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CIDERS: Cloud-Edge LLM Collaborative Learning via Accelerating Personalized Bilevel Optimization

Victor H. Chen, Hairui Yu, Stella K. Chung, Hong Yan

arXiv:2609.15664v1cs.DCcs.AI

TL;DR

Cloud-edge LLM collaboration must balance shared global knowledge with local personalization when data and operational constraints are distributed across cloud and edge. CIDERS combines a learnable backbone and client messenger through bilevel optimization and consensus-corrected updates. Its analysis establishes a sublinear convergence result, while the reported evaluation is bounded by specific protocol and measurement limitations.

  • Problem

    Cloud-centric systems leave edge-local data and timing constraints distributed while modifiable model parameters remain in the cloud, motivating coordinated local adaptation and global knowledge.

    Source: §I · pp. 1–2

  • Method

    CIDERS separates a learnable backbone from client messengers and uses consensus-corrected local updates to coordinate personalization with cloud-side learning.

    Source: A. Compressed Model Preparation · p. 5, B. Client Update Procedure · p. 6, §VII · p. 15

  • Results

    The analysis reports a comparable asymptotic convergence neighborhood to standard FL under heterogeneity, with consensus-variate correction mitigating heterogeneity-related asymptotic bias.

    Source: N. Discussion of the Convergence Result · p. 48

  • Takeaways & Limitations

    The paper presents CIDERS as a framework for continuous edge personalization coordinated with global consensus.

    Source: §VII · p. 15

    Runtime, peak memory, full network traffic, and provisioning costs were not measured on common hardware, so the reported component totals do not establish end-to-end system efficiency.

    Source: G. Integrated Discussion and Limitations · p. 15

Abstract

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Amid the rapid advancement of physical-world intelligence, cloud-edge collaborative large language models (LLMs) have emerged as a promising roadmap for practical LLM deployment. However, existing cloud-edge paradigms struggle to balance global consensus with local personalization, which fails to satisfy the need for a unified knowledge foundation on the cloud and domain-specific adaptation at the edge. To address this, we introduce, for the first time, a personalized bilevel optimization framework that formalizes cloud-edge LLM collaboration as a dual structure: the upper level optimizes edge-side personalization, while the lower level governs cloud-side knowledge transfer, reaching cloud-edge evolving in coordination. We then propose CIDERS, an efficient solver that decomposes the model into a learnable backbone and a messenger. While the cloud performs knowledge transfer to the learnable backbone, the key lies in embedding global trajectories into each local personalization step via consensus-variate correction to reconcile personalization with consensus. We provide a comprehensive theoretical analysis, including a geometric characterization of the local trajectory and a full convergence guarantee, revealing an explicit trade-off structure between personalization and global convergence. Extensive experiments demonstrate that CIDERS consistently outperforms competitive baselines on the compressed edge path, with 3.1x and 1.7x gains on mathematical reasoning and code generation, respectively, and a 10\% relative gain on instruction metrics. Mechanism experiments attribute these gains to early consensus-corrected coordination and task-aware distillation. Overall, CIDERS offers a viable path toward consensus-guided continuous personalization in cloud-edge LLM systems.

I. INTRODUCTION

Cloud-edge LLM systems must reconcile edge-local data and adaptation needs with cloud-held model parameters and shared knowledge. CIDERS formalizes that tension as personalized bilevel optimization, pairing edge messenger adaptation with cloud-side backbone distillation and consensus-guided updates.

I. INTRODUCTION

  • Cloud-centric deployment mismatches edge-local data, timing, and physical constraints with model parameters that remain in the cloud.The passages identify round-trip latency, privacy restrictions on offloading data, and distributed physical processes as relevant pressures.Source: §I · pp. 1–2

B. Cloud-Edge LLM Collaborative Learning via FL

  • CIDERS decomposes each edge model into a shared learnable backbone and local messenger, coordinating them through bilevel optimization and consensus-corrected local updates.Only compressed consensus increments are transmitted, allowing exact reconstruction of local displacements on the cloud.Source: §I · pp. 2–3

I. INTRODUCTION

  • CIDERS frames cloud-edge collaboration as personalized bilevel optimization: the upper level adapts messengers to heterogeneous clients, while the lower level distills a learnable backbone from the frozen full backbone.The formulation distinguishes personalization from conventional FL objectives focused on global accuracy.Source: §I · p. 2

A. Federated Learning and Personalization

  • Earlier FL and personalization methods provide foundations, but their conventional-model designs face parameter-scale, control-state, and distillation-coupling challenges in cloud-edge LLM settings.The cited related-work discussion covers methods for drift correction, parameter-efficient adaptation, and model decomposition.Source: A. Federated Learning and… · p. 3, B. Cloud-Edge LLM Collaborative… · p. 3

C. Discussion

  • CIDERS targets continuous edge personalization while keeping updates coordinated with global consensus through a unified bilevel optimization framework.The upper level optimizes messenger adaptation; the lower level aligns the backbone with the teacher and downstream task manifold.Source: C. Discussion · p. 4, §III · p. 5

III. PROBLEM FORMULATION

  • The problem formulation assigns client-specific adaptation to private heterogeneous edge data and backbone knowledge transfer to task-aware distillation on public data.The upper level optimizes a global messenger for rapid local adaptation, while the lower level trains the learnable backbone.Source: §III · p. 4

IV. METHODOLOGY

CIDERS prepares a compressed student backbone and client-side messenger, then adapts messengers locally using a consensus-corrected update. Clients send consensus increments, which support cloud reconstruction of their update trajectories.

A. Compressed Model Preparation

  • CIDERS decomposes the pretrained LLM into a top-layer messenger, frozen full backbone, and uniformly subsampled learnable backbone, with task-aware pre-alignment before client updates.The messenger captures domain-specific knowledge; the compact backbone is initialized through distillation on public data.Source: A. Compressed Model Preparation · p. 5

B. Client Update Procedure

  • Each client performs K local messenger updates using a consensus-variate-corrected geometric interpolation that balances local specialization with global alignment.The scaler γ ranges from pure local meta-gradient descent at γ → 0 to full alignment toward the global meta-initialization at γ = 1; a proximal term bounds divergence.Source: B. Client Update Procedure · pp. 5–6
  • Clients transmit only incremental consensus differences, allowing the cloud to recover their local messenger update trajectories from stored consensus state.The passage characterizes the upstream payload as a single low-dimensional consensus increment.Source: B. Client Update Procedure · p. 6

C. Server Update Procedure

The server reconstructs client update trajectories from consensus increments, aggregates messenger and consensus state, and then distills the learnable backbone with the aggregated messenger fixed.

Server reconstruction and aggregation

  • The server reconstructs each participating client’s averaged local messenger trajectory from its consensus increment and synchronized historical state.The reconstructed updates support global messenger and consensus-variate updates.Source: C. Server Update Procedure · p. 6

Server update procedure

  • The workflow describes the server’s alternating cycle of reconstructing updates, aggregating state, and performing task-aware backbone distillation.The cloud performs distillation before the next round’s broadcast.Source: C. Server Update Procedure · p. 6

Server reconstruction and aggregation

  • Global aggregation advances the messenger through a damped combination of the meta-initialization and local adaptations, while the consensus variate tracks an average of personalized gradients.The aggregation learning rate dampens noise from partial client participation, and the consensus variate supplies a reference direction for later client adaptations.Source: C. Server Update Procedure · p. 6

Learning the learnable backbone

  • With the messenger fixed, task-aware distillation trains the learnable backbone from the full backbone, establishing a task-aligned representation foundation for the next client update.The fixed messenger conditions this coordination and adaptation step.Source: C. Server Update Procedure · p. 6

V. THEORETICAL ANALYSIS

The analysis treats CIDERS as solving personalized bilevel optimization and establishes a sublinear convergence result under stated assumptions and step-size conditions. It also tracks the coupled errors through a Lyapunov function.

V. THEORETICAL ANALYSIS

  • The paper presents CIDERS as a theoretically robust solver that incorporates the global trajectory into personalized bilevel optimization.The convergence analysis is described as applying to the proposed bilevel structure.Source: §V · p. 7, A. Convergence Analysis · p. 7

A. Convergence Analysis

  • CIDERS is shown to attain sublinear convergence to a neighborhood of a stationary point for the personalized bilevel structure.The theorem states an ergodic joint-gap guarantee under regular assumptions and moderate conditions, with step size η = O(T^-1/2).Source: A. Convergence Analysis · p. 7
  • The analysis couples virtual messenger error, client drift, consensus lags, and the joint gap in a Lyapunov function.The objective includes an inexact learnable backbone alongside consensus-corrected local messenger steps.Source: A. Convergence Analysis · p. 7

B. Geometry of the Trajectory

The analysis characterizes how local personalization and global consensus shape the messenger’s trajectory, including how their relative influence changes over time.

B. Geometry of the Trajectory

  • The local trajectory couples a personalized meta-gradient, a consensus correction, and a proximal constraint that resists deviation from the starting point.This coupling forms a nonlinear dynamical process that makes analysis difficult.Source: B. Geometry of the Trajectory · p. 7
  • The correction shapes the trajectory’s direction, while proximity modulates its magnitude.The analysis describes the along-path average meta-gradient using σK = SK/K, with σK in (0, 1].Source: B. Geometry of the Trajectory · p. 7

1) Convex Combination:

  • Under the stated approximations, the reference direction lies on the segment between each client’s local starting gradient and the global average gradient.The consensus weight rises from 0 at cold start to γ at lock-in, with α controlling the transition speed.Source: 1) Convex Combination · p. 8

2) Quantifying Deviations.:

The ideal convex-combination geometry is approximate, with deviations tied to inner-loop length and client Hessian mismatch; proximal damping preserves span invariance and shifts the field toward personalization.

2) Quantifying Deviations.:

  • The true direction stays within an O(γLKηGbd) neighborhood of the ideal segment, and this deviation vanishes as Kη approaches zero.The bound scales linearly with γ, so stronger consensus alignment increases sensitivity to gradient-estimation error.Source: 2) Quantifying Deviations · p. 8
  • With large γ, long local steps can accumulate Hessian mismatch across clients rather than improve consensus.Even with ideally locked consensus, the executed field includes a Hessian remainder along the inner path.Source: 2) Quantifying Deviations · p. 8

3) The Case σK ‰ 1:

  • When σK < 1, the consensus-gap residual remains in the span of the heterogeneity gap, while proximal damping shifts the reference field toward personalization.The damped steady-state weight ψ∞ is closer to the local gradient than the undamped value 1 − γ.Source: 3) The Case σK ‰ 1 · p. 8

4) Quantitive Metric Design.:

CIDERS leads the reported compressed-deployment comparisons, while Output-KL and finite-difference diagnostics show task-selective effects and no consistent FO–FD advantage. Protocol C tracks how loss and consensus statistics evolve across data partitions and scheduled coefficients, including a PCA view of client-round statistics.

4) Quantitive Metric Design.:

  • Drift reduction (DR) measures cancellation of initial heterogeneity toward consensus, while personalization preservation (PPR) measures retention of the local orthogonal component.DR is 1 at full drift elimination and 0 with no drift reduction; PPR is 1 when the update aligns with the local reference and 0 when aligned with global consensus.Source: 4) Quantitive Metric Design · p. 8

A. Experimental Setup and Protocols

  • Evaluation separates Protocol A downstream performance and output alignment, Protocol B matched-domain personalization, and Protocol C non-IID consensus dynamics.Protocol C uses MetaMath data partitioned across eight clients and reports training loss and consensus-state statistics.Source: A. Experimental Setup and Protocols · pp. 8–9

B. Task Performance and Output Alignment

  • CIDERS ranks first on five of six Qwen2-1.5B metrics and all six Qwen2.5-3B metrics, with GSM8K margins over FedBiOT of 5.30 and 11.15 percentage points, respectively.On Qwen2.5-3B, Dolly PPL falls from 9.42 to 8.60 and Python, Java, and Go pass@1 improve by 2.4–4.3 percentage points; on Qwen2-1.5B, Java is 0.7 points lower. Table III reports compressed AdapEmu scores.Source: Table III · p. 10, B. Task Performance and Output Alignment · p. 10
  • For CIDERS-FO, recurring Output-KL raises GSM8K by 4.40 percentage points, Java and Go pass@1 by 3.05 and 1.83 points, and ROUGE-L by 0.0076, while Python falls 1.22 points.FD shows the same task-selective directions, while Dolly PPL changes little; the experiments do not identify the mechanism behind language-specific changes.Source: B. Task Performance and Output Alignment · p. 10, Figure 2 · p. 10

C. Finite Difference Diagnostics

  • At ηHVP 0.005, moderate and large corrections were 28.0% and 68.9%, versus 1.06% and 0.26% at 0.0005; FO is the default without a consistent FD advantage.The observation-level frequency shift does not establish an end-to-end speedup or explain the separate 150-round task outcomes.Source: Figure 3 · p. 11, C. Finite Difference Diagnostics · p. 11

D. PBO Dynamics under Data Heterogeneity

  • By rounds 15 and 30, loss reduction reached 74.0–76.0% and 86.8–88.5% of its total; E-RAE stabilized near 0.995–0.996 while loss continued falling.At round 150, loss at α=0.4 was 19.3% higher for CIDERS-FO and 26.9% higher for CIDERS-FD than at α=0.1; these differences reflect the realized partitions.Source: D. PBO Dynamics under Data Heterogeneity · pp. 11–12, Figure 4 · p. 12
  • Across scheduled coefficients, S-PPR remains high while S-DR becomes more negative at middle and high values; the profiles do not identify an optimal correction coefficient.Negative S-DR means the corrected direction is farther from the global reference than the uncorrected local direction under this proxy.Source: D. PBO Dynamics under Data Heterogeneity · p. 12, Figure 5 · p. 12
  • In a shared PCA of five standardized statistics, late-stage centroids generally cluster more tightly, but Client 7 remains displaced; the projection does not establish consensus-vector separation.The PCA complements time-series summaries rather than directly measuring personalization in task output.Source: D. PBO Dynamics under Data Heterogeneity · pp. 12–13, Figure 6 · p. 13

E. Personalized Messenger Evaluation

The cross-domain evaluation finds only a limited matched-domain benefit for CIDERS-FO, while the compression sweep exposes quality–storage trade-offs and the integrated discussion limits what the protocols can establish. The conclusion presents CIDERS as a consensus-guided personalization approach supported by theory and experiments.

E. Personalized Messenger Evaluation

  • CIDERS-FO’s matched-domain gap is 3.08 × 10⁻⁴, or 0.0176% of mean self NLL, and is small relative to target-domain variation.Its target-column means span 2.1001 NLL, versus 1.88 × 10⁻⁴ across source-row means; FedOT and FedBiOT have larger matched-domain gaps, but that does not imply lower absolute NLL.Source: E. Personalized Messenger Evaluation · p. 13

F. Compression and Deployment Tradeoffs

  • CIDERS has the lowest NLL in all six tested compression configurations, 9.9–22.2% below FedBiOT; its best tested NLL is 1.754 at six messenger layers and drop ratio 0.2.At six layers, raising the drop ratio from 0.2 to 0.5 cuts CIDERS LoRA payload by approximately 29.2% but raises NLL from 1.754 to 4.115; payload counts stored LoRA state, not total model storage or runtime memory.Source: F. Compression and Deployment Tradeoffs · p. 13, Table V · p. 14
  • CIDERS communicates approximately twice the counted messenger/consensus tensor volume of one-messenger baselines, making quality, storage, and communication a trade-off rather than an unconditional efficiency advantage.The communication totals exclude learnable-backbone downlink and protocol overhead.Source: F. Compression and Deployment Tradeoffs · p. 14, Table VI · p. 14

G. Integrated Discussion and Limitations

  • The protocols are descriptive rather than causal, and their different datasets and clients cannot establish a direct link between directional preservation and response quality.The complete compressed deployment path is evaluated under recorded configurations; runtime, peak memory, full network traffic, and provisioning costs were not measured on common hardware.Source: G. Integrated Discussion and Limitations · pp. 14–15

VII. CONCLUSION

  • CIDERS formalizes cloud-edge collaboration as personalized bilevel optimization and uses consensus-variate correction to coordinate local personalization with global consensus.The paper reports a geometric characterization of local trajectories, a convergence guarantee, and experiments on Qwen2.5-3B/1.5B across three protocols.Source: §VII · p. 15

APPENDIX DETAILED DERIVATIONS OF THEORETICAL ANALYSIS

The appendix derives bounds and recurrences used to characterize consensus-corrected local trajectories. It also examines how approximation assumptions and local-path effects qualify that characterization.

A. Derivation of Approximation A1

B. Derivation of Approximation A2

  • A second-order Taylor expansion shows the heterogeneity-gap approximation becomes locally valid for nonconvex losses as the global model converges.The derivation assumes Lipschitz-continuous Hessians and bounds the cross-round gap variation by a term that vanishes with the global update.Source: B. Derivation of Approximation A2 · pp. 17–18

C. Residual Recurrence

  • The appendix derives the consensus-gap recursion under its stated initialization and full-participation assumptions.The supplied derivation sets the initial local and global consensus variates equal and assumes all clients participate.Source: C. Residual Recurrence · pp. 18–19

D. Derivation of Proposition 2

  • Solving the recurrence ψ_{t+1} = (1 − α)ψ_t + α(1 − γ), with ψ_0 = 1, gives ψ_t = (1 − γ) + γ(1 − α)^t and steady state ψ* = 1 − γ.Source: D. Derivation of Proposition 2 · p. 19

E. Path-Averaging Error and Its Effect on the Residual

  • The convex-combination characterization holds with high accuracy when the local update step is small, as the path-induced deviation vanishes with Kη.The derivation bounds the path error and analyzes its propagation into the residual dynamics and reference direction.Source: E. Path-Averaging Error and Its Effect… · pp. 20–21

F. Hessian remainder along the local path.

  • Along the local path, curvature mismatch contributes a remainder scaling with γ and the displacement from the broadcast messenger.Its bound is γχ‖x − v̄‖ plus a second-order term in that displacement.Source: F. Hessian remainder along the local… · p. 21

G. The case σK ‰ 1: detailed derivation of the renormalized recurrence

The derivation extends the residual recurrence to compressed local updates and relates its steady state to the balance between global consensus and local gradients. It then states the bilevel objectives and sketches the algorithm’s client–server update flow.

G. The case σK ‰ 1: detailed derivation of the renormalized recurrence

  • With σ_K < 1, compression changes the displacement recurrence; its steady-state reference field lies closer to the local gradient than the undamped counterpart.The derivation assumes a small local update and fixed heterogeneity; when σ_K = 1, the recurrence reduces to the undamped case.Source: G. The case σK ‰ 1: detailed… · pp. 21–22

H. Problem Statement

  • The upper-level objective optimizes a global messenger for client adaptation, while the server’s lower-level objective learns a student backbone through task-aware knowledge distillation on public data.The local client loss is evaluated on private data; the lower-level student is conditioned on the current global messenger.Source: H. Problem Statement · pp. 22–23

I. Algorithm Abstraction

  • At each round, participating clients perform K local messenger updates with consensus correction, after which the server reconstructs displacements, aggregates them, and solves the lower-level distillation problem.The algorithm abstraction describes the framework as coordinating bilevel optimization while mitigating client drift under heterogeneous data.Source: §I · p. 23

J. Definitions

The definitions introduce quantities for tracking client deviation, consensus lag, and the lower-level solution used in the convergence analysis.

Client drift

  • Client drift E_t measures the average squared deviation of each client’s local messenger trajectory from the global messenger during K local steps.The trajectory consists of client i’s local messenger after each local step in round t.Source: J. Definitions · p. 23

Local consensus lag

  • Local consensus lag C_t measures the average squared difference between local consensus variates and true personalized meta-gradients evaluated at the global messenger.Source: J. Definitions · p. 23

Global consensus lag

  • Global consensus lag measures the deviation of the global consensus variate from the upper-level objective’s meta-gradient at the current global messenger and student backbone.Source: J. Definitions · p. 24

Lower-level minimizer

  • For a fixed messenger, S(v) denotes the exact minimizer of the lower-level task-aware knowledge-distillation objective.Source: J. Definitions · p. 24

K. Assumptions

The convergence analysis assumes smooth objectives, a Polyak–Łojasiewicz condition, bounded gradient variance and derivatives, and controlled heterogeneity and Hessian variation.

K. Assumptions

  • The client loss is jointly L-smooth, the upper-level meta-objective is β-smooth, and the lower-level distillation objective is Ly-smooth.The upper-level smoothness constant is defined in terms of the loss and its messenger Hessian; the lower-level smoothness is with respect to the backbone.Source: K. Assumptions · p. 24
  • The upper-level objective satisfies a Polyak–Łojasiewicz inequality, with bounded meta-gradient dissimilarity and gradient heterogeneity.The assumptions specify a positive PL parameter μ and constants bounding dissimilarity and heterogeneity.Source: K. Assumptions · pp. 24–25
  • Stochastic meta-gradients for client updates and stochastic gradients for the lower-level distillation objective have bounded variance.The assumptions assign separate variance bounds to the upper-level meta-gradients and lower-level TAKD gradients.Source: K. Assumptions · p. 25
  • Messenger gradients and Hessians are uniformly bounded, and the loss Hessian is Lipschitz-continuous across parameter pairs.The bounds apply across clients and messenger and backbone parameters.Source: K. Assumptions · p. 25

L. Basic Lemmas

The basic lemmas bound personalized meta-gradient dissimilarity and the upper-level gap, characterize lower-level approximation dynamics, and relate an inner update to the change in messenger gradients.

L. Basic Lemmas

  • Lemma 3 establishes a personalized meta-gradient dissimilarity bound and decomposes dissimilarity into approximation and client-gradient terms.The proof also relates the global gradient norm to the upper-level objective gap.Source: L. Basic Lemmas · pp. 25–26
  • After one inner messenger-gradient step, the gradient change is bounded by Lη_inner times the gradient norm.This follows from L-smoothness with the backbone fixed.Source: L. Basic Lemmas · p. 25
  • Lemma 4 states an upper-level gap inequality under assumptions A2 and A3, with Φ* defined as the minimum of Φ.The supplied passages do not state the inequality’s full expression.Source: L. Basic Lemmas · p. 25
  • Lemma 5 characterizes the expected squared distance to the exact minimizer after E lower-level distillation steps with a fixed messenger.Its proof uses smoothness and strong convexity for deterministic contraction, then accounts for stochastic-gradient variance and movement of the target minimizer under bounded minimizers.Source: L. Basic Lemmas · p. 26

M. Key Lemmas

This section establishes key lemmas and bounds for PERCE and CIDERS, covering local consensus deviation, update variance, client drift, and meta-objective progress. It then develops a Lyapunov evolution bound and states an ergodic-convergence result for the joint gap under explicit assumptions and weight and step-size conditions.

Lemma 6 (Local Consensus Deviation)

  • Lemma 6 establishes a client-wise local consensus deviation bound for every round t ≥ 0, with positive constants rG² and rB².Source: M. Key Lemmas · pp. 27–28

Lemma 7 (Variance of Reconstructed Updates)

  • Lemma 7 bounds the variance of reconstructed updates under Assumptions A3, A4, and A7; stochastic meta-gradients have variance at most σ², while residuals contribute additive convergence-rate terms.The reconstruction and proximal residuals do not appear in the lemma’s leading expression.Source: M. Key Lemmas · pp. 28–29

Lemma 8 (Server Messenger Update)

  • Lemma 8 bounds the expected squared displacement of the global personalization messenger between consecutive communication rounds under Assumptions A1–A8.The bound uses positive constants Λ, Ω, Υ, Ψ, and Γ.Source: M. Key Lemmas · p. 30

Lemma 9 (Evolution of Global Consensus Lag)

  • Lemma 9 gives an evolution bound for global consensus lag under Assumptions A1–A8, with positive constants specifying the bound.The proof derives it from the per-client consensus deviation result and the algorithm’s update rules.Source: M. Key Lemmas · p. 32

Lemma 10 (Evolution of Local Consensus Lag)

  • Lemma 10 establishes an evolution bound for local consensus lag under Assumptions A1–A8, with positive constants.Source: M. Key Lemmas · p. 34

Key Lemmas

  • Lemma 11 gives a fully explicit client-drift bound under Assumptions A1–A8.The proof derives the bound from the local displacement recursion, accounting for stochastic meta-gradient noise and smoothness-related remainder terms.Source: M. Key Lemmas · pp. 35–36
  • The client-drift proof uses a local displacement recursion with a consensus-variate correction and bounds the stochastic meta-gradient noise by its conditional variance.It also invokes smoothness and Hessian-Lipschitz control to bound the remainder in the practical meta-gradient.Source: M. Key Lemmas · pp. 36–37
  • Lemma 12 bounds expected one-round meta-objective progress under Assumptions A1–A8 and a step-size restriction.The bound relates progress to client drift, consensus-error quantities, and the cloud-side model’s distance from its reference point.Source: M. Key Lemmas · p. 41

M. Key Lemmas

  • The Lyapunov function combines virtual-messenger distance with client drift, consensus-related quantities, and the joint gap.Its definition weights the virtual-messenger distance, Et, Ct, rCt, and Jt.Source: M. Key Lemmas · p. 42
  • The Lyapunov analysis separates local-to-global meta-gradient deviation, consensus-variate deviation, and proximal residual; it bounds consensus deviation using client-drift terms.The consensus-variate deviation is bounded by 2(Ct + rCt).Source: M. Key Lemmas · pp. 43–44
  • The one-round Lyapunov bound decreases by δΦηJt, up to an additive term Dtot1.For sufficiently small η, the bound is V(t+1) ≤ V(t) − δΦηJt + Dtot1.Source: M. Key Lemmas · pp. 42, 47
  • Under Assumptions A1–A8 and stated Lyapunov-weight and upper-level step-size conditions, Theorem 14 establishes ergodic convergence for the joint gap Jt.The joint gap combines the expected lower-level objective gap and the expected upper-level distance.Source: M. Key Lemmas · p. 47

N. Discussion of the Convergence Result

CIDERS has a sublinear convergence rate whose asymptotic neighborhood reflects heterogeneity and lower-level approximation error. The discussion also identifies conditions on client participation, local steps, and backbone alignment that shape convergence.

N. Discussion of the Convergence Result

  • Larger client participation improves the convergence rate by reducing variance in the aggregated update.The leading rate balances the initial transient and asymptotic variance.Source: N. Discussion of the Convergence Result · p. 48
  • The leading convergence term is O(T^-1/2), matching the standard sublinear rate for stochastic methods under heterogeneous settings.The asymptotic neighborhood includes local gradient heterogeneity, the lower-level gap, and proximal second-moment terms.Source: N. Discussion of the Convergence Result · p. 48
  • Local steps can reduce communication but inflate client drift; the descent condition requires K < 1/(L2D1), with Kη sufficiently small to prevent divergence.Beyond the stated threshold, the Lyapunov descent inequality fails.Source: N. Discussion of the Convergence Result · p. 48
  • Before the next client-update round, the student backbone must be sufficiently aligned with the current messenger.The lower-level approximation error must not dominate upper-level progress; the cited condition constrains the number of distillation steps.Source: N. Discussion of the Convergence Result · p. 48
  • CIDERS matches best-known heterogeneous federated-learning rates, while its asymptotic neighborhood includes lower-level approximation bias that consensus correction helps mitigate.The discussion relates this mitigation to consensus variates reducing the heterogeneity gap.Source: N. Discussion of the Convergence Result · p. 48
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