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[Page no. 17] "by DSEBM, where Figure 5a includes all the samples and Figure 5b includes the normal samples only. As shown above, normal and anomalous samples are mixed in the range of [−8, −7.8]. For samples in this range, it is difﬁcult to use the energy derived from the latent representation to separate them. D RECONSTRUCTION FEATURES IN DAGMM In this section, we detail the discussion on reconstruction features. Why reconstruction features are important? We realize the importance of reconstruction features from our investigation on a private network security dataset. In this dataset, normal samples are normal network ﬂows, and anomalies are network ﬂows with spooﬁng attack. As it is difﬁcult to analyze the samples from their original space with 20 dimensions, we utilize deep autoencoders to perform dimension reduction. In this case, we are a little bit ambitious, and reduce dimensions from 20 to 1. In the reduced 1-dimensional space, for some of the anomalies, we are able to easily separate them from normal samples. However, for the rest, their latent representations are quite similar to the representations of the normal samples. Meanwhile, in the original space, they are actually quite different from the normal ones. Inspired by this observation, we investigate their L2 reconstruction error, and obtain the plot shown in Figure 1. In Figure 1, the red points in the top-right corner are the anomalies sharing similar representations with the normal samples in the reduced space. With the additional view from reconstruction error, it becomes easier to separate these anomalies from the normal samples. In our study, this concrete example motivates us to include reconstruction features into DAGMM. What are the guidelines for reconstruction feature selection? In practice, one can select recon- struction features by the following rules. First, for an error metric used to derive a reconstruction feature,"

[Page no. 9] "the best performance. We detail their exact conﬁguration in Appendix A. 4.4 ACCURACY Metric. We consider average precision, recall, and F1 score as intuitive ways to compare anomaly detection performance. In particular, based on the anomaly ratio suggested in Table 1, we select the threshold to identify anomalous samples. For example, when DAGMM performs on KDDCUP, the top 20% samples of the highest energy will be marked as anomalies. We take anomaly class as positive, and deﬁne precision, recall, and F1 score accordingly. In the ﬁrst set of experiment, we follow the setting in (Zhai et al. (2016)) with completely clean training data: in each run, we take 50% of data by random sampling for training with the rest 50% reserved for testing, and only data samples from the normal class are used for training models. Table 2 reports the average precision, recall, and F1 score after 20 runs for DAGMM and its baselines. In general, DAGMM demonstrates superior performance over the baseline methods in terms of F1 score on all the datasets. Especially on KDDCUP and KDDCUP-Rev, DAGMM achieves 14% and 10% improvement at F1 score, compared with the existing methods. For OC-SVM, the curse of dimensionality could be the main reason that limits its performance. For DSEBM, while it works reasonably well on multiple datasets, DAGMM outperforms as both latent representation and reconstruction error are jointly considered in energy modeling. For DCN, PAE-GMM, and DAGMM-p, their performance could be limited by the pre-trained deep autoencoders. When a deep autoencoder is well-trained, it is hard to make any signiﬁcant change on the reduced dimensions and favor the subsequent density estimation tasks. For GMM-EN, without the reconstruction constraints, it seems difﬁcult to perform reasonable density estimation. In terms of PAE, the single view of reconstruction error may not be sufﬁcient"

[Page no. 18] "its analytical form should be continuous and differentiable. Second, the output of an error metric should be in a range of relatively small values for the ease of training the estimation network in DAGMM. In the experiment of this paper, we select cosine similarity and relative Euclidean distance based on these two rules. For cosine similarity, it is continuous and differentiable, and the range of its output is [−1, 1]. For relative Euclidean distance, it is also continuous and differentiable. Theoretically, the range of its output is [0, +∞). On the datasets considered in the experiment, we observe that its output is usually a small positive value; therefore, we include this metric as one of the reconstruction features. In sum, as long as an error metric meets the above two rules, it could serve as a candidate metric to derive a reconstruction feature for DAGMM. 17  Published as a conference paper at ICLR 2018 E CASE STUDY: WHEN JOINT TRAINING OUTPERFORMS DECOUPLED TRAINING? In this section, we perform a case study to investigate what kind of samples beneﬁt more from the joint training applied in DAGMM over decoupled training. In the evaluation, we employ PAE-GMM as a representative for the methods that leverage decoupled training, and the following results are generated from one run on the KDDCUP dataset. PAE-GMM detect PAE-GMM miss DAGMM detect 34,285 12,038 DAGMM miss 1,640 926 Table 4: The comparison of anomaly detection results between DAGMM and PAE-GMM Table 4 presents the comparison between DAGMM and PAE-GMM in terms of their anomaly detection results. In the testing data of this run, there are 48, 889 anomalies in total, where 34, 285 of them are detected by both techniques, 926 anomalies can be detected by neither of them, 1, 640 can only be detected by PAE-GMM,"

[Page no. 4] "curse of dimensionality, it is difﬁcult to directly apply such methods to multi- or high- dimensional data. Traditional techniques adopt a two-step approach (Chandola et al. (2009)), where dimen- sionality reduction is conducted ﬁrst, then clustering analysis is performed, and the two steps are separately learned. One of the drawbacks in the two-step approach is that dimensionality reduction is trained without the guidance from the subsequent clustering analysis, thus the key information for clustering analysis could be lost during dimensionality reduction. To address this issue, recent works propose deep autoencoder based methods in order to jointly learn dimensionality reduction and clustering components. However, the performance of the state-of-the-art methods is limited by over-simpliﬁed clustering models that are unable to handle clustering or density estimation tasks for data of complex structures, or the pre-trained dimensionality reduction component (i.e., autoencoder) has little potential to accommodate further adjustment by the subsequent ﬁne-tuning for anomaly detection. DAGMM explicitly addresses these issues by a sub-network called estimation network that evaluates sample density in the low-dimensional space produced by its compression network. By predicting sample mixture membership, we are able to estimate the parameters of GMM without EM-like alternating procedures. Moreover, DAGMM is friendly to end-to-end training so that we can unleash the full potential of adjusting dimensionality reduction components and jointly improve the quality of clustering analysis/density estimation. In addition, one-class classiﬁcation approaches are also widely used for anomaly detection. Under this framework, a discriminative boundary surrounding the normal instances is learned by algorithms, 3  Published as a conference paper at ICLR 2018 such as one-class SVM (Chen et al. (2001); Song et al. (2002); Williams et al. (2002)). When the number of dimensions grows higher, such techniques usually suffer from suboptimal performance due to the curse of dimensionality. Unlike these methods, DAGMM estimates"

[Page no. 12] "difﬁcult to reduce the reconstruction error for a deep autoencoder of the identical structure by end- to-end training (e.g., around 1.13 in terms of per-sample reconstruction error for KDDCUP). In other words, the compression network and estimation network mutually boost each others’ performance during end-to-end training: the regularization introduced by the estimation network helps the deep autoencoder escape from less attractive local optima for better compression, while the compression network feeds more meaningful low-dimensional representations to estimation network for robust 11  Published as a conference paper at ICLR 2018 density estimation. In Appendix C, for readers of interest, we show the visualization of the latent representation learned by DSEBM. In summary, our experimental results show that DAGMM suggests a promising direction for density estimation and anomaly detection, where one can combine the forces of dimensionality reduction and density estimation by end-to-end training. In Appendix E, we provide another case study to discuss which kind of samples beneﬁt more from joint training in DAGMM for readers of interest. 5 CONCLUSION In this paper, we propose the Deep Autoencoding Gaussian Mixture Model (DAGMM) for unsu- pervised anomaly detection. DAGMM consists of two major components: compression network and estimation network, where the compression network projects samples into a low-dimensional space that preserves the key information for anomaly detection, and the estimation network evaluates sample energy in the low-dimensional space under the framework of Gaussian Mixture Modeling. DAGMM is friendly to end-to-end training: (1) the estimation network predicts sample mixture mem- bership so that the parameters in GMM can be estimated without alternating procedures; and (2) the regularization introduced by the estimation network helps the compression network escape from less attractive local optima and achieve low reconstruction error by end-to-end training. Compared with the pre-training strategy, the end-to-end training could be more beneﬁcial"

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