Title: Set Smoothness Unlocks Clarke Hyper-stationarity in Bilevel Optimization
Venue: NeurIPS 2025
Accepted: yes
Keywords: Bilevel optimization; Clarke stationarity; Weakly convex optimization; Set-valued functions
Abstract: Solving bilevel optimization (BLO) problems to global optimality is generally intractable. A common surrogate is to compute a hyper-stationary point—a stationary point of the hyper-objective function obtained by minimizing or maximizing the upper-level objective over the lower-level solution set. Existing methods, however, either provide weak notions of stationarity or require restrictive assumptions to guarantee the smoothness of hyper-objective functions. In this paper, we eliminate these impractical assumptions and show that strong (Clarke) hyper-stationarity remains computable even when the hyper-objective is nonsmooth. Our key ingredient is a new structural property, called set smoothness, which captures the variational dependence of the lower-level solution set on the upper-level variable. We prove that this property holds for a broad class of BLO problems and ensures weak convexity (resp. concavity) of pessimistic (resp. optimistic) hyper-objective functions. Building on this foundation, we show that a zeroth-order algorithm that computes approximate Clarke hyper-stationary points with non-asymptotic convergence guarantees. To the best of our knowledge, this is the first computational guarantee for Clarke-type stationarity in nonsmooth BLO.  Beyond this specific application, the set smoothness property emerges as a structural concept of independent interest, with potential to inform the analysis of broader classes of optimization and variational problems.


Peer reviews:
comments: SUMMARY:
This paper studies bilevel optimization. The authors introduce a novel notion of set smoothness to characterize the solution set and hyper objective. Using this set smoothness, the authors prove that under a PL-like assumption, the hyper objective is weak convex(concave), enabling the computing of a Clarke hyper stationary point. The authors propose a zero-th order algorithm to compute it non-asymptotically.

STRENGTHS AND WEAKNESSES:
The main strength of this paper is the theoretical contribution. The establishment of weak convexity (concavity) and computability of Clarke hyper stationary under PL-like condition is a novel and worthwhile contribution. The introduction of the set smoothness has technical novelty and implications for future work. 

The weakness of the paper lies in its proposed algorithm. It is a zero-th algorithm and may not be practically efficient. The authors also do not provide empirical results.

QUESTIONS:
Do you think the zero-th order algorithm have implications for first order algorithms, which is more practical? What is the major gap between them?

LIMITATIONS:
yes
TITLE: Official Review
CLARITY: 3
IMPACT: 4
ORIGINALITY: 3
DATE: 2025-06-10

comments: SUMMARY:
This paper proves that strong Clarke hyper-stationary points are computable for nonconvex-PL bilevel problems. The main results build on the following three arguments: 
1. The solution set of a smooth PL function satisfies the set smoothness introduced in this paper (Proposition 2). 
2. The hyper-objective is weakly convex/concave when the lower-level solution set is smooth (Proposition 1).
3. A zeroth-order method can find strong Clarke stationary points for a weakly convex/concave objective (Section 4).

All the above arguments are novel. I especially appreciate Proposition 1 and Proposition 2, which are highly non-trivial and I think they definitely contribute a deeper understanding of bilevel optimization and other fields of optimization theory.

STRENGTHS AND WEAKNESSES:
The strengths of this paper are clear:
1. The paper is well written.
2. Proposition 1 and Proposition 2 are very novel and both solve important theoretical problems.

I do not notice significant weaknesses; the only one I find is contained in the limitation part.

QUESTIONS:
I do not have additional questions.

LIMITATIONS:
One limitation is the dimension dependency in the upper bound, which comes from the use of the zeroth-order method. I can understand that the dimension dependency is hard to remove, but it is still a limitation compared to the existing dimension-free upper bounds (although they may require additional assumptions).
TITLE: Official Review
CLARITY: 3
IMPACT: 3
ORIGINALITY: 3
DATE: 2025-06-16

comments: SUMMARY:
The paper considers bilevel optimization problems in the non-convex/PL setting where the lower-level solution set is not necessarily a singleton. The authors introduce the notion of set smoothness and show that the solution set of the inner function in the considered setting is smooth. Then, they show that this smoothness implies the weak convexity (resp. the weak concavity) of the pessimistic (resp. optimistic) hyper-objective. With this regularity, they show that Clark stationarity can be obtained with a zeroth-order algorithm.

STRENGTHS AND WEAKNESSES:
### Strengths

- **S1**: The paper is very well-written.

- **S2**: The paper introduce the concept of set smoothness which offers new perspectives in analyzing bilevel optimization when the inner function is not strongly convex.

- **S3**: The authors establish regularity properties of the hyper-objective based on the introduced set smoothness.

- **S4**: The authors prove computability of Clark stationarity for non-smooth hyperobjectives.


### Weaknesses

- **W1**: The assumption on the convexity of $\mathcal{S}(x)$ is strong for a non-convex inner function.



#### Minor/Typo
- For clarity, it would be useful to recall the definition of Goldstein stationarity
- **Line 186**: "Lemma 2 (ii) and (iv)" -> "Lemma 3 (ii) and (iv)"
- **Line 496**: "$\varphi(x_2)$" -> "$\varphi_p(x_2)$"

QUESTIONS:
N/A

LIMITATIONS:
N/A
TITLE: Official Review
CLARITY: 4
IMPACT: 3
ORIGINALITY: 4
DATE: 2025-07-01

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper investigates unconstrained bilevel optimization problems in which the lower-level problem admits multiple solutions. The central contribution is the introduction of a novel set-smoothness condition on the lower-level solution mapping. The authors show that under this condition, the associated hyper-objectives -- whether defined optimistically or pessimistically -- exhibit weak concavity (or weak convexity in the pessimistic case). This structural property enables the use of subgradient-based methods to find approximate Clarke stationary points efficiently. To this end, the authors develop a zeroth-order algorithm tailored for the bilevel setting.\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths**\n\nTo the best of my knowledge, this is the first paper where the notion of set-smoothness has been introduced in the context of Bilevel optimization. The authors effectively leverage this idea to unify and strengthen several threads in nonsmooth optimization -- namely, Clarke stationarity, weak convexity, and zeroth-order methods -- to tackle a longstanding challenge in bilevel problems with multiple lower-level solutions. The presentation is concise and the logical development of the core ideas is clear and well-structured.\n\n\n**Weaknesses**\n\nThe main conceptual contribution is compelling, but the following must be clarified:\n\n- The core contribution of this submission lies in the introduction of the set-smoothness property, and I believe it might have significant future potential to advance understanding in multiple-solution settings — an area where progress has been notably limited. That said, to fully realize this potential, the submission should more clearly position the proposed condition within the broader context and articulate its implications in greater depth. In particular, it would be essential to discuss the algorithmic implication of results in Section 3 in greater details. \n\n\nAlso, the algorithmic developments could be more thoroughly distinguished and emphasized:\n\n- While the paper builds toward Clarke stationarity, the motivation for preferring Clarke subgradients over alternatives such as the Goldstein subdifferential could be more fully justified. Is this choice driven purely by analytical tractability, or does it yield practical benefits (e.g., stronger convergence guarantees, better empirical performance)?\n\n- The distinction between the proposed Algorithm 1 and the approach in [7] could be made more transparent. A line-by-line comparison or at least a clear summary of the differences would be helpful. If empirical performance comparisons are available, they would strengthen the case.\n\n- A discussion of how general the set-smoothness condition is would be valuable. For example, how does it relate to more common regularity conditions like error bounds or the Polyak-Łojasiewicz condition at the lower level? Are there natural examples where set-smoothness holds but other assumptions fail?\n\nQUESTIONS:\n- Has Definition 2 (set-smoothness) appeared in prior literature, perhaps in the context of set-valued analysis or variational analysis? Understanding any precedent could help position the contribution more broadly.\n\n- Could the proposed method or analysis be extended to stochastic Bilevel settings?\n\nLIMITATIONS:\nThis is a well-written and insightful paper that introduces a novel structural condition with meaningful implications for Bilevel optimization. While the algorithmic contributions are incremental, the conceptual clarity and technical rigor make it a valuable addition to the literature. I recommend acceptance.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 4, "DATE": "2025-07-02"}