Title: On the Complexity of Finding Stationary Points in Nonconvex Simple Bilevel Optimization
Venue: NeurIPS 2025
Accepted: yes
Keywords: bilevel optimization

nonconvex optimization
Abstract: In this paper, we study the problem of solving a simple bilevel optimization problem, where the upper-level objective is minimized over the solution set of the lower-level problem. We focus on the general setting in which both the upper- and lower-level objectives are smooth but potentially nonconvex. Due to the absence of additional structural assumptions for the lower-level objective—such as convexity or the Polyak–Łojasiewicz (PL) condition—guaranteeing global optimality is generally intractable. Instead, we introduce a suitable notion of stationarity for this class of problems and aim to design a first-order algorithm that finds such stationary points in polynomial time. Intuitively, stationarity in this setting means the upper-level objective cannot be substantially improved locally without causing a larger deterioration in the lower-level objective. To this end, we show that a simple and implementable variant of the dynamic barrier gradient descent (DBGD) framework can effectively solve the considered nonconvex simple bilevel problems up to stationarity. Specifically, to reach an $(\epsilon_f, \epsilon_g)$-stationary point—where $\epsilon_f$ and $\epsilon_g$ denote the target stationarity accuracies for the upper- and lower-level objectives, respectively—the considered method achieves a complexity of $\mathcal{O}(\max(\epsilon_f^{-\frac{3+p}{1+p}}, \epsilon_g^{-\frac{3+p}{2}}))$, where $p \geq 0$ is an arbitrary constant balancing the terms. To the best of our knowledge, this is the first complexity result for a discrete-time algorithm that guarantees joint stationarity for both levels in general nonconvex simple bilevel problems.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper studied a smooth nonconvex simple bilevel optimization problem and developed the stationary condition for it. They then proved that dynamic barrier gradient descent algorithm can converge to the defined stationary point in polynomial time. Specifically, to reach an $(\\epsilon_f, \\epsilon_g)$ stationary point, the algorithm needs $\\max(\\epsilon_f^{-\\frac{3+p}{1+p}},\\epsilon_f^{-\\frac{3+p}{2}})$ iterations. Numerical experiments validate the effectiveness of the proposed algorithm.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths: 1) The problem studied in this paper is significant for both simple bilevel optimization and general bilevel optimization. By removing convexity and PL conditions at the lower level, the authors provide a broader theoretical guarantee for stationarity. 2) The idea of decomposing the upper-level gradient into orthogonal and parallel components is interesting and innovative. 3) The proposed algorithm is supported by a strong convergence guarantee.  \n\nWeakness: (1) The main concern is the correctness—or perhaps overstatement—of Lemma 1. For a smooth function, if the local region is chosen to be sufficiently small (as in the proof, where the radius $r$ is on the order of the target error, i.e., very small), then almost any point could qualify as the \"near-local optimum\" under the authors’ definition. This is especially problematic when the definition’s validity depends heavily on $r$. Additionally, the conclusion of Lemma 1 seems counterintuitive: for a smooth function, a small gradient does not rule out the possibility of being a local maximum. From my perspective, a more robust definition of a near-local minimum should require the existence of a fixed-radius local region where the inequalities in Lemma 3.1 hold. \n\n(2) Another weakness is that I do not understand the second point of Lemma 3.1. It is rigorous to say that for any point with improvement on the lower-level objective, the upper-level objective changes is not to much since $\\delta$ is very small. But I do not see why the improvement on the lower-level will lead to the negative effect on the upper-level, as said in this paper. I quoted the statement here \"the upper-level objective cannot be significantly improved without negatively impacting\". \n\n(3) The above two concerns is on the definition of stationarity. I also do not understand the relations of existing stationary measure in (6), especially there is a strange (ii) in Line 176.\n\nQUESTIONS:\n1. Do you intend to show that the proposed stationarity condition implies (6) and (7)? If so, please provide a more detailed explanation.\n2. The domain of $\\lambda$ is unclear. I recommend distinguishing between cases where $\\lambda$ takes a finite value, i.e., $\\lambda \\in \\mathbb{R}_+$, and cases where it may include infinity, i.e., $\\lambda \\in (0, +\\infty]$. For example, in the line below Line 168, the KKT multiplier $\\lambda$ should be finite. However, in equation (3) and the algorithm design, $\\lambda$ can be infinite. \n3. Please refer to Weaknesses 1) and 2) for concerns related to Lemma 3.1.\n\nLIMITATIONS:\nThe main limitation lies in the potential overstatement of the conclusions drawn from Lemma 3.1.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-01"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper studies a simple bilevel optimization problem where both the upper- and the lower-level problems are non-convex. First, the authors introduce a novel notion of stationarity, tailored to the specific problem at hand, and establish its relationship to existing metrics in the bilevel optimization literature. Then, they apply the Dynamic Barrier Gradient Descent (DBGD) algorithm (originally developed for constrained optimization problems) to the nonconvex simple bilevel problem and establish discrete-time stationarity guarantees. We note that this represents the first discrete-time complexity bound that guarantees stationarity at both levels in such problems. Finally, a set of numerical experiments on toy examples and matrix factorization tasks illustrate DBGD method's superior performance over traditional penalty methods.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths\n- This paper deals with a challenging bilevel problem. It is a nonconvex simple bilevel problem where both the upper and lower-level objectives are nonconvex functions. \n- A suitable stationarity concept is introduced for the specific nonconvex simple bilevel problem, and its relationship to existing metrics in the bilevel optimization literature is established.\n- The paper provides the first discrete-time complexity bound that guarantees stationarity at both levels of a  nonconvex simple bilevel problem.\n\nWeaknesses\n- Since this is a simple bilevel problem, its applicability is somewhat limited. In fact, the only application discussed in detail is matrix factorization. The authors may wish to provide more information about the potential applications of the specific bilevel problem they are addressing, e.g. by providing specific example applications and their formulations.\n- The experimental evaluation is somehow limited. There is only one baseline, the penalty method and the experiments consist of toy problems and small scale matrix factorization tasks. Are there other bilevel methods that can be used for the solution of this problem? How about methods developed for generic (not necessarily) simple bilevel problems?\n\nQUESTIONS:\n- To situate the method within the existing literature, a table summarizing relevant results about simple bilevel optimization problems (method name, assumptions, optimality metric, complexity, etc.) could be helpful.\n- Is Assumption 3.1 (Local Error Bound) typically used in similar settings (ie simple nonconvex bilevel problems)? Is it a strong assumption? Are there common applications in which this assumptions holds?\n- The definition of stationarity for the proposed problem is a central contribution of this work and should be properly situated within the relevant literature. Has this particular stationarity metric, or a related one, been previously introduced or discussed in the bilevel optimization literature?\n- What approach did you use to tune the penalty parameter (of the penalty method baseline in the experiments) for obtaining the best possible performance and ensuring fair comparisons?\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 4, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-01"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper studies the complexity of finding stationary points in nonconvex simple bilevel optimization problems, where the upper-level objective is minimized over the solution set of a lower-level problem. The authors introduce a notion of \\((\\epsilon_f, \\epsilon_g)\\)-stationarity to measure convergence, capturing points where the upper-level objective cannot be improved without worsening the lower-level objective. They adopt a variant of the Dynamic Barrier Gradient Descent (DBGD) algorithm and establish the first discrete-time complexity guarantee for nonconvex simple bilevel problems, showing that DBGD achieves an \\((\\epsilon_f, \\epsilon_g)\\)-stationary point in \\(\\mathcal{O}(\\max(\\epsilon_f^{-\\frac{3+p}{1+p}}, \\epsilon_g^{-\\frac{3+p}{2}}))\\) iterations, where p \\geq 0 is a tunable parameter. Furthermore numerical experiments validate the algorithm's effectiveness compared to penalty-based methods.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n\nThis is a well-written paper that makes a valuable contribution by extending the framework of [46] to a non-convex setting for the low-level function. The authors provide a complete algorithmic solution and rigorously establish convergence guarantees, demonstrating both theoretical and practical advancements over prior work.\n\nWeaknesses: \n\nCould the authors elaborate on the additional technical difficulties arising from the non-convexity of the low-level function in their framework, compared to the convex formulation in [46]? Specifically:  \n1) How does non-convexity affect the theoretical analysis (e.g., convergence guarantees, optimality conditions)?  \n\n2) What algorithmic modifications were required to address these challenges?  \n\n3) Are there measurable performance trade-offs between the two approaches in practice?\n\nQUESTIONS:\nNA\n\nLIMITATIONS:\nNA", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 1, "CLARITY": 4, "IMPACT": 4, "ORIGINALITY": 4, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper studies the challenging problem of simple bilevel optimization where both the upper- and lower-level objective functions are smooth but potentially nonconvex. Because finding a global optimum in this setting is generally intractable, the work focuses on the more achievable goal of finding a near-stationary point. The authors first introduce a suitable notion of an $(\\epsilon_f, \\epsilon_g)$-stationary point, which intuitively identifies a solution where the upper-level objective cannot be substantially improved locally without causing a larger deterioration in the lower-level objective. This new metric consists of two conditions: the norm square of the lower-level gradient must be smaller than $\\epsilon_g$, and the norm square of the upper-level gradient + $\\lambda$ * the lower-level gradient must be smaller than $\\epsilon_f$.\n\nTo find such a point, the paper analyzes a discrete-time variant of the Dynamic Barrier Gradient Descent (DBGD) algorithm (previously proposed by Gong et al.) The core of the method is a first-order update step where the descent direction is a dynamic combination of the two gradients, controlled by a multiplier $\\lambda_k$ that is gradually increasing over iterations. The paper's main contribution is providing the first complexity result for a discrete-time algorithm that guarantees joint stationarity for both levels in this general nonconvex setting. Specifically, the algorithm is shown to find an $(\\epsilon_f, \\epsilon_g)$-stationary point with a complexity in $O(\\max (\\epsilon_f^{-\\frac{3+p}{1+p}}, \\epsilon_g^{-\\frac{3+p}{2}}) )$.\n\nI have checked most of the proofs in the main paper and the appendix. The main analysis is based on a cute application of the existing results from DBGD by Gong et al. Specifically, the results in Section 4 are the same as the derivation and results in Gong et al. in their Section 2, the analysis of DBGD. The major contribution of this paper starts in Section 5 by applying descent lemmas to both function $f$ and $g$ (Lemma 5.1). The discussion below Lemma 5.1 follows directly by weighting the terms in Lemma 5.1 and this leads to a $\\lambda_k$-dependent bound on the gradient norm. However, since the Lagrangian dual $\\lambda_k$ is increasing over iterations, this leads to additional discussion on Lemma 5.2 - 5.5 to ensure a $\\lambda_k$-independent bound. The bound is achieved by playing with the inequality to remove the $\\lambda_k$ term in Equation 17. This can be achieved by replacing $\\lambda_k$ by its upper bound. The rest of the analysis follows by doing some linear algebra tricks then. I like the analysis in Section 5 and the experimental results in Section 6. The experimental results verify that the proposed algorithm can achieve low gradient norm for both the lower level and the upper level. The baselines for comparison are relatively weak as it only uses fixed penalty parameter. I wonder if there are better baselines that the authors can compare with.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n- Clear explanation of the convergence analysis. I appreciate the completeness but I do think some of the proofs can be deferred to the appendix.\n- New setting in simple bilevel optimization problems where the lower-level objective is non-convex. To my knowledge, the general bilevel optimization problems with non-strongly-convex objective have been shown to have some impossibility results. It is good to see that the simple bilevel optimization problems can at least achieve some convergence result.\n\nWeaknesses:\n- Weak baselines in the experiments. Fixed penalty method is known to be not working.\n- Unclear lower bound for the convergence rate. It would be great if the authors can establish some lower bound / impossibility results for the non-convex simple bilevel optimization problems.\n- Most of the analyses are a direct application of DBGD by Gong et al. including the algorithm. The major difference between Gong et al.'s paper and this paper is just the problem setting. Gong et al. considers single level optimization with constraints, and this paper considers simple bilevel optimization that can be reformulated as constrained optimization by the value-based method. The technical novelty is weak.\n\nQUESTIONS:\nPlease respond to my questions in the strengths and weaknesses section.\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-03"}