1. Introduction

2. Methods

2.3. Loss Function of Our Network Architecture

In 2.1, we defined a temporal classifier for time series classification. In 2.2, we explained how to select time series samples based on the representativeness and uncertainty of the samples. Next, to enable the TLPM to accurately estimate the uncertainty of unlabeled data, we will define the loss function and explain the method of training the network. By inputting samples with labels into the classifier, we can obtain a loss value by comparing the true labels with the predicted ones. However, since unlabeled samples do not have corresponding labels, we cannot directly obtain the loss values for unlabeled data. Instead, we predict the loss value of a time-series instance through the TLPM. First, for the classifier, we define the output probability of $\mathbf{x}$ as $\widehat{\mathit{z}}=F_c\left(\mathit{x}\right)$, so the loss value for a single sample i is

$$l_c^i=L_c\left(\mathit{z},\widehat{\mathit{z}}\right)$$

(2)

$L_c$ represents the loss function of the classifier, where $\mathit{z}$ is the true label of the sample and $\mathit{z}=[z_1,\cdots ,z_T]$. For the TLPM $F_u$, its predicted loss is denoted as $l_u^i$, and $l_c^i$ is the corresponding actual loss value. Therefore, the loss of this module is defined as $L_u(l_c^i,l_u^i)$. The total loss is defined as:

$$L=L_c\left(\mathit{z},\widehat{\mathit{z}}\right)+\lambda L_u\left(l_c^i,l_u^i\right)$$

(3)

$\lambda$ is an adjustable parameter used to represent a scaling constant in the loss optimization process. Specifically, for the classifier, we use cross-entropy loss as its loss function. When a batch contains N temporal samples, the cross-entropy loss can be expressed as:

$$L_c=-{\displaystyle \frac{1}{NT}}\sum _{i=1}^N\sum _{t=1}^T{z}_t^{i}log\left({\widehat{z}}_t^i\right)$$

(4)

Defining the loss function for the TLPM is an important issue. During network training, while the classifier’s accuracy improves, its loss almost always decreases. It would be extremely challenging to train the TLPM using loss labels with large-scale variations. However, during network training, the relative loss values between temporal samples exhibit smaller variations. In other words, if we can compare loss functions between pairs of samples, we can discard the overall scale of loss. We designate a batch of trained data as N, which is an even number. Thus, this batch contains $N/2$ pairs of temporal sample data, denoted as ${\mathit{x}}_p=\{{\mathit{x}}_i,{\mathit{x}}_j\}$. We can then update the parameters of the TLPM by considering the difference between a pair of loss predictions, where the loss function is defined as:

$$L_u\left(l^p,\widehat{l^p}\right)=max(0,-sign\left(l_c^i-l_c^j\right)·(l_u^i-l_u^j))+\psi$$

(5)

Here, the symbol $\psi$ represents a predefined positive number, $l^p$ represents a pair of temporal samples$(i,j)$, and $sign(·)$ represents the sign function. For example, when $l_c^i>l_c^j$ and $-\left(l_u^i-l_u^j\right)+\psi <0$, the loss function value is 0, providing no feedback to the network. Feedback from the loss values is received by the module only when $l_c^i$ is larger than $l_c^j$ and $-\left(l_u^i-l_u^j\right)+\psi >0$, thereby increasing $l_c^i$ and decreasing $l_c^j$. When the network trains a batch of data $\eta$ with a size of N, the final loss function for the classifier and the TLPM is represented as:

$${\displaystyle \frac{1}{N}}\sum _{(\mathit{x},\mathit{z})\in \eta }{L}_c\left(\mathit{z},\widehat{\mathit{z}}\right)+\lambda {\displaystyle \frac{2}{N}}\sum _{({\mathit{x}}_p,{\mathit{z}}_p)\in \eta }{L}_u\left(l^p,\widehat{l^p}\right)$$

(6)

During training, the losses of both the classifier and the TLPM are simultaneously reduced. This means that the network parameters of these two modules are updated simultaneously, without requiring additional time or computational resources for separate steps.

3. Experiments and Results

4. Discussion

5. Conclusions

Abbreviations

References

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