A Single-Loop Gradient Algorithm for Pessimistic Bilevel Optimization via Smooth Approximation

We sincerely thank the reviewer for their time and comments. We are grateful for their acknowledgment of our work's primary contributions and appreciate the opportunity to clarify our work and address the concerns raised.

Below, we address each point in detail.

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1. On Related Work and Experimental Comparisons

We appreciate the reviewer's suggestions for related work and would like to clarify how our paper is positioned with respect to these references. We also take this opportunity to present new experimental results that further benchmark our method against relevant algorithms.

Discussion of Prior Literature:

The reviewer mentioned several references, a number of which were already cited and discussed in our manuscript.

• References [2, 4, 6, 7] in the reviewer's comments correspond to references [38, 34, 35, 11] in our paper, and their relationship to our work is discussed in the Related Work part.

• Regarding the other references suggested:

  • Reference [1]: This work studies first-order algorithms for optimistic bilevel optimization. Our paper, in contrast, focuses on the pessimistic formulation (PBO), which addresses a different class of problems.

  • Reference [3]: This paper applies PBO models to hyperparameter tuning. The algorithm it employs is the one proposed in our reference [11], which we have already discussed.

  • Reference [5]: This is a very recent work (published April 2025) that tackles a specific subclass of PBO where the upper-level objective is quadratic and the lower-level is linear. It proposes a single-level DC (Difference of Convex) program reformulation. The resulting algorithm is neither single-loop nor gradient-based in the same vein as our method, making it less directly comparable.

Furthermore, the reviewer suggested experiments on problems from [9, 10]. We have carefully examined these works and concluded they are not suitable for benchmarking our method.

• Reference [9]: The PBO applications in this work are from economics and involve integer variables, which falls outside the continuous optimization setting considered in our paper and the broader machine learning context.

• Reference[10]: This work focuses on the optimistic bilevel optimization model, which is fundamentally different from the PBO problem we study.

Given this, we respectfully maintain that our original manuscript provided a detailed discussion of the relevant existing literature.

Experimental Comparisons:

We wish to clarify that our original manuscript did contain experimental comparisons, specifically benchmarking our proposed SiPBA against the AdaProx-PD algorithm [2] on a synthetic example. To further strengthen our evaluation and address the reviewer's concern, we have now expanded these experiments to include additional important baselines: AdaProx-SG [2] and the Compact/Detailed Scholtes relaxation methods [3]. The comprehensive results for each method are summarized in our response to Reviewer ckLr, which demonstrate that SiPBA is not only significantly faster but also more robust, consistently converging to a valid solution across all runs.

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2. On the Theoretical Analysis

• Strong Concavity: The reviewer questioned the strong concavity of $\psi_k(x,y,z)$ (defined in Eq. (3)) with respect to $y$. This property follows directly from the strong concavity of the upper-level objective $F(x,y)$ with respect to $y$, which is an assumption stated in Assumption 1.

• Min-Max Interchange: The reviewer also raised a question about the interchange of minimization and maximization operators in Line 146: $\min_{z \in Y} \max_{y \in Y} \psi_k(x, y, z) = \max_{y \in Y} \min_{z \in Y} \psi_k(x, y, z)$. As we discussed in Lines 143-146, this equality holds due to the existence of a saddle point for $\psi_k(x, y, z)$ with respect to variables $(y,z)$. This is a direct consequence of classical min-max theorems (e.g., [VII, Theorem 4.3.1, 12] or [Theorem 3.1.29, 13]). There is no primal-dual gap in this step.

• Convergence of the Approximation: Finally, we would like to highlight Theorem 2.5, which establishes the convergence of our smoothing approximation to the original PBO problem. This theorem formally proves that the solution to the smoothed problem converges to a true solution of the original PBO problem as the smoothing parameter approaches zero, justifying the correctness of our smoothing framework.

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3. On Convergence Results

• Definition of Constants: The reviewer commented on the constants used in the merit function $V_k$. All terms, including $\underline{\phi}$, $a_k$ and $b_k$, are explicitly defined in the paper. Specifically, $\underline{\phi}$ is discussed in Lines 223-224, and the precise formulas for $a_k$ and $b_k$ are provided in the statement of Proposition 4.1 in Line 233.

• Convergence Guarantees: The reviewer's claim that our paper "only provides bounds on the minimum of the error terms" is a misinterpretation of our results. In Theorem 4.2, we establish the asymptotic convergence of the sequence generated by our algorithm, explicitly showing that key error terms approach zero:

  $$\lim_{k \rightarrow \infty} \frac{1}{\alpha_k} \|\|x^k - \mathrm{Proj}_X(x^k - \alpha_k \nabla \phi_k(x^k))\|\| = 0,$$ $$\lim_{k \rightarrow \infty} \|\| u^k - u_{k}^{\star}(x^k) \|\| = 0.$$

  Furthermore, establishing a convergence rate based on the metric $\min_{0 \le k < K} \frac{1}{\alpha_k^2} \|\|x^{k+1} - x^k\|\|^2$ is a standard and widely accepted practice for gradient-based methods applied to general non-convex optimization problems (see, e.g., [Theorem 10.15, 14]).

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4. On Algorithmic Complexity

We thank the reviewer for suggesting that we include a discussion of complexity. We will add the following analysis to the revised manuscript.

• Computational Complexity: At each iteration, SiPBA performs gradient evaluations ($\nabla_xF, \nabla_x f, \nabla_y F, \nabla_y f$) and several vector operations. Since the cost of gradient evaluation is typically linear in the variable dimension, the per-iteration computational complexity of SiPBA is $\mathcal{O}(n + m)$.

• Storage Complexity: The algorithm exhibits low memory requirements. To compute the next iterate, it only needs to store the variables from the preceding step $(x^k, y^k, z^k)$ and a few intermediate vectors. Consequently, the storage complexity is $\mathcal{O}(n + 2m)$.

This lean computational and memory footprint confirms that SiPBA is scalable and well-suited for practical, high-dimensional applications, representing a key advantage of our approach.

References

[1] J. Li, B. Gu, and H. Huang. A fully single loop algorithm for bilevel optimization without hessian inverse. In the AAAI Conference on Artificial Intelligence, 2022.

[2] B. Liu, M. Ye, S. J. Wright, P. Stone, and Q. Liu. BOME! Bilevel Optimization Made Easy: A Simple First-Order Approach. In Advances in Neural Information Processing Systems, 2022.

[3] M. A. Ustun, L. Xu, B. Zeng, and X. Qian. Hyperparameter Tuning Through Pessimistic Bilevel Optimization. In arXiv:2412.03666, 2024.

[4] J. Kwon and D. Kwon and S. J. Wright and R. D. Nowak A fully first-order method for stochastic bilevel optimization. In International Conference on Machine Learning, 2023.

[5] A. S. Strekalovsky. One way to solve pessimistic bilevel optimization problems. In Optimization, 2025.

[6] J. Kwon and D. Kwon and S. J. Wright and R. D. Nowak On Penalty Methods for Nonconvex Bilevel Optimization and First-Order Stochastic Approximation. In International Conference on Learning Representations, 2024.

[7] I. Benchouk, L. O. Jolaoso, K. Nachi, and A. B. Zemkoho. Scholtes relaxation method for pessimistic bilevel optimization. Set-Valued and Variational Analysis, 2025.

[8] J. Kwon, D. Kwon, and H. Lyu. On the complexity of first-order methods in stochastic bilevel optimization. International Conference on Machine Learning, 2024.

[9] L. Lampariello, S. Sagratella and O. Stein. The standard pessimistic bilevel problem. SIAM Journal on Optimization, 2019.

[10] Y. Zhang, G. Zhang, P. Khanduri, M. Hong, S. Chang and S. Liu. Revisiting and advancing fast adversarial training through the lens of bi-level optimization. International Conference on Machine Learning, 2022.

[11] B. Zeng. A practical scheme to compute the pessimistic bilevel optimization problem. INFORMS Journal on Computing, 2020.

[12] JB. Hiriart-Urruty and C. Lemaréchal. Convex analysis and minimization algorithms I: Fundamentals. In Springer science and business media, 2013.

[13] Y. Nesterov. Lectures on convex optimization. In Springer International Publishing, 2018.

[14] A. Beck. First-order methods in optimization. In SIAM, 2017.

Thanks for your detailed response, which solves most of the raised issues! And I've raised the rate accordingly.

However, some questions still remain as follows.

(1) EXP: Current empirical results, including the responses, are kind of limited, which may not be sufficient to verify the motivation for "SiPBA is scalable and well-suited for practical, high-dimensional applications, representing a key advantage of our approach". Moreover, could the authors provide the empirical results on higher dimensional cases?

(2) Theory: Since the convex-concave in Assumptions 1,2 may not be satisfied in practice [1], could the author provide with some ML examples satisfying the assumptions on convex/concave PBO settings as in this paper if possible? Or some relevant PBO works with similar assumptions?

I would be grateful if these concerns could also be well addressed.

[1] A global optimization algorithm for generalized semi-infinite, continuous minimax with coupled constraints and bi-level problems

[2] BOME! Bilevel Optimization Made Easy: A Simple First-Order Approach

In this experiment setting, the dimension of the upper-level variable $w$ (the classifier parameters) is approximately 3,000. The lower-level variable $\hat{x}$, representing adversarial examples for a set of 500 emails, has a dimension well over 100,000. This experiment showcases SiPBA's ability to handle high-dimensional applications.

We sincerely thank the reviewer for their insightful comments and constructive suggestions. We are grateful for their acknowledgment of our work's primary contributions: proposing a gradient-based method for Pessimistic Bilevel Optimization (PBO) that avoids second-order derivatives and the need for an inner-loop optimization solver.
Below, we address the specific concerns raised.

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1. On the Weaknesses of Numerical Experiment

We thank the reviewer for this feedback. We have revised the experimental section to improve clarity, particularly the spam classification setup, which we detail in our response to Question 3.

We agree that including more baseline comparisons is valuable. Accordingly, we have expanded our synthetic experiments to include additional methods. As detailed in our response to Reviewer ckLr, these new results further highlight the effectiveness of our proposed method, SiPBA. We will incorporate these results into the revised manuscript.

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2. On the Typo on Page 3

We appreciate the reviewer for the attention to detail. This typo has been corrected in the revised manuscript.

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3. On the Additional Bilinear Term in the Smoothing Approach

We appreciate this insightful question. The bilinear term is indeed crucial for the theoretical guarantees of our proposed method.

While adding a quadratic term to ensure strong convexity is a common technique in optimization, its specific form in our work is not arbitrary. Its inclusion is theoretically motivated and essential for the convergence of our smoothing approximation. As noted in the manuscript, this coupling term is critical for establishing the key inequality in the proof of Lemma 2.2:

$$\rho_{k}(f(\bar{x}, z_{k}^{\star}(\bar{x})) - f(\bar{x}, y_{k}^{\star}(\bar{x}))) + \frac{\sigma_k}{2} \|\| z_{k}^{\star}(\bar{x}) - y_{k}^{\star}(\bar{x})\|\|^2 \le 0.$$

This result forms the foundation of our subsequent convergence analysis, underpinning both Proposition 2.3 and Theorem 2.5.

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4. On the "SVM" and "Logistic" Models in Table 2

We apologize for the lack of clarity and thank the reviewer for the opportunity to elaborate on our experimental design for the spam classification task. The original experiments were designed to compare the performance of the PBO model trained using SiPBA with standard single-level models implemented in scikit-learn. For clarity, we restate the PBO formulation considered in the spam classification task:

$$\min_{w\in\mathbb{R}^n} \max_{\hat{x}}l(w,\hat{x},y) + \lambda_1 \mathrm{Reg}(w) \quad \text{s.t.} \hat{x} \in \arg\min_{x'\in\mathcal{X}} l'(w,x') + \lambda_2 \|\|\varphi(x') - \varphi(x)\|\|^2,$$

where $w$ denotes the classifier parameters, $(x,y)$ represents the vectorized training data, $l$ (resp. $l'$) corresponds to the classifier (resp. adversarial generator) loss, $\mathrm{Reg}(\cdot)$ denotes the regularization term, and $\varphi(\cdot)$ characterizes the feature mapping.

We perform a two-part empirical comparison:

• PBO solvers: We compare the PBO model above trained using SiPBA (with $\varphi(\mathbf{x})$ as the top $k$ principal components) to the same model trained using the SQP-based method with $\varphi(\mathbf{x}) = \mathbf{x}$, as the setting in [2].
• Single-level baselines: We also compare the SiPBA-trained PBO model against a standard single-level model:

$$\min_{w} l(w,x,y) + \lambda_1 \mathrm{Reg}(w),$$

implemented using the scikit-learn library.

For both $l$ and $l'$, we consider hinge loss and cross-entropy loss, leading to the following method names:

• SiPBA-Hinge / SiPBA-CE: Our proposed PBO solver using SiPBA.
• SQP-Hinge / SQP-CE: SQP-based method from [3].
• Single-Hinge / Single-CE: Standard single-level classifiers corresponding to SVM and Logistic Regression.

The results over ten runs are summarized in the table below, which indicate that the PBO model (either trained with SQP or SiPBA) exhibits superior cross-domain performance compared to the single-level models. Moreover, the SiPBA-trained models achieve the best overall accuracy and F1 score.

Table: Accuracy (Acc) and F1 score (F1) on four spam corpora, training on TREC06, TREC07, EnronSpam or LingSpam.

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This clarification and the full results will be incorporated into Section 5.2 of the revised manuscript.

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References

[1] Z. Guan, D. Sow, S. Lin, and Y. Liang. AdaProx: A novel method for bilevel optimization under pessimistic framework. Conference on Parsimony and Learning, 2025.
[2] I. Benchouk, L. O. Jolaoso, K. Nachi, and A. B. Zemkoho. Scholtes relaxation method for pessimistic bilevel optimization. Set-Valued and Variational Analysis, 2025.
[3] M. Brückner and T. Scheffer. Stackelberg games for adversarial prediction problems. In International Conference on Knowledge Discovery and Data Mining, 2011.

We thank the reviewer for their insightful feedback. We are grateful for their acknowledgment of our work's primary contribution: proposing a single-loop gradient-based method for Pessimistic Bilevel Optimization (PBO). We believe this is a valuable contribution to a relatively underexplored area.

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1. On the Strong Concavity Assumption

The reviewer is correct that the strong concavity assumption on the upper-level objective $F(x,y)$ is primarily used to ensure the uniqueness of the saddle point and the differentiability of the smoothed value function $\phi_{\rho,\sigma}(x)$. Without this, $\phi_{\rho,\sigma}(x)$ may become non-differentiable.

One way to weaken this assumption to mere concavity in $y$ would be to add a regularization term, such as $-\delta_k \|\|y\|\|^2$ with vanishing $\delta_k$. We opted for the strong concavity assumption to maintain clarity and focus. Our goal was to present the core ideas of our new smoothing approximation and single-loop algorithm as clearly as possible. Introducing an additional hyperparameter would complicate the theoretical analysis and could obscure the central contributions of this work. We therefore made a deliberate choice to prioritize clarity in this paper, leaving the exploration of this relaxation for future work.

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2. On the Novelty of the Proposed Method

We appreciate the reviewer for the recognition of the novelty of our smoothing strategy. While we agree that penalty methods and first-order updates are established techniques in optimization, we wish to clarify that their synthesis and application to create a single-loop, gradient-based method for PBO are novel.

Specifically, these techniques are not merely combined, but are strategically integrated. The penalty strategy is a key component within our smoothing approximation framework, and the first-order updates are subsequently enabled by this new smoothing. It is this complete, integrated methodology that constitutes the main contribution and allows for an efficient single-loop PBO algorithm.

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3. On the Intuition Behind the Smoothing Approach

We thank the reviewer for this valuable question. The reviewer correctly notes that adding a quadratic term to ensure strong convexity/concavity is a common technique. Our key innovation, however, lies in the introduction of an additional bilinear term. To our knowledge, no prior work has incorporated both a quadratic and a bilinear term for smoothing a bilevel optimization problem.

This bilinear term is not arbitrary; it is theoretically essential for the convergence of our smoothing approximation. As noted in the manuscript, without this coupling term, we cannot establish the key inequality in the proof of Lemma 2.2:

$$\rho_{k}(f(\bar{x}, z_{k}^{\star}(\bar{x})) - f(\bar{x}, y_{k}^{\star}(\bar{x}))) + \frac{\sigma_k}{2} \|\| z_{k}^{\star}(\bar{x}) - y_{k}^{\star}(\bar{x})\|\|^2 \le 0.$$

This result forms the foundation of our subsequent convergence analysis, underpinning both Proposition 2.3 and Theorem 2.5.

The work in [28] considers a different setting and relies on an additional assumption on the lower-level Hessian (there exists a constant $\kappa > 0$ such that $\lambda_{\min}(\nabla^2_{yy} f(x,y)) > \kappa$ for all $\nabla_y g(x,y) \neq 0$), which our method does not require. This additional condition is what permits their use of a simpler smoothing term. Our approach, by design, is more general; the inclusion of the bilinear term is the critical element.

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4. On the Comparison with [28] and Discussion on Convergence Rates

We appreciate the reviewer for this question. Our work differs from [28] in several aspects:

• Assumptions: [28] requires stricter assumptions, including Lipschitz continuous Hessians and a strictly positive definite lower-level Hessian, none of which are needed for our method.

• Methodology and Metric: AdaProx [28] follows a different path by reformulating the PBO problem via KKT conditions of the maximum subproblem, leading to a constrained single-level problem that is structurally distinct from our smoothing-based formulation. Consequently, their stationarity metric is different from ours, and their method relies on second-order information.

Given these significant differences in assumptions and formulation, a direct theoretical comparison of the convergence rates would not be meaningful.

For the question regarding the convergence rates of terms $\|\|x^{k+1} - x^k\|\|^2/\alpha_k^k$ and $\|\|u^k - u_{k}^{\star}(x^k)\|\|^2$ stated in Theorem 4.2, as it requires that $8p + 8q \le s$. Therefore, the convergence rate of the term $\|\|x^{k+1} - x^k\|\|^2/\alpha_k^2$ is at most $\mathcal{O}(1/K^{1-16p-16q})$. This is strictly slower than the $\mathcal{O}(1/K^{1-6p-7q})$ rate of the term $\|\| u^k -u^{\star}_k(x^k)\|\|^2$. We will add this clarification to the revised manuscript.

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5. On the Convergence Notion in Theorem 4.2

The reviewer's understanding is correct. We can indeed show from Theorem 4.2 that $y_k$ will eventually be close to the solution of $\max_{y \in \mathcal{S}(x_k)} F(x_k,y)$, and therefore $F(x^k,y^k) - \max_{y \in \mathcal{S}(x_k)} F(x_k,y)) \rightarrow 0$. More rigorously, under the assumptions of Theorem 4.2, we can establish that

$$\lim_{k \to \infty} y_k^{\star}(x_k) = y^*(\bar{x}),$$

where $y^*(\bar{x})=\underset{y\in \mathcal{S}(\bar{x})}{\arg\max} F(\bar{x},y)$. We will add this result to the revised manuscript.

Regarding other convergence notions, the choice of a stationarity metric is tied to the specific reformulation an algorithm is based on. As the reviewer correctly notes, our metric is based on the reformulation in (2) and its smooth approximation in (6), whereas the metric for AdaProx [28] is based on a constrained single-level reformulation with KKT conditions of the max subproblem. Establishing a formal relationship between these distinct metrics is a non-trivial direction for future research, as it would likely require additional assumptions to bridge stationarity conditions of different problem formulations.

We thank the reviewer for their insightful comments and constructive suggestions. We are grateful for their acknowledgment of our work's primary contributions: proposing a gradient-based method for Pessimistic Bilevel Optimization (PBO) that avoids second-order derivatives and the need for an inner-loop optimization solver. We believe this is a valuable contribution to a relatively underexplored area.

Below, we address the specific concerns raised.

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1. On the Lower-Level Problem as a Soft Constraint

Regarding the question of whether the lower-level optimization acts as a soft constraint, the reviewer's intuition is correct. This result is established in Theorem 3.6 of [1]. The theorem shows that if the lower-level problem constraint is relaxed to $f(x,y) \le \inf_y f(x,y) + \epsilon$, the solution to this relaxed problem converges to the solution of the original PBO as the violation $\epsilon \rightarrow 0$. We will add this interpretation in the revised manuscript.

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2. On the "Single-Loop" Nature of the Algorithm

We appreciate the opportunity to clarify our "single-loop" claim. Our use of this term emphasizes that our algorithm does not require an inner loop to iteratively solve the minimax problem in Eq. (4) to approximate the gradient of $\phi_{\rho,\sigma}$ at each iteration. We agree that this advantage relies on the assumption that the projections onto the feasible sets $X$ and $Y$ are computationally simple (i.e., have a closed-form solution or can be computed efficiently). We will add a statement to the manuscript to explicitly clarify this condition for the single-loop claim.

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3. On the Justification for Assumption 3

We agree that Assumption 3 is not a commonly used assumption in bilevel optimization. In fact, Assumption 3 is equivalent to the closedness of the function’s epigraph, and it is a common assumption for ensuring the existence of a function's minimum (see, e.g., Theorem 1.9 of [2]).

We recognize that verifying Assumption 3 directly can be challenging. For this reason, we provide a more practical sufficient condition immediately following the assumption: the inner semi-continuity of the lower-level solution map $\mathcal{S}(x)$. This condition holds for all problem instances considered in our numerical experiments under mild assumptions, thereby satisfying Assumption 3. We will expand this discussion in the revised manuscript to make the motivation and verification of this assumption clearer.

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4. On the Comparison of Non-Asymptotic Convergence Rates

We thank the reviewer for raising this important point. To the best of our knowledge, AdaProx [3] is the only other algorithm for PBO with a non-asymptotic convergence rate analysis. However, a direct theoretical comparison of our rate with that of AdaProx is not straightforward for several fundamental reasons:

• Different Assumptions: AdaProx requires additional assumptions, including Lipschitz continuity of the Hessians for both objective functions and the strict positive definiteness of the lower-level Hessian, none of which are required by our method.
• Different Methodology and Metric: AdaProx reformulates the PBO problem using a relaxed KKT system, which leads to a different stationary metric for measuring convergence.

Given these differences, a direct comparison of the theoretical convergence rates would not be meaningful.

To provide a comprehensive numerical comparison, we have expanded our numerical evaluation on the synthetic example from Section 5.1 to include AdaProx-SG and the Compact/Detailed Scholtes (Scholtes-C/D) relaxation methods [4]. We ran SiPBA, AdaProx-PD, We report the minimum and maximum relative errors of the outputs, the number of valid runs (achieving $\epsilon_{\text{rel}} < 10^{-4}$), and the average runtime (to reach $\epsilon_{\text{rel}} < 10^{-4}$) of those valid runs. The results over10 times from random initial points, summarized in the table below, demonstrate that SiPBA is not only significantly faster but also more robust, consistently converging to a valid solution.

Table: Performance comparison of SiPBA, AdaProx variants, and Scholtes relaxations on synthetic experiment ($n=100$).

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5. On Experimental Scope and Practical Applications

We appreciate the feedback regarding the experimental results and the practical applications of PBO. A key motivation for our work is to provide an efficient and accessible algorithm for PBO, thereby encouraging its adoption in a wider range of machine learning applications.

To further demonstrate the practical utility of SiPBA, we have added a new experiment on deep hyper-representation using real-world datasets. The problem is formulated as:

$$\min_{\theta\in\Theta}\max_{w}\frac{1}{m_1}\|\|f(X_{val},\theta)w-\mathbf{y}_{val}\|\|^2,$$ $$\text{s.t.} w\in\underset{w^\prime\in\mathcal{W}}{\arg\min} \frac{1}{m_2}\|\|f(X_{train},\theta)w^\prime-\mathbf{y}_{train}\|\|^2,$$

where $f(\cdot, \theta)$ represents a neural network parameterized by $\theta$, and $w$ corresponds to a linear layer. We adopt the LeNet-5 architecture [5] as the feature extractor $f(\cdot, \theta)$ and evaluate performance on the MNIST and FashionMNIST datasets. Each dataset is randomly split into 50,000 training samples, 10,000 validation samples, and 10,000 test samples, with performance evaluated based on test accuracy. The accuracy on test data are reported below:

Table:Test accuracy (%) on MNIST and Fashion MNIST.

These results indicate that the PBO model optimized by SiPBA exhibits superior generalization performance, further demonstrating the promise of PBO in machine learning applications.

Actually, a growing body of work has explored the potential of PBO in various machine learning applications recently. We will add a discussion to the revised manuscript. Specifically, adversarial learning has been formulated as a PBO problem [6,7], and decision-focused learning in contextual optimization has been addressed within the PBO framework [8,9]. The use of PBO for hyperparameter optimization has also been investigated [10]. Outside the machine learning domain, PBO has found applications in many practical scenarios, including but not limited to demand response management [11], rank pricing and second-best toll pricing [12,13], production-distribution planning [14], and gene knockout model in biological systems [15].

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References

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[2] R. T. Rockafellar, and R. J. Wets. Variational analysis. Springer Berlin Heidelberg,1998.

[3] Z. Guan, D. Sow, S. Lin, and Y. Liang. AdaProx: A novel method for bilevel optimization under pessimistic framework. Conference on Parsimony and Learning, 2025.

[4] I. Benchouk, L. O. Jolaoso, K. Nachi, and A. B. Zemkoho. Scholtes relaxation method for pessimistic bilevel optimization. Set-Valued and Variational Analysis, 2025.

[5] D. Sow, K. Ji, and Y. Liang. On the convergence theory for hessian-free bilevel algorithms. Advances in Neural Information Processing Systems, 2022.

[6] M. Brückner and T. Scheffer. Stackelberg games for adversarial prediction problems. International Conference on Knowledge Discovery and Data Mining, 2011.

[7] D. Benfield, S. Coniglio, M. Kunc, P. T. Vuong and A. B. Zemkoho. Classification under strategic adversary manipulation using pessimistic bilevel optimisation. In arXiv:2410.20284, 2024.

[8] V. Bucarey, S. Calderón, G. Muñoz, and F. Semet. Decision-focused predictions via pessimistic bilevel optimization: A computational study. Integration of Constraint Programming, Artificial Intelligence, and Operations Research, 2024.

[9] D. Jiménez, B. K. Pagnoncelli and H. Yaman. Pessimistic bilevel optimization approach for decision-focused learning. In arXiv:2501.16826, 2025.

[10] M. A. Ustun, L. Xu, B. Zeng, and X. Qian. Hyperparameter Tuning Through Pessimistic Bilevel Optimization. In arXiv:2412.03666, 2024.

[11] T. Kis, A. Kovács, and C. Mészáros. On optimistic and pessimistic bilevel optimization models for demand response management. Energies, 2021.

[12] H.I. Calvete, C Galé, A Hernández and J. A. Iranzo. A novel approach to pessimistic bilevel problems. An application to the rank pricing problem with ties. In Optimization, 2024.

[13] X. Ban, S. Lu, M.Ferris and H. X. Liu. Risk averse second best toll pricing. In Transportation and Traffic Theory 2009: Golden Jubilee, 2009.

[14] Y. Zheng, G. Zhang, J. Han and J. Lu. Pessimistic bilevel optimization model for risk-averse production-distribution planning. In Information Sciences, 2016.

[15] B. Zeng. A practical scheme to compute the pessimistic bilevel optimization problem. INFORMS Journal on Computing, 2020.