A New Approach to Backtracking Counterfactual Explanations: A Unified Causal Framework for Efficient Model Interpretability

Thank you for your feedback on our paper. We look forward to exploring your suggestions further to enhance our work.

We do not dismiss the backtracking distribution. As stated in Equation (15) of our paper, our method can be viewed as a special case of Backtracking Counterfactual Explanations by using the backtracking conditional distribution presented there.

Regarding your questions about our experiments, please refer to our Answer for Q1, Answer for Q3, and Answer for Q4 in our rebuttal to Reviewer 7Lfk.

Concerning your question about generalizing our problem to cases where the input SCM is not fully known or is non-invertible: suppose that $\mathbf{F}(\cdot)$ is non-invertible and not fully known. In this scenario, after observing $x$, the posterior over $U$ is no longer a point mass. Instead, we can compute the distribution $U \mid x$ using the available causal relationships and the prior $P(U)$. An interesting scenario arises when the causal graph among variables is known, yet the specific functional forms are not. Following the approach in [1], we can assume a probability distribution (e.g., a Gaussian process) over each causal function. By combining this with the prior on $U$, we can compute the posterior distribution $U \mid x$. In this case, both $\mathbf{F}(\cdot)$ and $U \mid x$ become random, leading to the following optimization:

\begin{split} \arg\min_{u^{\mathrm{CF}}} \quad & E_{F} \left[d_X\left(x,F(u^{\mathrm{CF}})\right)\right] + \lambda \ E_{U \mid x}\left[d_U\left(U, u^{\mathrm{CF}} \right)\right] \ \quad \text{s.t.} \quad & E_{F} \left[h\left(F(u^{\mathrm{CF}})\right)\right] \ge \alpha \end{split}

In practice, heuristic methods are used to solve these optimization problems, as discussed in [1].

Can you better clarify the implications of the various assumptions made in Theorem 6.1 when they do not hold?

It should be mentioned that, to the best of our knowledge, Theorem 5.1 in our paper is the first result that relates backtracking and interventional counterfactuals, and it holds independent significance. Additionally, Theorem 6.1 is, as far as we know, the first result that connects causal algorithmic recourse with backtracking explanation methods.

Assumptions of linearity and convexity in our models are necessary because we require the vector optimization (Equation 20) to be convex. This convexity ensures that by varying $\lambda$, we can capture all Pareto optimal solutions, thereby guaranteeing that we also obtain the desired Pareto optimal solution in Equation (18). However, as mentioned in the paper (lines 295–305), even without any additional assumptions—relying solely on the ANM assumption—we can always find a better solution than causal algorithmic recourse among the Pareto optimal points of the vector optimization (Equation 20). The assumptions of linearity and convexity ensure that we can capture this Pareto optimal point for some value of $\lambda$. That said, many Pareto optimal points of a non-convex vector optimization can still be reached, and our required Pareto optimal solution (the solution of Equation (18)) might be one of them. In essence, linearity and convexity assumptions ensure we can capture all the Pareto optimal points and, consequently, our desired one.

Can you better clarify the potential challenges arising when optimizing Equation 24 with respect to classical differentiable recourse?

When optimizing Equation (24) generally, there is no need to predefine a feasible set of features. However, if we want to restrict the optimization to a specific subset of variables that we intend to change, we can select them and solve the optimization accordingly. This approach contrasts sharply with causal algorithmic recourse, where one must actively search for the optimal subset of variables to intervene on from all feasible options. Their method relies on combinatorial optimization; however, when we fix a set of feasible variables in our method, it essentially reduces to a hyperparameter choice. In other words, if one claims to solve causal algorithmic recourse while simply fixing the feature set instead of optimizing to identify the optimal variables for intervention, the solution lacks the clear intuition behind an interventional counterfactual.

[1] Karimi et al., Algorithmic recourse under imperfect causal knowledge: a probabilistic approach. Advances in Neural Information Processing Systems, 33:265–277, 2020.

Thank you for your thorough analysis and constructive feedback on our paper. We appreciate the opportunity to clarify the points raised and to provide additional insights into our research.

Answer for Q1: In our quest for inputs that are actionable for the user, it is crucial to take the user profile into account. When the user’s initial features are given by x = (female, 24, \$4308, 48), this suggests that a 48-month loan repayment is appropriate for the user. However, if we adjust this feature vector solely by reducing the repayment duration, the repayment becomes significantly more challenging for the user, thereby making the explanation less actionable.

To put it quantitatively, repaying \$4308 over 48 months corresponds to a monthly payment of \\$89.75. Any explanation that deviates considerably from this monthly rate is less actionable. Counterfactual Explanations (Wachter et al., 2017) and Causal Algorithmic Recourse (Karimi et al., 2021) yield a monthly repayment of about \$131.3 (i.e., \\$4308 divided by 32.8 months). In contrast, our solution results in a monthly repayment of \$123.8 when λ = 1 (i.e., \\$4087 divided by 33 months) and \$112.2 when λ = 1.2 (i.e., \\$3736 divided by 33.3 months). Thus, while Counterfactual Explanations and Causal Algorithmic Recourse increase the monthly repayment by 46.3%, our approach leads to increases of 37.8% and 25.0% for λ = 1 and λ = 1.2, respectively, making our recommendation more actionable for the user.

Answer for Q2: According to our optimization formulation (see Equation 26), by “significant departure” from the input we refer to the L1 norm difference, namely, $\left\| \mathbf{x} - \mathbf{x}^{\mathrm{CF}} \right\|_1$. This metric captures both the number of features modified and the aggregate relative change across those features. As you correctly mentioned, the Counterfactual Explanations method just minimizes this L1 norm, resulting in the smallest overall deviation from the original input. However, as highlighted in the Problem Definition point 3 (line 102), it is also essential to incorporate causal relationships among the input features. Consequently, our solution seeks an alternative that is not only close to the original observation in terms of the L1 norm but is also consistent with the underlying causal graph, thereby providing an actionable insight for the user.

Answer for Q3: The following results demonstrate the performance of the Deep Backtracking Explanation method when identical noise is introduced:

(female, 27.17, \$2696, 35.8)

(female, 27.14, \$2752, 35.7)

(female, 27.18, \$2831, 35.6)

As shown, the deviations are minimal in this method.

Given that the causal algorithmic recourse method employs combinatorial optimization—which is typically sensitive to noise—we initially expected the introduction of noise to yield more dramatic changes in this method.

It is important to note that when the counterfactual explanation only modifies the sink nodes of our causal DAG (like the example in our paper), the outcomes of the counterfactual explanation and the causal algorithmic recourse methods become identical. This is because interventions on these sink nodes are sufficient for interventional counterfactuals in causal algorithmic recourse optimization. In such cases, adding noise is less likely to alter the set of required interventions, so causal algorithmic recourse would be more robust to noises.

However, as the added noise alters the causal graph, the solution of the causal algorithmic recourse shifts slightly. Under the same noise conditions, the new solutions of causal algorithmic recourse are:

(female, 24, \$4353, 32.7)

(female, 24, \$4192, 32.9)

(female, 24, \$4432, 32.6)

Although these shifts are not dramatic, we anticipate that in more complex cases—where combinatorial optimization plays a larger role—the impact of noise could be more pronounced.

Answer for Q4: For a high-risk individual with attributes (male, 23, \$15,672, 48), our solution for explaining the model's behavior and providing actionable insights—using$\lambda = 1.2$—is (male, 23, \\$15,116, 33.7). Under the same noise conditions described in our paper, our method yields the following results:

(male, 23, \$15,253, 33.6)

(male, 23, \$15,055, 33.8)

(male, 23, \$14,953, 33.9)

For another high-risk individual with attributes (female, 24, \$7,408, 60), our solution with$\lambda = 1.2$is (female, 24, \\$6,273, 30.9). When the same noise is applied, the method produces:

(female, 24, \$6,214, 31.0)

(female, 24, \$6,504, 30.7)

(female, 24, \$6,371, 30.8)

Similarly, for another high-risk individual (male, 27, \$14,027, 60), our solution for$\lambda = 1.2$is (male, 27, \\$13,149, 37.4). Under identical noise conditions, our method yields:

(male, 27, \$13,077, 37.5)

(male, 27, \$12,984, 37.6)

(male, 27, \$13,274, 37.3)

These results underscore the stability of our approach across different input instances.

Thank you for your insightful remarks regarding our paper. We truly appreciate your thoughtful feedback and look forward to exploring your suggestions further to enhance our work.

The current paper presents a method that is limited to additive noise models.

First, we note that our method applies to any case where $\\mathbf{F}(\\cdot)$ is invertible- a condition that is more general than the Additive Noise Models (ANM). While we require the ANM assumption to prove our theorems and to establish the connection between our method and causal algorithmic recourse, the practical application of our method only necessitates that $\\mathbf{F}(\\cdot)$ is invertible.

Why is it bad that a method recommends reducing loan duration alone?

For this question, please first refer to Answer for Q1 of our rebuttal to Reviewer 7Lfk.

Additionally, we observe similar behavior in other instances. For example, for a high-risk individual with attributes (male, 27, \$14,027, 60), our solution for$\lambda = 1.2$is (male, 27, \\$13,149, 37.4). In comparison, the solution provided by Counterfactual Explanations and Causal Algorithmic Recourse is (male, 27, \$14,027, 36.6), while the Deep Backtracking Explanations method yields (male, 31.9, \\$11,626, 41.1).

For another high-risk individual with attributes (female, 24, \$7,408, 60), our solution with$\lambda = 1.2$is (female, 24, \\$6,273, 30.9). In this case, the solution from Counterfactual Explanations and Causal Algorithmic Recourse is (female, 24, \$7,408, 29.8), and the Deep Backtracking Explanations method produces (male, 30.1, \\$4380, 35.5).

We believe these results underscore the practical relevance of our approach in delivering actionable insights.

For your question about robustness, please refer to Answer for Q3 and Answer for Q4 of our rebuttal to Reviewer 7Lfk.

Assumptions of linearity and convexity in models is increasingly limiting in the modern age of machine learning.

It should be mentioned that, to the best of our knowledge, Theorem 5.1 in our paper is the first result to relate backtracking and interventional counterfactuals, and it holds independent importance. Additionally, Theorem 6.1 is, as far as we know, the first result that relates causal algorithmic recourse with backtracking explanation methods.

Assumptions of linearity and convexity in our models are necessary because we require the vector optimization (Equation 20) to be convex. This convexity ensures that by varying $\\lambda$, we can capture all Pareto optimal solutions, thereby guaranteeing that we also obtain the desired Pareto optimal solution in Equation 18. However, as mentioned in the paper (lines 295–305), even without any additional assumptions—relying solely on the ANM assumption—we can always find a better solution than causal algorithmic recourse among the Pareto optimal points of the vector optimization (Equation 20). We need the assumptions of linearity and convexity to ensure that we can capture this Pareto optimal point for some $\\lambda$. That said, many Pareto optimal points of a non-convex vector optimization can still be reached, and our required Pareto optimal solution (the solution of Equation 18) might be one of them. For example, consider [1, Figure 4.9]. In the figure, all Pareto optimal points are shown as a bold line. Although the problem is non-convex, we can capture $f_0(x_1)$ and $f_0(x_2)$ with $\\lambda_1$ and $\\lambda_2$, respectively; however, there is no value of $\\lambda$ that can capture $f_0(x_3)$. We know that one of the points on the bold line is our desired point (Equation 18) under the ANM assumption. The linearity and convexity assumptions ensure that we can capture all the Pareto optimal points and, as a result, our desired one.

[1] Boyd, Stephen P., and Lieven Vandenberghe. Convex Optimization Book. Cambridge university press, 2004.

Thank you for your kind feedback and positive review of our work. We appreciate your recognition of the theoretical contributions, experimental design, and overall rigor of our paper.

Regarding your question on how our algorithm would perform on large state-of-the-art models, we acknowledge that this is an important aspect to consider. As mentioned in lines 422–428 of section 10 of our paper, we recognize the significance of evaluating our method on more complex models. While our current experiments were conducted on a benchmark dataset for algorithmic recourse, we plan to extend our investigations to include large-scale, state-of-the-art models in future work. A comprehensive evaluation in this context remains an exciting direction for our future research, and we are eager to explore how our approach performs in real-world, complex model scenarios.

Thank you again for your encouraging feedback.