Generative Models for Counterfactual Explanations

Abstract

Counterfactual explanations have emerged as an efective method of explaining machine learning models. These explanations elucidate how to tweak the model input in order to flip its output. Generative approaches serve as a tool for creating meaningful counterfactuals for complex problems, where other methods fail or require too much computation. This work presents an overview of generative approaches and their applications in the generation of counterfactual explanations. We highlight the prevailing challenges, such as diversity and distinction from adversarial examples, and identify open questions with future research directions, such as ensuring the stability of counterfactuals and automatic reasoning with counterfactual explanations.

Keywords

• eol>Counterfactual Explanations
• Generative Models
• Explainable AI

1. Introduction

Counterfactual explanations clarify complex system decisions by answering "what if" scenarios, showing how minimal input changes can lead to diferent outcomes [ 1 ]. This is crucial in Machine Learning (ML), where understanding the rationale of a model is as important as the decision itself [ 2 ]. By examining hypothetical alternatives, counterfactual explanations make ML models’ decision-making more transparent and comprehensible.

Despite growing interest in counterfactual explanations, there is a gap in the literature on the generative methods used to create them. Variational Autoencoders (VAEs) [ 3 ], Generative Adversarial Networks (GANs) [ 4 ], and Denoising Difusion Probabilistic Models (DDPMs) [ 5 ] are notable for generating counterfactuals, especially for complex data modalities such as images, where tweaking uninterpretable features fall short. However, existing surveys often overlook the generative aspects or high-dimensional data scenarios [ 6, 7, 8 ]. Our work addresses this gap by focusing on generative models for counterfactual explanations in complex data, ofering a comprehensive understanding of their capabilities and limitations.

In this paper, we explore the common use cases of generative models for counterfactual explanations and highlight primary challenges. We categorize methods by their generative techniques and examine modifications to standard processes to meet counterfactual requirements. Our discussion aims to stimulate further research by identifying key challenges and potential directions for advancing generative methods in counterfactual explanations. While decision boundary data space perturb latent space counterfactual generation is often seen through the lens of causal generative modeling [ 9 ], we focus on noncausal approaches.

2. Counterfactual Explanations

Counterfactual explanations are essential for explainability in machine learning [ 6, 1 ]. Formally, given a dataset with corresponding labels , a counterfactual for a query sample q ∈ is an alternative sample cf ∈ that results in a diferent outcome ∈ under a predictive model : → , providing insight into model behavior. The quality of a counterfactual explanation depends on several conditions [ 1 ]. Proximity and validity require a counterfactual to be as similar as possible to the query sample q but lead to a diferent desired outcome. Meanwhile, plausibility and actionability demand that the suggested modifications be meaningful and practical to users. For example, suggesting a reduction in age is impractical, as age cannot be reversed.

Counterfactuals and adversarial examples. To further understand counterfactual explanations, it is useful to compare them with adversarial examples, since both involve modifying the original samples to change the output of a model. Counterfactual explanations and adversarial examples modify the original samples to change the output of a model but difer in their objectives. Counterfactuals introduce semantically reasonable changes to provide meaningful insights, while adversarial examples use subtle, imperceptible perturbations to mislead the model. Distinguishing between them is challenging. Thus, it is crucially important for counterfactual approaches to ensure modifications that are perceptible and semantically significant [ 10, 11 ].

a) c)

VAE

Optimization methods Latent perturbations Modality-specific VAEs

DDPM b)

GAN Explicit guidance Textual inversion

Semantic decomposition Cycle-consistency

GANs

StyleGAN modifications domains like images or time-series is challenging. In tabular data, ensuring properties like validity and feasibility is straightforward. However, high-dimensional data with non-interpretable features poses significant dificulties. Generative models capable of approximating data distributions ofer a promising solution. Figure 1 illustrates common uses of VAEs, GANs, and DDPMs for counterfactual generation. These models can satisfy counterfactual conditions through specific modifications and regularizations, such as incorporating distance metrics for validity or controllable generation techniques for actionability.

3. Variational Autoencoders

VAEs are useful for counterfactual generation by approximating data density (|), ensuring modifications remain within the data distribution (see the Appendix A for details). Since VAEs can produce interpretable latent factors, they are useful for counterfactual explanations. Our taxonomy for VAE-based counterfactual generation methods includes optimization methods, which employ optimization of an expression defining a good counterfactual while using VAE to stay in the data manifold; latent perturbations, which encode a sample into a latent space, modify it, and decode to obtain a counterfactual; we also mention some modality-specific VAEs to highlight the versatility of this approach.

Optimization methods.

This is the most common approach to use density approximation for counterfactual explanations [ 1, 6 ]. They optimize an expression involving a classifier , desired outcome , classifier loss ℓ, and a cost function that enforces desired properties, balanced by : ℒcf = ℓ(︁ (︀ (|))︀ , ︁) + (︀ q, (|))︀ , cf = arg min ℒcf (1) The counterfactual explanation is derived by cf ∼ (|cf). Despite using the learnable posterior (|), stochastic optimization process can result in being outside the prior (), making generating actionable counterfactuals a challenge.

Latent perturbations. [ 12 ] introduced a conditional VAE with factorized encoder and decoder. It uses mixture priors for clustering in the latent space. Counterfactuals are generated by small perturbations to the latent representation and reconstruction through the decoder, ensuring proximity and high-density data alignment. [ 13 ] proposed an approach to generate non-trivial, diverse explanations by varying less influential latent factors.

Modality-specific VAEs. [ 14 ] presented a framework for counterfactual explanations for graph ML models using a conditional graph VAE. It handles graph data challenges and generalizes to out-of-distribution graphs. [ 15 ] focused on anomaly detection in multivariate time-series, segmenting the latent space into general and salient components with supervised contrastive loss. Counterfactuals replace the salient component with a healthy latent prototype, estimated using kernel density estimation.

4. Generative Adversarial Networks

The capability of GANs to generate high-quality samples makes them useful for realistic counterfactuals. However, they face challenges with unstable training and mode collapse, limiting diversity [ 4, 16, 17, 18 ]. We categorize GAN-based approaches into four groups: Conditional GANs, where the generator combines latent noise , encoded features from the original sample q, and the class label to generate counterfactuals: cf = (, q, ); Semantic decomposition, which involves segmenting original images into meaningful regions and treating each region individually during generation, these approaches utilize the individual editing of semantically distinct regions to enhance actionability; Cycle-consistency GANs, which produce a counterfactual and its reversal (as a counterfactual with respect to the counterfactual), aligning the reversal with the original sample; and StyleGAN modifications , which use StyleGAN modifications for detailed and high-quality counterfactuals.

Conditional GANs. [19] introduced a GAN-based approach for counterfactual generation by finding the latent encoding of a query image and using a class-conditional GAN to produce three instances: a reconstructed original, a modified image, and a change mask. The final counterfactual blends the original and modified images based on the predicted mask. [ 20] used typical conditional GAN training with additional constraints to ensure validity and counterfactual properties. [21] modified the GAN architecture, training the generator to output residuals instead of complete data points. [22] used an external classifier as a discriminator to improve robustness against adversarial attacks.

Semantic decomposition. [23] decomposed counterfactual generation into three components: background, foreground, and object mask, using a conditional GAN for each. [24] used semantic maps and embeddings to generate counterfactuals. [25] combined conditional GANs with saliency maps to target specific regions for modification.

Cycle-consistency GANs. [26] used cycle-consistency loss to ensure coherence and reversibility of counterfactual changes. [27] added latent concept vectors for disentangled concept learning. [28] enforced cycle-consistency between original and counterfactual latent embeddings.

StyleGAN modifications. StyleGAN [29] uses latent style vectors for image generation. [30] combined StyleGAN vectors with classifier outputs for counterfactuals. [ 31] integrated a style vector with a CLIP [32] embedding to allow user-defined modifications in natural language.

5. Denoising Difusion Probabilistic Models

DDPMs have emerged as the state-of-the-art generative approach, particularly useful for counterfactual generation (see the Appendix C for details). We classify these approaches into two main groups: explicit guidance and textual inversion methods. Explicit guidance methods exploit an external diferentiable classifier (|) to replace the original score function ∇ log () with a conditional one ∇ log (|) to generate specific counterfactuals. Textual inversion is a technique used in text-conditioned generative models, where unique trainable embeddings are assigned to images that share common concepts. These embeddings are optimized so that, when used as conditioning inputs, the generative model produces images that are similar to the original ones, efectively capturing and reproducing the shared concepts. Counterfactual generation here involves modifications of the original concept embedding. Explicit guidance. Advances in DDPMs have led to innovative applications in counterfactual generation, such as DiME [33] and DVCE [34]. These methods use classifier-guidance during the difusion process. Since difusion models operate with noised samples, DVCE employs a one-step denoising approximation, while DiME uses multiple iterations. [35] emphasized the significance of techniques for handling noised samples, such as gradient cone projections and intermediate denoising steps. [36] introduced a two-stage approach using classifier feedback for initial image modifications, followed by iterative denoising with DDPM. [ 37] utilized the latent difusion method from [ 38], known for computational eficiency by operating in a lowerdimensional latent space. The introduced consensus guidance mechanism filters gradients to ensure plausible counterfactual changes.

DDPMs gained wide application in medical image processing. One significant task addressed with counterfactual generation is the medical anomaly detection, which involves identifying disease-specific regions within samples. [ 39] and [40] used reverse difusion with classifier guidance to generate healthy counterparts of pathological images. [41] combined DDIM [42], DDPM, and saliency maps for precisely targeted modifications. [ 43] explores counterfactual generation for fMRI data, using a transformer-only model for long-range dependencies and a modified difusion sampling process for enhanced eficiency.

Textual inversion. Textual inversion [44] learns distinctive tokens for specific classes or concepts, enabling refined control over the generation process. [ 45] used this technique for counterfactual generation by combining learned concept tokens with additional counterfactual shift. [46] applied textual inversion to visual counterfactuals by learning concept embeddings and prompts for predefined objects or classes. In contrast to the previous approaches, this method implemented counterfactual modifications as transitions between concepts.

6. Challenges, open questions, and future directions

6.1. Challenges

Dealing with adversarial perturbations. All methods for counterfactual explanations vary significantly in their approach to ensuring semantically reasonable modifications. Some strategies use discriminative models that are adversarially resilient [34]. Other approaches apply changes within a structured latent space [ 37, 28, 13 ], resulting in more feasible and meaningful modifications. Another promising method involves the use of a structured latent space with representations of concepts as individual entities present in a sample [47, 48], learned with or without supervision [46, 45, 27].

Diversity of generated explanations. In high-dimensional data, generating diverse counterfactual explanations is crucial but rarely addressed [ 13 ]. Multiple plausible changes can lead to the same desired outcome, making it a complex task to consider all possible directions. However, too many diverse explanations can overwhelm users and hinder decision-making, requiring a balance between variety and interpretability. The challenge is to combine potential alterations, identify the most reasonable and relevant ones, and allow users to choose specific directions of change without being inundated. This diversity of counterfactuals, vital for comprehensive explainability, remains an open area for research and development. However, providing users with multiple diverse explanations can invoke problems related to the Rashomon Efect [ 49], where diferent potentially contradictory explanations may cause potentially contradictory interpretations of the same phenomena.

6.2. Open questions

Are counterfactuals stable? The stability of counterfactuals — ensuring minimal changes in the input sample do not lead to disproportionate changes in the output—remains a critical challenge. Research in tabular data has revealed vulnerabilities, where slight manipulations in the input can drastically alter the counterfactual [50]. This raises the question of whether generative approaches, increasingly used for counterfactual explanations, exhibit similar instability or ofer more robust solutions. Understanding and improving the stability of these models is crucial for their reliability and trustworthiness in practical applications.

How to evaluate generated counterfactuals? Evaluating counterfactual explanations in AI is challenging due to a disconnect between theoretical metrics and real-world applicability. Traditional metrics, focusing on conditions such as proximity to factual instances or diversity, do not necessarily translate to practical utility for end-users [51]. Research [52] has shown that while counterfactual explanations may satisfy theoretical criteria, their impact on user trust and understanding is inconsistent. To address this, recent approaches integrate users and domain experts into the evaluation process [34, 53]. This expert-in-the-loop methodology aligns theoretical constructs with practical realities, especially in critical areas like healthcare, providing a more comprehensive evaluation of counterfactual explanations’ utility. However, a universal benchmark for comparing diferent methods is still missing.

6.3. Future directions

Modalities and multimodal counterfactual explanations. Much of the current research

on counterfactual explanations focuses on image data. Extending these methodologies to other modalities, such as graphs, time-series, audio, and video is an important challenge. In addition, there is a gap in methods capable of handling multimodal data, which is increasingly prevalent in practical applications. Developing techniques that can generate counterfactuals across various modalities, or even within multimodal contexts, is a critical area for future exploration.

Grounding and reasoning in counterfactual explanations. The typical process of gener

ating counterfactuals often lacks detailed explanations for why certain changes lead to specific outcomes, leaving users to interpret these changes on their own, which can lead to misunderstandings. Recent advances in Large Language Models (LLMs) with multimodal capabilities ofer a promising solution to this issue [ 54]. Integrating advanced LLMs with counterfactual generation can enhance user comprehension by providing reasoning for suggested changes. This aligns with the concept of Evaluative AI [55]. Eforts like [ 56] highlight the versatility of LLMs in improving model explainability, especially with tabular data. Combining various forms of explanations, including semi-factuals [57], with multimodal LLMs conditioned on specific scenarios, could lead to more comprehensive and transparent AI systems.

7. Conclusion

This work highlights the crucial role of generative models in producing counterfactual explanations for high-dimensional data. We emphasize the need for semantically rich, intuitive explanations and robust user-centered evaluation describing existing approaches. We discussed future research directions, which include application of counterfactuals across diverse data modalities and integrating them with LLMs and other explanatory methods.

Acknowledgment

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A. VAE Background

Introduced by [ 3 ], VAE has become a significant tool in deep learning for training latent variable models through variational inference. A VAE comprises an encoder and a decoder, with its primary objective typically being the minimization of the reconstruction error of data samples. The encoder, parameterized by trainable parameters , approximates a variational distribution (|), where is a latent variable with a prior distribution (), often chosen as the standard Gaussian distribution (0, ). The decoder models (|). VAEs are trained by maximizing the Evidence Lower Bound (ELBO), a variational lower bound of the exact log-likelihood: log () ≥

ELBO = E(|)[log (|)] − ︀( (|) ‖ ())︀ , (2) where is Kullback–Leibler divergence.

B. GAN Background

(3) (4) In contrast to VAEs, GANs [ 4 ] operate on a diferent principle, as they do not explicitly learn the likelihood of data samples. Instead, GANs employ two neural networks: a generator and a discriminator . The generator maps input noise, sampled from a distribution () = (0, ), to the data space with the objective of learning the distribution of the generator on the data samples. The discriminator , on the other hand, aims to estimate the probability that a given data sample originated from the actual data distribution data. The training of involves distinguishing real samples drawn from data and generated samples from . Concurrently, is trained to maximize the probability of its generated samples being misclassified by , efectively minimizing log(1 − (())). This training process sets up a minimax zero-sum game between and , where each network continuously improves its performance in response to the other, leading to the generation of increasingly realistic samples: min max

E∼ data()[log ()] + E∼ ()[log(1 − (()))] . Compared to VAEs, GANs sample the latent code from the same prior distribution during both training and inference, and are capable of generating more complex and high-fidelity data samples [29].

C. DDPM

Background

In recent years, Denoising Difusion Probabilistic Models (DDPMs) have solidified their position as a leading framework in generative modeling [ 5 ]. The central component of DDPMs is an iterative process where noise is incrementally added to a data sample 0, transforming it into a complete noise sample ∼ ( ) = (0, ): (|− 1) = (√︀1 − − 1, ), where { 1, . . . , } is a predefined variance schedule. This is coupled with a learned reversal of this process: (5) (6) (7) (8) (9) This procedure results in training a so-called score function ∇ log (). The resultant sampling procedure is executed by initial sampling of ∼ (0, ), followed by a sequence of − 1 ∼ (− 1|) until the final 0 ∼ (0|1). This technique, involving multiple trainable denoising iterations, empowers DDPMs to produce outputs that are both highly detailed and diverse, setting them apart in the generative modeling arena.

A notable feature of DDPMs, with significant potential for counterfactual generation, is the development of a classifier guidance mechanism [ 5 ]. This approach diverges from traditional generative models, which typically necessitate explicit conditioning during the training phase. Guided difusion introduces a paradigm where an unconditional generative model is first trained. Subsequently, this model is adapted for conditional sampling via the integration of an auxiliary classifier model by replacing (− 1|) with

(− 1|) = ︀( (, ), Σ (, ))︀ , where (, ) and Σ (, ) are predicted by models parameterized with learnable parameters . Since DDPMs, similar to VAEs, represent latent variable models, they are trained by optimizing the following variational lower bound:

, (− 1|, ) ∝ (− 1|)(|, ), where (|, ) is an external model to be explained. This strategy provides remarkable lfexibility in conditional generation but introduces a pivotal challenge: the classifier must be either trained on noise-augmented samples to align with the DDPM’s intermediate stages, or a denoising mechanism should be applied prior to classifier usage.

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