Title: A Single-Loop Gradient Algorithm for Pessimistic Bilevel Optimization via Smooth Approximation
Venue: NeurIPS 2025
Accepted: yes
Keywords: Bi-level Optimization

Pessimistic Bilevel Optimization

Gradient Method

Single-loop

Value Function

Smooth Approximation
Abstract: Bilevel optimization has garnered significant attention in the machine learning community recently, particularly regarding the development of efficient numerical methods. 
While substantial progress has been made in developing efficient algorithms for optimistic bilevel optimization, the study of methods for solving Pessimistic Bilevel Optimization (PBO) remains relatively less explored, 
especially the design of fully first-order, single-loop gradient-based algorithms. This paper aims to bridge this research gap. We first propose a novel smooth approximation to the PBO problem, using penalization and regularization techniques. Building upon this approximation, we then propose SiPBA (Single-loop Pessimistic Bilevel Algorithm), a new gradient-based method specifically designed for PBO which avoids second-order derivative information or inner-loop iterations for subproblem solving. We provide theoretical validation for the proposed smooth approximation scheme and establish theoretical convergence for the algorithm SiPBA. Numerical experiments on synthetic examples and practical applications demonstrate the effectiveness and efficiency of SiPBA.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper presents a novel approach to solving Pessimistic Bilevel Optimization (PBO) problems by proposing a single-loop gradient-based algorithm named SiPBA. The core technique involves constructing a smooth approximation of the PBO problem using penalization and regularization methods. This approximation transforms the original non-smooth PBO into a smooth optimization problem, enabling the application of efficient gradient-based methods. The authors provide theoretical validation for the proposed smooth approximation scheme and establish convergence guarantees.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths\n\n1. Algorithmic Innovation. The paper successfully relaxes the original PBO problem to a strongly convex-strongly concave formulation, enabling the development of a single-loop first-order gradient algorithm. \n   \n2. Theoretical and Empirical Validation. The authors provide rigorous theoretical analysis, including convergence rates and stationarity conditions. Additionally, the empirical results demonstrate the effectiveness and efficiency.\n\n3. The paper is well-structured and clearly presented. The technical details are well-explained, and the mathematical derivations are thorough.\n \nWeaknesses\n\n1. Related Work. This paper lacks a detailed discussion of existing work on first-order optimization algorithms and single-loop methods for both Optimistic and Pessimistic Bilevel Optimization [1-7]. This omission makes it difficult to assess the novelty and improvements over existing methods. Additionally, the lack of experimental comparisons with other PBO algorithms limits the evaluation of the proposed method's performance.\n\n[1] A Fully Single Loop Algorithm for Bilevel Optimization without Hessian Inverse\n\n[2] BOME! Bilevel Optimization Made Easy: A Simple First-Order Approach\n\n[3] Hyperparameter Tuning Through Pessimistic Bilevel Optimization\n\n[4] A fully first-order method for stochastic bilevel optimization\n\n[5] One way to solve pessimistic bilevel optimization problems\n\n[6] On penalty methods for nonconvex bilevel optimization and first-order stochastic approximation\n\n[7] Scholtes Relaxation Method for Pessimistic Bilevel Optimization\n\n2. Theoretical Gaps. The paper constructs an approximation function that ensures strong convexity in \\(z\\) and strong concavity in \\(y\\). While this is a simple strategy for problem relaxation, I am not quite sure about the strong concavity property. Moreover, the min-max problem is transformed into a max-min problem using Sion's theorem. Such relaxation is similar to conjugate gradient method, which require statements on the prima-dual gap. The authors are kindly suggested to provide a theoretical bound on the gap between the optimal solutions of the original and approximated problems.\n\n3. Convergence Results. The convergence results presented in Proposition 4.1 and Theorem 4.2 are somewhat weak. The derivation of the merit function \\(V_k\\) includes additional constant terms that are not thoroughly discussed. I’ve checked partial proof. It is unclear how these terms are introduced or whether they depend on specific assumptions or parameter choices. \n\nAdditionally, the paper only provides bounds on the minimum of the error terms \\min_k \\|x^{k+1}-x^k\\|, which does not strongly support the algorithm's convergence guarantees. A more detailed analysis of the asymptotic or expected convergence rates of V_{k+1} - V_k, x^k-x^* or x^{k+1} - x^k would be beneficial.\n\n4. The authors are encouraged to analyze the algorithmic (computational and storage) complexity [8], which is crucial for comparison among existing algorithms, and understanding the algorithm's practical applicability and performance in large-scale problems.\n\n[8] On the complexity of first-order methods in stochastic bilevel optimization\n\n5. The experimental section lacks comparisons with other existing PBO optimization algorithms . Some commonly investigated problems in PBO [9,10] are also suggested in the experiments, instead of merely on simple synthetic data.\n\n[9] The Standard Pessimistic Bilevel Problem\n\n[10] Revisiting and Advancing Fast Adversarial Training Through The Lens of Bi-Level Optimization\n\nQUESTIONS:\nSee the above weakness.\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 3, "REVIEWER_CONFIDENCE": 5, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-06-22"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper presents a numerical method for solving pessimistic bilevel optimization (PBO) problems, a class of bilevel optimization that has received less attention in the literature compared to its optimistic counterpart. The authors introduce a novel smoothing technique for the PBO value function, leveraging penalization and regularization. This approach transforms the value function into a smooth function corresponding to a strongly convex-strongly concave minimax problem.\n\nBuilding upon this smoothed formulation, the authors propose a fully first-order, single-loop algorithm for solving PBO.  The paper provides a rigorous theoretical convergence analysis of the proposed algorithm. Furthermore, the effectiveness and efficiency of the method are validated through a series of numerical experiments on both synthetic and real-world problems.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths :\n1. This paper addresses PBO, a challenging and less-explored area of bilevel optimization. The inherent difficulty of PBO, which includes a maximization problem at the upper level, has limited the development of efficient algorithms. This work makes a valuable contribution by focusing on single-loop, first-order methods for this problem class.\n\n2. The proposed methodology is both clear and sound. The authors introduce a novel technique to construct a smooth approximation of the PBO problem using penalization and regularization. The paper provides a detailed analysis to justify the convergence of this approximation to the true PBO problem, a contribution that may inspire further research in this domain.\n\n3. Based on the smoothed approximation, the paper presents a single-loop algorithm that relies solely on first-order information. This design choice makes the proposed algorithm simple to implement and computationally efficient, rendering it suitable for large-scale applications.\n\n4. The authors provide a non-asymptotic convergence analysis for the proposed algorithm. This rigorous theoretical support substantiates the algorithm's validity and strengthens the paper's overall contribution.\n\n5. The numerical experiments demonstrate the practical effectiveness and efficiency of the proposed algorithm.\n\nWeaknesses:\nThe primary area for improvement lies in the numerical experiments. Several descriptions of the experimental setup and results lack clarity. Furthermore, the empirical evaluation is somewhat limited due to the small number of baseline algorithms used for comparison. Although the literature on PBO algorithms is not extensive, several existing methods could have been included, at least in the experiments on synthetic data. A more comprehensive comparison would provide a stronger case for the advantages of the proposed approach.\n\nQUESTIONS:\n1. On page 3, in lines 119 and 122, the word \"and\" appears to be duplicated. Please correct this for clarity.\n\n2. Regarding the regularized function defined in Equation (3), could the authors elaborate on the motivation for including the coupling term $\\langle y, z\\rangle $? What is the specific role of this term? The reviewer understands the need to ensure the function is strongly convex with respect to $z$. Would the quadratic regularization term alone be insufficient to achieve this, or does the coupling term serve an additional, distinct purpose in the analysis or performance of the method?\n\n3. Could the authors precisely define the \"SVM\" and \"Logistic\" models listed in Table 2 and explain how they relate to the PBO model formulated in Equation (16)? The basis for comparison in this experiment is unclear. Please specify which algorithms are being compared. Was the proposed method benchmarked against other PBO algorithms, or does the comparison involve other methodologies?\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 5, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-27"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper addresses pessimistic bilevel optimization and introduces a single-loop algorithm based on a smooth approximation of a reformulated value function. The authors establish asymptotic convergence with respect to the smoothing parameter in the approximation and also provide a non-asymptotic convergence rate.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n\n1. The algorithm is single-loop and gradient-based, which is easy to implement. \n\n2. The idea of smoothing seems effective here. \n\nWeaknesses: \n\n1. The upper-level function $F(x, y)$ is also assume to be strongly convex in $y$. This assumption is usually not needed in (optimistic) bilevel optimization. I think the main reason for this assumption is that the unique $y_{\\rho, \\sigma}^* (x), z_{\\rho, \\sigma}^* (x)$ is needed in gradient of value function $ \\phi_{\\rho, \\sigma}^* (x)$. Is there any way to remove this assumption?\n\n2. The penalty method and the associated first-order updates are not particularly novel. However, the smoothing technique introduced here appears to be new.\n\nQUESTIONS:\n1. What is the intuition behind this smoothing approach, which incorporates both a quadratic term and a bilinear term? Does this technique have precedent in the literature? What would happen if we only add $\\sigma ||z||^2$ in the smoothing? Can the non-asymptotic results in Section 4 still hold? It seems in [28], they only used something similar to $\\sigma ||z||^2$ for smoothing.\n\n2. How do the results compare to those in reference [28]? Can authors add a remark under Theorem 4.2 on the choice of $s, p, q$ so that $||x^{k+1} - x^k||^2 / \\alpha_k^2$ and $||u^k - u_k^* (x^k)||^2$ converge in the same order of rate?\n\n3. The convergence notion in Theorem 4.2 is tied to the reformulation in (2) and smoothing approximation. Does this convergence imply $F(x^k , y^k)$ is close to $\\min_{y \\in S(x^k)} F(x^k , y)$? Are there other commonly used notions of convergence, and what is the relationship between these notions?\n\nLIMITATIONS:\nN.A", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-03"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis work focuses on addressing the relatively underexplored area of Pessimistic Bilevel Optimization (PBO) by proposing a single-loop, fully first-order algorithm. Specifically, it introduces a smooth approximation to the PBO problem through penalization and regularization, and builds on this to develop SiPBA, a gradient-based method that avoids second-order derivatives and inner-loop optimization. Theoretical analysis is provided to justify the approximation and prove the convergence of SiPBA.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n\n1. The studied problem is interesting.\n\n2. The presentation in this work is good.\n\nWeaknesses:\n\n1. The recent work on Optimistic Bilevel Optimization has shown that the lower-level optimization problem is a soft constraint to the upper-level ones. Whether the lower-level optimization problem is also a soft constraint in Pessimistic Bilevel Optimization? Since some approximation and smoothing tricks are used in this manuscript, and the soft constraint means that this constraint can be violated to some extent without resulting in a meaningless solution.\n\n2. Although the framework proposed in this work is single-loop. Each step in this framework is computationally complex since multiple gradient projection steps are used. I understand that the gradient projection step is efficient if the feasible region set is simple, for instance, the feasible region for some dual variables. However, the projection set in this algorithm is for the main variables, which are usually complex and non-convex. Operating the projection in each iteration needs to solve a non-convex optimization problem. Thus, more discussions on whether the proposed method is fully single-loop need to be added.\n\n3. The Assumption 3 is not a widely-used assumption in bilevel optimization, can you provide more discussions on this assumption? Additionally, whether the assumption is practical? Can you provide more results, e.g., experimental results, to support this assumption?\n\n4. Can you compare the non-asymptotic convergence rates of the proposed method with the existing bilevel optimization methods? \n\n5. The experimental results are limited. It seems that PBO has limited applications in machine learning. Can you provide more experimental results on additional application? More discussions on the applications of PBO in the introduction can also improve this manuscript. The reviewer will appreciate this work not only for its theoretical analysis of the optimization problem, but also for the practical relevance of the studied problem, which has broad applications in machine learning (since NeurIPS is a ML conference).\n\nQUESTIONS:\nMy questions are based on the weakness:\n\nIn the recent literature on Optimistic Bilevel Optimization, it has been shown that the lower-level optimization problem can be viewed as a soft constraint to the upper-level. In this work, since some approximation and smoothing techniques are adopted, does the lower-level optimization problem in Pessimistic Bilevel Optimization also act as a soft constraint to the upper-level problem?\n\nAlthough the proposed framework is claimed to be single-loop, each iteration involves multiple gradient projection steps. Given that the projection in this work is applied to the main variables (which are often non-convex and complex), does this still qualify the framework as fully single-loop in practice? Could the authors provide more discussion or clarification on this point?\n\nAssumption 3 appears to be uncommon in the context of bilevel optimization. Could the authors elaborate on the motivation and practicality of this assumption? In particular, are there empirical results or real-world scenarios that support its validity?\n\nHow does the proposed method compare with existing bilevel optimization methods in terms of non-asymptotic convergence rates? Can the authors provide a theoretical or empirical comparison?\n\nThe experimental validation appears limited, and the practical applications of Pessimistic Bilevel Optimization in machine learning are not well demonstrated. Could the authors include more experiments on additional tasks or provide further discussion on the potential applications of PBO in the introduction?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-07-04"}