{"id":"cc82e622-6e72-44c5-8101-4917e12d98a6","arxiv_id":"2608.12704","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"The paper introduces MOMEHA and MB-MOMEHA, Hessian-free single-loop algorithms that converge to relaxed Pareto-stationary points for multi-objective bilevel problems with nonconvex lower levels.","lead":"An algorithm for multi-objective bilevel optimization that allows the inner problem to be nonconvex, using a Moreau-envelope reformulation and smooth Tchebycheff scalarization. The paper gives convergence guarantees for deterministic and stochastic variants and tests them on meta-learning and neural architecture search.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Moreau-envelope reformulation is not equivalent to the original bilevel problem under weak convexity; Theorems 1-2 solve a stationarity surrogate, so the central claim about problem (1) is unsupported.","rationale":"The reader's weakest assumption is exactly the Moreau-envelope equivalence issue, and I agree it is the most load-bearing concern. The deterministic and stochastic convergence theorems are stated about the Moreau-envelope stationarity problem, and the paper's claimed contribution to nonconvex lower-level bilevel optimization depends on that problem being the original bilevel problem (1). Without equivalence, Theorems 1 and 2 describe convergence to Pareto stationarity of a superset of the feasible points of (1), so the method can return a point that is infeasible for the original problem. This is not a cosmetic gap: for weakly convex lower levels, stationarity is substantially weaker than global optimality, and the proposed counterexample check would settle whether the equivalence holds. A secondary issue, which the reader also noted, is that the stochastic estimator in Algorithm 2 (lines 6 and 9) uses a ratio of exponentials with per-objective mini-batches; its expectation is not the gradient of the smooth Tchebycheff scalarization, so Assumption 4 is not satisfied by the described plug-in estimator. That issue is real but affects only the stochastic theorem, whereas the equivalence problem also invalidates the deterministic contribution. Because the central claim as stated does not hold without either an additional assumption restoring equivalence or a substantial reframing of the contribution, I would move the verdict from CONDITIONAL to REJECT in the current form.","tokens_in":37470,"tokens_out":13203,"duration_ms":137263,"concrete_test":"Construct the minimal deterministic counterexample: let g(x,y) = -y^2 + y^4/4 (rho_y = 2 weakly convex), gamma < 1/4, m = 1, and F(x,y) = x^2 + (y-1)^2. At y = 0, gradient_y g = 0, so (0,0) is feasible for (6), but y = 0 is a local maximum, not a global minimizer of g (global minima are at y = +/- sqrt(2)). Check whether the cited Theorem A.2 of Liu et al. [2024] applies to this g; if it does, the asserted equivalence (5) <-> (6) is false. Alternatively, run MOMEHA from a starting point near y = 0 with preference w = 1; if it converges to y = 0 or another non-global stationary point, the claim that the algorithm solves (1) fails. If the authors' equivalence proof relies on an unstated condition, that condition must be stated and added to Assumptions 1-3, and the theorems re-proved for the original problem (1).","verdict_should_be":"REJECT","load_bearing_attack":"The paper's central claim that MOMEHA and MB-MOMEHA solve the nonconvex-lower-level MOBL problem (1) rests on the equivalence asserted in Section 4.1 between the Moreau-envelope constrained problem (5) and the stationarity-constrained problem (6). The paper cites Theorem A.2 of Liu et al. [2024] for the claim that, for rho_y-weakly convex g with gamma in (0, 1/(2 rho_y)), (5) is equivalent to (6). This equivalence is false in general: for weakly convex g, gradient_y g(x,y) = 0 is necessary but not sufficient for y in argmin_y g(x,y). The constraint g(x,y) - upsilon_gamma(x,y) <= 0 means y is the unique minimizer of theta maps to g(x,theta) + (1/(2 gamma)) ||theta - y||^2, which is equivalent to stationarity of g at y, not to global minimality. Thus the feasible set of (6) strictly contains the feasible set of (1); for example, a local maximum of a weakly convex lower-level objective satisfies gradient_y g = 0. The epsilon_c-epsilon_s-Pareto stationarity guarantees in Theorems 1 and 2, and Definition 6, are defined for the relaxed problem (16), i.e., for the enlarged stationarity set, not for the original bilevel problem (1). Consequently, without an additional hypothesis (e.g., lower-level convexity, a Polyak-Lojasiewicz condition, or a global-minimizer characterization), the abstract and Section 1 claim of solving (1) is not supported; the algorithms are guaranteed only for a surrogate problem.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes MOMEHA and its stochastic variant MB-MOMEHA for multi-objective bilevel optimization with a nonconvex (weakly convex) lower-level problem. The methods combine a Moreau-envelope reformulation of the lower-level problem with a smooth Tchebycheff scalarization, yielding single-loop, Hessian-free, preference-guided algorithms. The main theoretical claims (Theorems 1 and 2) are non-asymptotic convergence to ε_c-ε_s-Pareto stationary points of the reformulated problem, with rates O(T^{-1/2}) in the deterministic setting and O(T^{-(1/8-δ)}) in the stochastic setting. Experiments on few-shot meta-learning and neural architecture search are reported and compared with several prior methods.","tokens_in":37816,"tokens_out":12263,"duration_ms":116841,"significance":"If the reformulation were equivalent to the stated bilevel problem, the results would be a substantial advance: they would remove the lower-level convexity assumption for multi-objective bilevel learning, provide the first momentum-based stochastic convergence guarantee for the Moreau-envelope Hessian-free framework, and demonstrate preference-guided Pareto front exploration. The analysis is detailed and largely self-contained, with a Lyapunov argument in the stochastic case, and the experimental comparison covers relevant benchmarks. However, the equivalence between the reformulation and the original problem (1) is not established for weakly convex lower levels, so the significance is currently limited to a stationarity-constrained surrogate problem rather than the global-minimizer bilevel problem stated in the title and abstract.","major_comments":[{"comment":"The claimed equivalence between the Moreau-envelope constrained problem (5) and the stationarity-constrained problem (6) is false for weakly convex lower-level objectives. Stationarity ∇_y g(x,y)=0 is necessary but not sufficient for y ∈ argmin_y g(x,y). A concrete counterexample is g(y) = -y^2/2 + y^4/4, which is 1-weakly convex; y=0 is a stationary point (indeed a local maximum) but the global minima are at y=±1. Since the constraint g(x,y)-v_γ(x,y)≤0 is equivalent to ∇_y g(x,y)=0 under the stated assumptions, the feasible set of (6) is strictly larger than the feasible set of (1). Consequently, the ε_c-ε_s-Pareto stationarity guarantees in Theorems 1 and 2 and Definition 6 are guarantees for the relaxed surrogate problem (16), not for the original MOBL problem (1). The abstract and Section 1 claim of solving (1) is therefore unsupported unless an additional hypothesis is imposed, such as lower-level convexity, a Polyak-Łojasiewicz condition, or another condition guaranteeing that stationarity implies global minimality.","section":"Section 4.1, Eqs. (5)-(6)"},{"comment":"The equivalence between the normal-cone formulation (14) and the multiplier formulation (15) omits the complementarity condition p·(g(x,y)-v_γ(x,y)-ε_c)=0. Without this condition, (15) can be satisfied at points strictly inside the relaxed feasible set with p>0, which is not equivalent to the normal-cone condition in (14). The convergence proofs appear to construct points on the boundary of the relaxed feasible set (where ε_c equals the actual constraint violation), so the omission is repairable, but Definition 6 as stated is not precise and should be amended.","section":"Section 5.1, Definition 6"}],"minor_comments":[{"comment":"There are several typographical errors, including 'satiesfies' in Definition 6, 'staionarity' in Section 4.1, 'supplemeTntary' in Section 6, and 'has been begun' in the abstract.","section":"Throughout"},{"comment":"The step-size intervals are written as [α_x, α_x] and [α_y, α_y]; the overbars appear to be missing from the upper endpoints, making the notation ambiguous.","section":"Theorem 1 and Theorem 2"},{"comment":"In the definition of L_υ and in the chain of inequalities in Eq. (23), the second term inconsistently uses L_θ in one place and L_g in another; the constants should be defined consistently.","section":"Lemma 2"},{"comment":"The equivalence between (14) and (15) should explicitly state that the multiplier p is nonnegative and that complementarity holds, or restrict the equivalence to boundary points of the relaxed feasible set.","section":"Definition 6"},{"comment":"The definition of \\hat{d}_{θ,t} in the algorithm box has an unbalanced parenthesis; the reader must infer the intended grouping of terms.","section":"Algorithm 2"}],"recommendation":"major_revision","confidential_remarks":"The referee agrees with the skeptical assessment that the reformulation gap in Section 4.1 is load-bearing. The authors should either add assumptions that make stationarity of the lower level coincide with global minimality, or revise the claims to present the contribution as solving the Moreau-envelope stationarity surrogate and clearly state the relationship to the original bilevel problem. Given the substantial algorithmic machinery, detailed proofs, and experimental work, this issue is repairable within the scope of a major revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things worth knowing. The paper really does propose a new single-loop, Hessian-free algorithm for multi-objective bilevel problems with nonconvex lower-level objectives, and the deterministic convergence analysis is deep and mostly coherent. But the central claim that MOMEHA solves the original bilevel problem (1) is not supported: the Moreau-envelope reformulation replaces the lower-level argmin by the stationarity condition ∇_y g=0, which is not equivalent to global minimality under weak convexity. Theorems 1 and 2 establish convergence to a relaxed Pareto stationarity of this surrogate problem, not of (1). The stochastic variant has a second issue: the smoothed Tchebycheff gradient estimator is not unbiased, so the momentum analysis applies to an idealized estimator the algorithm does not actually use.\n\nWhat is genuinely useful: this is the first method in the cited set with nonconvex lower level, single-loop structure, and no Hessian; the smooth Tchebycheff scalarization gives preference-guided Pareto exploration; and the potential-function argument with the auxiliary variable θ is intricate, with deterministic rates O(T^{-1/2}) matching single-objective MEHA. The ε_c-ε_s Pareto stationarity concept is a sensible way to handle infeasible penalty iterates, and the paper is transparent about using it.\n\nThe soft spots are not cosmetic. Section 4.1 asserts an equivalence with Theorem A.2 of Liu et al., but under ρ_y-weak convexity, ∇_y g=0 can hold at local maxima, so the feasible set of (6) strictly contains the argmin-feasible set. The stress-test note is right on this. Without an additional condition (PL inequality, convexity, or a global-minimizer characterization), the abstract overclaims. I’d ask the authors to recast the contribution as convergence for the Moreau-envelope surrogate, or supply a hypothesis that closes the equivalence gap.\n\nThe stochastic analysis has a matching hole: Assumption 4 assumes unbiased estimates of ∇F_w^(STCH), but Algorithm 2’s mini-batch ratios are biased because numerator and denominator share samples. The error recursions in Lemma 8 don’t cover the actual estimator. That is fixable with independent batches or a correction term, but as written it is a real gap.\n\nExperiments are illustrative only: one seed, no released code, and WC-MHGD is excluded for non-convergence. Minor relative to the theory issues.\n\nBottom line: this deserves serious refereeing, and I’d engage with a revised version. I would not cite the current arXiv version for solving nonconvex MOBL; I might cite it for the surrogate stationarity framework once the claims are scoped properly.","headline":"A substantial algorithmic extension whose central equivalence claim fails under weak convexity, so both convergence theorems currently solve a surrogate problem rather than the stated one.","tokens_in":38345,"tokens_out":4022,"would_cite":false,"duration_ms":39648,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["90C29","90C26","90C30","90C15"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper claims that multi-objective bilevel optimization with a nonconvex lower level is solvable by a single-loop, Hessian-free algorithm family that converges to Pareto-stationary points, deterministically at rate $O(T^{-1/2})$ and…","keywords":["multi-objective bilevel optimization","nonconvex lower level","Moreau envelope","Hessian-free algorithm","smooth Tchebycheff scalarization","Pareto stationarity","stochastic momentum","single-loop algorithm"],"falsifier":"Run MOMEHA on the scalar bilevel problem $\\min_{x,y} (x-1)^2+y^2$ subject to $y\\in\\arg\\min_y(-y^2+y^4)$. The lower-level minimizers are $y=\\pm 1/\\sqrt{2}$, not the stationary point $y=0$; if the reported $\\varepsilon_c$-$\\varepsilon_s$-Pareto stationary point is at $y=0$, the reformulation has certified a point that is not a true bilevel solution.","tokens_in":37262,"feed_emoji":"⚖️","tokens_out":7653,"duration_ms":74631,"temperature":0.7,"pith_summary":"This paper takes on multi-objective bilevel optimization, where an upper-level decision must balance several conflicting objectives, each evaluated at the solution of a nested lower-level problem, in the regime where that lower level is nonconvex. Previous multi-objective bilevel methods required convex or strongly convex lower levels, which excludes many modern machine-learning applications. The paper claims that a Moreau-envelope reformulation together with smooth Tchebycheff scalarization turns the problem into a single-level constrained problem that can be attacked by alternating gradient steps, with no Hessian computation and no inner loop. For the deterministic method MOMEHA and the stochastic momentum variant MB-MOMEHA, it proves convergence to a relaxed Pareto-stationarity concept at rates $O(T^{-1/2})$ and $O(T^{-(1/8-\\delta)})$, respectively, and reports better Pareto fronts than existing methods on few-shot meta-learning and neural architecture search.","feed_headline":"Hessian-free method tackles nonconvex bilevel multi-objective problems","feed_subtitle":"Single-loop algorithms provably reach Pareto-stationary points, with a stochastic momentum variant included.","key_machinery":"The load-bearing mechanism is the Moreau envelope $\\upsilon_\\gamma(x,y)=\\min_\\theta\\{g(x,\\theta)+\\frac{1}{2\\gamma}\\|\\theta-y\\|^2\\}$, which converts the lower-level problem into an explicit scalar constraint $g(x,y)-\\upsilon_\\gamma(x,y)\\le 0$, together with the smooth weighted Tchebycheff scalarization $F_w^{\\mathrm{STCH}}(x,y)=\\frac{1}{\\mu}\\log\\sum_{i=1}^m\\exp(\\mu w_i(f_i(x,y)-z_i))$, which encodes user preferences and enables Pareto-front exploration. The algorithms maintain an auxiliary variable $\\theta$ that tracks the envelope minimizer with a single gradient step per iteration, so the overall scheme is single-loop and Hessian-free. The stationarity measure is a relaxed concept, $\\varepsilon_c$-$\\varepsilon_s$-Pareto stationarity, defined through the normal cone of the $\\varepsilon_c$-relaxed feasible region, because penalty iterates can remain slightly infeasible with respect to the envelope constraint.","core_discovery":"The paper's central claim is that the Moreau-envelope reformulation of bilevel programs, which replaces the lower-level argmin by the explicit constraint $g(x,y)-\\upsilon_\\gamma(x,y)\\le 0$ and, under weak convexity, by the stationarity condition $\\nabla_y g(x,y)=0$, can be combined with smooth weighted Tchebycheff scalarization to produce preference-guided, Hessian-free, single-loop algorithms for multi-objective bilevel problems with nonconvex lower levels. For the deterministic algorithm MOMEHA it establishes convergence to an $\\varepsilon_c$-$\\varepsilon_s$-Pareto stationary point of the relaxed constrained problem at rate $O(T^{-1/2})$. For the stochastic momentum variant MB-MOMEHA it establishes the same stationarity concept at rate $O(T^{-(1/8-\\delta)})$, with the envelope constraint violation driven to $O(T^{-1/16}\\sqrt{\\ln T})$ under an increasing penalty. The paper further claims that the stochastic result is the first momentum-based convergence proof for the Moreau-envelope Hessian-free framework, and reports that the methods outperform existing baselines on Pareto-front quality in few-shot meta-learning and neural architecture search.","pith_inferences":["The paper leaves implicit, but important, that the equivalence between $\\nabla_y g(x,y)=0$ and the true lower-level argmin is not guaranteed for weakly convex lower levels; without strong convexity, the certified stationary point may not be a lower-level global minimizer, so empirical results should be read as solving the reformulated problem rather than the original bilevel problem as stated.","The smooth Tchebycheff scalarization with finite $\\mu$ only approximates the true Tchebycheff scalarization, so reported Pareto fronts may depend noticeably on the choice of $\\mu$; a natural editorial extension would be to report front sensitivity to $\\mu$ alongside the proposed stationarity measure.","A testable extension beyond the paper is to benchmark MOMEHA and MB-MOMEHA on synthetic nonconvex bilevel problems with known Pareto fronts, where the exact trade-off surface is available, to separate reformulation error from optimization error.","The stochastic rate $O(T^{-(1/8-\\delta)})$ suggests that variance-reduced or adaptive momentum variants could push the stochastic guarantee closer to the deterministic $O(T^{-1/2})$ rate, but the paper does not claim this."],"forward_implications":["Multi-objective bilevel problems with nonconvex lower levels become tractable with single-loop, Hessian-free updates, removing the two main computational bottlenecks of prior methods.","User-specified preference vectors translate into concrete points on the Pareto front, so practitioners can trade off objectives without re-solving the entire problem.","The deterministic convergence rate $O(T^{-1/2})$ matches the single-objective Moreau-envelope Hessian-free baseline, so the multi-objective extension adds no extra convergence cost in the full-gradient setting.","In the stochastic setting, momentum-based mini-batch updates converge to the same stationarity concept, making the method applicable to data-driven tasks with noisy gradients.","The $\\varepsilon_c$-$\\varepsilon_s$-Pareto stationarity measure provides a well-defined stopping criterion even when iterates remain slightly infeasible with respect to the envelope constraint."],"supporting_citations":[{"why":"Supplies the single-loop, Hessian-free Moreau-envelope framework whose deterministic analysis this paper extends, and whose stochastic momentum gap the paper claims to fill.","marker":"Liu et al. [2024]"},{"why":"Provides the Moreau-envelope reformulation and the gradient formula for $\\nabla\\upsilon_\\gamma$ used in the algorithm updates.","marker":"Gao et al. [2023]"},{"why":"Supplies the smooth weighted Tchebycheff scalarization that gives the algorithms preference-guided Pareto-front exploration.","marker":"Lin et al. [2024]"},{"why":"Provides the preference-based weighted Chebyshev multi-hyper gradient descent baseline that requires strong convexity and Hessian information, which the new method removes.","marker":"Zhang et al. [2026a]"},{"why":"Supplies the WC-penalty method for general convex lower levels, the closest existing preference-based method that still assumes convexity in the lower level.","marker":"Zhang et al. [2026b]"},{"why":"Provides the stochastic multi-objective gradient correction method (MoCo), a nested-loop, Hessian-based baseline for stochastic multi-objective bilevel learning.","marker":"Fernando et al. [2022]"},{"why":"Supplies the weak-convexity facts used to infer $(\\rho_x,\\rho_y)$-weak convexity from gradient Lipschitz continuity of the lower-level objective.","marker":"Beck [2017]"}],"fun_headline_variants":["Hessian-free bilevel optimizer cracks nonconvex multi-objective tasks","Moreau envelope yields single-loop Hessian-free bilevel solver","Nonconvex lower levels? MOMEHA delivers Pareto-stationary points","Stochastic momentum boosts Hessian-free bilevel optimization","Multi-objective bilevel made tractable without Hessians"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that, for the weakly convex lower-level functions considered, the stationarity condition $\\nabla_y g(x,y)=0$ pins down the lower level's true best response; a stationary point of a weakly convex function need not be a minimizer, and if that gap is wide the reformulated problem and the original bilevel problem are different problems.","fun_headline_variants_meta":{"raw":{"variants":["Hessian-free bilevel optimizer cracks nonconvex multi-objective tasks","Moreau envelope yields single-loop Hessian-free bilevel solver","Nonconvex lower levels? MOMEHA delivers Pareto-stationary points","Stochastic momentum boosts Hessian-free bilevel optimization","Multi-objective bilevel made tractable without Hessians"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000858,"raw_usage":{"total_tokens":3773,"prompt_tokens":1039,"completion_tokens":2734,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":655,"completion_tokens_details":{"reasoning_tokens":2645}},"tokens_in":655,"tokens_out":2734,"duration_ms":19692,"temperature":1.0,"reasoning_tokens":2645,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T04:10:47.241276+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run MOMEHA on the scalar bilevel problem $\\min_{x,y} (x-1)^2+y^2$ subject to $y\\in\\arg\\min_y(-y^2+y^4)$. The lower-level minimizers are $y=\\pm 1/\\sqrt{2}$, not the stationary point $y=0$; if the reported $\\varepsilon_c$-$\\varepsilon_s$-Pareto stationary point is at $y=0$, the reformulation has certified a point that is not a true bilevel solution.","supporting_citations":[{"cited_title":"A first-order multi-gradient algo- rithm for multi-objective bi-level optimization","cited_arxiv_id":null,"evidence_quote":"Supplies the smooth weighted Tchebycheff scalarization that gives the algorithms preference-guided Pareto-front exploration."}],"review_version":1}