}$ is illustrated in Fig. 11, where β₅ = 1 and β₆ = 1. It is worth noting that there is no conceptual neuron in the preceding four hidden layers, so the corresponding costs of these layers are all zero. Hence, the values of β₁, β₂, β₃ and β₄ are not cared according to Equation (8). The result shows that the deeper architecture $𝔸_{4}$ boosts the accuracy of the hidden layer to exceed 90% in identifying the 10 original concepts. And the accuracy of the output layer is reached above 99%. As a comparison with architecture $𝔸_{3}$ , the deeper architecture $𝔸_{5}$ is also tried. The conceptual label “ankle boot” is also eliminated in the conceptual neurons of $𝔸_{5}$ . Several experiments demonstrate that the architecture $𝔸_{5}$ easily falls into a local optimal solution that has a high accuracy of output layer but a low accuracy of the hidden layer if it uses the parameters of β₅ = 1 and β₆ = 1, just like the case of $𝔸_{3}$ , see Fig. 9. Under the parameters of β₅ = 10 and β₆ = 1, architecture $𝔸_{5}$ improves the situation to a certain degree. Nevertheless, it is still not very stable as $𝔸_{3}$ , which result is illustrated in Fig. 12. Maybe it is due to the deeper layers and more parameters of $𝔸_{5}$ . In order to achieve a more stable solution, a tricky method is to replace the sigmoid activation function with the softmax in the last hidden layer of $𝔸_{5}$ . In this case, $𝔸_{5}$ also has similar performance with $𝔸_{4}$ that boosts the accuracy of the hidden layer and the accuracy of the output layer above 90% and 99% respectively. A trade off happens in the first 10 training epoches that the accuracy of the hidden layer is increasing while the accuracy of the output layer is decreasing, as illustrated in Fig. 13. It demonstrates that architecture $𝔸_{5}$ has also learnt the effective representation of “ankle boot” through the free neuron.

Fig.11

The illustration of the training result of the architecture $𝔸_{4}$ , β₅ = 1 and β₆ = 1 (β₁, β₂, β₃ and β₄ are not cared). The accuracy of last hidden layer is above 90%, and the accuracy of output layer is above 99%. Compared with the experimental results of previous shallow architectures, it shows an accuracy boost.

Fig.12

The illustration of the training result of architecture $𝔸_{5}$ , β₅ = 10 and β₆ = 1 (β₁, β₂, β₃ and β₄ are not cared). The output layer has a high accuracy above 99%, while the last hidden layer just achieves an accuracy just about 80%. The accuracy reduction of last hidden layer demonstrates that the free neuron has not converged to an effective representation of “ankle boot”.

Fig.13

The illustration of a training result of the architecture $𝔸_{5}$ replaced the activation function of last hidden layer by softmax, β₅ = 10 and β₆ = 1 (β₁, β₂, β₃ and β₄ are not cared). The accuracy of last hidden layer is still above 90%, whilst the accuracy of output layer is still above 99%. There is a trade off that the accuracy of output layer is decreasing while the accuracy of hidden layer is increasing in the first 10 training epoches. After 10 training epoches, the accuracy of hidden layer and the accuracy of output layer are increasing simultaneously.

5 Discussion

There are numerous different representation spaces for an object, which can be recognized in many ways. The purpose of CADNNs is to align the representation space of DNNs with some human concepts, so as to enhance the interpretability. Optimization algorithms cannot always find an effective representation space, particularly in the case of lacking training dataset. The introduction of conceptual neurons in CADNNs can promote the architectures converged to a desired representation space. On the other hand, the free neurons can learn the representations that are not contained in the conceptual neurons, so as to keep the accuracy of the output layer. Moreover, the architectures of CADNNs can be used to explore the latent representations for the recognition tasks. It is easy to check the effectiveness of some concepts used for recognition of certain objects in CADNNs. The interpretable CADNNs is very appropriate for the transfer learning tasks that learn some representations from one task and use the representations to other tasks. However, there is a problem that few existing training data sets have hierarchical labels used for training the CADNNs. One solution is to directly build new data sets that contain hierarchical labels, while the other solution is to utilize a number of correlative existing data sets to form hierarchical labels to train CADNNs.

The valuable human knowledge, formed by a long evolutional history, is constructed by a complex network of numerous effective concepts. It should be promising that infusing the effective human-formed concepts into computer systems to boost artificial intelligence. Moreover, maybe there is a key to answering what is the essence of human consciousness. However, the way of organizing the numerous concepts into a dynamical network is still unclear so that it needs more explorations. Although the work of CADNNs may be a primitive exploration, there is no harm in proposing the perspective that serves as a modest spur to induce someone to come forward with his or her valuable contributions.

6 Conclusion

This paper proposes a kind of Deep Neural Networks (DNNs), termed as Conceptual Alignment Deep Neural Networks (CADNNs). There are conceptual neurons used for learning representations of human-formed concepts in the hidden layers of CADNNs. Although added the extra constrains of some conceptual neurons, CADNNs can guarantee the performance compared with the DNNs. Meanwhile, the conceptual neurons can align the representation space with human-formed concepts in CADNNs, as shown in the experiments. Experiments also demonstrate that the free neurons of CADNNs could learn effective representations aligned with human-formed concepts in some cases. Moreover, hyper-parameters could be used to trade off the interpretability of the hidden layers and the accuracy of the output layer. However, the results are not always converged to expected solutions that have both good interpretability and high accuracy. The method of choosing appropriate hyper-parameters and activation functions is still need to be researched. There is also a challenge to make the free neurons form some new effective concepts based on known concepts that people can understand. Constraints and improved training methods for free neurons will be the future work. Furthermore, the CADNN framework could also extend to other structures, such as Recurrent Neural Networks (RNN). More explorations are needed to make CADNNs become more comprehensive and dynamical.

Footnotes

Acknowledgments

This work is supported in part by the National Natural Science Foundation of China under Grant Numbers 61632009 and 61472451, the Guangdong Provincial Natural Science Foundation under Grant Number 2017A030308006, and the High Level Talents Program of Higher Education in Guangdong Province under Funding Support Number 2016ZJ01.

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Conceptual alignment deep neural networks

Abstract

Keywords

1 Introduction

2 Related work

3 Methods

3.1 General framework

4.1 Dataset

Table 1 The labels and the examples of Fashion-MNIST dataset

4.2.1 Hierarchical label generation

Table 2 The high-level concepts and the label definition Label Concept Original labels 0 Top 0 (T-shirt), 2 (Pullover), 3 (Dress), 4 (Coat), 6 (Shirt) 1 Bottom 1 (Trouser) 2 Shoe 5 (Sandal), 7 (Sneaker), 9 (Ankle boot) 3 Bag 8 (Bag)

6 Conclusion

Footnotes

Acknowledgments

References

Table 1
The labels and the examples of Fashion-MNIST dataset

Table 2
The high-level concepts and the label definition

Label Concept Original labels

0 Top 0 (T-shirt), 2 (Pullover), 3 (Dress), 4 (Coat), 6 (Shirt)

1 Bottom 1 (Trouser)

2 Shoe 5 (Sandal), 7 (Sneaker), 9 (Ankle boot)

3 Bag 8 (Bag)