Detecting Anomalies with Autoencoders on Data Streams

Abstract

Autoencoders have achieved impressive results in anomaly detection tasks by identifying anomalous data as instances that do not match their learned representation of normality. To this end, autoencoders are typically trained on large amounts of previously collected data before being deployed. However, in an online learning scenario, where a predictor has to operate on an evolving data stream and therefore continuously adapt to new instances, this approach is inadequate. Despite their success in offline anomaly detection, there has been little research leveraging autoencoders as anomaly detectors in such a setting. Therefore, in this work, we propose an approach for online anomaly detection with autoencoders and demonstrate its competitiveness against established online anomaly detection algorithms on multiple real-world datasets. We further address the issue of autoencoders gradually adapting to anomalies and thereby reducing their sensitivity to such data by introducing a simple modification to the models’ training approach. Our experimental results indicate that our solution achieves a larger gap between the losses on anomalous and normal instances than a conventional training procedure.

1 Introduction

Autoencoders (AEs) were found to enable the unsupervised learning of useful non-linear data representations. Consequently, they sparked significant progress in many areas of machine learning, including the area of unsupervised anomaly detection that deals with the identification of unusual events.

For this purpose, AEs are typically trained on large amounts of normal, non-anomalous data instances, reducing the networks’ reconstruction error for this kind of data. Due to the absence of anomalies in the training data as well as the constraints imposed on AEs, losses for anomalies remain at a higher level and can therefore be distinguished from losses of normal instances. However, collecting and fitting large amounts of data can be an obstacle in many real-world applications. Data might for example not be available from the get-go but rather arrive little by little in small chunks or even in continuous streams of individual samples. In many cases, a model may also be subject to distributional changes over time in the form of concept drift [27]. This is especially true for anomaly detection, as the definition of what constitutes an anomaly is not definitive [31] and may for instance depend heavily on the temporal context in which the data appears. As a result, incremental model updates can often be required to adapt the model to such variations after the initial training.

Based on this observation, an anomaly detector operating on a continuous and potentially infinite stream of individual samples should fulfill the requirements proposed by [4] to

• R1: be able to process a single instance at a time,

• R2: be able to process each instance in a limited amount of time,

• R3: use a limited amount of memory,

• R4: be ready to predict at any time,

• R5: be able to adapt to changes in the data distribution.

To establish a unified view, we additionally formalize the task of online Anomaly Detection (AD) in Sect. 2.

As feed-forward neural networks, AEs inherently feature a fixed time (R2) complexity and, due to their fixed structure, a constant memory (R3) consumption with respect to the number of processed examples. Since they are conventionally optimized with gradient-based methods, AEs also allow processing individual examples (R1) to adapt to the most recent data instance (R5) and predicting at any time (R4). Even their tendency to forget previously learned data patterns [19] when being exposed to new tasks does not necessarily inhibit the performance of AE for AD. This can even be beneficial that forgetting old concepts or distributions of normality may often even be desired, as the sudden reappearance of data that would have previously been considered normal could be deemed as abnormal.

In conclusion, AEs seem well suited for the task of online AD. Nevertheless, only a handful of studies attempted to leverage the remarkable representation learning capabilities of AEs to detect anomalies in data streams. Rather, previous contributions focused on adapting conventional offline learning AD algorithms to fulfil the requirements of online learning (see e.g. [15, 24, 26]). In this work, we present an AE-based anomaly detector for the above requirements and demonstrate its ability to outperform previous state-of-the-art anomaly detectors on multiple real-world datasets. Further, we address the problem of contaminated training data as studied by [5]. Unlike in traditional offline learning, the training data of an online AD will inevitably be contaminated with anomalous instances, since the ability to remove such data from the data stream would make the model redundant in the first place. Thus, depending on the proportion and arrival of anomalies and other external circumstances, reconstruction losses for abnormal data can be expected to decrease to a greater extent than when training on non-contaminated data, degrading the model’s detection accuracy [32]. To mitigate this effect on our approach, we propose an improved optimization objective for AE-based AD, that leads to increases in discrimination performance. To the best of our knowledge, we are the first to address the issue of training data contamination in the context of AE-based online AD.

To enable to enable the usage of our results in future applications, we implemented our approach as an extension¹ to the online learning framework River [17]². In conclusion, we provide the following contributions:

• C1: A formalization of streaming AD.

• C2: A competitive approach for AE-based anomaly detection on data streams.

• C3: An alternative optimization technique to improve the performance of AEs anomaly detectors under the influence of anomalous training examples.

• C4: An in-depth evaluation of our approaches with established real-world datasets.

• C5: A Python package (See footnote 1) extending River [17] that facilitates the reproducibility and reuse of our work.

2 Problem Statement

When using an offline learning approach AEs infringe most of the defined requirements. Predictions for new data instances have to be calculated immediately at the time of their arrival to conform with R1. The requirements R4 and R5 pose the problem that online AD models also need to continuously adapt to their stream of inputs. In addition, data streams often have high frequency, requiring efficient resource utilization, as required by R2 - R4. Therefore, the separation of training and testing stages as performed in most offline scenarios is not applicable when evaluating data streams. Accordingly, we define the problem of anomaly detection as follows:

Definition 1

Incremental Anomaly Detection

Let $\mathcal {S}^+=\{(\boldsymbol{x}^{(i)},y^{(i)}) \, | \, \forall i \in \mathbb {N}\}$ be a potentially infinite sequence of tuples each consisting of a feature-vector $\boldsymbol{x}^{(i)} \in \mathbb {R}^d , \, d \in \mathbb {N}$ and a corresponding anomaly label $y^{(i)} \in \{0,1\}$. Further, let $\mathcal {S}=\{(\boldsymbol{x}^{(i)},y^{(i)}) \, | \, \forall i \in \{1, \dots , I\}\}$ be a sub-sequence of previously observed instances. Let $\mathcal {A} = \{\boldsymbol{x}^{(i)} \,|\,y^{(i)}=1\, i \in \{1, \dots , I\}\}$ be the set of anomalies in $\mathcal {S}$, where $\frac{|\mathcal {A}|}{I} \ll 0.5$.

Also, let $\mathcal {S}_{train}$ and $\mathcal {S}_{valid}$ be disjoint subsets of $\mathcal {S}$. Given a validation protocol $\mathcal {V}$ and an arbitrary anomaly detection function $g: \mathbb {R}^d \rightarrow \{0, 1\}$ that was trained on $\mathcal {S}_{train}$, the objective of incremental anomaly detection can then be defined as

$$\begin{aligned} \min _{g} \mathcal {V}(\mathcal {L}_c,\mathcal {S}_{valid},g(\, \cdot \,; \mathcal {S}_{train})), \end{aligned}$$

(1)

where $\mathcal {L}_c: \{0, 1\} \times \mathbb {R} \rightarrow \mathbb {R}$ denotes a classification loss that quantifies the dissimilarity of the true label $y^{(i)}$ and the prediction $\hat{y}^{(i)} = g(\boldsymbol{x}^{(i)})$.

As a validation protocol $\mathcal {V}$ we employ a test-then-train [4] protocol that is sensitive to concept drifts. In a test-then-train evaluation, models are first validated based on a performance metric $\mathcal {L}_c$ and then trained on each instance in $\mathcal {S}$ at arrival-time. Although our definition of incremental AD includes supervised approaches, we limit our analysis to unsupervised AD, where only the feature instances $\boldsymbol{x}^{(i)}$ are available for training.

3 Related Work

In this section, we depict the work related to our approach by presenting online learning approaches for the AD task as well as offline learning AD approaches that are based on AE. Finally, we summarize previous work on online learning and AE-based AD.

3.1 Offline Anomaly Detection

Anomaly Detection is a very active area of research that has produced a variety of different approaches for identifying anomalous data examples. Early studies focused on statistical measures in this context (see [21]). Machine learning approaches have relied on the assumption that anomalies are found in low-density areas. Highly successful examples of such approaches include One Class Support Vector Machines (OC-SVMs) [25], which learn a hyperplane that separates high-density normal data from low-density anomalous data, and Local Outlier Factor (LOF) [6], which scores data points based on the relative densities of their neighborhoods. Rather than directly assessing differences in density, Isolation Forests [25] separate anomalies with an ensemble of trees that iteratively splits the data along random attribute thresholds. Since the algorithm assumes that anomalies are easily separable from the rest of the data, they can be identified as instances isolated by a small number of splits. According to [14], Isolation Forests provide better scalability compared to previous techniques such as OC-SVMs and LOF.

In more recent years, significant progress in the area of AD has been achieved with deep learning models. Some of the most successful model types in this regard are AEs. Inspired by the functioning principle of PCA-based AD, Sakurada and Yairi [23] proposed to exploit the limited generalization of AE feature mappings, which lead to higher reconstruction losses when facing previously unobserved data patterns. Numerous studies subsequently improved the original AD algorithm’s performance by introducing advanced model variations. Zhou and Paffenroth [29] for instance introduced a robust AE that iteratively separates noise from the underlying clean data whereas Gong et al. [11] mapped the AE’s representations to a fixed number of memorized vectors to avoid learning representations that generalize to anomalous data.

3.2 Online Anomaly Detection

Despite the success of AEs in offline learning, only a few studies investigated their usage as online anomaly detectors for data streams.

Mirsky et al. [16] adapted the concept of AE ensembles [7] to the task of online learning and added a secondary downstream AE which reconstructs the loss values of the ensemble, forming the Kit-Net model. To assign data features to individual AEs in the ensemble, Kit-Net requires initial training data.

In the realm of online AD previous research has been focused on adaptations of well-established conventional machine learning techniques. Particularly popular in this context are streaming variants of LOF and IsolationForest. Based on Isolation Forests, Tan et al. [26] for instance proposed the Half Space Tree (HST) AD approach that builds an ensemble of trees by iteratively halving random attribute subspaces of an initial data subset. Other tree-based methods include RS-Forests [28] and Robust Random Cut Forest (RRCF) [12]. The xStream anomaly detector introduced by [15] elaborates on the concept of iterative subspace cuts by applying random matrix projections before dividing examples based on randomly selected features that were created in the process. LOF-based streaming AD techniques include Incremental Local Outlier Factor (iLOF) [20] and its advancements, MiLOF [24] and DILOF [18], both of which aim to overcome the loss of density data that would result from simply removing the oldest instances to maintain fixed memory and run time usage.

There are several studies on training AEs and other neural networks in a more general online learning setting [1, 22, 30]. However, none have investigated the online learning of AEs in the specific context of AD, which differs considerably from most conventional scenarios, since the often targeted minimization of the reconstruction loss may even be disadvantageous in the case of AD if losses for anomalies are affected by it.

To overcome the high sensitivity of neural networks to the learning rate, Baydin et al. [3] proposed Hypergradient Descent (HD) optimization algorithms, which simultaneously optimize network parameters as well as the learning rate using gradient descent. We evaluate the HD modification of Stochastic Gradient Descent (SGD) for our approach in Sect. 6.

4 Streaming Anomaly Detection with Autoencoders

According to Definition 1, we define an online AD system as depicted in Fig. 1. We define an anomaly detector as $g = \tau \circ f$, where $f: \mathbb {R}^d \rightarrow \mathbb {R}$ is an anomaly scoring function that judges the abnormality of its input in the form of an anomaly score $z^{(i)}$, and $\tau : \mathbb {R} \rightarrow \{0, 1\}$ is a thresholding function that decides whether to label the current input as an anomaly given $z^{(i)}$. Since the choice of an appropriate threshold mostly depends on the characteristics of the particular application, e.g., the relative cost of false-positive versus false-negative classifications, we omit the thresholding problem and focus on the computation of anomaly scores in this work.

Fig. 1.

Conceptual overview of proposed AE-anomaly scorer f within an incremental anomaly detector g.

While arbitrary AD models can implement the scoring function f, we aim to improve upon the performance of existing techniques by introducing basic-AE and Denoising AE (DAE) online anomaly scorers as well as a Probability Weighted AE (PW-AE) which we specifically develop to address the issue of contamination. In the following, we briefly explain the functional concept of these models. In accordance with R1, we begin the score calculation for the latest sample with calling the reconstruction function $r_{\boldsymbol{\theta }}$ on the model input $\boldsymbol{x}^{(i)}$ by performing a forward pass through the AE, generating a reconstruction $\boldsymbol{\hat{x}}^{(i)}$. Subsequently, we calculate the reconstruction loss $l^{(i)}=\mathcal {L}(\boldsymbol{x}^{(i)}, \boldsymbol{\hat{x}}^{(i)})$. To accommodate for fluctuations of average reconstruction losses, we scale $l^{(i)}$ by applying the post-processing function $\pi$, yielding an anomaly score $z^{(i)}$. For simplicity, we scale the losses by the average of a sliding window $\mu _{\omega }$ to obtain the anomaly score $z^{(i)}$. We further use the loss value $l^{(i)}$ to optimize the AE’s parameters $\boldsymbol{\theta }$ for every streaming instance by calculating a parameter update $\varDelta ^{(i)} \boldsymbol{\theta }$ with an optimization function opt, which we subsequently subtract from the current model parameters $\boldsymbol{\theta }$. For the optimization function opt any gradient-based optimization technique can be used, although we will subsequently assume standard SGD for the sake of simplicity. In Algorithm 1, we describe our approach towards online AD with the basic AE variant. For this type of model, we calculate the weight updates $\varDelta ^{(i)}\boldsymbol{\theta }$ according to the conventional SGD approach as

$$\begin{aligned} \varDelta ^{(i)}\boldsymbol{\theta } = \gamma \nabla _{\boldsymbol{\theta }} \, l^{(i)}, \end{aligned}$$

(2)

where $\gamma$ represents the learning rate and $\nabla _{\boldsymbol{\theta }} \, l^{(i)}$ the gradient of $l^{(i)}$ with respect to the model parameters $\boldsymbol{\theta }$. Apart from calculating a new reconstruction and training loss using dropout, we use the same update rule for the DAE as for the base variant. Through training with corrupted input data, DAEs were shown to be able to achieve more robust representations and thus better AD accuracy in previous studies [23].

From a probabilistic point of view, SGD optimizes the objective function

$$\begin{aligned} \min _{\boldsymbol{\theta }} \mathbb {E}_{(\boldsymbol{x},y) \sim \hat{p}_{\mathcal {S}}}\mathcal {L}(\boldsymbol{x},r_{\boldsymbol{\theta }}(\boldsymbol{x})), \end{aligned}$$

(3)

where $\boldsymbol{x}$ follows the empirical distribution defined by previously observed data $\hat{p}_{\mathcal {S}}$. Due to the fact, that Equation (3) rewards decreasing loss values for both normal and anomalous instances, minimizing the above objective does not directly correspond with maximizing the usefulness of an AE-model for the purpose of AD. Therefore, to establish a closer relationship between the training objective of the AE and its usefulness as an anomaly detector, we propose the alternative objective function

$$\begin{aligned} \max _{\boldsymbol{\theta }} \mathbb {E}_{(\boldsymbol{x},y) \sim \hat{p}_{\mathcal {S}}}\mathcal {L}(\boldsymbol{x},r_{\boldsymbol{\theta }}(\boldsymbol{x}) \, | \, y=1) - \mathcal {L}(\boldsymbol{x},r_{\boldsymbol{\theta }}(\boldsymbol{x}) \, | \, y=0), \end{aligned}$$

(4)

that is equivalent to the maximization of the expected margin between the scores of anomalous- and normal data. Empirically, this objective estimates to

$$\begin{aligned} \begin{aligned}&\max _{\boldsymbol{\theta }} \frac{1}{I} \sum _{i=1}^I y^{(i)}\mathcal {L}(\boldsymbol{x}^{(i)},r_{\boldsymbol{\theta }}(\boldsymbol{x}^{(i)})) - (1-y^{(i)})\mathcal {L}(\boldsymbol{x}^{(i)},r_{\boldsymbol{\theta }}(\boldsymbol{x}^{(i)}))\\ \Leftrightarrow&\min _{\boldsymbol{\theta }} \frac{1}{I} \sum _{i=1}^I (1-2y^{(i)})\mathcal {L}(\boldsymbol{x}^{(i)},r_{\boldsymbol{\theta }}(\boldsymbol{x}^{(i)})). \end{aligned} \end{aligned}$$

(5)

Most AE-based AD applications assume that training is performed on data, where the number of anomalous instances ($y^{(i)} = 1$) is close to zero. Under this circumstance Equation (5) approximates the standard AE optimization goal. However, this assumption is not necessarily valid for every phase of a data stream. Anomalies can, for example, concentrate in small time windows and therefore negatively affect the performance of a model trained to optimize the objective in Equation (3), as losses decrease for such instances. In the following, we derive an improved update rule from the objective defined in Equation (5). We refer to DAEs trained with this update rule as PW-AEs.

Since the true value of $y^{(i)}$ is unknown at the time of training, we substitute $y^{(i)}$ with an estimate of the probability $\hat{p}(y^{(i)}=1)$, that we calculate by assuming $l^{(i)} \sim \mathcal {N}(\mu _{\omega }, \sigma _{\omega })$, with $\mu _{\omega }$ and $\sigma _{\omega }$ being the average and standard deviation of a sliding window $\omega$ of previous losses. Under these assumptions, we estimate

$$\begin{aligned} \hat{p}_{\mathcal {N}}(y^{(i)}=1) = \Upphi (\frac{l^{(i)} - \mu _{\omega }}{\sigma _{\omega }}), \end{aligned}$$

(6)

where $\Upphi$ denotes the cumulative standard-normal distribution function. Although reconstruction losses cannot be accurately represented by a normal distribution, they could improve the sensitivity of AD to anomalies. To incorporate the prior knowledge that on average $p(y^{(i)}=1) \ll 0.5$ (see Definition 1), we calculate the final probability estimate $\hat{p}(y^{(i)} = 1)$ as

$$\begin{aligned} \hat{p}(y^{(i)} = 1)=\hat{p}_{\mathcal {N}}(y^{(i)}=1) * \frac{1}{2p_0}, \end{aligned}$$

(7)

where $p_0$ is a hyperparameter that determines the value of $\hat{p}_{\mathcal {N}}(y^{(i)} = 1)$ for which the weight update $\varDelta ^{(i)}$ assumes a value of 0. Using this probability estimate, the weight update for optimizing the objective in Equation (5) is given by

$$\begin{aligned} \begin{aligned} \varDelta ^{(i)} \boldsymbol{\theta }&= (1 - 2\hat{p}_{\mathcal {N}}(y^{(i)}=1) * \frac{1}{2p_0}) \gamma \nabla _{\boldsymbol{\theta }} l^{(i)} \\&= (1 - \frac{\hat{p}_{\mathcal {N}}(y^{(i)}=1)}{p_0}) \gamma \nabla _{\boldsymbol{\theta }} l^{(i)} \end{aligned} \end{aligned}$$

(8)

5 Experiments

We evaluated our approaches (AE, DAE, and PW-AE) based on 3 commonly used real-world datasets in comparison with state-of-the-art approaches for online AD in the following experiments:

Performance. We compare all approaches based on the area-under-the-curve measures of their Receiver Operating Characteristic (ROC-AUC) and Precision Recall (PR-AUC) curves as well as their runtimes.

Contamination Robustness. To evaluate the robustness against the degree of contamination, we present performance measures for different proportions of anomalies.

Parameters. Finally, we present an evaluation based on different design choices for the latent space, the optimizer, and the learning rate of AD.

In the following, we describe the used datasets and the experimental setting in which we evaluated our approach along with established online AD algorithms.

5.1 Data Streams

We conducted our experiments on incremental data streams which we emulated using the Covertype, Shuttle³ and Creditcard [8] real-world data streams. The key characteristics of the selected streams are presented in the following.

• Covertype originally consists of 7 classes representing different types of forest cover. Due to its non-stationary data distribution, Covertype is frequently used to emulate data streams in online learning (see e.g. [24, 26]. We modified the dataset according to the standard procedure from previous work on AD (e.g. [26]), where the most frequent class is selected as normal and the rarest class as anomalous, resulting in an anomaly share of 0.96%.⁴ We removed the remaining classes and categorical attributes, leaving a total of 10 numerical attributes. We then grouped the anomalous samples in short sequences of 2 to 9 samples and randomly distributed them throughout the data stream.

• Shuttle originally contains 8 classes. According to [26], we used the first class as normal data and samples of the remaining classes, except for the excluded fourth class, as anomalous. The share of anomalies in this modified dataset is 7.15%. Originally in time order, the dataset donor later randomized the order of data examples causing the dataset to exhibit a largely stationary data distribution [26].

• Creditcard [8] consists of credit card transactions performed in September 2013 out of which only 0.17% were marked as fraudulent, causing a highly imbalanced class distribution. Due to the confidentiality of the original transactions, the data is provided in the form of the first 28 principal components as well as the timestamp and monetary amount of each transaction. Since Creditcard features stream-typical characteristics such as concept drift, as well as a close relation to real-world fraud detection, the dataset was intensively investigated regarding online AD (see [8]).

Due to their different properties regarding the occurrence of concept drifts as well as their share of anomalous data, the selected data streams cover the necessary range of possible streaming-AD scenarios.

5.2 Setup

In this section, we present the experimental setup, necessary to obtain the results presented in Sect. 6. While the proposed AD algorithms can, in theory, be executed using any AE architecture and optimization algorithm, we used shallow networks with coupled encoder- and decoder weights and Scaled Exponential Linear Unit (SELU) activations along with basic SGD optimization for the sake of simplicity and computational efficiency. For calculating the models’ reconstruction errors, we used a smooth Mean Absolute Error (MAE) function, which we selected due to its advantage of being less prone to outliers compared to Mean Squared Errors (MSEs) [9].

For the performance evaluation, we used a learning rate of 0.02 for AE and DAE, and a higher value of 0.1 for the PW-AE to account for the adaptive reduction of its learning rate. Like [16], we determined the network width by using a latent layer ratio specifying the number of units in the hidden layer relative to the number of input features to account for the varying dimensionality between data streams. We evaluated the basic AE with a latent layer ratio of $10\%$ and, due to the regularization induced by applying 10% dropout, we chose $100\%$ for the DAE and PW-AE models. For the $p_0$ parameter of the latter variant, we used a value of 0.9.

We evaluate the proposed incremental AE anomaly detectors along with several well-established online anomaly detectors such as iLOF [20], HST [26], RRCF [12], xStream [15] and Kit-Net [16]. Except for xStream, we executed all reference algorithms using the authors’ suggested parameter values and restricted the sizes of all sliding windows to a maximum of 250 data points. We ran xStream with an improved configuration due to poor performance with its default setting. For Kit-Net, we used the first 100 examples in each data stream to map the input features to the individual AEs.

6 Results

In this section, we present the results for the experiments depicted in Sect. 5. Figure 2 shows an example of the distribution of individual anomaly scores generated by different AD models for Shuttle. Denote that the value range of the anomaly scores differs significantly between the individual models, which can be remedied by adjusting the value range of the anomaly threshold. While all models yield higher average scores for anomalous examples, it can be seen that the AE models provided the most significant gap between scores of anomalous and normal samples throughout most of the data stream segment. The ROC-AUC and PR-AUC scores shown in Table 1 reflect this observation. While the tree-based RRCF and HST come close to the AE models in terms of ROC-AUC, the latter clearly outperformed all other approaches in terms of PR-AUC. For Covertype and Creditcard, we found similar results in that the DAE and PW-AE models yielded significantly higher PR-AUC values than the baseline approaches while being among the highest performing models in terms of ROC-AUC. Their high performance on all three of the investigated datasets highlights the models’ ability to accurately identify anomalies for both data with and without concept drift and different degrees of class imbalance. While providing overall slightly less accurate anomaly scores, the basic AE also managed to exceed the baseline models’ PR-AUC for all datasets except for Covertype. The AEs were among the most computationally efficient models, using only a fraction of the runtime (R2 and R4) of most competing algorithms. Although being slower than Kit-Net on Covertype and Shuttle, the AE models’ runtimes were still within the same order of magnitude. For Creditcard, the runtimes were, on the other hand, significantly lower, which made the AEs on average faster than the competing algorithms when considering all datasets. Since the memory requirement of an AE is influenced by the choice of architecture rather than by data instances occurring over time, the memory consumption is constant and can be intentionally limited/selected by adjusting the network’s size. When comparing individual AE models, the DAE and PW-AE demonstrated a distinct advantage in terms of accuracy metrics. Especially for Covertype, the alternative model architecture led to significantly higher PR-AUC scores, which came at the cost of longer runtimes due to the additional forward pass needed to reconstruct the corrupted input for each training step. This advantage is also apparent by the distribution of scores displayed in Fig. 3, where the scores of anomalous data examples separate from the scores of normal data at a much higher rate at the start of the data stream. We achieved further improvements in discriminative performance by using the PW-AE update rule for Creditcard and - likely due to its higher grade of contamination - especially for Shuttle. With regard to R5, Fig. 3 shows, that the PW-AE adapted more quickly to normal samples in Shuttle than the DAE due to its higher default learning rate. While the DAE’s losses briefly collapse to a single range of values, the PW-AE’s losses remain almost perfectly linearly separable throughout the whole stream segment and separate at a much faster rate after the sudden increase of normal losses due to a faster adaptation to normal data and even a slight increase in losses for anomalies. Since the PW-AE’s training procedure reinforces the model’s concept of abnormality, its benefit likely depends on the accuracy of the underlying base model, which is supported by the performance gains on Shuttle for which the DAE and AE models are already able to label the majority of data accurately. Nevertheless, the distribution of anomaly scores demonstrates that our alternative PW-AE training approach can significantly widen the gap between the scores of anomalous and normal instances and therefore improve the separability of anomalies under the influence of contaminated training data as reflected by its increases in ROC-AUC, and PR-AUC.

Fig. 2.

Distribution of anomaly scores as a function of the number of processed instances on Shuttle. Anomalies are drawn in red. (Color figure online)

Table 1. Average benchmarking results for 10 random seeds. Best results are displayed in bold. Runtimes are given in core-minutes of an Intel(R) Xeon(R) Platinum 8180M CPU.

Fig. 3.

Distribution of unscaled AE anomaly scores concerning the number of processed instances on Shuttle. Anomalies are drawn in red.

Regarding the robustness of different contamination shares, Fig. 4 shows that the PW-AE appears to benefit from higher levels of contamination. As the proportion of anomalies randomly distributed in the Covertype or Shuttle data increases, the benefit of the PW-AE training procedure increases, leading to an increase in the ROC-AUC of the model compared to the architecturally identical DAE trained at a fixed learning rate. This observation supports the premise that our modified procedure can mitigate the effects of training on anomalous data.

Fig. 4.

ROC-AUC scores for varying anomaly percentages, which we generated by randomly sampling and inserting anomalies into the data. To allow for larger anomaly percentages, we used sampling with replacement. Each dataset created in this manner consisted of 50,000 examples.

Fig. 5.

ROC-AUC scores for variable learning rates $\gamma$ averaged over the first 50,000 samples of each dataset.

Fig. 6.

ROC-AUC scores for AE anomaly detectors with varying latent layer ratios on the first 50,000 samples of each dataset.

To investigate the effect of the parameterization, we ran evaluations for learning rates ranging from 0.001 to $0.001 \cdot 2^{8}$ and different optimizers, the results of which are illustrated in Fig. 5. When using conventional SGD, the basic- and denoising-AE on average produced the highest ROC-AUC for learning rates between 0.01 and 0.04, with both increases and decreases resulting in declines. Due to its adaptively decreasing step sizes, the PW-AE appears to benefit from larger base learning rates than the models using conventional SGD updates. While yielding marginally lower ROC-AUC at learning rates below 0.05, the PW-AE’s average ROC-AUC for higher learning rates exceeded those of the best-performing AE and DAE, indicating that the PW-AE’s performance advantage must be caused by its alternative training approach rather than a difference in learning rate. The optimal learning rates are relatively consistent across datasets, supporting the transferability of our results. Overall, the Adam [13] and HD-SGD [3] optimization algorithms did not provide significant advantages over basic SGD, since neither approach was able to improve the ROC-AUC or contribute to the robustness with respect to the base learning rate. Considering the additional computational complexity of Adam and HD-SGD compared to plain SGD, the latter is therefore likely the best suited out of the investigated optimization algorithms for online AD.

To investigate the impact of the network width on AD performance, we performed test-then-train evaluations with latent layer ratios between 10% and 200%. As displayed in Fig. 6 all model variants achieved ROC-AUC scores close to or greater than 0.96 for all investigated latent layer ratios, indicating that the performance of the proposed approach is rather robust to the choice of network width. In terms of differences between model variants, it appears that the basic AE favors lower latent layer ratios in some cases, which its lack of regularization could explain.

7 Conclusion and Future Work

In this work, we formalized the task of online AD based on predefined requirements for AD on streaming data, for which we introduced an AE-based framework. Our experimental results on three real-world datasets showed the proposed approach to yield significantly more accurate predictions than established baseline models, as is evident by their at least competitive or often superior ROC-AUC scores and their PR-AUC scores, which in the case of the most performant PW-AE and DAE regularly exceeded the scores of the next best non-AE model by more than $20\%$. Using only a fraction of the processing time required by most baseline models, the proposed models were also among the most computationally efficient models.

To mitigate the drawback of the conventional AE training objective in the case of contaminated training data, we further proposed an alternative AE objective for AD, which we used to derive a modified simple yet effective modification of the SGD update rule that reduces the learning rate for data instances that the AE deems to have a high chance of being an outlier. Our experiments indicate that the PW-AE models using this technique can reduce losses on normal data while maintaining higher losses on anomalous data compared to the models.

To investigate the effect of different design choices on the predictive performance of the proposed approach, we performed additional experiments, in which the online AE models showed relatively high robustness towards the choice of the learning rate and the network width (see Fig. 5 and 6). All in all, our experimental results also suggest that the proposed approach provides a promising basis for the development of even more effective online AD techniques. In this work, we showed a promising approach for online AD based on AEs that outperforms existing methods and can be further improved by tuning the model’s learning rate according to an estimate of the anomaly probability as demonstrated by our PW-AE. While we calculated this estimate under the assumption that the AE’s losses are normally distributed, a more tailored approach would most likely lead to more accurate estimates and therefore decrease the risk of selecting an incorrect learning rate. If available, label information could also be exploited to more precisely adjust the learning rate in a semi-supervised approach.