{"id":"840feef4-5a9f-48ec-b200-77c6e9b5e14c","arxiv_id":"2411.15926","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"A modified proximal bundle method with a fixed absolute accuracy null-step test is shown via Frank-Wolfe duality to have O(ε^{-4/5} log^{2/5}(1/ε)) iteration complexity.","lead":"This paper analyzes a modified proximal bundle method (MPB-FA) for minimizing the sum of a smooth convex function and a piecewise linear convex function, and derives an O(ε^{-4/5}) iteration complexity. The result is a significant theoretical improvement over prior O(ε^{-2}) guarantees for related bundle methods.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3.15's ε^{-4/5} rate is contradicted by its own asymptotic identities; corrected balancing gives ε^{-8/9}, matching the abstract.","rationale":"The central claim is Theorem 3.15's O(ε^{-4/5}) complexity. I looked for the least secure condition needed for it. The reader flagged the transfer of the static-polytope angle lemma (Lemma 3.9) to the changing-Moreau-envelope setting; that step is worth checking, but Lemma 3.9's claim 2 only requires the previously chosen vertex to lie in the current model and the current FCFW iterate to be optimal for the current objective, so the identified gap may be repairable. A more decisive problem is that the proof's own asymptotic bookkeeping does not close. The two identities used to simplify (3.33) are inconsistent with the formulas in Lemmas 3.6 and 3.10: one has (αρ)^{-1}=Ω(1/ρ) instead of O(ρ), and the other has \\bar μ_{ψ,ρ}^{-1}=Ω(1/ρ^2) instead of O(1/ρ). Once these are corrected, balancing the dominating terms gives ρ=ε^{1/9} and total complexity O(ε^{-8/9} log(1/ε)^{2/9}) or a closely related exponent, not O(ε^{-4/5}). Notably, the paper's abstract states O(ε^{-8/9}), so the manuscript is internally inconsistent about its own main theorem. This does not destroy the paper's contribution: even ε^{-8/9} improves on the prior O(ε^{-2}) bound, and the dual Frank-Wolfe correspondence and bundle-management insight remain valuable. But the theorem as stated cannot be accepted without correcting either the asymptotic estimates or the stated optimal ρ. A conditional acceptance, contingent on a corrected derivation and a consistent abstract, is appropriate; I therefore keep the reader's conditional verdict unchanged.","tokens_in":42,"tokens_out":20988,"duration_ms":415656,"concrete_test":"Independently re-derive the asymptotic balance in Theorem 3.15 using the exact expressions for α from Lemma 3.10 and for \\bar μ_{ψ,ρ} from Lemma 3.6, and re-minimize the displayed total bound (3.33) over ρ. If, as the formulas show, (αρ)^{-1}=Θ(ρ^{-1}) and \\bar μ_{ψ,ρ}^{-1}=Θ(ρ^{-2}), the optimized exponent is ε^{-8/9}, confirming that the O(ε^{-4/5}) claim is a bookkeeping artifact and that the theorem should be restated to match the abstract.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Theorem 3.15 contains two false asymptotic assertions on which the claimed exponent depends. After (3.32), the paper states that (αρ)^{-1}=O(ρ/(2L_g)) as ρ→0. But Lemma 3.10 defines α = -1/2·sqrt(1/L_g^2+4/ρ^2)+1/(2L_g)+1/ρ, which tends to 1/(2L_g), so (αρ)^{-1}=Θ(1/ρ), not O(ρ). Second, Lemma 3.6 defines 1/2·\\bar μ_{ψ,ρ}^{-1} = D_b + 3M_f^2/(8ρ) + 6M_f(·) + 2L_g((4M_f/ρ+·)^2+1), so \\bar μ_{ψ,ρ}^{-1}=Ω(1/ρ^2), contradicting the \"O(1/ρ)\" used in Theorem 3.15. Consequently, the null-step bound in Lemma 3.6 is O(ρ^{-3} log(1/ε)), not O(ρ^{-2} log(1/ε)), and the serious-steps-with-null-steps factor has coefficient K=O(1/ρ), not O(1). Balancing ρ‖x0-x*‖^2/ε against the corrected 1/(ρ^{7/2}√ε) log(1/ε) term yields ρ=ε^{1/9} and O(ε^{-8/9} log(·)), which is the exponent in the paper's own abstract, not the O(ε^{-4/5}) advertised in Theorem 3.15. The headline improvement is therefore internally inconsistent with the constants derived in Lemmas 3.6 and 3.10.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies MPB-FA, a proximal bundle method variant for minimizing the sum of a smooth convex function g and a piecewise linear convex function f. The main contributions are: (i) an extension of the linear convergence of Kelley's method from positive homogeneous f to general piecewise linear convex f when g is smooth and strongly convex; (ii) a dual interpretation of the null steps of MPB-FA as a Fully Corrective Frank-Wolfe algorithm applied to a Moreau-envelope dual problem; (iii) complexity bounds for MPB-FA, with a claimed improved rate of O(epsilon^{-4/5} log^{2/5}); and (iv) numerical experiments on bundle-management policies. The dual correspondence in Section 3.2 and the first complexity analysis in Theorem 3.7 are carefully derived, and the paper gives a useful new perspective on cut management after serious steps.","tokens_in":28991,"tokens_out":10876,"duration_ms":89814,"significance":"If the improved rate were established, the paper would substantially advance the theory of proximal bundle methods: the best known rate for related fixed-proximal-parameter bundle variants is O(epsilon^{-2} log), so an exponent below 2 would be a notable result. The paper also contains genuinely useful elements: the duality between null steps of MPB-FA and FCFW on the Moreau envelope is clearly developed, the extension of Kelley's linear convergence to non-homogeneous piecewise linear objectives is valuable, and the active-cut bundle management experiments give concrete empirical guidance. However, the central advertised improvement in Theorem 3.15 is not supported by the proof as written, because two asymptotic identities used in the balancing argument are incorrect and the geometric transfer in Lemma 3.12 has a gap. The paper therefore needs substantial revision before its main claim is reliable.","major_comments":[{"comment":"The advertised O(epsilon^{-4/5}) rate is not supported by the proof's own asymptotics. The text after (3.32) claims (alpha*rho)^{-1} = O(rho/(2L_g)) as rho tends to 0, but Lemma 3.10 defines alpha = -1/2*sqrt(1/L_g^2 + 4/rho^2) + 1/(2L_g) + 1/rho, which tends to 1/(2L_g), so (alpha*rho)^{-1} = Theta(1/rho). Also, the text below (3.33) states that \\bar{\\mu}_{\\psi,\\rho}^{-1} = O(1/rho), whereas (3.15) contains a term 2L_g((4M_f/rho + ...)^2 + 1), giving \\bar{\\mu}_{\\psi,\\rho}^{-1} = Omega(1/rho^2). Consequently K in the proof is O(1/rho), not O(1), and the null-step bound in Lemma 3.6 is O(rho^{-3} log(1/epsilon)), not O(rho^{-2} log(1/epsilon)). Balancing rho ||x0-x*||^2/epsilon against the resulting 1/(rho^{7/2} sqrt(epsilon)) log(1/epsilon) term yields rho = epsilon^{1/9} and O(epsilon^{-8/9}) up to logarithmic factors, matching the abstract's exponent rather than the epsilon^{-4/5} claimed in Theorem 3.15. The theorem's parameter selection and rate statement therefore need to be corrected.","section":"Theorem 3.15, after (3.32)-(3.33)"},{"comment":"Lemma 3.12 applies the static-polytope angle lemma (Lemma 3.9, claim 2) to a sequence of Frank-Wolfe problems whose objectives Psi_k change after every serious step because the proximal center x_k is updated. Lemma 3.9 is proved for a fixed objective psi and a fixed polytope, and claim 2 requires the direction d_1 to have been explored in the same FCFW run on the same objective. In Lemma 3.12, \\tilde{y}_k is a gradient direction for Psi_{k-1}, while optimality is claimed for Psi_k; Lemma 3.11 bounds the difference between gradients at different points and different proximal centers, but it does not show that \\tilde{y}_k is a direction of the current objective Psi_k at the appropriate iterate. The premise sin(theta') < gamma/(3D) therefore does not, as written, certify optimality for Psi_k. This is load-bearing: Lemma 3.12 is the mechanism that makes consecutive serious steps cheap, and without it the improved rate collapses to the O(epsilon^{-2} log(1/epsilon)) bound of Theorem 3.7.","section":"Lemma 3.12"}],"minor_comments":[{"comment":"The opening abstract states an O(epsilon^{-8/9}) iteration complexity, while the full-text abstract, Section 1.3, Section 1.4, Theorem 3.15, and the conclusion state O(epsilon^{-4/5} log^{2/5}). These are different exponents and the inconsistency must be resolved, especially in light of the corrected balancing in Theorem 3.15.","section":"Abstract vs. Theorem 3.15"},{"comment":"The abstract refers to a modification of the serious-step test, while Algorithm 3.1 and Section 3.1 describe a null-step test. The terminology should be made consistent throughout.","section":"Section 3.1"},{"comment":"The displayed formula for the logarithmic factor in (3.14) contains a fraction that simplifies awkwardly; please check the expression \\log(D^4/(\\delta^2 \\bar{\\mu}_{\\psi,\\rho}^2)) and ensure it matches the substitution from Theorem 2.7.","section":"Lemma 3.6"}],"recommendation":"major_revision","confidential_remarks":"The discrepancy between the abstract's epsilon^{-8/9} and the body's epsilon^{-4/5} suggests the body may have been updated without a corresponding update of the proof. The geometric transfer issue in Lemma 3.12 is the deeper concern; if it cannot be repaired, the improved-rate contribution may be limited to the elementary O(epsilon^{-2} log) analysis of Theorem 3.7. I would encourage the editor to ask for a full revision addressing both the algebraic balancing and the geometric transfer before considering publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The first half of this paper is genuinely solid and useful; the second half, the headline complexity claim, does not currently hold up. Section 2's extension of linear convergence for Kelley's method from positive homogeneous to general piecewise linear f (Theorem 2.7) is clean and a real contribution. The dual correspondence in Section 3.2, interpreting MPB-FA's null steps as FCFW on the Moreau envelope of the dual, is the most valuable part -- it gives a reusable lens and provides theoretical grounding for the active-cut bundle policy, which the numerical experiment supports (preliminary but suggestive).\n\nThe problems are in the proof of Theorem 3.15, and they are load-bearing. Two asymptotic claims in that proof are false. After (3.32), the paper says (αρ)^{-1}=O(ρ/(2L_g)), but α from Lemma 3.10 tends to 1/(2L_g) as ρ→0, so (αρ)^{-1} is actually Θ(1/ρ). The paper also uses \\bar μ_{ψ,ρ}^{-1}=O(1/ρ), but the expression in Lemma 3.6 contains a (4M_f/ρ)^2 term, making it Θ(1/ρ^2). These errors flatter the null-step bound (it should be O(ρ^{-3} log(1/ε)), not O(ρ^{-2})) and the coefficient K (should be Θ(1/ρ), not O(1)). Rebalancing honestly gives ρ=ε^{1/9} and O(ε^{-8/9} log(1/ε)), which matches the paper's own abstract but directly contradicts the ε^{-4/5} advertised in the main text. The paper literally contains two different rates.\n\nThere is also a subtle gap in Lemma 3.12: it applies the static-polytope angle lemma (Lemma 3.9) to a sequence of Frank-Wolfe problems with changing Moreau envelopes. The assertion that a near-repeat of an earlier search direction certifies optimality for the new problem is not proved and looks non-trivial. This is fixable, but it needs to be addressed head-on.\n\nBottom line: the framework deserves referee time, and the corrected ε^{-8/9} rate would still be a meaningful improvement over the existing O(ε^{-2}). But the main theorem as stated is not supported by the proof. I would send this to review with a request for major revision, and I would not cite the ε^{-4/5} result until the algebra is fixed and the rates are reconciled.","headline":"Solid dual Frank-Wolfe framework, but Theorem 3.15's ε^{-4/5} rate is undone by two asymptotic errors; corrected balancing gives ε^{-8/9}, matching the paper's own abstract.","tokens_in":29553,"tokens_out":10252,"would_cite":false,"duration_ms":82244,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["90C25","90C30","90C60","90C46"],"pacs":[],"model":"deepseek-v4-flash","headline":"A modified proximal bundle method whose null steps mirror a Frank-Wolfe algorithm on the dual's Moreau envelope solves smooth-plus-piecewise-linear convex problems in $O(\\varepsilon^{-4/5} \\log^{2/5}(1/\\varepsilon))$ iterations, improving…","keywords":["proximal bundle method","Frank-Wolfe algorithm","iteration complexity","nonsmooth convex optimization","Moreau envelope","piecewise linear convex function","Kelley's cutting-plane method","dual correspondence"],"falsifier":"Run MPB-FA on $\\min_x \\tfrac12\\|x\\|^2 + \\max_i(a_i^T x + b_i)$ with known optimum, set $\\delta=\\varepsilon/2$, and record the angles between FCFW search directions at consecutive null steps; if a pair satisfies $\\sin\\theta<\\gamma/(3D)$ while the Frank-Wolfe gap for the current dual problem (3.8) is still positive, Lemma 3.12 fails. Alternatively, measure total iterations for $\\varepsilon=10^{-3},10^{-4},10^{-5}$: growth steeper than $\\varepsilon^{-4/5}$ (e.g. roughly $\\varepsilon^{-1}$) would contradict the claimed rate.","tokens_in":28391,"feed_emoji":"📉","tokens_out":10273,"duration_ms":84073,"temperature":0.7,"pith_summary":"The paper studies MPB-FA, a proximal bundle method variant with a fixed absolute-accuracy null-step test and a fixed proximal parameter, applied to minimizing the sum of a smooth convex function and a convex piecewise linear function. It establishes that the algorithm's null steps are exactly dual to a Fully Corrective Frank-Wolfe algorithm run on the Moreau envelope of the dual problem. Through this lens the paper proves an $O(\\varepsilon^{-4/5} \\log^{2/5}(1/\\varepsilon))$ iteration complexity, a substantial improvement over the best-known $O(\\varepsilon^{-2})$ guarantee for related bundle variants. A byproduct is a linear convergence result for Kelley's cutting-plane method when the smooth term is strongly convex and the piecewise linear term is general rather than positive homogeneous. The analysis also gives a theoretical reason to keep active cuts in the bundle after serious steps.","feed_headline":"Nonsmooth optimization rate cut to O(ε^-4/5) iterations","feed_subtitle":"A proximal bundle variant mirrors Frank-Wolfe in the dual, beating the O(ε^-2) standard.","key_machinery":"The load-bearing object is the dual correspondence (Lemmas 3.1-3.3): the bundle subproblem $\\min_x g(x)+f_k(x)+\\frac{\\rho}{2}\\|x-x_k\\|^2$ is dual to $\\min_{(w,\\beta)\\in\\operatorname{conv}(V^k)} M_{\\rho,\\varphi}(w-\\rho x_k)-\\beta$, with $y_{k+1}=-\\nabla M_{\\rho,\\varphi}(w_{k+1}-\\rho x_k)$. The argument that carries the improved rate is Lemma 3.9, a static-polytope angle lemma: if an FCFW search direction is within angle $\\theta$ with $\\sin\\theta<\\gamma/(3D)$ of a previously explored direction, then the current iterate is already optimal for the fixed problem, where $\\gamma$ is the pyramidal width and $D$ the diameter of the polytope. Lemmas 3.11-3.12 transfer this to the changing-objective setting by bounding the change in $\\nabla M_{\\rho,\\varphi}$ between consecutive proximal centers in terms of their distance, so a close center pair forces the angle condition and hence a consecutive serious step. The primal progress inequality (3.13) then bounds the number of non-consecutive serious steps, and Lemma 3.6 bounds null-step sequences logarithmically, yielding the total $O(\\varepsilon^{-4/5} \\log^{2/5}(1/\\varepsilon))$ complexity.","core_discovery":"The central discovery is that the sequence of null steps of MPB-FA, i.e. iterations where no progress to the prox-center is accepted, coincides iterate-for-iterate with Fully Corrective Frank-Wolfe applied to the problem $\\min_{(w,\\beta)\\in\\operatorname{conv}(V)} M_{\\rho,\\varphi}(w-\\rho x_k)-\\beta$, where $\\varphi = g^*(-\\cdot)$ and $\\operatorname{conv}(V)$ encodes the piecewise linear structure of $f$. Because each minimization step of the bundle method is the primal of this dual problem, the null-step test $f(y_{k+1})-f_k(y_{k+1})\\le\\delta$ equals the Frank-Wolfe gap of that dual problem. Using a geometric lemma on polytopes, namely that two nearly parallel search directions of FCFW cannot both be non-optimal, the paper shows most serious steps are immediately followed by another serious step, and the remainder are controlled by the primal progress of the proximal center. The result is the iteration bound of Theorem 3.15: at most $\\rho\\|x_0-x^*\\|^2/\\varepsilon + O((\\rho^{-3/2}/\\sqrt{\\varepsilon})\\log(1/\\varepsilon))$ iterations, minimized to $O(\\varepsilon^{-4/5} \\log^{2/5}(1/\\varepsilon))$ by setting $\\rho=\\varepsilon^{1/5} \\log^{2/5}(1/\\varepsilon)$.","pith_inferences":["Beyond the paper: Lemma 3.9's angle-repetition principle, a near-repeated FCFW direction certifies optimality on a polytope, looks like a general property of Frank-Wolfe on polytopes; it may yield similar improved rates for other FW variants, or with inexact linear minimization oracles.","Beyond the paper: the same proof machinery should carry to a general convex $f$ approximated by a growing cut bundle, with the rate governed by the geometry of the cut polytope; this is testable by replacing the LMO with a cut-generation oracle.","Beyond the paper: the angle condition fires exactly when the Frank-Wolfe gap is zero, so it could be used as a cheap inner optimality certificate to halt the bundle subproblem before the $\\delta$-threshold is met.","Beyond the paper: the abstract states $O(\\varepsilon^{-8/9})$ while Theorem 3.15 and the conclusions state $O(\\varepsilon^{-4/5} \\log^{2/5}(1/\\varepsilon))$; the body is consistent in its favor, so the abstract rate appears stale and should be read as such."],"forward_implications":["Worst-case iteration count for minimizing a smooth convex plus convex piecewise linear function drops from $O(\\varepsilon^{-2})$ to $O(\\varepsilon^{-4/5} \\log^{2/5}(1/\\varepsilon))$ when the proximal parameter is set to $\\rho=\\varepsilon^{1/5} \\log^{2/5}(1/\\varepsilon)$.","Kelley's cutting-plane method converges linearly when $g$ is smooth and strongly convex and $f$ is any convex piecewise linear function, extending the prior positive-homogeneous result.","The active-cut bundle-management policy, keeping active cuts after serious steps rather than all cuts or a single cut, is the one the improved rate requires, and the numerical experiments show it balances bundle size and runtime best.","In the high-dimensional, low-accuracy regime $n\\gg 1/\\varepsilon$, the earlier $O(\\varepsilon^{-2} \\log(1/\\varepsilon))$-type bounds can be tighter than the new geometry-dependent rate, as the paper notes.","The dual view places MPB-FA in the family of inexact augmented Lagrangian methods whose subproblems are solved by FCFW, giving a concrete bridge between bundle methods and Frank-Wolfe theory."],"supporting_citations":[{"why":"Supplies the linear convergence of Fully Corrective Frank-Wolfe on polytopes and the pyramidal-width constant $\\gamma$ that the angle lemma and the improved rate rely on.","marker":"[27]"},{"why":"Gives the prior linear convergence of Kelley's method for positive homogeneous $f$, which the paper extends to general piecewise linear $f$.","marker":"[48]"},{"why":"Provides the unified proximal bundle analysis, the fixed-accuracy null-step test, and the $O(\\varepsilon^{-2} \\log(1/\\varepsilon))$ baseline that Theorem 3.15 improves.","marker":"[33]"},{"why":"Introduces the null-step test variant and the optimal complexity for a fixed-proximal-parameter bundle, the direct ancestor of MPB-FA.","marker":"[32]"},{"why":"Establishes away-step Frank-Wolfe linear convergence under generalized geometric strong convexity, adapted in Lemma 2.6 for the dual problem.","marker":"[8]"},{"why":"Establishes the $O(\\varepsilon^{-3})$ rate for the classic proximal bundle method, the complexity that this line of work seeks to improve.","marker":"[24]"},{"why":"Gives optimal rates for the proximal bundle method with fixed proximal parameter, providing the $O(\\varepsilon^{-2})$ context for the improvement.","marker":"[14]"}],"fun_headline_variants":["Proximal bundle mirrors Frank-Wolfe for faster nonsmooth rates","Nonsmooth optimization: bundle method converges in O(ε^-4/5) steps","Fixed-accuracy bundle method beats standard with Frank-Wolfe insight","New analysis links bundle and Frank-Wolfe, improves complexity to ε^-4/5","Bundle method's null steps map to Frank-Wolfe, yielding better bounds"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The $O(\\varepsilon^{-4/5})$ rate collapses unless a near-parallel pair of Frank-Wolfe search directions in the sequence of changing proximal problems certifies optimality for the current problem; the paper proves this for a fixed polytope (Lemma 3.9) and transfers it to changing Moreau envelopes (Lemma 3.12), but the transfer step is asserted rather than fully demonstrated.","fun_headline_variants_meta":{"raw":{"variants":["Proximal bundle mirrors Frank-Wolfe for faster nonsmooth rates","Nonsmooth optimization: bundle method converges in O(ε^-4/5) steps","Fixed-accuracy bundle method beats standard with Frank-Wolfe insight","New analysis links bundle and Frank-Wolfe, improves complexity to ε^-4/5","Bundle method's null steps map to Frank-Wolfe, yielding better bounds"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000333,"raw_usage":{"total_tokens":1969,"prompt_tokens":1182,"completion_tokens":787,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":798,"completion_tokens_details":{"reasoning_tokens":683}},"tokens_in":798,"tokens_out":787,"duration_ms":7232,"temperature":1.0,"reasoning_tokens":683,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T13:55:00.436275+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run MPB-FA on $\\min_x \\tfrac12\\|x\\|^2 + \\max_i(a_i^T x + b_i)$ with known optimum, set $\\delta=\\varepsilon/2$, and record the angles between FCFW search directions at consecutive null steps; if a pair satisfies $\\sin\\theta<\\gamma/(3D)$ while the Frank-Wolfe gap for the current dual problem (3.8) is still positive, Lemma 3.12 fails. Alternatively, measure total iterations for $\\varepsilon=10^{-3},10^{-4},10^{-5}$: growth steeper than $\\varepsilon^{-4/5}$ (e.g. roughly $\\varepsilon^{-1}$) would contradict the claimed rate.","supporting_citations":[{"cited_title":"Lacoste-Julien and M","cited_arxiv_id":null,"evidence_quote":"Supplies the linear convergence of Fully Corrective Frank-Wolfe on polytopes and the pyramidal-width constant $\\gamma$ that the angle lemma and the improved rate rely on."},{"cited_title":"Limited Memory Kelley's Method Converges for Composite Convex and Submodular Objectives","cited_arxiv_id":"1807.07531","evidence_quote":"Gives the prior linear convergence of Kelley's method for positive homogeneous $f$, which the paper extends to general piecewise linear $f$."},{"cited_title":"Liang and R","cited_arxiv_id":null,"evidence_quote":"Provides the unified proximal bundle analysis, the fixed-accuracy null-step test, and the $O(\\varepsilon^{-2} \\log(1/\\varepsilon))$ baseline that Theorem 3.15 improves."},{"cited_title":"Liang and R","cited_arxiv_id":null,"evidence_quote":"Introduces the null-step test variant and the optimal complexity for a fixed-proximal-parameter bundle, the direct ancestor of MPB-FA."},{"cited_title":"Beck and S","cited_arxiv_id":null,"evidence_quote":"Establishes away-step Frank-Wolfe linear convergence under generalized geometric strong convexity, adapted in Lemma 2.6 for the dual problem."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the $O(\\varepsilon^{-3})$ rate for the classic proximal bundle method, the complexity that this line of work seeks to improve."},{"cited_title":"Diaz and B","cited_arxiv_id":null,"evidence_quote":"Gives optimal rates for the proximal bundle method with fixed proximal parameter, providing the $O(\\varepsilon^{-2})$ context for the improvement."}],"review_version":1}