-During operation of a system including a deep neural network (DNN), new input values to the DNN that were not included in the initial dataset retained during development may be given. In such a case, the DNN may be additionally trained with the new input values; however, that additional training may reduce the accuracy of the DNN in regard to the initial dataset. It is therefore necessary to evaluate the influence of the additional training on the accuracy for the initial dataset. However, evaluation by testing all the input values included in the initial dataset takes time. In this paper, we newly propose a method for quickly evaluating the influence on the accuracy for the initial dataset. As for the proposed method, the gradient of the parameter values (such as weight and bias) for the initial dataset is extracted by running the DNN before the additional training. Then, after the additional training, its influence on the accuracy with respect to the initial dataset is calculated from the gradient and update differences of the parameter values. To show the feasibility of the proposed method, results of experiments with the MNIST dataset are presented, Accordingly, it is confirmed that the calculation amount of the proposed method after the additional training depends on the number of parameters; therefore, even if the number of input values included in the initial dataset is enormous, the influence of additional training can be evaluated quickly. Index Terms-machine learning, neural networks, regression analysis
In recent years, the introduction of machine-learning technologies in various industrial fields has been advancing. Among those technologies, deep neural networks (DNNs) are being popularly applied. In addition to replacing human tasks, in some fields, DNNs are demonstrating better ability than people. DNNs are trained with a dataset. A dataset is composed of pairs of input values to DNNs and their corresponding expected output values. Part of the dataset is used as training dataset for training DNNs, and the rest of the dataset is used as test dataset to evaluate the trained DNNs. In the training of DNNs, when input values included in the training dataset are input into DNNs, values of parameters such as weights and bias are adjusted so that the possibility of obtaining the expected output values increases. After the training is completed, the test dataset is used to measure the probability of obtaining the expected output value as expected. This probability—called accuracy—is an indicator showing the validity of the developed DNNs. However, the accuracy measured during development is only that with respect to the initial dataset retained at that time. That is to say, when input values that are not included in the initial dataset are given, the output values are not exclusively those expected; consequently, the accuracy of the DNN during operation is lower than that during its development.
When the accuracy of DNNs decreases, the DNNs may be
subjected to additional training during operation [
In this paper, a method for quickly evaluating the influence of additional training on the accuracy for the initial dataset— after completion of the additional training—is proposed. In regard to the proposed method, the gradient of the parameter values is extracted by executing the DNNs with input values in the initial dataset before the additional training. Then, after the additional training, its influence on the accuracy for the initial dataset is calculated from the gradient and the update differences of parameter values resulting from the additional training. The results of experiments using the proposed method with the MNIST dataset confirmed that the amount of calculation to be performed after additional training does not depend on the number of input values in the initial dataset; instead, it depends on the number of parameters used in the DNN. As a result, even if the number of input values in the initial dataset is huge, applying the proposed method makes it possible to evaluate the result of additional training quickly. Moreover, the result of evaluating the influence of additional training by the proposed method was compared with the result of evaluating it by test execution. On the basis of the results of that comparison, the utility of the proposed method was also confirmed.
For an arbitrary DNN, denoted as N , handling a classification problem of class c(c > 1), I is taken as the number of neurons making up N , and each neuron (in any layer) is denoted as ni(1 i I ). Also, Ji is taken as the number of parameters used to calculate the value of ni, and the parameter itself is denoted as [wi;1; :::; wi;j ; :::; wi;Ji ]. Note that if ni1 and ni1 are included in different layers, Ji1 and Ji2 can be different. Then, a vector obtained by combining the parameters of all the neurons is defined as W = [w1;1; :::; w1;J1 ; w2;1; :::; w2;J2 ; :::; wI;1; :::; wI;JI ]. In the case of a multilayer perceptron, for example, these parameters correspond to weights and biases. Note that the formula for calculating the value of each neuron by using these parameters is not prescribed in this paper.
For an arbitrary input value xm, the expected output value corresponding to xm is expressed as t(xm). t(xm) represents the identifier of the classification class to which xm belongs. Moreover, when xm is input to N , the output value returned by N is expressed as y(xm), which corresponds to the c-dimensional vector [y1m; :::; ykm; :::; ycm]. The ym value k for each dimension represents the probability that the input value xm belongs to class k. If ym has the largest value in k [y1m; :::; ykm; :::; ycm], class k is denoted as f st(y(xm)). That is, 8ykm ykm yfmst(y(xm)) holds. When f st(y(xm)) = t(xm) holds, it is said that N can correctly classify xm. Likewise, if ykm has the second-largest value, class k is denoted as snd(y(xm)). That is, 8ykm ykm 6= yfmst(y(xm)) ) ykm ysmnd(y(xm)) holds.
The flow of additional training assumed in this paper is shown in Fig. 1.
In this flow, untrained DNN N 1 is trained first by using the training dataset of initial dataset. The result of that training is evaluated by using test dataset of the initial dataset. After that, the resulting DNN, N0, starts to be used. During its use, additional dataset 1 is newly acquired. Then, N0 is additionally trained using the training dataset that is a subset of additional dataset 1, and the result is evaluated in the same way using the corresponding test dataset. The resulting DNN, N1, achieves sufficient accuracy for additional dataset 1. However, N1 does not always maintain sufficient accuracy for the initial dataset. Accordingly, it is necessary to evaluate the influence of the additional training on the accuracy for the initial dataset. That necessity holds for the following additional trainings. As for additional training s(1 s), if a dataset whose influence should be evaluated is expressed as X^s, Xs is defined as ^ follows:
Definition 1: ^ Xs = {I D
I D [ ∪ss0=11 ADs0 (s = 1) (otherwise) where I D and ADs0 represent the initial dataset and additional dataset of additional training s0 respectively.
As for the proposed method, prior to additional training 1, preprocessing is performed on DNN N0. It is supposed that N0 deals with a classification problem with class c(c > 1). Hereafter, as for the expected output value t(xm) for an arbitrary input value xm to N0, positive supervisor ps(t(xm)) and negative supervisor ns(t(xm)) are defined as follows: Definition 2: ps(t(xm)) ns(t(xm)) = = t(xm) {snd(y(xm)) f st(y(xm)) (f st(y(xm)) = t(xm)) (otherwise) where y(xm) represents the output value returned by N0 when xm is input to N0. At that time, as for all input values included in arbitrary-input-value set X1 X^1, positive loss that is loss under the assumption that positive supervisor is the supervisory signal of the training, and negative loss that is loss under the assumption that negative supervisor is the supervisory signal of the training are calculated. Average positive loss for X1 is denoted as P L(X1). Similarly, average negative loss is denoted as N L(X1). For an arbitrary loss function expressed as L(y(xm); t(xm)), P L(X1) and N L(X1) are defined as follows:
P L(X1) = N L(X1) = 1 ∑X1 L(y(xm); ps(t(xm))) M xm 1 ∑X1 L(y(xm); ns(t(xm)))
M xm where M represents the number of input values included in X1. Next, the gradient of parameter W0 = [w1;1; :::; wi;j ; :::; wI;JI ] of N0 with respect to P L(X1) (obtained as previously described) is calculated. The gradient of P L(X1), called positive gradient, is expressed by ∇P L(X1). Similarly, the gradient of N L(X1), called negative gradient, is expressed by ∇N L(X1). ∇P L(X1) and ∇N L(X1) are defined as follows:
Definition 4:
After ∇P L(X1) and ∇N L(X1) are created for N0, additional training 1 is executed, and DNN N1 is created. Parameter W1 of N1 is compared with parameter W0 of N0, and the update difference of the parameter W = W1 W0, is acquired. Then, positive influence P I(X1; W ) and negative influence N I(X1; W ) given to X1 by updating the parameter is calculated as follows:
Definition 5:
P I(X1; N I(X1;
W ) = ( ∇P L(X1)) ( W )
W ) = ( ∇N L(X1)) ( W )
P I(X1; W ) approximates the amount by which P L(X1) is decreased by updating parameter W0 to W1. Likewise, N I(X1; W ) corresponds to the approximate value of the decrease in N L(X1). If W is denoted as [ w1;1; :::; wi;j ; :::; wI;JI ], P I(X1; W ) corresponds to ∑i;j ( @P@Lw(iX;j1) wi;j ). For example, in the case that loss function L is cross entropy, the relation between the value of the decrease of in P L(X1) due to updating wi;j by wi;j , and its approximate value @P@Lw(iX;j1) wi;j is shown in Fig. 2.
Here, it is assumed that L is the loss function used for the initial training to create N0 from N 1. The initial training aims to reduce the loss calculated based on L (called training loss hereafter), in which the expected output value is the supervisory signal of the training. As a result, it is assumed that the accuracy of the DNN in regard to X1 is improved. In that case, the value of the decrease in training loss is considered to be an index for evaluating the change of accuracy with respect to X1. According to Definition 2, since the positive supervisor is the same as the expected output value, P L(X1) coincides with the training loss. It can therefore be assumed that P I(X1; W ), which is an approximate value of the decrease in P L(X1), can be taken as an index for evaluating the change in accuracy with respect to X1.
It can be said that the value of the decrease in P L(X1) functions as an index indicating ”How close the output value is to the expected output value.” However, after training has progressed to a certain extent and the output value sufficiently approaches the expected output value, it is considered that not only ”How close the output value is to the expected output value” but also ”How far it has deviated from an erroneous output value” will influence accuracy. Accordingly, as for the proposed method, the ”dimension with the maximum value except for the dimension indicated by the expected output value” among the elements of y(xm) (which is the output value of N0) is focused on. Therefore, the output value indicating that dimension is defined as negative supervisor. Since the loss N L(X1) is calculated based on the negative supervisor, the decrease in the loss N L(X1) represents ”how close the output value is to the erroneous output value.” Hence, N L(X1) is considered to be related to the decrease in accuracy. Since N I(X1; W ) corresponds to the approximate value of the decrease in N L(X1), it can therefore be assumed that N I(X1; W ) also serves as an index for evaluating the change in accuracy. Based on the above considerations, the following IN F (X1; W ) can be used as an index for evaluating change in accuracy with respect to X1.
Definition 6: IN F (X1;
W ) = P I(X1;
W )
N I(X1;
W )
Among the formulas in Definitions 2 to 6 for determining IN F (X1; W ), the formulas in Definitions 2 to 4 are calculated after the end of the initial training and before the start of additional training 1. Although the formulas in Definitions 5 and 6 are calculated after additional training 1, since the amount of calculation depends on the number of elements of W0 (not the number of input values contained in X1), even if the size of X1 is enormous, change in accuracy can be evaluated in a short time. In the case of a multilayer perceptron, since the number of elements of W0 is given as O(I2) for number of neurons I constituting N0, the calculation order of the proposed method after additional training 1 is given as O(I2).
The results of experimentally applying the proposed method
to the MNIST dataset [
In this experiment, it is assumed that 50 additional trainings will be conducted after the initial training. It is also assumed that 46,666 image data, which equals two-thirds of the 70,000 image data provided as the MNIST dataset, are retained as initial dataset. It is supposed that the remaining 23,334 data are acquired equally between additional trainings. Namely, the number of data in each additional dataset is given by 23; 334=50 ; 466. Moreover, the ratio of the training dataset to the test dataset in each dataset (initial, additional 1, and so on) is set to 6:1 (as in the MNIST dataset). The DNN used is a multilayer perceptron with two hidden layers (composed of 1,000 neurons each) in addition to the input and output layers.
We evaluate the influence of additional training for each classification class in this experiment. Accordingly, in additional training s(1 s 50), a set of input values with the same classification class k(0 k 9) is created from X^s as Xsk. The proposed method is then applied for each Xsk and IN F (Xsk; W ) was calculated. Moreover, to evaluate the effectiveness of IN F , the difference value of actual accuracy in regard to Xsk between before and after the additional training is measured. Furthermore, the following times were calculated: (i) the time taken to evaluate the influence by applying the proposed method and (ii) the time taken to evaluate the influence by executing the DNN with all the input values included in X^s. Note that in the case of the proposed method, the formulas to be calculated after the additional training are those in Definitions 5 and 6. In the initial training and all subsequent additional trainings, cross entropy is used as loss function L. The experiment was performed on a Windows 10® PC equipped with two Intel® Core™ i7-8700 3.2-GHz processor with 6 cores, 16-GB memory.
Time taken to apply the proposed method and time taken to run the DNN with all the input values contained in X^s are plotted in Fig. 3.
Fig. 3 indicates that calculation time of test execution increases as number of additional training since the number of input values in X^s increases. On the contrary, in the case of the proposed method, it is confirmed that the evaluation can be performed in almost the same time regardless of the increase in number of data. It is evident from the fact that the calculation order of the formulas in Definitions 5 and 6 is
Fig. 3. Calculation times given as O(I2) as mentioned in Section III-B. The amount of calculation corresponding to the formulas in Definitions 2 to 4 increases in accordance with number of data; even so, those calculations are carried out before additional training, so their results are not included in the time for evaluating the results of additional training.
Values of IN F obtained as a result of the experiment (on the left y-axis) and the difference value of actual accuracy (right) in regard to Xsk between before and after the additional training are plotted in Fig. 4.
It is clear from Fig. 4 that the IN F value and the difference value of actual accuracy have similar waveforms. In other words, it can be said that when the IN F value increases or decreases as compared with its previous value, it is highly likely that the difference value of accuracy also increases or decreases as compared with its previous time. It can be supposed from this result that it is possible to utilize IN F as an index for evaluating the change in accuracy. For example, for the graphs with k = 0, the value of the second IN F acquired in the additional training is smaller than that of the first additional training. In that case, it can be predicted that the difference value of the accuracy of the second additional training becomes smaller than that of the first additional training. (However, whether it increases by a smaller value than the first additional training or it turns into a decrease cannot be predicted.) For all classification classes, relative change (increase or decrease) of IN F was compared with relative change (increase or decrease) of difference value of accuracy, and they agreed with a probability of about 89.2%.
According to the above-presented experimental results, the feasibility of the proposed method is confirmed. Since it is confirmed that the proposed method does not depend on the number of retained input values, the more the number of retained input values is, the more useful the proposed method is to evaluate the results of additional training than testing all the retained input values.
As for the proposed method, by examining whether IN F increases or decreases compared with its previous value, it is possible to evaluate whether the difference value of accuracy also increases or decreases compared with its previous value. Moreover, because IN F value and the difference value of actual accuracy have similar waveforms, how much accuracy changes after additional training can be roughly predicted from the waveform of IN F . For example, if IN F increases more than it previously increased, the difference value of accuracy will also increase more than it previously increased. However, the result of the evaluation by the proposed method is not always accurate. Also, with the proposed method, it is not possible to evaluate the precise value of accuracy after additional training.
In view of these advantages and disadvantages, it is thought that the proposed method will be useful in a system that repeats additional training of DNNs to find a better DNN during operation. In the case of such a system, it should be promptly decided whether the system adopts the DNN additionally trained, tries another additional training, or performs rollback to DNNs before the additional training. Examples of such systems include rescue robots and stock trading systems. These systems tend to prefer adapting to changes in the environment as soon as possible by repeating additional training not to lose opportunities to save people or gain profits. Otherwise, even if the precise value of accuracy is evaluated by running on all the data, using the proposed method in advance is advantageous in that ”triage” becomes possible. For example, it is possible to execute tests with input values from classification classes that are highly likely to degrade accuracy.
Because the number of layers, neurons, and parameters are generalized in the definition of DNNs and types of activation functions are not specified, the proposed method can be applied to any neural networks of multilayer perceptron topology. For the same reason, it is thought that the proposed method gives similar results when applied to a convolutional neural network.
In the graphs for k = 0 shown in Fig. 4, as for the 4th and 8th additional trainings, accuracy drops due to the additional training. In that case, it is considered that P L increases, so IN F representing the decrease in P L should be a negative value. However, as shown in Fig. 4, IN F may have a positive value, the reason for which is explained below. As explained in Section III-B, P I is an approximate value of decrease in P L. First, it is supposed that wi;j is added to wi;j (which is an element of parameter W ) and that the decrement value of P L in that case is represented by P L. When wi;j > 0, P I > 0 and P L > 0; however, due to the property of the cross entropy function used as L, P L < P I holds. On the contrary, if wi;j < 0, although P I < 0 and P L < 0, for the same reason, P L < P I. That is, regardless of the updating direction of wi;j , P I becomes larger than P L. For the above reasons, the value of P I tends to be greater than the actual value of the decrease in P L. Similar errors in N I also occur. However, the additional training is conducted so that P L decreases, so it is conceivable that decrease in P L tends to be larger than that in N L in many cases. In that case, the error caused by P I is larger than that caused by N I. As a result, the value of IN F tends to be larger than the actual value of the decrease in P L.
To the author’s knowledge, no research results on a method
for evaluating additional training results have been published.
However, in a similar manner to our research presented in
this paper, Kirkpatrick et al. [
Regarding the problem that the accuracy of the initial dataset is decreased by the additional training of DNNs, a method for quickly evaluating the influence of additional training is proposed. The computational complexity of the proposed method depends on the number of parameters of DNNs (such as weight and bias), namely, not on the amount of data in the initial dataset. Therefore, even if the amount of data included in the initial dataset is enormous, applying the proposed method makes it possible to evaluate the result of additional training quickly. Moreover, it is clear from the results of an experiment with the MNIST dataset that the method is useful in the case of a system that requires quickness— rather than accuracy—of evaluation, such as when additional training of DNNs is performed during operation of the system and further additional training is performed according to the evaluation results. As for future work, the proposed method will be evaluated using datasets other than MNIST. Moreover, improving the means of creating I N F will help in the search for more accurate evaluation.