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arxiv: 2605.06328 · v1 · submitted 2026-05-07 · 🧮 math.OC

FAB: A First-Order AB-based Gradient Algorithm for Distributed Bilevel Optimization over Time-Varying Directed Graphs

Yaoshuai Ma , Xiao Wang , Wei Yao , Jin Zhang This is my paper

Pith reviewed 2026-05-08 08:22 UTC · model grok-4.3

classification 🧮 math.OC
keywords distributed bilevel optimizationtime-varying directed graphsPush-Pull algorithmfirst-order methodsnon-asymptotic convergencevalue function penalty
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The pith

A first-order algorithm integrates Push-Pull communication with value-function penalties to solve distributed bilevel optimization over time-varying directed graphs.

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

The paper develops a fully first-order distributed gradient algorithm called FAB for bilevel optimization problems where agents must reach consensus across networks whose directed links change over time. Bilevel tasks nest one optimization inside another and appear in hyperparameter tuning, data cleaning, and reinforcement learning, yet standard distributed methods struggle with both the nesting and the consensus bias from unbalanced dynamic graphs. By combining the Push-Pull (AB) strategy for information exchange with a penalty term drawn from the lower-level value function, the method avoids second-order information and manual tuning. The authors prove non-asymptotic convergence rates under standard smoothness assumptions and, as a byproduct, supply the first convergence rate for the Push-Pull algorithm itself in nonconvex single-level distributed optimization on such graphs.

Core claim

The central claim is that the FAB algorithm, formed by embedding the Push-Pull communication protocol inside a value-function-based penalty framework, converges at a non-asymptotic rate to stationary points of the distributed bilevel problem over time-varying directed graphs, while a simplified version of the same analysis yields an explicit rate for Push-Pull in the nonconvex single-level case.

What carries the argument

The integration of the Push-Pull (AB) strategy, which maintains two separate weight sequences to achieve consensus on unbalanced directed graphs, with a value-function penalty that converts the bilevel structure into a single penalized objective.

If this is right

  • The method directly applies to distributed hyperparameter tuning and data hyper-cleaning without requiring second-order derivatives or centralized coordination.
  • It supplies the first explicit convergence rate for the Push-Pull algorithm on nonconvex single-level problems over time-varying directed graphs.
  • The same analysis framework can be reused to derive rates for other first-order distributed methods that handle nested objectives.
  • Practical tasks such as distributed reinforcement learning become feasible on dynamic networks with the same communication cost as single-level methods.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The value-function penalty may extend to other nested optimization structures beyond bilevel problems in multi-agent systems.
  • The approach suggests that similar penalty reductions could simplify analysis of federated bilevel learning under changing topologies.
  • Further work could test whether the same rates hold when communication is stochastic or when agents use asynchronous updates.
  • The resolved open question on Push-Pull rates opens the door to using that protocol as a building block for more complex distributed learning tasks.

Load-bearing premise

The sequence of directed graphs must satisfy joint connectivity and balance conditions over time so that consensus is eventually reached despite individual graphs being unbalanced and time-varying.

What would settle it

Numerical runs of the algorithm on a periodic sequence of directed graphs whose union graph is not strongly connected, checking whether the iterates fail to approach any stationary point of the bilevel problem.

Figures

Figures reproduced from arXiv: 2605.06328 by Jin Zhang, Wei Yao, Xiao Wang, Yaoshuai Ma.

Figure 1
Figure 1. Figure 1: Test accuracy comparison between single-level baselines and FAB (ours) over time-varying directed graphs. The baselines use a fixed regularization parameter λ = 0.2 selected via grid search, while the dotted curve for FAB (associated with the right vertical axis) depicts the evolution of λ during training. The results demonstrate that bilevel optimization enhances the effectiveness of AB/Push–Pull. 1.2. Co… view at source ↗
Figure 2
Figure 2. Figure 2: Performance of distributed policy evaluation for re￾inforcement learning over a dynamic network topology, under varying edge connection parameter ν and gradient noise level ω. proposed FAB to conduct policy evaluation in multi-agent RL by attacking Bellman equations. Let S be the state space, π be the policy of interest, and V π (s) denotes the value of being in state s under policy π. As in Wang et al. (2… view at source ↗
Figure 3
Figure 3. Figure 3: Performance on Fashion-MNIST under varying label cor￾ruption rates. Subfigures (a), (c), and (d) illustrate test accuracy vs. epochs under corruption rates cr ∈ {0.4, 0.5, 0.6}, respectively, while subfigure (b) presents test accuracy vs. runtime. Fixed pa￾rameters: connectivity ν = 0.5 and heterogeneity ρ = 0.5. the IMDB (Maas et al., 2011) dataset to enhance its capa￾bility in complex sentiment analysis … view at source ↗
Figure 7
Figure 7. Figure 7: Impact of the model size on the Fashion-MNIST data hyper-cleaning task using an MLP architecture. We vary the num￾ber of neurons in the hidden layer within {64, 128, 256, 512} to control the parameter size. The bars represent the average commu￾nication cost per iteration (including fluctuations), peak memory footprint, and runtime per epoch. Centralized_FAB Static_FAB PushSum_FAB PushPull_SOBA FAB 0 25 50 … view at source ↗
Figure 8
Figure 8. Figure 8: Performance of algorithms that integrate different tech￾niques for distributed policy evaluation in RL over time-varying directed graphs. Comparison of the different integrations of techniques. We compare the proposed FAB with four variants that inte￾grate different techniques: (1) Centralized FAB: The cen￾tralized version of FAB, which is equivalent to F2SA (Kwon et al., 2023); (2) Static FAB: The static … view at source ↗
Figure 9
Figure 9. Figure 9: Robustness against label noise over time-varying directed graphs. Test accuracy comparison of single-level baselines (subgradient-push (Nedic & Olshevsky ´ , 2015), Push-DIGing (Nedic et al. ´ , 2017), Push-Pull (Nedic et al. ´ , 2025)) against our proposed FAB algorithm under corruption rates (cr) of 0.2 and 0.4 . For the baselines, we employ fixed regularization parameters: (a) and (c) use λ = 0.2 (selec… view at source ↗
Figure 10
Figure 10. Figure 10: Performance comparison on the distributed RL task under varying noise levels. We observe the convergence behavior with noise standard deviation ω set to 1, 2, and 3, respectively. Fixed parameters: network connectivity ν = 0.3. B.2. Distributed Policy Evaluation for RL In this subsection, we provide additional experimental results to further evaluate the robustness and efficiency of the proposed FAB algor… view at source ↗
Figure 11
Figure 11. Figure 11: Performance comparison on the distributed RL task under varying data heterogeneity levels. We observe the convergence behavior with the network connectivity ν set to 0.5, 0.3, and 0.1, respectively. Fixed parameters: noise levels ω = 1. Centralized_FAB Static_FAB PushSum_FAB PushPull_SOBA FAB 0 5000 10000 15000 20000 25000 30000 Iterations 0.0 0.2 0.4 0.6 0.8 1.0 x k x * / x 0 x * (a) Convergence w.r.t. i… view at source ↗
Figure 12
Figure 12. Figure 12: Performance of different bilevel algorithms on the RL task. We benchmark FAB against four variants: (1) Centralized FAB (non-distributed environment); (2) Static FAB (non-dynamic communication); (3) PushSum FAB (non-Push-Pull mechanism); (4) PushPull SOBA (non-fully first-order baseline). cost of higher communication overhead, as shown in (d). Specifically, while FAB incurs the highest communication cost,… view at source ↗
Figure 13
Figure 13. Figure 13: Performance on Fashion-MNIST under varying network connectivity ν. Subfigures (a)-(c) illustrate the test accuracy, while (d) presents the communication cost per epoch of the compared algorithms. Fixed parameters: corruption rate cr = 0.4 and heterogeneity parameter ρ = 0.5. 18 view at source ↗
Figure 14
Figure 14. Figure 14: Runtime performance evaluation on IMDB with BERT under varying label corruption rates. The network connectivity is fixed at ν = 0.5 and the heterogeneity parameter at ρ = 0.5. 19 view at source ↗
Figure 15
Figure 15. Figure 15: Performance comparison on the distributed RL task. We consider different scales of agents (n = 100 and n = 1000) and noise levels (ω = 2 and ω = 3). The connectivity is fixed at ν = 0.05. baselines, the Push-Pull mechanism exhibits better time efficiency as n increases. It is worth noting that while Push-SAGA and Push-SGD show nearly identical convergence behavior in terms of iterations, their runtime per… view at source ↗
Figure 16
Figure 16. Figure 16: Illustration of the time-varying network topology. The system evolves periodically with a cycle of 30 iterations. (a) Phase I: An Augmented ER graph combining a fixed cycle (blue solid lines) for strong connectivity and random edges (orange dashed lines) for fast mixing. (b) Phase II: A sparse Directed Ring to minimize communication cost. (c) Phase III: A Reversed Ring to balance information flow. C.1. Hy… view at source ↗
read the original abstract

Distributed optimization over time-varying directed graphs has shown promising performance in addressing challenges posed by complex communication constraints in real-world scenarios. In many practical settings, however, the direct application of distributed optimization algorithms encounters additional difficulties, most notably hyperparameter tuning, which our empirical observations suggest can be effectively mitigated by integrating bilevel optimization. Motivated by these findings, we study distributed bilevel optimization over time-varying directed networks, a problem that remains largely unexplored due to the compounded challenges arising from consensus bias in dynamic unbalanced communication and the nested optimization structure. In this work, we propose a fully first-order distributed gradient-based algorithm that integrates the Push-Pull (also known as AB) communication strategy with a value function-based penalty method and establish its non-asymptotic convergence properties. Notably, a simplified variant of our analysis framework for nonconvex single-level distributed optimization establishes a convergence rate for the Push-Pull algorithm, thereby resolving an open question concerning its convergence over time-varying directed graphs. Empirical evaluations across diverse tasks, including hyperparameter tuning, data hyper-cleaning, and reinforcement learning, validate the effectiveness and efficiency of the proposed algorithm.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper introduces the FAB algorithm for distributed bilevel optimization over time-varying directed graphs. It combines the Push-Pull (AB) strategy with a value function-based penalty method to create a fully first-order distributed gradient-based algorithm. The authors establish non-asymptotic convergence properties for this algorithm. A simplified analysis framework is used to provide a convergence rate for the Push-Pull algorithm in the context of nonconvex single-level distributed optimization, thereby addressing an open question regarding its convergence over time-varying directed graphs. The approach is validated through empirical studies on hyperparameter tuning, data hyper-cleaning, and reinforcement learning tasks.

Significance. If the claimed non-asymptotic convergence rates hold under the stated assumptions, this work would make a notable contribution to distributed optimization by providing the first such guarantees for bilevel problems on time-varying directed graphs and resolving the open question on Push-Pull convergence. The integration of bilevel optimization to mitigate hyperparameter tuning issues is practically relevant. The first-order nature and handling of unbalanced dynamic networks are strengths. The empirical validation across diverse tasks supports the practical utility.

minor comments (2)
  1. [Abstract] The abstract asserts non-asymptotic convergence without providing a proof sketch or explicit assumptions; while the full manuscript likely contains these, a brief mention could improve accessibility.
  2. [Empirical Evaluations] The experiments validate the algorithm but could include more details on the graph topologies used to demonstrate the time-varying directed nature.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive review, the recognition of our contributions to distributed bilevel optimization over time-varying directed graphs, and the recommendation for minor revision. We are pleased that the work's novelty in providing non-asymptotic convergence guarantees and resolving the open question on Push-Pull convergence has been acknowledged.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper introduces a novel first-order algorithm combining the established Push-Pull (AB) strategy with a value-function penalty for distributed bilevel optimization over time-varying directed graphs, then derives non-asymptotic convergence under standard smoothness and connectivity assumptions. A simplified variant of this analysis is used to obtain a rate for single-level nonconvex Push-Pull, presented as resolving an open question. No step reduces by construction to a fitted parameter, self-definition, or load-bearing self-citation chain; the central claims rest on fresh analysis rather than renaming or smuggling prior results. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard assumptions from distributed optimization and bilevel literature plus the new convergence analysis; no invented entities are introduced.

free parameters (1)
  • step sizes and penalty parameter
    Typical tunable parameters in gradient and penalty methods that must satisfy conditions for the stated non-asymptotic rates.
axioms (2)
  • domain assumption Time-varying directed graphs satisfy connectivity and balance conditions sufficient for consensus despite being unbalanced and dynamic.
    Invoked to handle consensus bias in the dynamic unbalanced communication setting.
  • domain assumption Objective functions are sufficiently smooth and satisfy standard assumptions for bilevel optimization.
    Required for the value-function penalty and gradient-based analysis to hold.

pith-pipeline@v0.9.0 · 5503 in / 1266 out tokens · 76374 ms · 2026-05-08T08:22:59.332289+00:00 · methodology

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Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages

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    Xin, R., Pu, S., Nedi ´c, A., and Khan, U

    IEEE, 2019. Xin, R., Pu, S., Nedi ´c, A., and Khan, U. A. A general framework for decentralized optimization with first-order methods.Proceedings of the IEEE, 108(11):1869–1889, 2020. Yang, S., Zhang, X., and Wang, M. Decentralized gossip- based stochastic bilevel optimization over communication networks. InAdvances in Neural Information Processing System...

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    Network and setup:The system comprises n= 10 agents connected via a default topology

    To perform binary classification, we append a linear classification head to the output representations of the backbone, mapping the extracted features to the2sentiment classes. Network and setup:The system comprises n= 10 agents connected via a default topology. We vary the connection probability ν∈ {0.1,0.5} . The dataset is partitioned using a Dirichlet...

    work page 2023

  3. [3]

    Contraction of the unregularized estimatorˆzk: ˆzk+1 −y ∗(ˆxk+1) 2 ≤(1 +δ k,1) " (1 +δ k,2)qz,k(ηk z ) ˆzk −y ∗(ˆxk) 2 + 1 + 1 δk,2 (ηk z )2 2S(tk z , βk) + 6λ2L2 g,1 n min(αk) Vk D # + L2 g,1 µ2 1 + 1 δk,1 ˆxk+1 −ˆxk 2 .(42)

    work page

  4. [4]

    The weight vectors αk and βk are specified in Lemmas D.4 and 30 First-Order AB-based Gradient Algorithm for Distributed Bilevel Optimization D.5, respectively

    Contraction of the regularized estimatorˆyk: ˆyk+1 −y ∗ λ(ˆxk+1) 2 ≤(1 +δ k,3) " (1 +δ k,4)qy,k(ηk y) ˆyk −y ∗ λ(ˆxk) 2 + 1 + 1 δk,4 (ηk y)2 2S(tk y, βk) + 8(L2 f,1 +λ 2L2 g,1) n min(αk) Vk D # + 9L2 g,1 µ2 1 + 1 δk,3 ˆxk+1 −ˆxk 2 .(43) where Vk D is defined in (30), S(tk ξ , βk) is defined in (29). The weight vectors αk and βk are specified in Lemmas D.4...

    work page 2018

  5. [5]

    In this section, we derive the recurrence relationships for these quantities, showing that they contract linearly up to a perturbation term controlled by the step sizes

    The consensus of tracking variables: We must verify that the auxiliary tracking variables accurately estimate the global gradients (i.e.,S→0). In this section, we derive the recurrence relationships for these quantities, showing that they contract linearly up to a perturbation term controlled by the step sizes. We define thestacked weighted average vector...

    work page 2025

  6. [6]

    Coefficient of the drift term ∥ˆxk+1 −ˆxk∥2.Collecting all terms involving the decision variable difference, we obtain the coefficient: C(b) Φ,1 =− 1 α β 1 2ηkx + L∗,1 2 +C b,1 2L2 g,1 ληkz q βµ2 +C b,2 72L2 g,1 ληky q βµ2

    work page

  7. [7]

    40 First-Order AB-based Gradient Algorithm for Distributed Bilevel Optimization

    Coefficient of the unregularized estimation error∥ˆzk −y ∗(ˆxk)∥2.Grouping the terms associated with the lower-level variablez, the coefficient is: C(b) Φ,2 = ηk x 2 6λ2L2 g,1n2α2β αβ −C b,1ληk z q β + 2 c Cb,3α βη k x 2 + 56Cb,4 γ e λ2L2 g,13ηk x 2 4n2λ2L2 g,1 + 2 c Cb,3α βη k z 2 + 56Cb,4 γ e λ2L2 g,13ηk z 2 2n2λ2L2 g,1. 40 First-Order AB-based Gradient...

    work page

  8. [8]

    Coefficient of the regularized estimation error∥ˆyk −y ∗ λ(ˆxk)∥2.Similarly, for the variabley, the coefficient is: C(b) Φ,3 = 12ηk xλ2L2 g,1n2α2β αβ −C b,2ληk y q β + 2 c Cb,3α βη k x 2 + 56Cb,4 γ e λ2L2 g,13ηk x 2 4n2λ2L2 g,1 + 2 c Cb,3α βη k y 2 + 56Cb,4 γ e λ2L2 g,13ηk y 2 8n2λ2L2 g,1

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  9. [9]

    Coefficient of the consensus error n α Vk D.The coefficient for the consensus error consists of the negative contraction term from the graph topology and positive perturbations: C(b) Φ,4 = 1 α β 18λ2L2 g,1 ηk x 2 +C b,1 12 λq β ηk z λ2L2 g,1 +C b,2 32 λq β ηk y λ2L2 g,1 −C b,3c + 336Cb,4 γ e λ2L2 g,1 + 2 c Cb,3α β+ 168C b,4 γ e λ2L2 g,1 36λ2L2 g,1(ηk x 2 ...

    work page

  10. [10]

    Coefficient of the tracking errorV k S.The coefficient for the gradient tracking error is: C(b) Φ,5 = 3 α β ηk x +C b,1 4 λq β ηk z +C b,2 4 λq β ηk y −C b,4e + 2 c Cb,3α β+ 168C b,4 γ e λ2L2 g,1 (ηk x 2 +η k y 2 +η k z 2 )

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  11. [11]

    Coefficient of the gradient norm∥∇ xL∗ λ(ˆxk)∥2.Finally, the coefficient for the descent term is: C(b) Φ,6 = 2 c Cb,3α β+ 168C b,4 γ e λ2L2 g,1 4n2ηk x 2 − ηk x 2 α β. By ensuring that the step-sizes and the penalty parameter λ satisfy conditions (79)–(81), we guarantee that the coefficients C(b) Φ,1, C(b) Φ,2, C(b) Φ,3, and C(b) Φ,5 are non-positive, whi...

    work page 2018

  12. [12]

    Term∥ˆxk+1 −ˆxk∥2: The coefficient is C(s) Φ,1 :=− 1 2ηkxαβ + Lf,1 2

    work page

  13. [13]

    46 First-Order AB-based Gradient Algorithm for Distributed Bilevel Optimization

    TermD(x k, αk): The coefficient is C(s) Φ,2 := 3L2 f,1n βα2 ηk x − Cs,1c n α +C s,2 24γL2 f,1 e α + Cs,1 2 c αβ n α +C s,2 12γL2 f,1 e ! 12nL2 f,1 α (ηk x)2. 46 First-Order AB-based Gradient Algorithm for Distributed Bilevel Optimization

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  14. [14]

    TermS(t k x, βk): The coefficient is C(s) Φ,3 := 3 αβ ηk x − Cs,2e +C s,1 2 c (ηk x)2αβ n α +C s,2 12γL2 f,1 e (ηk x)2

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  15. [15]

    Term∥∇ xF(ˆxk)∥2: The coefficient is C(s) Φ,4 :=− αβ 2 ηk x + Cs,1 2 c αβ n α +C s,2 12γL2 f,1 e ! 4n2(ηk x)2. Under these conditions in (103), the coefficients simplify to the desired bounds, yielding: Φ(s)(ˆxk+1)−Φ (s)(ˆxk)≤ − ηk x 4 αβ ∇xF(ˆxk) 2 − Cs,1c 4 n α D(xk, αk).(105) Convergence rate of Push-Pull Theorem E.7.Suppose that Assumptions 3.1 and 3....

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