Paper: An Entropic Risk Measure for Robust Counterfactual Explanations (ICLR-2024)

Reviewer comment (Writing):
Readability and understandability could be improved by describing the final algorithm in more detail and in a more structured way (e.g., using pseudo-code). Right now it is only described in textual form in the paragraph "Algorithmic Strategy".

Author response:
**(3 of 3) Regarding readability and understandability**: We acknowledge the importance of a readable and understandable manuscript. We are committed to incorporating your suggestions.

Reviewer comment (Evaluation):
What about the computational complexity? How does it compare to the other methods? This is only briefly mentioned in the end.

Author response:
**(1 of 3) Regarding the gradient and computation complexity:** The algorithm does not use the actual gradients but only the estimates. Our approach shares comparable computational complexity with T-Rex. As discussed in Section 5, it is important to note that the reliance of our method on sampling and gradient estimation introduces a downside, namely, an elevated level of computational complexity.

Reviewer comment (Theory):
Are true gradients always needed? This assumes access to the model. What about black-box models? This is also only briefly mentioned in the end.

Author response:
**(1 of 3) Regarding the gradient and computation complexity:** The algorithm does not use the actual gradients but only the estimates. Our approach shares comparable computational complexity with T-Rex. As discussed in Section 5, it is important to note that the reliance of our method on sampling and gradient estimation introduces a downside, namely, an elevated level of computational complexity.

Reviewer comment (Evaluation):
Plausibility and actionability are also very important aspects in recourse — how could these be added to the proposed method?

Author response:
**(2 of 3) Regarding the inclusion of additional metrics:** We appreciate the reviewer's insight regarding the inclusion of additional metrics like actionability, plausibility, and fairness. This consideration is acknowledged in the final sentence of the first paragraph of Appendix C, where we recognize the importance of addressing these factors beyond explainability and robustness. Dealing with multi-criteria optimization involves techniques from multi-objective optimization, and we plan to delve into the necessary trade-off analysis using multi-criteria optimization in future research. Thank you for sharing this with us. We make sure to explain these thoroughly in an earlier part of the paper.

Reviewer comment (Evaluation):
The paper is not able to convince the reader that using this risk measure is an improvement compared to the state-of-the-art methods.

Author response:
**(2 of 5) Regarding the benchmarking**: Our main contribution is to demonstrate a theoretical connection between our method (entropic-risk-based approach) and the worst-case robust counterfactuals (min-max optimization), e.g. ROAR. Our experiments simply aim at showcasing that our method performs at par with the state-of-the-art, SNS algorithm, while being grounded in the solid theoretical foundation of large deviation theory, without relying on any postulations; enabling a unifying theory for robust counterfactuals. We acknowledge the existence of several intriguing techniques in robust counterfactuals that might surpass our strategy in specific scenarios. However, the practical implications of model change remain uncertain. Our fundamental innovation lies in making a theoretical contribution that facilitates a nuanced trade-off with robustness.

Reviewer comment (Evaluation):
While the approach provides a nice theoretical connection to the worst-case approach, the experiments actually show that the SNS method leads to a much better trade-off between costs and robustness. While the SNS method has only a small reduction in robustness, the reduction in costs is significant.

Author response:
**(2 of 5) Regarding the benchmarking**: Our main contribution is to demonstrate a theoretical connection between our method (entropic-risk-based approach) and the worst-case robust counterfactuals (min-max optimization), e.g. ROAR. Our experiments simply aim at showcasing that our method performs at par with the state-of-the-art, SNS algorithm, while being grounded in the solid theoretical foundation of large deviation theory, without relying on any postulations; enabling a unifying theory for robust counterfactuals. We acknowledge the existence of several intriguing techniques in robust counterfactuals that might surpass our strategy in specific scenarios. However, the practical implications of model change remain uncertain. Our fundamental innovation lies in making a theoretical contribution that facilitates a nuanced trade-off with robustness.

Reviewer comment (Writing):
p.4 first paragraph: l is defined on M\times X but takes only one input M(x).

Author response:
**(5 of 5) Regarding Minor issues**: Thank you for bringing these to our attention. We have promptly corrected them.

Reviewer comment (Writing):
Example 1 and 2 need explanation (or citation).

Author response:
**(5 of 5) Regarding Minor issues**: Thank you for bringing these to our attention. We have promptly corrected them.

Reviewer comment (Writing):
p.7, something is missing in the sentence "However, in cases where our setup permits us to assume that both the distributions of m(X_i) and M(x') characteristics".

Author response:
**(5 of 5) Regarding Minor issues**: Thank you for bringing these to our attention. We have promptly corrected them.

Reviewer comment (Novelty):
Why should I use your method while the SNS method provides much better tradeoff between costs and robustness?

Author response:
**(2 of 5) Regarding the benchmarking**: Our main contribution is to demonstrate a theoretical connection between our method (entropic-risk-based approach) and the worst-case robust counterfactuals (min-max optimization), e.g. ROAR. Our experiments simply aim at showcasing that our method performs at par with the state-of-the-art, SNS algorithm, while being grounded in the solid theoretical foundation of large deviation theory, without relying on any postulations; enabling a unifying theory for robust counterfactuals. We acknowledge the existence of several intriguing techniques in robust counterfactuals that might surpass our strategy in specific scenarios. However, the practical implications of model change remain uncertain. Our fundamental innovation lies in making a theoretical contribution that facilitates a nuanced trade-off with robustness.

Reviewer comment (Novelty):
My primary concern is about the novelty and significance of the proposed method.

Author response:
**(1 of 6) Regarding novelty and significance of the proposed method**: The novelty of our measure is analogous to the significance of **Renyi mutual information over mutual information** [Mironov (2017); Baharlouei (2020)] that arises in other domains such as fairness, privacy, communications, etc. Reyni measures introduce a tunable $\alpha$ parameter to bridge between worst case and average case. In the limit of $\alpha$ going to infinity, the measure converges to the worst-case expression (maximum), while in the limit of $\alpha$ going to 1, the measure converges to an average-case expression (mutual information).

Similarly, the significance of our contribution lies in the insights that our measure offers for the theoretical understanding of robust counterfactuals, along with the novel **unification** it brings to various approaches within this domain. Our metric has a ‘knob’, the risk parameter $\theta$, that can be adjusted to trade-off between risk-constrained and worst-case/adversarially robust approaches, depending on where the user wants to be. **While risk-constrained accounts for general model changes in an expected sense, the adversarial robustness prioritizes the worst-case perturbations to the model, thus having a higher cost.** **Our approach enables one to tune “how much” a user wants to prioritize for the worst-case model changes, by trading off cost.** The extreme value of the knob (risk parameter $\theta$) maps our measure back to a min-max/adversarial approach. 

Notably, (i) Without such a tunable knob, it is not possible to have the flexibility to tradeoff between Risk-Constrained and Worst-Case. **To the best of our knowledge, existing Risk-Constrained measures, e.g., T-Rex, SNS, etc. do not have such an analogous tunable parameter that enables them to mathematically converge to Adversarial (Worst-Case) Robustness.** To clarify, they do have other hyperparameters but varying them would not lead to the mathematical equivalence to **Adversarial (Worst-Case) Robustness**. (ii) It also enables us to characterize the Pareto-Optimal front and balance cost and risk by varying the tunable risk parameter $\theta$ which is not present in T-Rex-like algorithms.

[Mironov (2017)] Mironov, Ilya. "Rényi differential privacy." 2017 IEEE 30th Computer Security Foundations Symposium (CSF). IEEE, 2017; 

[Baharlouei(2019)]  Sina Baharlouei, Maher Nouiehed, Ahmad Beirami, Meisam Razaviyayn: Rényi Fair Inference. ICLR 2020

Reviewer comment (Experiments):
Experiments are not very comprehensive. More recent baselines and datasets are needed.

Author response:
**(2 of 6) Regarding the experiments**: Our main contribution is to demonstrate a theoretical connection between our method (entropic-risk-based approach) and the worst-case robust counterfactuals (min-max optimization), e.g. ROAR. Our experiments simply aim at showcasing that our method performs at par with the state-of-the-art while being grounded in the solid theoretical foundation of large deviation theory, without relying on any postulations; enabling a unifying theory for robust counterfactuals. We acknowledge the existence of several intriguing techniques in robust counterfactuals that might surpass our strategy in specific scenarios. However, the practical implications of model change remain uncertain. Our fundamental innovation lies in making a theoretical contribution that facilitates a nuanced trade-off with robustness.

Reviewer comment (Novelty):
The paper is relatively well-written. However, it is not clear about the novelty or significance of the proposed method compared to previous work. The authors have included some experimental details, but they have not provided code for the paper.

Author response:
**(1 of 6) Regarding novelty and significance of the proposed method**: The novelty of our measure is analogous to the significance of **Renyi mutual information over mutual information** [Mironov (2017); Baharlouei (2020)] that arises in other domains such as fairness, privacy, communications, etc. Reyni measures introduce a tunable $\alpha$ parameter to bridge between worst case and average case. In the limit of $\alpha$ going to infinity, the measure converges to the worst-case expression (maximum), while in the limit of $\alpha$ going to 1, the measure converges to an average-case expression (mutual information).

Similarly, the significance of our contribution lies in the insights that our measure offers for the theoretical understanding of robust counterfactuals, along with the novel **unification** it brings to various approaches within this domain. Our metric has a ‘knob’, the risk parameter $\theta$, that can be adjusted to trade-off between risk-constrained and worst-case/adversarially robust approaches, depending on where the user wants to be. **While risk-constrained accounts for general model changes in an expected sense, the adversarial robustness prioritizes the worst-case perturbations to the model, thus having a higher cost.** **Our approach enables one to tune “how much” a user wants to prioritize for the worst-case model changes, by trading off cost.** The extreme value of the knob (risk parameter $\theta$) maps our measure back to a min-max/adversarial approach. 

Notably, (i) Without such a tunable knob, it is not possible to have the flexibility to tradeoff between Risk-Constrained and Worst-Case. **To the best of our knowledge, existing Risk-Constrained measures, e.g., T-Rex, SNS, etc. do not have such an analogous tunable parameter that enables them to mathematically converge to Adversarial (Worst-Case) Robustness.** To clarify, they do have other hyperparameters but varying them would not lead to the mathematical equivalence to **Adversarial (Worst-Case) Robustness**. (ii) It also enables us to characterize the Pareto-Optimal front and balance cost and risk by varying the tunable risk parameter $\theta$ which is not present in T-Rex-like algorithms.

[Mironov (2017)] Mironov, Ilya. "Rényi differential privacy." 2017 IEEE 30th Computer Security Foundations Symposium (CSF). IEEE, 2017; 

[Baharlouei(2019)]  Sina Baharlouei, Maher Nouiehed, Ahmad Beirami, Meisam Razaviyayn: Rényi Fair Inference. ICLR 2020

Reviewer comment (Novelty):
In Theorem 1, I acknowledge that Lemma 1 is a result of previous work. Therefore, Lemma 2 seems to be the main technical contribution. However, its significance remains unclear to me.

Author response:
**(4 of 6) Regarding Theorem 1**: The significance of Theorem 1 lies in its utilization of Vardhan's theorem (Lemma 1) in conjunction with Lemma 2 in this specific setup. While robust counterfactuals have been studied, Theorem 1 enables us to bridge the gap between the two existing classes of algorithms in this space, namely worst-case min max and average-case risk-constrained. Lemma 2 played a crucial role in bridging the gap between Vardhan's theorem and the main objective of the proof. Lemma 2 shows that supremum over a (compact set) probability simplex is equal to the maximum. This might be in agreement with intuition but not trivial.

Reviewer comment (Theory):
The probabilistic guarantees in Theorem 2 are based on the assumption of Lipschitz continuity, and it might limit the application of this paper.

Author response:
**(5 of 6) Regarding Theorem 2**: We agree with the reviewer and have acknowledged the dependence on the Lipschidtz constant and its implications in the limitation section in the appendix.

Reviewer comment (Novelty):
Some citations should be mentioned:  
1. Counterfactual Plans under Distributional Ambiguity, ICLR22  
2. Provably Robust and Plausible Counterfactual Explanations for Neural Networks via Robust Optimisation, ACML23  
3. Distributionally Robust Recourse Action, ICLR23  
4. On Minimizing the Impact of Dataset Shifts on Actionable Explanations, UAI23

Author response:
**(6 of 6) Regarding references**: Thank you for pointing out the references. We included a brief note and discussed these references in our paper.