{"id":"30175807-06c3-493e-9478-6b9a896d4bc7","arxiv_id":"2607.27632","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"A federated first-order algorithm for constrained trilevel optimization is applied to distributed robust coreset selection, with a claimed O(ε^{-3/2}) rate to ε-stationarity.","lead":"This paper formulates distributed robust coreset selection as a trilevel optimization problem and proposes F2CTO, a federated first-order projected-gradient algorithm with an O(ε−3/2) convergence guarantee. The work matters because it unifies coreset selection, robustness, and distributed learning into one framework for edge/IoT settings.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The O(ε^{-3/2}) convergence proof is for exact value functions, but Algorithm 1 uses finite R and R̂ approximations with no error bound, so the rate for the implemented method does not follow as stated.","rationale":"The reader's weakest assumption precisely identifies the same load-bearing gap: Assumption 1 is unverified for value-function-containing objectives, and the finite-step approximations in Algorithm 1 are not accounted for in the proof. I agree that the core formulation and experiments are meaningful, and the empirical results are promising. However, the paper's main advertised contribution—the O(ε^{-3/2}) non-asymptotic rate for the proposed distributed constrained trilevel method—is not actually proven for the algorithm as implemented. This is a correctness risk in the central theoretical claim, not merely a missing reference or a stylistic issue. Because the gap is potentially fixable by adding an approximation-error analysis or by treating R as an inner-loop parameter with explicit dependence in the rate, a conditional verdict is appropriate: the paper should be accepted only if the authors supply the missing bound or revise the claim to match the proven statement. No fraud or misconduct is implied; the issue is purely internal to the proof structure.","tokens_in":20113,"tokens_out":4008,"duration_ms":47434,"concrete_test":"Re-derive Lemma 2 for the implemented updates in Eqs. (7)-(15) without replacing the R-step value-function approximations by exact minimizers. Add the missing approximation term E_R = (1/N) Σ_i ||∇L_i^R - ∇L_i||^2 to the descent inequality and check whether it can be bounded by O(η^2) or O(1/R) independently of T. If E_R cannot be controlled without sending R → ∞, then Theorem 1 does not cover Algorithm 1 as stated; a small N=1 linear-quadratic instance with known exact value functions can be used to numerically confirm that the stationarity gap depends materially on R.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central theoretical claim is Theorem 1's O(ε^{-3/2}) iteration complexity for Algorithm 1. The proof, however, analyzes the penalized objective L in Eq. (10) as if the three value functions V_{3,i}, V_{2,i}^{(1)}, and V_2^{(2)} were exact. In the algorithm, these value functions are replaced by the outputs of R and R̂ projected-gradient steps in Eqs. (7)-(9), and no term bounds the resulting bias ||V^R - V|| or ||∇V^R - ∇V||. Consequently, Lemma 2's descent inequality (Eq. (20)) and Theorem 1's rate (Eq. (21)) apply to a different, exact-value-function algorithm, not to the F2CTO procedure that is actually run. This is compounded by Assumption 1, which assumes L-smoothness of a penalized objective containing minima of smooth but nonconvex loss functions; such value functions are not generically smooth, and no argument is given that Assumption 1 holds for Eq. (10). Even if Assumption 1 were granted, the finite-R bias remains unquantified. Since the abstract's headline contribution is precisely the O(ε^{-3/2}) convergence guarantee, this gap is load-bearing.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes F2CTO, a federated first-order algorithm for a constrained trilevel optimization formulation of distributed robust coreset selection. The authors model the problem as Eq. (1), reformulate it via a hierarchical composite value-function approach into Eq. (6), and then introduce a penalty-based objective Eq. (10). Algorithm 1 alternates between refining approximate value functions (Eqs. (7)-(9)) and performing projected gradient updates (Eqs. (11)-(15)). The central theoretical claim is Theorem 1, which states an O(ε^{-3/2}) iteration complexity for finding an ε-stationary point, with a corresponding communication complexity in Theorem 2. Experiments on continual learning benchmarks and a large-scale Edge-IIoT scenario compare F2CTO against several recent baselines.","tokens_in":20517,"tokens_out":7424,"duration_ms":80957,"significance":"The problem formulation is novel and timely: robust coreset selection has not previously been cast as a distributed, level-wise constrained trilevel problem, and the paper makes a credible case for the motivating applications. If the convergence result were established for the implemented algorithm, the O(ε^{-3/2}) rate would be a meaningful contribution to the multilevel optimization literature, which lacks distributed constrained trilevel methods. The empirical results are also encouraging: F2CTO outperforms the compared baselines by a noticeable margin on several robustness metrics. However, the advertised theoretical guarantee is not currently supported by the supplied proof, and this gap is load-bearing for the paper's main claim.","major_comments":[{"comment":"Algorithm 1 replaces the value functions V_{3,i}, V_{2,i}^{(1)}, and V_{2}^{(2)} in Eq. (10) with the outputs of R and R̂ projected-gradient steps (Eqs. (7)-(9)). The proof of Lemma 2 and Theorem 1 treats these value functions as exact and never bounds the approximation error ||V^R - V|| or ||∇V^R - ∇V||. Consequently, the O(ε^{-3/2}) rate is proved for an exact-value-function algorithm, not for the procedure actually executed. Since R and R̂ are fixed finite constants in the experiments (Table IV), the advertised rate for F2CTO does not follow. The authors must quantify the finite-step bias and include it in Eqs. (20)-(21), or explicitly restrict the theorem to the exact-value-function idealization and qualify the claim. Additionally, the update in Eq. (11) requires ∇_{α_i} L_i, which includes ∇_{α_i} V_{2}^{(2)}({α}); the manuscript does not specify how this derivative is obtained or a","section":"Sec. III-C2, Sec. IV (Eqs. (7)-(9), (10), (20)-(21))"},{"comment":"Assumption 1 assumes that the penalized objective L_i in Eq. (10) has an L-Lipschitz continuous gradient. However, L_i contains value functions that are minima of smooth nonconvex loss functions over compact sets (e.g., V_{3,i} in Eq. (2)) and a globally coupled value function V_{2}^{(2)} in Eq. (5). Such value functions are not generically smooth, and no argument is provided that Eq. (10) satisfies Assumption 1. This is not a minor technicality: the descent inequality in Lemma 2 and the summation in Eq. (35) rely on L-smoothness at every point. The authors should either prove Assumption 1 under explicit conditions on the loss functions or replace it with verifiable assumptions (e.g., uniqueness and differentiability of the lower-level minimizers via Danskin-type arguments).","section":"Sec. IV, Assumption 1"},{"comment":"Even granting Assumption 1, the stationarity guarantee is only for the penalized objective Eq. (10) with fixed penalty parameters ρ_1, ρ_2, ρ_3. The paper never connects an ε-stationary point of Eq. (10) to feasibility or ε-stationarity of the original constrained trilevel problem Eq. (6). A fixed-penalty minimizer can violate the constraints f(·)-V(·) ≤ 0. To make the result meaningful for the original problem, the authors must either state a correspondence as ρ → ∞ (e.g., with bounded multipliers) or show that the penalty parameters used in the experiments are exact for the instances considered. Without such a connection, the theorem addresses a surrogate objective, not the problem claimed in the abstract.","section":"Sec. IV, Theorem 1 and Eq. (10)"}],"minor_comments":[{"comment":"The text around Fig. 2 contains apparent character-encoding artifacts (e.g., the 'uni00000029...' fragment), making part of the experimental section difficult to read. The manuscript should be recompiled with a clean encoding.","section":"Sec. V"},{"comment":"Theorem 2 states C_comm(ε)=O(d/ε^{3/2}), but Theorem 1 gives T(ε) ~ max{L^3, c/ε^{3/2}}. The communication bound should include the L^3 term unless an additional assumption such as ε ≤ L^{-2} is made.","section":"Sec. IV, Theorem 2"},{"comment":"The claim that distributed robust coreset selection 'admits no downward polynomial-time reduction' and is 'inherently intractable' is stated without proof and is not used in the algorithm or analysis. Please provide a precise reference or soften the statement.","section":"Sec. II.A"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The short version: this is a genuine new problem formulation with a plausible algorithm and solid, consistent experiments; the theory does not support the headline rate for the algorithm that is actually run.\n\nWhat is new and good: the constrained trilevel formulation for distributed robust coreset selection — three levels, level-wise constraints, two second-level subproblems — is new, and the hierarchical composite value-function reformulation (inner value function plus two outer value functions) is a real twist. The 'first distributed constrained trilevel method' claim checks out: AFTO and DTZO handle unconstrained trilevel only. The experiments are the paper's strength: three datasets, three attack types, an ablation, a 200-worker scalability test, and communication-round plots. Table III shows consistent gains over reasonable baselines, and the memory/time comparison against a hypergradient method is informative.\n\nWhere it is soft. The load-bearing issue: Theorem 1 proves convergence for the penalized objective in Eq. (10) with exact value functions, but Algorithm 1 uses finite-step approximations from Eqs. (7)-(9). The proof never bounds the bias from finite R and hat-R, so the O(epsilon^{-3/2}) rate does not follow for the implemented method. This is not a cosmetic gap — the descent inequality in Lemma 2 uses L-smoothness of the exact composite, while the algorithm computes gradients of the approximate composite, and no term controls the difference.\n\nTwo more issues, in order of severity. Assumption 1 simply assumes L-smoothness of a penalized objective that contains value functions of box-constrained nonconvex problems; that is not generically true and no argument is supplied. And even granting the penalty analysis, an epsilon-stationary point of the fixed-penalty objective is never shown to be near-feasible or near-stationary for the original problem in Eq. (1), so the 'solves constrained trilevel' claim is only partially backed. Minor: the values of lambda and the rho penalty parameters are not reported, and there is no code or data link.\n\nWho this is for: people working on multilevel or federated optimization and on coreset selection. The theory needs real work — bound the finite-step value-function error under explicit regularity conditions, justify or replace Assumption 1, and clarify the penalty-to-feasibility step — but the formulation and experiments are worth referee time. I would not accept the theory claims as they stand, but I would send this to review; the gaps are addressable.","headline":"Genuinely new constrained-trilevel formulation with solid experiments, but the headline O(epsilon^{-3/2}) rate is proven for an exact-value-function variant, not the algorithm that is actually run.","tokens_in":20932,"tokens_out":5869,"would_cite":true,"duration_ms":61827,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["90C26","90C30"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper claims the first distributed, first-order method for trilevel optimization with level-wise constraints, proves a non-asymptotic convergence rate of O(ε^-3/2), and applies it to robust coreset selection over federated networks wit","keywords":["distributed optimization","trilevel optimization","constrained optimization","coreset selection","robust optimization","federated learning","value-function method","projected gradient descent"],"falsifier":"On a small synthetic trilevel instance with known smoothness constant L, set R=1 and R-hat=1, run F2CTO for T iterations, and measure the projected gradient norm ||G^t||. If it plateaus above the predicted O(T^{-2/3}) envelope, or fails to reach ε within the bound of Eq. (21), the finite-step approximation is not controlled.","tokens_in":20012,"feed_emoji":"📡","tokens_out":4113,"duration_ms":38770,"temperature":0.7,"pith_summary":"The paper tries to establish that distributed robust coreset selection — choosing a small, adversarially robust subset of data across a federated network without sharing raw data — can be formulated as a trilevel optimization problem with constraints at each level, and that this hard problem can be solved by a single-loop first-order algorithm with a non-asymptotic convergence guarantee of O(ε^-3/2) to an ε-stationary point. If true, it would be the first distributed method for constrained trilevel optimization, and it would give privacy-preserving continual learning and intrusion detection a principled way to select robust coresets. The key move is a hierarchical composite value-function reformulation that turns the nested problem into a single-level penalized objective, plus a distributed alternating projected gradient scheme.","feed_headline":"Federated optimizer converges at O(ε^-3/2) on trilevel tasks","feed_subtitle":"First distributed algorithm for constrained trilevel optimization, applied to coreset selection.","key_machinery":"The hierarchical composite value-function reformulation: the third-level perturbation problem and the two second-level subproblems are each replaced by value-function constraints of the form f - V ≤ 0, where V is the optimal value of the lower-level minimization. This converts the nested structure into a single-level constrained problem, which is then penalized and solved with distributed alternating projected gradients. The mechanism that carries the proof is the one-step descent inequality (Lemma 2), which uses smoothness of the penalized objective and a local-global projection gap bound.","core_discovery":"The central claim is that the hierarchical dependencies among coreset selection, adversarial robustness, and distributed learning can be expressed as a trilevel problem with level-wise constraints, and that the resulting problem is solvable in a federated manner by F2CTO. The method replaces each lower-level optimality condition with a value-function constraint, builds a penalty objective, and alternates projected gradient steps on the four variable blocks, with a master node aggregating model parameters. The proof shows the stationarity gap shrinks as O(T^{-2/3}), giving an O(ε^{-3/2}) iteration complexity and O(d ε^{-3/2}) communication complexity — the same order as unconstrained trilevel","pith_inferences":["A direct extension would be to bound the error introduced by truncating the inner value-function loops at R and R-hat steps; an adaptive schedule that grows those loops could make the theoretical guarantee match the implemented algorithm.","The same hierarchical value-function penalty construction applies to any trilevel program with convex level-wise constraints, so the method may generalize to other nested robust-learning tasks such as robust meta-learning or robust prompt tuning.","The empirical gains over the paper's own bilevel ablations hint that the full trilevel coupling, not merely the extra perturbation levels, drives robustness; comparing against a version with fixed random perturbations would isolate that effect."],"forward_implications":["Distributed robust coreset selection gains a rigorous formulation and a first-order algorithm with the same convergence rate as unconstrained trilevel methods.","Federated networks can select coresets without transmitting raw data, improving worst-case robustness in continual learning and intrusion detection.","The O(ε^-3/2) iteration complexity and O(d/ε^{3/2}) communication complexity make the method a candidate for large-scale cross-device IoT settings.","The framework extends beyond coreset selection to any trilevel problem with box- or sphere-type constraints at each level."],"fun_headline_variants":["First federated trilevel optimizer for robust coresets","F2CTO: distributed trilevel optimization at O(ε^-3/2)","Federated trilevel optimizer solves robust coreset selection","Distributed constrained trilevel optimization for coresets"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The proof assumes the penalized objective built from the true optimal value functions is smooth, but the algorithm only runs a few inner gradient steps per outer iteration; if that inner-loop shortcut changes the objective's geometry, the rate guarantee does not apply to the code that is actually run.","fun_headline_variants_meta":{"raw":{"variants":["First federated trilevel optimizer for robust coresets","F2CTO: distributed trilevel optimization at O(ε^-3/2)","Federated trilevel optimizer solves robust coreset selection","Distributed constrained trilevel optimization for coresets"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000551,"raw_usage":{"total_tokens":2501,"prompt_tokens":813,"completion_tokens":1688,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":557,"completion_tokens_details":{"reasoning_tokens":1618}},"tokens_in":557,"tokens_out":1688,"duration_ms":12358,"temperature":1.0,"reasoning_tokens":1618,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T04:19:24.514836+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"On a small synthetic trilevel instance with known smoothness constant L, set R=1 and R-hat=1, run F2CTO for T iterations, and measure the projected gradient norm ||G^t||. If it plateaus above the predicted O(T^{-2/3}) envelope, or fails to reach ε within the bound of Eq. (21), the finite-step approximation is not controlled.","supporting_citations":[],"review_version":1}