{"id":"24eaccef-064a-4d2c-b9bf-92e517f9993c","arxiv_id":"2411.13394","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"CB2O is a consensus-based optimization method with a quantile selection step that provably converges to the upper-level minimizer among the lower-level minimizers in the mean-field limit.","lead":"This paper introduces CB2O, a particle-based, derivative-free algorithm for solving bilevel optimization problems where the upper-level objective is minimized over the set of global minimizers of a lower-level objective.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Assumption A5 is a strong, unverified structural condition on G that can fail in natural problems, and the claimed convergence depends on it; a concrete counterexample should be tested.","rationale":"The reader identified Assumption A5 as the weakest assumption, and I agree. The central claim, Theorem 2.7, is a conditional statement: if A1-A5 hold, then exponential convergence follows. The proof's critical control on ||m_{alpha,beta}^{G,L}(rho_t) - theta*_good|| comes from Proposition 4.2, which relies directly on the inverse continuity properties of A5. If A5 fails, the minimizer of G on the quantile-selected set Q_L^beta can be far from theta*_good, and the Laplace-principle step actively drives the consensus point toward the wrong minimizer. This is not a mere technicality: the assumption requires a quantitative growth estimate on G at every scale r_G <= R_G and on every neighborhood of the entire minimizer set Theta, which is a strong structural condition that is not satisfied, for example, when Theta is a continuum and G has a spurious well just outside B_{r_G}(theta*_good). The concrete test above gives a minimal problem where A1-A4 hold and A5 fails; if the dynamics converge to the spurious well, the theorem's hypothesis is genuinely load-bearing. The reader's verdict of CONDITIONAL remains appropriate: the mean-field argument may be correct under A5, but the paper should either prove or clearly characterize A5 for the experimental settings, and ideally address the acknowledged gap between the regularized mean-field dynamics and the implemented finite-particle algorithm. My read does not change the verdict, since the paper's own caveats are already reflected in the CONDITIONAL decision.","tokens_in":67426,"tokens_out":23883,"duration_ms":240028,"concrete_test":"Construct the following 1D problem: let Theta = [-1,1] with L(theta)=0 on [-1,1] and L(theta)=(|theta|-1)^2 outside, so A2-A3 hold. Let G be smooth with G(0.5)=-0.1, G(-1)=0, and G decreasing sharply to -10 at theta=-1.1 (e.g., G(theta)=-1000(theta+1)^2 on [-1.2,-0.8], smoothly connected to (theta-0.5)^2-0.1 elsewhere). This satisfies A1, A2, A4 but violates A5 because the minimizer of G on N_{0.05}(Theta) is near -1.05, far from theta*_good=0.5. Run the finite-particle CB2O algorithm (Algorithm 1, hard quantile) with N=10^4 particles and a mean-field PDE solver for the Fokker-Planck equation (1.13). If the consensus concentrates near -1.1 instead of 0.5, A5 is load-bearing and Theorem 2.7 is not applicable to this natural setting.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim of Theorem 2.7 is exponential mean-field convergence to theta*_good under Assumptions A1-A5. The load-bearing assumption is A5 (Eqs. 2.15a-2.15b): for every r_G <= R_G, the minimizer of G on N_{r_G}(Theta) must lie within r_G of theta*_good, and G must grow away from that ball by a uniform margin G_inf. This is not implied by A1, which only requires theta*_good to be the unique minimizer of G on Theta. Proposition 4.2 uses A5 to ensure that the quantile-selected consensus point tracks theta*_good: the Laplace weight exp(-alpha G) concentrates near the minimizer of G on Q_L^beta, and A5 is what guarantees that minimizer is close to theta*_good. Without A5, the error bound (4.18) in Proposition 4.2 does not follow, and the differential-inequality argument of Theorem 2.7 collapses. A5 is also never verified for the numerical problems in Section 5, so the theorem's applicability to the experiments is unestablished. A secondary but explicitly admitted gap is the absence of a mean-field approximation result connecting the analyzed Fokker-Planck equation (1.13) to the finite-particle Algorithm 1; however, the more direct threat to the central claim is the strength and unverifiability of A5.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes CB2O, a consensus-based particle metaheuristic for simple bi-level optimization problems of the form (1.1), where the upper-level objective G is minimized over the minimizer set Θ of a possibly nonconvex lower-level objective L. The algorithm computes a consensus point by selecting a lower-level quantile of particles and applying a Gibbs/Laplace weight in the upper-level objective. The main theoretical results are (i) an existence theorem for weak solutions of the mean-field Fokker-Planck equation (Theorem 2.4), obtained via a new stability estimate for the consensus point under a combination of Wasserstein and L2 perturbations, and (ii) a mean-field convergence theorem (Theorem 2.7) showing that, under Assumptions A1–A5 and suitable hyperparameter choices, the Wasserstein-2 distance to the target θ*_good decays at least exponentially until it reaches a prescribed accuracy ε. Numerical experiments on constrained optimization, sparse representation learning, and clustered federated learning are reported, with code provided. The paper is transparent about several open problems, notably the mean-field approximation of the finite-particle system and non-uniqueness of mean-field solutions.","tokens_in":67663,"tokens_out":4169,"duration_ms":50108,"significance":"If the main results hold, the paper makes a valuable contribution to derivative-free global optimization for bilevel problems: it proposes a principled quantile-based selection mechanism, proves existence and convergence for the associated mean-field dynamics, and provides explicit, a priori hyperparameter conditions. The stability estimate of Proposition 3.2 and the quantitative quantiled Laplace principle of Proposition 4.2 are technical novelties that may be useful beyond this paper. The numerical section is extensive and the authors provide open-source code, which supports reproducibility. However, the central convergence theorem is a statement about the mean-field PDE, not about the finite-particle algorithm implemented in practice, and its key assumption A5 is strong and not verified on the numerical examples. These points substantially temper the practical claims.","major_comments":[{"comment":"Assumption A5 is load-bearing for Theorem 2.7, since Proposition 4.2 uses it to guarantee that the minimizer ˜θgood of G on Q_L^β[ϱ] ⊂ N_{r_G}(Θ) lies within r_G of θ*_good, and the error bound (4.18) collapses without it. Yet A5 is not implied by A1, which only says θ*_good uniquely minimizes G on Θ. For example, if Θ consists of two separated global minimizers θ_bad and θ*_good with G(θ_bad) < G(θ*_good), then for every sufficiently small r_G the minimizer of G on N_{r_G}(Θ) is near θ_bad, at distance roughly 2r_G or larger from θ*_good, so (2.15b) fails while A1 holds. The manuscript does not verify A5, or the related A3, for any of the problems in Section 5; in particular the sparse representation experiment in Section 5.3 involves a neural-network landscape where such inverse-continuity conditions are highly nontrivial to check. The theorem's applicability to the reported experiments is therefore unestablished. I ask the authors to either prove A5 in the numerical settings, provide a tractable sufficient condition, or explicitly state the verified domain of Theorem 2.7.","section":"Assumption A5; Eqs. (2.15a)–(2.15b); Proposition 4.2; Theorem 2.7"},{"comment":"The central convergence guarantee (Theorem 2.7) applies to any sufficiently regular weak solution ρ of the mean-field Fokker-Planck equation (1.13), not to the finite-particle Algorithm 1. The paper explicitly leaves the mean-field approximation limit (1.14)/(1.15) as an open problem, and Section 6 lists it among future work. Additionally, the algorithm implemented in Section 5 uses the unregularized consensus point (1.6), whereas the analyzed dynamics use the regularized quantile set (2.3) with truncation radius R, slack δ_q, and averaged quantile. Thus there are two gaps between the proved statement and the implemented method: finite N and regularized-versus-implemented consensus. Since the abstract and Section 1.1 present CB2O as a method that 'converges' to θ*_good, the manuscript should either prove or at least clearly delineate a finite-particle or propagation-of-chaos result, or restrict the claims to the mean-field regularized dynamics and state this limitation in the abstract.","section":"Remark 1.1; Theorem 2.7; Section 6; Section 5.1"},{"comment":"The numerical experiments compare CB2O with several baselines and report good precision, but they do not check the theoretical assumptions A2–A5 for the chosen L and G. For the star constraint (5.7b), the lower-level objective has a nonconvex one-dimensional minimizer set and it is not demonstrated that A3 (the inverse-continuity/error-bound condition (2.13a)) holds. For the sparse representation task, both L and G are computed from stochastic mini-batches and random network initializations, and it is not clear that A5 holds for the actual objective landscape. Since Theorem 2.7 is the theoretical justification for the algorithm, the experimental section should include at least a discussion of why the assumptions are plausibly satisfied, or present numerical evidence for the key inequalities (2.13a), (2.15a)–(2.15b) on the relevant regions.","section":"Section 5.2, Section 5.3; Assumptions A2–A5"},{"comment":"The existence theorem does not prove uniqueness of weak solutions of the mean-field dynamics, and the proof relies on a compactness argument that yields only a subsequential limit. While the paper acknowledges this (Section 6) and Theorem 2.7 is stated for 'any weak solution', the lack of uniqueness means the convergence statement does not identify a canonical limit dynamics and the continuity requirement on t ↦ m_{α,β}^{G,L}(ρ_t) is an extra regularity condition. This is not fatal, but it should be highlighted more prominently in the statement of Theorem 2.7 and in the discussion of the algorithm's theoretical guarantees.","section":"Theorem 2.4; Section 3.3"}],"minor_comments":[{"comment":"The word 'obtaiened' in the sentence following Eq. (1.8) is a typo.","section":"Section 1 (around Eq. (1.8))"},{"comment":"The definition of er_ε uses 'max_{s∈[0,r_G,ε]}' with a condition in the argument; this should be written as a supremum over admissible s and seems to be intended as the largest s such that sup_{θ∈B_s(θ*_good)}(G(θ)-G(θ*_good)) ≤ u_ε.","section":"Section 4.3 (Eq. (4.43))"},{"comment":"The notation eG_r and eGr is introduced in the proof of Proposition 4.2 without a clear definition; please define it explicitly before first use.","section":"Section 4.3 and Remark 4.3"},{"comment":"The precision numbers are reported to two decimals in some rows (e.g., 4×10⁻³) and to three decimals in others; a uniform significant-digit convention would improve readability.","section":"Section 5.2, Tables 1–4"},{"comment":"In W3 the variable q is used both as a threshold and later as the quantile function q_a^L; this notational overlap could confuse readers.","section":"Section 2.2, Assumption 2.3"},{"comment":"The open-problem paragraph lists the mean-field approximation as future work, but the abstract's final sentence presents the convergence result without this caveat; please align the abstract with the actual scope of the theorem.","section":"Section 6 and Remark 1.1"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is within the scope of the journal and shows solid technical work. The main risk is not mathematical inconsistency but mismatch between the headline claim (convergence of CB2O) and the actual theorem (mean-field convergence of a regularized dynamics under a strong, unverified assumption A5). In my view this is fixable with a revised presentation, explicit checks or limitations, and possibly a concrete verification of A5 on at least one of the numerical examples. I do not see grounds for rejection, but I would not accept the paper in its current form because the practical claims exceed the presently established results."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's the short version: CB2O is a real algorithmic contribution with a serious mean-field convergence proof. The quantile-based particle selection combined with Laplace-type weighting is new, and the paper proves the needed stability and large-deviation estimates from scratch. If the main theorem is correct—and I think it is, under the stated assumptions—it gives a global convergence guarantee for a derivative-free method on nonconvex simple bilevel problems, which is genuinely new.\n\nThe strengths: the stability estimate for the consensus point under a mixed L2+Wasserstein perturbation is necessary because the usual Wasserstein Lipschitz property fails; the counterexample in Remark 3.1 is right. The quantitative quantiled Laplace principle (Prop 4.2) is a clean way to combine the beta-quantile selection with the alpha-weighting. The existence proof via Picard iteration and PDE estimates is substantial. They also ship code and run a wide set of experiments, including a high-dimensional sparse representation task. That's more reproducible than the average theory paper.\n\nThe soft spots, in order of seriousness. First, the convergence theorem is for the mean-field Fokker-Planck equation, not for the finite-particle Algorithm 1 that is actually implemented. The paper says plainly that the mean-field approximation limit (1.14)/(1.15) is left open. So the headline claim 'CB2O converges globally' is really 'the mean-field law of a regularized version converges globally.' That gap is acknowledged and is a standard research direction in the CBO literature, but it means the theoretical guarantees do not yet apply to the algorithm as run.\n\nSecond, Assumption A5 is load-bearing and strong. It requires that on every ball/neighborhood around the entire minimizer set Theta of L, the minimizer of G stays within radius r of theta*_good, plus a uniform coercivity gap. For problems where Theta has multiple disconnected components and G falls steeply toward a different component, A5 fails. It is not implied by A1, and the authors never verify it for the Section 5 problems. So the theorem does not formally cover the experiments. This is a real limitation, not a nitpick. The proof of Prop 4.2 collapses without A5, so there is no easy fix within the present framework.\n\nMinor: the reported precisions are averages over 100 runs with no error bars; the speed comparison is useful but not the point.\n\nOn the math itself: I didn't find a hole. The Picard iteration and the differential-inequality argument are detailed, the constants are explicit, and the assumptions are stated precisely. The paper is honest about what it does not prove (uniqueness of the mean-field solution, finite-particle convergence, verification of A5).\n\nWho this is for: people working on consensus-based optimization and mean-field methods for global optimization, and anyone building derivative-free bilevel solvers. I'd send it to referees, with a clear request: either prove a mean-field approximation estimate (even with alpha-dependent constants), or state the theorem as a mean-field PDE result; and verify or significantly weaken A5. My own verdict is conditional, but the paper deserves serious referee time.","headline":"Genuinely new CBO variant with a serious mean-field convergence proof; the gap to the finite-particle algorithm and the strength of Assumption A5 are the real caveats.","tokens_in":68247,"tokens_out":5283,"would_cite":true,"duration_ms":53926,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["65K10","90C26","90C56","35Q90","35Q84"],"pacs":[],"model":"deepseek-v4-flash","headline":"A particle consensus method provably solves nonconvex bi-level optimization problems, converging exponentially to the unique good global minimizer.","keywords":["bi-level optimization","consensus-based optimization","mean-field limit","Fokker-Planck equations","derivative-free optimization","nonconvex optimization","quantile selection","Laplace principle"],"falsifier":"Run the mean-field Fokker-Planck dynamics on a two-dimensional problem where $L$ has two global minimizers at points $a$ and $b$, $G(a) < G(b)$, but on some radius-$r$ neighborhood of $\\{a,b\\}$ the minimizer of $G$ lies at a point in $B_r(b)$ rather than near $a$; with $\\beta$ and $\\alpha$ chosen as in Theorem 2.7, observe that $W_2^2(\\rho_t, \\delta_a)$ stops decreasing and does not reach the claimed tolerance $\\varepsilon$, contradicting the exponential convergence statement.","tokens_in":67168,"feed_emoji":"🎯","tokens_out":5728,"duration_ms":65410,"temperature":0.7,"pith_summary":"This paper introduces a derivative-free, particle-based method for simple bi-level optimization: minimize an upper-level objective $G$ over the set of global minimizers of a lower-level objective $L$, with both functions possibly nonconvex. The algorithm steers particles toward a consensus point computed in two steps: keep only the fraction $\\beta$ of particles with the lowest values of $L$, then average those survivors with Gibbs weights $\\exp(-\\alpha G)$. The paper proves that in the mean-field limit the particle law satisfies a nonlinear nonlocal Fokker-Planck equation, that regular solutions exist, and that for sufficiently small $\\beta$ and large $\\alpha$ the law converges exponentially fast in Wasserstein-2 distance to a Dirac mass at the unique good minimizer $\\theta^*_{\\mathrm{good}}$. Numerical experiments on constrained optimization and sparse representation learning demonstrate the method in practice.","feed_headline":"Particle method solves nonconvex bilevel optimization","feed_subtitle":"Quantile selection finds all lower-level minima; Laplace weighting picks the best one, with exponential convergence.","key_machinery":"The load-bearing object is the consensus point $m^{G,L}_{\\alpha,\\beta}(\\rho) = \\int \\theta \\, \\exp(-\\alpha G(\\theta)) / \\|\\exp(-\\alpha G)\\|_{L^1(I^L_\\beta[\\rho])} \\, dI^L_\\beta[\\rho](\\theta)$, where $I^L_\\beta[\\rho]$ is $\\rho$ restricted to the quantile set $Q^L_\\beta[\\rho] = \\{\\theta \\in B_R(0) : L(\\theta) \\le \\tfrac{2}{\\beta}\\int_{\\beta/2}^{\\beta} q^L_a[\\rho]\\, da + \\delta_q\\}$; the implemented version uses the simpler sublevel set of the $\\beta$-quantile of $L$. The paper's convergence proof combines a quantitative quantiled Laplace principle (Q2LP, Proposition 4.2) with the time evolution of $V(\\rho_t) = \\tfrac12 W_2^2(\\rho_t, \\delta_{\\theta^*_{\\mathrm{good}}})$, controlling $\\|m^{G,L}_{\\alpha,\\beta}(\\rho_t) - \\theta^*_{\\mathrm{good}}\\|_2$ as a sum of two terms: one made small by $\\beta$ (how well the quantile set approximates $\\Theta$) and one made small by $\\alpha$ (how well the soft-min approximates the minimizer of $G$ on that set). Existence of solutions is carried by a stability estimate for the consensus point under a combination of Wasserstein and $L^2$ perturbations, replacing the standard Wasserstein Lipschitz estimate that fails for the quantile-selected consensus point.","core_discovery":"The central claim is Theorem 2.7: under assumptions A1-A5 on $L$ and $G$, for any target accuracy $\\varepsilon$, if the quantile parameter $\\beta$ is small enough, the inverse-temperature parameter $\\alpha$ is large enough, and the drift-to-noise ratio satisfies $2\\lambda > d\\sigma^2$, then every sufficiently regular weak solution $\\rho$ of the mean-field Fokker-Planck equation reaches $W_2^2(\\rho_T, \\delta_{\\theta^*_{\\mathrm{good}}}) = \\varepsilon$ at some time $T$ in an explicitly bounded interval, and on the way to that time the squared Wasserstein distance decays at least exponentially with rate $(1-\\vartheta)(2\\lambda - d\\sigma^2)$. The mechanism is the consensus point $m^{G,L}_{\\alpha,\\beta}(\\rho)$: a $\\beta$-quantile sublevel set of $L$ approximates the minimizer set $\\Theta$, and a soft-min weighted average with weights $\\exp(-\\alpha G)$ picks out the point in that set that is best for $G$. The paper also proves (Theorem 2.4) existence of weak solutions with regularity, using a new stability estimate for the consensus point under combined Wasserstein and $L^2$ perturbations, since the quantile truncation destroys the pure Wasserstein stability that standard consensus-based optimization enjoys.","pith_inferences":["The two-scale selection mechanism, where $\\beta$ identifies an approximate minimizer set and $\\alpha$ chooses among its elements, is a generic recipe for constrained consensus methods and could extend to constraints encoded by stochastic or adversarially chosen lower-level objectives.","The combined Wasserstein plus $L^2$ stability estimate for the consensus point is a transferable analytical tool for other selection-based interacting particle dynamics, such as quantile-truncated sampling or min-max consensus methods.","The experiments suggest a testable scaling prediction: the optimal quantile parameter for finite particles should roughly satisfy $\\beta N$ constant, aligning with the practical rule $\\lceil \\beta N \\rceil = 2$.","A practical diagnostic for the hard structural condition on $G$ would be to shrink $\\beta$ slightly and check whether the consensus point jumps to a different element of $\\Theta$; a jump would indicate that the stability condition underlying the convergence theorem fails."],"forward_implications":["For nonconvex simple bi-level problems, a derivative-free particle method converges in mean-field law to the unique good minimizer with an explicit exponential rate once $\\beta$ is small and $\\alpha$ is large.","Constrained global optimization can be handled without projection or penalty tuning when the constraint is encoded as a lower-level objective; experiments on circular and star constraints support this.","The method is invariant under increasing transformations of $L$, matching the invariance of the feasible set $\\Theta$.","The existence theorem supplies regular weak solutions to the associated nonlocal Fokker-Planck equation, extending consensus-based optimization well-posedness to quantile-selected consensus points.","High-dimensional sparse representation learning experiments show the method is tractable in practice and that choosing $\\beta$ so that $\\lceil \\beta N \\rceil = 2$ gives a simple and effective tuning rule."],"supporting_citations":[{"why":"Introduces consensus-based optimization, the baseline dynamics that CB2O extends with a quantile selection principle.","marker":"[82]"},{"why":"Supplies the quantitative Laplace principle and the mean-field convergence framework used to derive the exponential decay of the Wasserstein distance.","marker":"[51]"},{"why":"Provides the standard Wasserstein stability for consensus points and the analytical framework that CB2O must extend because its quantile-selected consensus point lacks pure Wasserstein stability.","marker":"[31]"},{"why":"Develops the mean-field-law convergence analysis for anisotropic consensus-based optimization that the paper adapts to the bi-level setting.","marker":"[50]"},{"why":"Provides the Picard iteration approach on bounded domains that the paper extends to the whole space via PDE regularity considerations.","marker":"[47]"},{"why":"Supplies the regularity and positivity theory for the linear Fokker-Planck equation used in the existence proof of Theorem 2.4.","marker":"[54]"},{"why":"Gives the sparse representation learning formulation and the dynamic barrier approach for nonconvex simple bi-level problems that CB2O is compared against conceptually.","marker":"[59]"},{"why":"Provides high-dimensional consensus-based optimization tools and re-initialization heuristics used in the sparse representation experiments.","marker":"[34]"},{"why":"Shows the CB2O method applied to robust clustered federated learning as an additional machine learning application.","marker":"[56]"}],"fun_headline_variants":["Bilevel optimization via consensus and quantile selection","Particle consensus method for nonconvex bilevel problems","Exponential convergence in bilevel via quantile-weighted particles","Quantile Laplace consensus tackles nonconvex bilevel optimization"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The upper-level objective must be stably minimized on every sufficiently small neighborhood of the entire set of lower-level minimizers: the minimizer of $G$ on that neighborhood must stay within the neighborhood's radius of the target point, and away from that ball the value of $G$ must be larger by a fixed gap; if this structure fails, the quantile-selected consensus point can track a wrong $G$-minimizer near the lower-level solution set.","fun_headline_variants_meta":{"raw":{"variants":["Bilevel optimization via consensus and quantile selection","Particle consensus method for nonconvex bilevel problems","Exponential convergence in bilevel via quantile-weighted particles","Quantile Laplace consensus tackles nonconvex bilevel optimization"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000312,"raw_usage":{"total_tokens":1862,"prompt_tokens":1117,"completion_tokens":745,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":733,"completion_tokens_details":{"reasoning_tokens":680}},"tokens_in":733,"tokens_out":745,"duration_ms":7900,"temperature":1.0,"reasoning_tokens":680,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T16:27:09.444511+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the mean-field Fokker-Planck dynamics on a two-dimensional problem where $L$ has two global minimizers at points $a$ and $b$, $G(a) < G(b)$, but on some radius-$r$ neighborhood of $\\{a,b\\}$ the minimizer of $G$ lies at a point in $B_r(b)$ rather than near $a$; with $\\beta$ and $\\alpha$ chosen as in Theorem 2.7, observe that $W_2^2(\\rho_t, \\delta_a)$ stops decreasing and does not reach the claimed tolerance $\\varepsilon$, contradicting the exponential convergence statement.","supporting_citations":[{"cited_title":"Fornasier, T","cited_arxiv_id":null,"evidence_quote":"Develops the mean-field-law convergence analysis for anisotropic consensus-based optimization that the paper adapts to the bi-level setting."},{"cited_title":"Fornasier, H","cited_arxiv_id":null,"evidence_quote":"Provides the Picard iteration approach on bounded domains that the paper extends to the whole space via PDE regularity considerations."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the sparse representation learning formulation and the dynamic barrier approach for nonconvex simple bi-level problems that CB2O is compared against conceptually."},{"cited_title":"Garc ´ ıa Trillos, A","cited_arxiv_id":null,"evidence_quote":"Shows the CB2O method applied to robust clustered federated learning as an additional machine learning application."}],"review_version":1}