W1: No mention of limitations or future work. Are there any edge cases that might be problematic for EAP-GP? Can we make EAP-GP cheaper to run (since it seems to be ~4x more expensive than EAP-IG)? Etc.

A1: Thank you for the thoughtful comments. We acknowledge that EAP-GP introduces additional computational cost (∼4× slower than EAP-IG, as shown in Table 3), primarily due to the dynamic construction of model-dependent integration paths. In future work, we plan to explore efficiency improvements such as low-rank approximations, early stopping based on gradient convergence, and path simplification techniques.

As for limitations, EAP-GP may underperform in cases where gradients near corrupted inputs are highly unstable or noisy—especially in tasks with weak supervision signals. We view this as an opportunity for further robustness improvements, such as incorporating gradient smoothing or hybrid attribution strategies. We will add a dedicated section at the end of the paper to discuss these limitations and future directions.

W2: Unclear if the ground-truth circuits used for comparison (Syed et al) provide a good signal or not on the quality of the circuits found automatically. I.e., if these ground-truth circuits are not very faithful, then EAP-GP having less precision/recall than EAP-IG might be good.

A2: Thank you for the insightful suggestion. In the context of circuit discovery, the primary goal of automatic methods is to approximate both the process and the quality of manual circuit discovery. Since both approaches rely on the same evaluation metrics (e.g., logit difference) to identify circuits, it is natural to validate their consistency through qualitative analysis. By qualitatively comparing the circuits discovered by automatic methods with those obtained manually, we can assess whether they capture the same interpretable mechanisms. This offers further evidence that automatic circuit discovery can faithfully reproduce the insights of manual analysis and scale to more complex settings, making it a practical and effective alternative.

W3: A qualitative analysis of some of the circuits found (e.g., for IOI) would be nice. Does EAP-GP find attention heads or MLPs that are interesting or relevant for the circuit and haven't been analysed before?

A3:Thank you for the insightful suggestion. We conducted a qualitative comparison of the circuits discovered by EAP, EAP-IG, and EAP-GP on the IOI task. Below are representative circuits extracted by each method.

EAP: input->a0.h1<v>,input->m0,a0.h1->m0,a0.h1->m1,m0->m1,a0.h1->m2,m0->m2,m1->m2,m0->m3,m2->m3,m1->a6.h9<v>,a0.h1->a7.h10<v>,m0->a7.h10<v>,m2->a7.h10<v>,m0->a8.h8<v>,m0->a8.h11<v>,a6.h9->m8,a7.h10->m8,a8.h8->m8,a8.h11->m8,a0.h1->a9.h1<v>,m0->a9.h1<v>,m1->a9.h1<v>,m2->a9.h1<v>,m3->a9.h1<v>,a7.h10->m9,a8.h11->m9,m8->m9,a9.h1->m9,m0->a10.h7<v>,a7.h10->m10,a8.h11->m10,m8->m10,a9.h1->m10,m9->m10,m8->m11,m9->m11,m10->m11,m8->logits,a9.h1->logits,m9->logits,a10.h7->logits,m10->logits,m11->logits

EAP-IG: input->a0.h1<v>,input->a0.h10<v>,input->m0,a0.h1->m0,a0.h10->m0,m0->m1,m0->m2,m1->m2,m0->m3,m1->m3,m0->a5.h1<v>,m0->a5.h5<v>,a0.h1->a6.h9<v>,m0->a6.h9<v>,m1->a6.h9<v>,a0.h1->a7.h10<v>,m0->a7.h10<v>,m2->a7.h10<v>,m0->a8.h8<v>,m0->a8.h11<v>,a5.h1->m8,a6.h9->m8,a7.h10->m8,a8.h8->m8,a8.h11->m8,a0.h1->a9.h1<v>,m0->a9.h1<v>,m1->a9.h1<v>,m2->a9.h1<v>,m3->a9.h1<v>,m8->a9.h1<q>,a5.h5->m9,a6.h9->m9,a7.h10->m9,a8.h11->m9,m8->m9,a9.h1->m9,a6.h9->m10,a7.h10->m10,m8->m10,a9.h1->m10,m9->m10,m8->m11,m9->m11,m10->m11,m8->logits,a9.h1->logits,m9->logits,m10->logits,m11->logits

EAP-GP: input->a0.h1<v>,input->a0.h10<v>,input->m0,a0.h1->m0,a0.h10->m0,a0.h1->m1,m0->m1,m0->m2,m1->m2,m0->a5.h1<v>,m0->a5.h5<v>,a0.h1->a6.h9<v>,m0->a6.h9<v>,m1->a6.h9<v>,a0.h1->a7.h10<v>,m0->a7.h10<v>,m2->a7.h10<v>,m0->a8.h8<v>,m0->a8.h11<v>,a5.h1->m8,a5.h5->m8,a6.h9->m8,a7.h10->m8,a8.h8->m8,a8.h11->m8,a0.h1->a9.h1<v>,m0->a9.h1<v>,m1->a9.h1<v>,m2->a9.h1<v>,m8->a9.h1<q>,a5.h5->m9,a6.h9->m9,a7.h10->m9,a8.h11->m9,m8->m9,a9.h1->m9,a6.h9->m10,a7.h10->m10,m8->m10,a9.h1->m10,m9->m10,m8->m11,m9->m11,m10->m11,m8->logits,a9.h1->logits,m9->logits,m10->logits,m11->logits

The circuits discovered by EAP, EAP-IG, and EAP-GP on the IOI task demonstrate progressively more advanced patterns of information flow and integration. EAP identifies a sparse, mostly linear circuit centered on a single attention head, resulting in limited feature fusion and low redundancy. EAP-IG builds upon this by introducing multiple attention heads and localized modular interactions, enabling richer internal connectivity and the early use of query-based aggregation. EAP-GP further advances these capabilities by leveraging its gradient-guided path integration design to identify attribution paths that avoid saturated or misleading gradients. As a result, it discovers circuits involving a broader set of attention heads across multiple layers, centralizes information through a key convergence node (e.g., m8), and forms deep, multi-branch structures with extensive cross-layer integration. This reflects EAP-GP's ability to uncover more faithful and causally grounded circuits. As a result, the discovered structures exhibit significantly higher structural complexity, redundancy, and robustness. These properties enable the final logits to be influenced by diverse and complementary sources of information.

W4: Unclear what the practical applications of having more faithful circuits for AI Safety are. Also, circuit discovery might not scale to bigger SOTA models.

A4:Thank you for raising these concerns. We believe that improving circuit faithfulness has important implications for AI safety. Faithful circuits allow researchers to uncover the true causal mechanisms behind model behavior, which is essential for tasks such as auditing, detecting spurious correlations, and ensuring behavioral alignment in high-stakes applications (e.g., content moderation, planning, or question answering).

Importantly, much of mechanistic interpretability is grounded in attribution analysis, and our work reveals saturation as a previously underappreciated failure mode of gradient-based attribution in circuit discovery. This issue may also affect other attribution-based tools widely used in AI safety and interpretability, highlighting the need for methods that produce faithful and robust causal explanations. Without such faithfulness, interventions or explanations may be misleading or ineffective.

Regarding scalability, earlier circuit discovery approaches relied heavily on manual inspection. Recent advances, including our proposed EAP-GP, demonstrate that gradient-based methods are capable of scaling to larger models. We acknowledge that challenges remain for frontier-scale systems, but view our work as a meaningful step toward scalable and reliable mechanistic interpretability.

Minor issues and typos:

A5: Thank you for your careful reading and constructive suggestions. We will revise the paper accordingly. Specifically, we will reference the compute analysis in Table 3 to highlight the ~4× cost of EAP-GP, rename Section 6 to “Evaluation,” include AUC values in Fig. 4, and standardize terminology to “clean input” and “corrupted input” for clarity. We will also clarify the definition of circuit faithfulness earlier in the text, fix the typo in line 125 (“graph”), add a citation or reference for the IOI dataset at its first mention, and remove the duplicated “iteratively” in line 211. Your feedback is much appreciated and will help improve the overall clarity and consistency of the paper.

Q1: Improve motivation of the work: What is the theory of impact for this work? What is the utter goal of having more faithful circuits?

A6: Thank you for the Question. Please refer to our response to W4 for details.

Q2:Can we make EAP-GP faster so that it's more competitive against EAP-IG?

A7: Thank you for the Question. Please refer to our response to W1 for details.

Q3: Are there any limitations on using manually found ground-truth circuits as a part of a quantitative experiment?

A8: Thank you for the Question. Please refer to our response to W2 for details.

Q4:Does it make sense to do a qualitative analysis of the circuits found by these techniques?

A9: Thank you for the Question. Please refer to our response to W3 for details.

W1: how ubiquitous the saturation effects are?

A1: Thank you for the Question. Please refer to our response to Q2 for details.

W2: further discussion of the greater-than and gender-bias results is warranted

A2: Thank you for the Question. Please refer to our response to Q1 for details.

W3: Writing: The "zero-gradients problem" is introduced early on, but not explicitly defined at that point. This will confuse some readers, so please define.

A3: Thank you for the suggestion. We will revise the manuscript to include an explicit definition of the “zero-gradients problem” when it is first introduced, to avoid potential confusion for readers.

W4: limitations are discussed at the end of the results section, but should be put in a limitations section, with a disclaimer that the method is sensitive to hyperparameters

A4: Thank you for the suggestion. Please refer to our response to L1 for details.

Q1: Why does EAP-GP not still produce performance gains at very high sparsities if these tasks are basically solved by simple feature extraction?

A5: Thank you for your thoughtful question. We agree that tasks like Greater-Than and Gender-Bias are largely solvable via simple feature extraction mechanisms. As shown in Figure 3 and Table 4 of our paper, the faithfulness scores of all three methods (EAP, EAP-IG, EAP-GP) are indeed relatively close under high sparsity regimes in these tasks. However, there are a few important considerations explaining why EAP-GP does not yield further substantial gains in such cases:

Ceiling Effect due to Task Simplicity: These tasks involve relatively shallow reasoning and are dominated by localized features (e.g., profession-to-pronoun or numeric increments). Consequently, even a small number of edges are sufficient to recover most of the task-relevant signal. Thus, all methods tend to converge to similarly high faithfulness scores under high sparsity, leaving limited headroom for improvement.

Circuit Saturation vs. Attribution Saturation: EAP-GP is designed to mitigate saturation effects in attribution scores, not to discover more complex circuits where none are needed. In tasks that do not require long-range or compositional reasoning, even naive edge selection strategies may yield circuits that already match the model's behavior well. Hence, while EAP-GP still helps ensure robustness, its advantage becomes less pronounced.

Sensitivity to Sparsity Thresholds: At extremely high sparsity (e.g., >99.5%), all methods can become sensitive to small attribution variations, as shown in Figure 2. EAP-GP’s step size dynamics (see Section 6.2) may also amplify noise due to gradient instability near the baseline, which can slightly hurt performance compared to moderate sparsity.

In summary, the marginal gains of EAP-GP in simple tasks under high sparsity are bounded not by its capability, but by the simplicity of the task structure itself. We believe this aligns with the broader finding that gradient saturation effects are more critical in complex tasks such as IOI and Hypernymy, where EAP-GP shows the most substantial improvements.

Q2: How ubiquitous is the saturation effect within a model, across models, across tasks?

A6: We thank the reviewer for raising this important point regarding the generality of saturation effects. Our investigation reveals that the saturation effect is a widespread and recurring phenomenon, as detailed below:

Within a Model: As shown in Figure 1 and Section 4, we observe saturation effects on edge gradients even within a single circuit in the IOI task. Specifically, the straight-line path used by EAP-IG frequently traverses regions where the gradient of the loss ∂L/∂x becomes nearly zero. These regions occur at both the start and end of the interpolation path, regardless of the specific edge being evaluated. This indicates that saturation is not limited to individual edges but affects broad regions of the activation space.

To address this, we conducted additional experiments on the IOI task to measure the average gradient norm across all circuit edges during attribution. As shown in the supplementary table, EAP-IG tends to follow integration paths that pass through saturated regions, resulting in low gradient magnitudes and less accurate attributions. In contrast, our proposed EAP-GP avoids these saturated areas by adaptively selecting informative paths, leading to higher gradient norms and more faithful circuit discovery.

Across Tasks: Figure 3 and Table 4 show that saturation effects are especially detrimental in complex reasoning tasks such as IOI, Hypernymy, and SVA. In these tasks, EAP-IG shows a noticeable drop in faithfulness compared to EAP-GP as sparsity increases, suggesting that saturation limits its ability to select faithful edges. In contrast, in simpler tasks like Greater-Than and Gender-Bias, where feature extraction is more direct, the saturation effect has less impact—but is still measurable (e.g., Table 4 shows 1–2% gains by EAP-GP at 99% sparsity).

Across Models: As shown in Figures 5 and 6 (Appendix A.2), the saturation effect persists across GPT-2 Small, Medium, and XL. In particular, EAP-GP consistently outperforms EAP-IG across all model sizes, demonstrating that the issue is not alleviated by larger capacity or more parameters. In fact, larger models often exhibit more pronounced saturation due to their increased nonlinearity and overparameterization, further motivating the need for path-aware methods like EAP-GP.

In conclusion, saturation is a general and systematic problem that arises across architectures, tasks, and layers. Our method mitigates this problem by dynamically adjusting the gradient path to avoid such regions, which proves effective and robust in a variety of experimental settings.

L1:Limitations are discussed at the end of the results section, but should be put in a limitations section, with a disclaimer that the method is sensitive to hyperparameters

A7: Thank you for the suggestion. We agree that the paper would benefit from a clearer presentation of its limitations. To address this, we will move the current limitations discussion from the end of Section 6.2 into a new section titled Limitations and Future Work. This section will also mention that EAP-GP is somewhat sensitive to its main hyperparameter k, which controls the number of integration steps.

Q1: What do you think the better performance of EAP-GP can bring for the interpretability community?

A1: Thank you for the constructive question. We believe the improved performance of EAP-GP has several important implications for the interpretability community:

More Reliable Circuits Under Sparsity Constraints: EAP-GP consistently discovers more faithful circuits under high sparsity levels (see Table 4), particularly in complex reasoning tasks. This enables researchers to analyze much smaller subgraphs without sacrificing interpretability fidelity, which is critical for scaling interpretability to larger models.

Recognizing Saturation as a General Challenge: By identifying and addressing saturation effects, EAP-GP highlights a previously under-discussed failure mode in mechanistic interpretability methods. While saturation has been noted in other contexts, to our knowledge, this is the first work to explicitly observe and address it within circuit discovery. This finding may prompt a broader re-examination of gradient-based attribution tools used throughout mechanistic interpretability.

Towards Path-Adaptive Attribution Methods EAP-GP offers a new perspective on attribution by introducing a general design principle: adapting the integration path to the model’s local gradient landscape. This perspective encourages future research into dynamic, model-aware attribution strategies that are better equipped to handle common issues such as saturation and vanishing gradients.

Q2: Can any misconceptions be refuted, or previously overlooked points revealed, based on the improved performance? You do not need to do new experiments, but please pointing out some possible directions.

A4: Thank you for the insightful suggestion. We conducted a qualitative comparison of the circuits discovered by EAP, EAP-IG, and EAP-GP on the IOI task. Below are representative circuits extracted by each method.

EAP: input->a0.h1<v>,input->m0,a0.h1->m0,a0.h1->m1,m0->m1,a0.h1->m2,m0->m2,m1->m2,m0->m3,m2->m3,m1->a6.h9<v>,a0.h1->a7.h10<v>,m0->a7.h10<v>,m2->a7.h10<v>,m0->a8.h8<v>,m0->a8.h11<v>,a6.h9->m8,a7.h10->m8,a8.h8->m8,a8.h11->m8,a0.h1->a9.h1<v>,m0->a9.h1<v>,m1->a9.h1<v>,m2->a9.h1<v>,m3->a9.h1<v>,a7.h10->m9,a8.h11->m9,m8->m9,a9.h1->m9,m0->a10.h7<v>,a7.h10->m10,a8.h11->m10,m8->m10,a9.h1->m10,m9->m10,m8->m11,m9->m11,m10->m11,m8->logits,a9.h1->logits,m9->logits,a10.h7->logits,m10->logits,m11->logits

EAP-IG: input->a0.h1<v>,input->a0.h10<v>,input->m0,a0.h1->m0,a0.h10->m0,m0->m1,m0->m2,m1->m2,m0->m3,m1->m3,m0->a5.h1<v>,m0->a5.h5<v>,a0.h1->a6.h9<v>,m0->a6.h9<v>,m1->a6.h9<v>,a0.h1->a7.h10<v>,m0->a7.h10<v>,m2->a7.h10<v>,m0->a8.h8<v>,m0->a8.h11<v>,a5.h1->m8,a6.h9->m8,a7.h10->m8,a8.h8->m8,a8.h11->m8,a0.h1->a9.h1<v>,m0->a9.h1<v>,m1->a9.h1<v>,m2->a9.h1<v>,m3->a9.h1<v>,m8->a9.h1<q>,a5.h5->m9,a6.h9->m9,a7.h10->m9,a8.h11->m9,m8->m9,a9.h1->m9,a6.h9->m10,a7.h10->m10,m8->m10,a9.h1->m10,m9->m10,m8->m11,m9->m11,m10->m11,m8->logits,a9.h1->logits,m9->logits,m10->logits,m11->logits

EAP-GP: input->a0.h1<v>,input->a0.h10<v>,input->m0,a0.h1->m0,a0.h10->m0,a0.h1->m1,m0->m1,m0->m2,m1->m2,m0->a5.h1<v>,m0->a5.h5<v>,a0.h1->a6.h9<v>,m0->a6.h9<v>,m1->a6.h9<v>,a0.h1->a7.h10<v>,m0->a7.h10<v>,m2->a7.h10<v>,m0->a8.h8<v>,m0->a8.h11<v>,a5.h1->m8,a5.h5->m8,a6.h9->m8,a7.h10->m8,a8.h8->m8,a8.h11->m8,a0.h1->a9.h1<v>,m0->a9.h1<v>,m1->a9.h1<v>,m2->a9.h1<v>,m8->a9.h1<q>,a5.h5->m9,a6.h9->m9,a7.h10->m9,a8.h11->m9,m8->m9,a9.h1->m9,a6.h9->m10,a7.h10->m10,m8->m10,a9.h1->m10,m9->m10,m8->m11,m9->m11,m10->m11,m8->logits,a9.h1->logits,m9->logits,m10->logits,m11->logits

The circuits discovered by EAP, EAP-IG, and EAP-GP on the IOI task demonstrate progressively more advanced patterns of information flow and integration. EAP identifies a sparse, mostly linear circuit centered on a single attention head, resulting in limited feature fusion and low redundancy. EAP-IG builds upon this by introducing multiple attention heads and localized modular interactions, enabling richer internal connectivity and the early use of query-based aggregation. EAP-GP further advances these capabilities by leveraging its gradient-guided path integration design to identify attribution paths that avoid saturated or misleading gradients. As a result, it discovers circuits involving a broader set of attention heads across multiple layers, centralizes information through a key convergence node (e.g., m8), and forms deep, multi-branch structures with extensive cross-layer integration. This reflects EAP-GP's ability to uncover more faithful and causally grounded circuits. As a result, the discovered structures exhibit significantly higher structural complexity, redundancy, and robustness. These properties enable the final logits to be influenced by diverse and complementary sources of information.

Weaknesses:

W1: Figure 1 (right) is misleading: this method does not end up at corrupted x'

A1: Thank you for your careful reading and insightful comments. We agree that Figure 1 is currently misleading, and we appreciate you pointing this out.

In our actual setup, the start and end points of the integration path are explicitly fixed as the clean input x and the corrupted input x', respectively, to ensure the path lies between them. The intermediate values of $\gamma$ are computed by minimizing the loss

and are then used for attribution analysis.

Therefore, Figure 1 only includes intermediate points along the integration path, excluding the clean and corrupted inputs themselves. The plotted values represent the gradients at those intermediate points. We will revise Figure 1 to more accurately reflect this setup and include an explicit clarification in the manuscript, as suggested. Thank you again for highlighting this important issue.

W2: No explicit exploration of whether the method is better due to dealing with saturation, or some other reason

A2: Thank you for raising this important point. We agree that the claim regarding gradient saturation needs empirical support.

To address this, we conducted additional experiments on the IOI task to measure the average gradient norm of all edges within a selected circuit during attribution. As shown in the supplementary table, EAP-IG tends to follow integration paths that pass through saturated regions, resulting in low gradient magnitudes and less accurate attributions. In contrast, our proposed EAP-GP avoids these saturated areas by adaptively selecting informative paths, leading to higher gradient norms and more faithful circuit discovery.

W3: Syed et al.'s benchmark is not representative of modern models

A3: Thank you for the insightful suggestion. We recognize the importance of rigorous and representative benchmarks such as the Mechanistic Interpretability Benchmark (MIB). According to the NeurIPS 2025 policy, “For the purpose of the reviewing process, papers that appeared online after March 1st, 2025 will generally be considered 'contemporaneous' in the sense that the submission will not be rejected on the basis of the comparison to contemporaneous work.” Since MIB was released in April 2025, we consider it contemporaneous work for the purposes of this submission.

In the context of circuit discovery, our primary objective is to approximate both the process and quality of manual circuit identification. Given that both manual and automated approaches rely on shared evaluation metrics (e.g., logit difference), qualitative analysis as exemplified by Syed et al.'s benchmark remains a key tool for assessing alignment.

While we acknowledge MIB’s potential as a rigorous evaluation framework, its current implementation supports only a limited set of HuggingFace-compatible models (e.g., Gemma-2 2B, LLaMA-3.1-8B). Our EAP-GP method is built on a custom tracing framework with edge-level hook mechanisms, which are not directly compatible with MIB’s interface. Additionally, intervention-based evaluation on MIB-supported models requires substantial computational resources that exceed our current hardware capacity.

That said, we fully recognize the value of MIB and plan to gradually adapt our implementation for compatibility with it in future work. We will also incorporate a discussion of this contemporaneous benchmark and its relevance in the final version.

W4: Prior art on saturation: AtP* from early 2024

A4: Thank you for the suggestion. We respectfully argue that AtP is not a suitable quantitative baseline for EAP-GP, due to the following key differences:

1. Different Saturation Problems and Solutions: While both AtP∗ and EAP-GP address saturation effects, they do so in fundamentally different contexts. AtP∗ targets softmax saturation in attention layers and addresses it via QK-Fix, which recomputes attention weights before applying gradient-based approximation. In contrast, EAP-GP tackles gradient saturation along arbitrary edges (including MLPs and cross-layer paths) through gradient projection and rescaling, enabling recovery of deeply embedded causal edges. The saturation types and corrective mechanisms are not comparable.

2. Granularity Mismatch（Component vs. Edge）： AtP∗ provides component-level attribution, assigning importance scores to attention heads or MLP units. EAP-GP, on the other hand, performs edge-level attribution, identifying specific information flows between components. Since our goal is to extract interpretable subgraphs by selecting high-contributing edges, AtP∗ cannot serve as a directly comparable baseline.

3. No Public Implementation: As of submission, the full implementation of AtP∗ has not been open-sourced. Although the methodology is described, faithful and efficient reproduction remains challenging. In contrast, all our baselines are fully reproducible, ensuring a fair evaluation framework.

Despite the above, we devoted considerable effort to re-implement AtP∗ based on the paper description and conducted a qualitative comparison on the IOI dataset. The top components identified by AtP∗ are as follows:

Top 10 Attention Heads (contribution score):

Layer 10, Head 7: 0.0424 Layer 11, Head 10: 0.0359 Layer 9, Head 9: 0.0219 Layer 5, Head 5: 0.0147 Layer 10, Head 0: 0.0116 Layer 9, Head 6: 0.0110 Layer 11, Head 2: 0.0069 Layer 10, Head 10: 0.0067 Layer 10, Head 6: 0.0060 Layer 6, Head 9: 0.0049

Top 5 MLP Layers (mean contribution):

Layer 0: 0.0189 Layer 5: 0.0069 Layer 10: 0.0069 Layer 7: 0.0066 Layer 9: 0.0064

These components substantially overlap with those identified in the causal circuit recovered by EAP-GP: input, m0, a3.h0, m4, a5.h5, a6.h9, a7.h9, a8.h6, a8.h10, a9.h9, a10.h0, a10.h6, a10.h7, a10.h10, m10, a11.h2, a11.h10, logits

Specifically, 10 out of AtP∗’s top 10 attention heads (100%) appear in the EAP-GP circuit, and 2 out of the top 5 MLP layers (40%) are also shared.This strong alignment confirms the consistency of the two methods at the component level, while EAP-GP additionally reveals the directed causal structure via fine-grained edge-level attribution.

Questions:

Q1-3:The paper should compare with AtP*, which also deals with saturation; explain why it's OK for the line to not be between x and x' and instead be something else, and empirically show that the method works because of why they say it works.

A5: Thank you for the Question. Please refer to our response to W1-4 for details.

Q4: What is the variability between runs in Figures 3 and 4? It is impossible to say if the current method is an improvement if there is a lot of variation based on e.g. random seeds and the exact selection of training points.

A6: Thank you for pointing this out. We agree that assessing the variability across runs is essential for evaluating the robustness of our method. We have conducted additional experiments using different random seeds and training samples to assess the variability across runs. The standard deviation of performance across five runs is approximately 3% for IOI, around 3% for SVA and Hypernymy, and about 1% for Gender-Bias, Country-Capital, and Greater-Than. For instance, the following table reports the detailed faithfulness scores on the IOI task:

In the revised version of the paper, we will include error bars in Figures 3 and 4 to reflect this variability and report detailed standard deviations in the appendix to ensure transparency and reproducibility.

Q5: Are the quantities in Figure 3 evaluated on test datasets? They should be.

A7: Thank you for the question. In our setting, Figure 3 presents the evaluation results on the test datasets.

Minor questions

Q6: Where exactly does the 17.7% figure for the "improvement up to 17.7%" in the abstract come from? Is it the largest gap between EAP-GP and the next method in Figure 3?

A8: Yes, the 17.7% improvement refers to the largest performance gap between EAP-GP and the next-best method shown in Figure 3. We also provide the detailed faithfulness scores of the discovered circuits in the appendix for reference.

Q7: How did you choose the circuit edge for Fig. 1? In L165 it just says "we analyze a circuit edge".

A9: Thank you for the question. In Figure 1, We report the average gradient norm of all edges within a selected circuit, rather than focusing on a single edge. We will clarify this in the manuscript to avoid confusion.

Q8: Showing the sparsity in %s close to 100% in Fig. 2 and 3 was confusing to me. I'm used to low % (so the % is the amount of nonzero things) or straight up counting. That way you can also space your datapoints in a log-space.

A10: Thank you for the suggestion. We will consider using nonzero ratios spacing for the sparsity axis to improve clarity.

Q9: It would be nice if the caption of Fig. 1 linked to Section 4, which discusses it, since it appears 3 pages later. I was looking for an explanation of the figure in the same and next page and could not find it, leaving me a bit confused.

A11: Thank you for the suggestion. We will revise the Figure 1 caption to reference Section 4 for clarity.

Q10: use \left( and \right) in equations like eq. 7 to make the brackets as high as the thing they're bracketing (the fraction)

A12: Thank you for the suggestion. We will update the equations to use properly sized brackets with \left and \right.

@Reviewer pkLe: You have raised substantial concerns, and the authors have attempted to address your concerns in their rebuttal. Could you please let us know whether these updates have changed your overall assessment of the paper?