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REVIEW 3 major objections 3 minor 44 references

First-order Constrained Trilevel Optimization Over Distributed Networks for Robust Coreset Selection

T0 review · 3 major / 3 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read 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

desk verdict 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. read the letter →

arxiv 2607.27632 v1 pith:UQG6X5AD submitted 2026-07-30 cs.LG

Yang Jiao , Kaixuan Jiao , Kai Yang , Nadjib Aitsaadi , Ilhem Fajjari , Renwei (Richard) Li This is my paper · ORCID
classification cs.LG MSC 90C2690C30
keywords distributedoptimizationtrilevelconstrainedcoresetselectionrobustfederatedlearningvalue-functionmethodprojectedgradientdescent
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 3 minor

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.

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 (3)
  1. [Sec. III-C2, Sec. IV (Eqs. (7)-(9), (10), (20)-(21))] 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
  2. [Sec. IV, Assumption 1] 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).
  3. [Sec. IV, Theorem 1 and Eq. (10)] 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.
minor comments (3)
  1. [Sec. V] 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.
  2. [Sec. IV, Theorem 2] 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.
  3. [Sec. II.A] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity in the derivation chain; the convergence proof is self-contained under its stated smoothness assumption.

full rationale

The paper's central claim is the O(epsilon^{-3/2}) iteration complexity of F2CTO for an epsilon-stationary point. The derivation chain is: formulate the trilevel constrained problem (Eq. (1)); reformulate it by hierarchical value functions (Eqs. (2)-(6)); penalize the constraints to obtain a single-level objective L (Eq. (10)); run projected gradient updates (Eqs. (11)-(15)); and prove, under Assumption 1, a descent lemma (Lemma 2, Eq. (20)) leading to the stated complexity (Theorem 1, Eq. (21)). None of these steps defines the conclusion in terms of its own inputs: the stationarity gap (Definition 1) is a standard projected-gradient measure, Lemma 2 is proved from L-smoothness by a telescoping argument, and the rate follows from the step-size choice eta = T^{-1/3}. Assumption 1 assumes L-smoothness of the penalized objective containing value functions; this is a strong and possibly unjustified regularity assumption for minima of nonconvex losses, but an assumption is not the same as assuming the target theorem. Self-citations to the authors' prior AFTO/DTZO work are used for definitions, baselines, and proof framework, but Theorem 1 is not obtained by citing those papers; the proof is carried out in Section VII. The most serious concern is that Algorithm 1 approximates the value functions with R and Rhat projected-gradient steps (Eqs. (7)-(9)) while the theory analyzes the exact value functions in Eq. (10), with no bound on ||V^R - V|| or ||nabla V^R - nabla V||. That is a correctness/rigor gap between the implemented algorithm and the analyzed idealization, not a circularity: it does not make the theorem equivalent to its assumptions. Hence no specific circular reduction can be exhibited, and the circularity score is 0.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

The central claim depends on four hand-chosen parameter families (step sizes, λ, penalties ρ, inner-loop steps) and on three unverified structural assumptions about smoothness, surrogate equivalence, and value-function approximation. No new physical or conceptual entities are introduced.

free parameters (4)
  • step sizes η_α, η_q, η_w, η_p = Dataset-dependent values in Table IV
    Hand-selected per dataset; theory requires all equal to T^{-1/3}, which the experiments do not use.
  • regularization coefficient λ = Not reported
    Controls the smoothed top-K term in the first-level objective Eq. (1); no selection procedure or value given.
  • penalty parameters ρ_1, ρ_2, ρ_3 = Not reported
    Required by Eq. (10); they determine how close the penalized solution is to the constrained trilevel problem and are absent from both the theory and the experimental table.
  • value-function refinement steps R and Rhat = R=10; Rhat=1 or 3 depending on dataset
    Finite-step approximations to inner/outer value functions; no convergence guarantee in T for these inner loops.
assumptions (4)
  • ad hoc to paper L_i in Eq. (10) admits an L-Lipschitz continuous gradient (Assumption 1)
    The theorem's only structural assumption. L_i contains value functions of nested argmin problems, which are generally nondifferentiable; the paper asserts smoothness rather than deriving conditions on the loss ℓ that would guarantee it.
  • ad hoc to paper Solving the penalized objective Eq. (10) with fixed ρ's is an adequate substitute for the constrained trilevel problem Eq. (6)
    No theorem links stationary points of the penalty surrogate to feasible or stationary points of the original problem; the paper states the penalty approach is 'inspired by' soft constraints but gives no equivalence or ρ→0 analysis.
  • ad hoc to paper Finite-step projected-gradient approximations (R, Rhat) can stand in for exact value functions
    Algorithm 1 sets p*≈p^R etc. The convergence proof still uses exact V terms and contains no approximation-error term.
  • domain assumption Existence of lower-level solutions and compactness of W, P, Q, A
    Used to define the value functions and the finite constant B_w; standard but unstated. The loss ℓ must be continuous on compact domains.

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Pith. "Pith review of First-order Constrained Trilevel Optimization Over Distributed Networks for Robust Coreset Selection." pith.science (2026). https://pith.science/paper/UQG6X5AD

@misc{pith2026260727632,
  author       = {Pith},
  title        = {Pith review of: First-order Constrained Trilevel Optimization Over Distributed Networks for Robust Coreset Selection},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UQG6X5AD}},
  note         = {Machine review of arXiv:2607.27632}
}
abstract

With the rapid advancement of the Internet of Things (IoT), massive amounts of data are generated across distributed edge networks. Training models on full data incurs significant computational overhead and storage bottlenecks, rendering coreset selection a critical paradigm. Furthermore, given the privacy-sensitive nature of local data and the escalating demand for model robustness in real-world deployments, developing an effective distributed optimization framework for robust coreset selection is vital, yet remains largely unexplored. To this end, this work first characterizes the hierarchical dependencies among coreset selection, robust optimization, and distributed learning, and formulates the distributed robust coreset selection as a trilevel optimization problem with level-wise constraints. Furthermore, to effectively solve the trilevel problem in a distributed manner, the \underline{F}ederated \underline{F}irst-order \underline{C}onstrained \underline{T}rilevel \underline{O}ptimization (F$^2$CTO) is proposed, which synergistically integrates a hierarchical composite value-function reformulation and a distributed alternating projected gradient algorithm. To the best of our knowledge, F$^2$CTO is the first method developed for distributed robust coreset selection, as well as the first distributed optimization approach for trilevel optimization problems with level-wise constraints. Additionally, we prove that the proposed method achieves a non-asymptotic convergence rate of $\mathcal{O}(\epsilon^{-3/2})$ for finding an $\epsilon$-stationary point. Extensive empirical evaluations on reliable continual learning demonstrate the effectiveness and efficiency of the proposed F$^2$CTO.

Figures

Figures reproduced from arXiv: 2607.27632 by the authors.

Figure 1
Figure 1. Illustration of the proposed distributed robust coreset selection framework. The framework is formulated as a trilevel optimization problem. At the first [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. The comparisons between F2CTO and its two bilevel variants. BCSR Greedy FedCS GCFL ACS DTZO AFTO F 2CTO 45 50 55 60 65 Robustness (%) (a) Robustness against FGSM attack BCSR Greedy FedCS GCFL ACS DTZO AFTO F 2CTO 45 50 55 60 65 Robustness (%) (b) Robustness against PGD attack [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Results on large-scale IoT setting. we assess adversarial robustness against three representative attacks: FGSM, PGD-10, and AutoAttack, following [43]. The comparisons between the proposed F2CTO and the state-of￾the-art methods across various datasets are presented in Table III, it is seen that the F2CTO outperforms all compared meth￾ods. This improvement can be attributed to two main factors: (1) In contrast to th… view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: Comparisons between the proposed F2CTO and the hypergradient￾based method in terms of running time (100 iterations) and memory usage. mization methods AFTO and DTZO, which are designed for unconstrained trilevel optimization, the proposed F2CTO is the first distributed…
Figure 5
Figure 5. Figure 5: F2CTO with periodic communication strategy (I = 5). TABLE IV DETAILED EXPERIMENTAL SETTINGS. Dataset N ηα ηq ηw ηp c1 c2 c3 εF εP εA R Rˆ S-CIFAR 10 5e-2 4e-3 5e-2 4e-3 4/255 5 4/255 4/255 4/255 4/255 10 1 P-MNIST 5 2e-2 8e-3 5e-3 8e-3 40/255 5 40/255 25/255 40/255 20/…

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Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.