Machine learning applications in fields such as financial accounting or the healthcare industry have to meet high transparency requirements for user acceptance and to meet the growing number of regulatory standards. Counterfactual explanations as a rather easy to interpret concept of local explanations combined with the generative power of Variational Autoencoder (VAE) and their ability to learn distributions of latent representations can ofer information to fulfill the needs of machine learning experts and non-expert users at the same time. Most current studies leveraging the power of deep generative models for counterfactual generation focus on vision data. We focus on anomaly detection applications on real world tabular data in the two high-risk fields of financial accounting and healthcare. We give an overview on constructions of counterfactual explanations and a categorization of current approaches to produce counterfactual explanations. We are investigating supervised extensions of the VAE for simultaneous classification and counterfactual generation. Therefor we explore the connection between diferent approaches of probabilistic modelling and separability properties in latent space. We discuss their applicability to anomaly detection and evaluation criteria.
Generative neural networks as a special form of deep learning have changed the way of data driven application in many fields, such as computer vision, robotics, and natural language processing. Specifically medical and financial industries ML-based approaches for decision support have to meet high requirements of transparency. Explainability methods are constructed to help the user of ml-methods to interpret the results of black-box models. In this PhD project we focus on counterfactual explanations as an local explanability approach and investigate, how this concept can be used to answer the users question: “What Should I have done diferently to change the outcome of the model prediction?” We investigate how counterfactual explanations can be integrated into the VAE method to generate exogenous counterfactuals in the context of anomaly detection.
The VAE was introduced by [
An explanation is called interpretable, when it describes the internals of a system in a way that is understandable to humans. The success of this goal is tied to the cognition, knowledge, and biases of a decision maker. For an explanation to be interpretable, it must give descriptions that are simple enough for a decision maker to understand using a vocabulary that is meaningful to the them. Note, that we will not measure the interpretability of a model. But we measure, how well the construction goals of the counterfactual are met by the approach used.
An overview over current approaches for calculating counterfactual explanations is given in [5]. The approaches can be categorized into model agnostic and model specific approaches. We will look at concepts specific to VAE.
This project aims to achieve the following goals: 1. get an overview on counterfactual construction objectives (CE-objectives) and categorize, how they can be integrated into a VAE training procedure, 2. develop metrics to evaluate, how good the CE-objectives can be achieved, 3. develop the best way to integrated the CE-objectives into the VAE-objective specifically for anomaly detection and 4. apply our method(s) on real world anomaly detection use cases from the healthcare and ifnancial sector.
Counterfactual explanations (CE) have a long history in philosophy, but [3] first mathematically formulated the counterfactual explanation as solution of an optimization problem for machine learning.
Definition 4.1 (Counterfactual explanation). Given a classifier that outputs the decision = () for an instance , a counterfactual explanation consists of an instance ′ such that the decision for on is diferent from , i.e., () ̸= and such that the diference between and ′ is minimal.
Since the first formulation, the requirements on this concept have evolved to meet practical considerations. In [5] these CE-objectives are formulated as: 1. Validity: A counterfactual ′ should actually changes the classification outcome. 2. Proximity: Given a distance function in the domain of , the distance between and ′ should be as small as possible. 3. Minimality: There should not be any other valid counterfactual example ′′ such that the number of diferent attribute value pairs between and ′ is higher than the number of diferent attribute value pairs between and ′′. 4. Plausibility: The counterfactual ′ should not be labeled as an outlier with respect to the instances in . 5. Diversity: Let = {′1, . . . ′} be a set of (valid) CE for the instance . The CE should be formed by diverse CE, i.e., while every CE ′ ∈ should be minimal and close to , the diference among all the CE in should be maximized. 6. Actionability: A CE ′ is actionable if all the diferences between and ′ refers only to actionable (mutable) features. This requirement links the concept of counterfactual explanation to the concept of algorithmic recourse. 7. Causality: Let be a directed acyclic graph (DAG) where every node models a feature and there is a directed edge from to if contributes in causing . The DAG describes the known causalities among features. Thus, given a DAG , a counterfactual ′ respects the causalities in iif ∀′ = (, ) ∈ ,′ such that the node in has at least an incoming/outcoming edge, the value maintains any known causal relation between and the values 1 , . . . , , where the features 1, . . . , identifies the nodes connected with in .
The VAE was introduced in [
• a value () is generated from some conditional distribution * (|).
We assume the prior * () and likelihood * (|) come from parametric families of distributions () and (|), and that their probability density functions (PDF(s)) are diferentiable almost everywhere w.r.t. both and . The true parameters * as well as the values of the latent variables () are unknown. Where the integral of the marginal likelihood () = ∫︀ () (|) is intractable, the true posterior density (|) = (|) ()/ () is intractable. Let us introduce a surrogate (|) ≈ (|) to approximate the intractable true posterior. We refer to (|) as probabilistic encoder and (|) as probabilistic decoder.
The VAE-objective aims to maximize the so called evidence lower bound (ELBO). ℒ(, ; ) = − ((|)|| ()) + E∼ (|) [log (|)] , (1) where is the Kullback-Leibler-divergence (KLD).
By using a neural network architecture for encoding and decoding and neural network
optimization (e.g. adam) [
1 1 2 2 3 3 ∼ (0, ) 1 2 = + 1 2 3 1 2 3 input layer parameter layer hidden layer output layer
In the current literature we can see modifications to the VAE architecture, to produce CE. There are modifications used to increase the interpretability of the network architecture. For example the use of invertable mappings form feature space to latent space (see [6] or [7]) or to encode a hierarchical latent sequence () to capture more complex causal structures in latent space (see [8]). There are modifications to the VAE-objective to include CE-objectives. This can be included by regularization (see [7] or by conditioning the prior distribution to control the CE candidate generation(see [9]). If the model is not explicitly trained to generate counterfactuals, the training objective aims to separate the diferent classes and use some perturbation (see [ 10]), projection (see [6]), interpolation (see [8]) or sampling technique (see [11]) to generate CEs. Due to reference space limitations, the overview of current literature is not comprehensive, but all current methods can be categorised into one of the following tree categories: • Include the CE-objectives in the VAE-objective and train the model to produce a CE. • Train the VAE to separate the classes. Use perturbation or projection technique to produce valid candidate(s). Select the candidate optimal under the CE-objectives. • Train diferent models on partitioned data. Use interpolation or sampling technique to produce valid candidates. Select the candidate optimal under the CE-objectives. Since the methods are designed for diferent data types, use data specific model modifications and try to meet diferent CE-objectives, a comprehensive comparison using the literature so far is not possible. None of the methods is designed or evaluated for anomaly detection use-cases with rare anomaly data.
After reviewing and grouping current approaches on counterfactual generation in VAE, we develop ideas on how to integrate the CE-objectives specifically for the anomaly detection usecase. Currently we are investigating the complementary supervised VAE approach introduced in [12]. We want to investigate it’s ability to separate anomalies from normal data and using it, to generate CE candidates. The concept in [12] is to train the VAE with normal prior and a loss function with so called standard KLD on a dataset with normal data. For training on seen anomaly data a complementary prior and corresponding KLD is chosen. The complementary set VAE approach follows the assumptions, that anomalies are regarded as the complementary set of the normal set and the normal region and the anomalous region are both mutually exclusive and collectively exhaustive. With defining () to be the PDF for normal data [12] construct () to be PDF of the anomalous data. It is constructed to satisfy the relationship () = 1
(max (′) − ()) ′ ′ where ′ is a norming constant such that (· ) is PDF. This construction satisfies the property of the complementary set, but ′ is infinity because the mass explodes. To ensure () is a PDF, we multiply () that is wide enough for each dimension. Then the density function is where is a finite normalizing constant
1 () = ()(max (′) − ())
′ ⏟ = ∫︁ ∞ −∞
⏞ =:*() *().
Using this as a prior, [12] expand the conventional unsupervised VAE into a supervised one to distinguish anomalies in the latent space. We choose the Standard Gaussian distribution (2) (3) (4) and max (; 0, 1) = √2 ′
=: 1 = ∫︁ ∞ −∞ (; 0, 2) · ( − (; 0, 1)) = 1
− ︃( √︂
1 2 + 1 )︃ .
The parameter 2 determines the width of the distribution. The multi-dimensional version is derived as a product of each dimension composed of the one-dimensional version. The function (; 0, 2) and equation (3) as a prior for the representation of normal samples ∼ (; 0, 1) with PDF (; 0, 1).
We construct the one-dimensional PDF () of the representation for anormal samples using the prior for the normal data representation (; 0, 1), the bounding Gaussian density (; 2) = 1 where the constants in this equation are described as (5) (6) (7) (8) complementary KLD can be approximately calculated as ((|; )||()) ≃ √︂ −
2 2 + 1 exp ︂(
− 2 )︂ 2( 2 + 1) + 2 + 2
22 log(2 + 1) − log + log + log(√︀2 + 1 − 1) (a) Standard KLD (b) Complementary KLD for s = 5 This VAE is trained alternating on a batch of normal data with standard prior and KLD and a batch of anormal data with complementary prior and complementary KLD. It is based on the idea, that both models share parameters. We want to investigate this training setup, because it can be used to detect also unseen anomalies. We have implemented the training procedure and want test it on synthetic data with given distributions, to validate this training setup.
We aim to give an overview and discuss diferent evaluation metrics for current approaches and for our model extensions. In next research steps we evaluate the training procedure for complementary set VAE, produce and evaluate CE-candidates and develop and evaluate diferent approaches to integrate CE-objectives in the VAE-objective. Specifically we aim to investigate modifications in conditioning the prior distribution and the causal model of the VAE, investigate modifications in optimization, e.g multi-criteria optimization, investigate diferent training settings and develop and discuss an evaluation scheme for synthetically generated data and for real world use case data. [2] Jinwon An, Sungzoon Cho, Variational autoencoder based anomaly detection using reconstruction probability, 2015. URL: http://dm.snu.ac.kr/static/docs/tr/snudm-tr-2015-03.pdf. [3] S. Wachter, B. Mittelstadt, C. Russell, Counterfactual explanations without opening the black box: Automated decisions and the gdpr, SSRN Electronic Journal (2017). doi:10. 2139/ssrn.3063289. [4] N. Burkart, M. F. Huber, A survey on the explainability of supervised machine learning, Journal of Artificial Intelligence Research 70 (2021) 245–317. URL: https://www.jair.org/ index.php/jair/article/view/12228. doi:10.1613/jair.1.12228. [5] R. Guidotti, Counterfactual explanations and how to find them: literature review and benchmarking, Data Mining and Knowledge Discovery (2022) 1–55. doi:10.1007/ s10618-022-00831-6. [6] W. Zhang, B. Barr, J. Paisley, An interpretable deep classifier for counterfactual generation, in: D. Magazzeni, S. Kumar, R. Savani, R. Xu, C. Ventre, B. Horvath, R. Hu, T. Balch, F. Toni, S. T. Kumar (Eds.), Proceedings of the 3rd ACM International Conference on AI in Finance (ICAIF’22), Association for Computing Machinery, New York, NY, 2022, pp. 36–43. doi:10.1145/3533271.3561722. [7] Deep structural causal models for tractable counterfactual inference, 2020. [8] B. Barr, M. R. Harrington, S. Sharpe, C. B. Bruss, Counterfactual explanations via latent space projection and interpolation, 2021. URL: https://arxiv.org/abs/2112.00890. arXiv:2112.00890. [9] M. Pawelczyk, K. Broelemann, G. Kasneci, Learning model-agnostic counterfactual explanations for tabular data, in: Proceedings of The Web Conference 2020, WWW ’20, Association for Computing Machinery, New York, NY, USA, 2020, p. 3126–3132. URL: https://doi.org/10.1145/3366423.3380087. doi:10.1145/3366423.3380087. [10] R. Balasubramanian, S. Sharpe, B. Barr, J. Wittenbach, C. B. Bruss, Latent-cf: A simple baseline for reverse counterfactual explanations, 2021. URL: https://arxiv.org/abs/2012. 09301. arXiv:2012.09301. [11] X. Xiang, A. Lenskiy, Realistic counterfactual explanations with learned relations, 2022.
URL: https://arxiv.org/abs/2202.07356. arXiv:2202.07356. [12] Y. Kawachi, Y. Koizumi, N. Harada, Complementary set variational autoencoder for supervised anomaly detection, in: 2018 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 2018, pp. 2366–2370.