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FedSplit (Operator Splitting for Fast Federated Optimization)

Algorithms AIMech:FedSplitOperatorSplittingForFastFederatedOptimization DRAFT

Aliases: FedSplit, Peaceman-Rachford Federated Operator Splitting, FedSplit: An Algorithmic Framework for Fast Federated Optimization

An operator-splitting federated optimization algorithm (Pathak & Wainwright 2020, Algorithm 1) that reformulates finite-sum federated minimization min_{x in R^d} sum_{j=1}^m f_j(x) over the consensus subspace E = {(x_1,...,x_m) : x_1 = ... = x_m} as the monotone inclusion 0 in nabla F(x) + N_E(x) and applies Peaceman-Rachford reflection splitting T = (1/2) * (I + refl_{I_E} o refl_{s F}), updating client auxiliary variables via z_j^{(t+1/2)} = prox_{s f_j}(2 x^{(t)} - z_j^{(t)}), reflected local states z_j^{(t+1)} = z_j^{(t)} + 2 (z_j^{(t+1/2)} - x^{(t)}), and the server consensus iterate x^{(t+1)} = (1/m) sum_{j=1}^m z_j^{(t+1)}.

Identity Notes & Concept Scope Distinct from FedAvg / multi-step FedGD(s, e) and FedProx (AIMech:FedProxProximalHeterogeneousFederatedOptimization), whose deterministic fixed points fail to satisfy sum_{j=1}^m nabla f_j(x^*) = 0 when client objectives are heterogeneous (Propositions 1 and 2), from FedDyn (AIMech:FedDynFederatedLearningBasedOnDynamicRegularization, which maintains a server dual-like state h^t for partial client participation using ADMM-like linear penalties), and from FedPD (AIMech:FedPdFederatedPrimalDualOptimizationWithAdaptivityToNonIidData, which targets non-convex primal-dual augmented Lagrangian consensus with randomized communication skipping).

Claims & Grounded Evidence

4 Claims

Mechanism #r410-fedgd-and-fedprox-fixed-point-inconsistency-vs-fedsplit-correctness

Propositions 1 and 2 prove that deterministic multi-step FedGD(s, e) with e > 1 local steps has fixed points satisfying sum_{i=1}^e sum_{j=1}^m nabla f_j(G_j^{i-1}(x^*)) = 0 (Eq. 8) and deterministic FedProx has fixed points satisfying sum_{j=1}^m nabla M_{s f_j}(x^*) = 0 (Eq. 14, zeros of the sum of Moreau envelope gradients), neither of which coincides with the global minimizer x_ls^* even on heterogeneous least-squares objectives (Eqs. 11, 14), whereas Proposition 3 proves that every FedSplit fixed point satisfies sum_{j=1}^m nabla f_j(x^*) = 0 for any step size s > 0.

supports Source: FedSplit: An algorithmic framework for fast federated optimization · Locator: Section 2.1-2.3 (Propositions 1, 2, and 3, Eqs. 8-19) & Appendix A.1-A.3

Mechanism #r410-sqrt-kappa-linear-contraction-under-exact-and-approximate-proximal-solvers

Under l_*-strong convexity and L^*-smoothness (condition number kappa = L^* / l_*), Theorems 1 and 3 and Corollary 1 prove that FedSplit with step size s = 1 / sqrt(l_* L^*) contracts geometrically as ||z^{(t+1)} - z^*||_2 <= rho^t ||z^{(1)} - z^*||_2 + (2 / (1 - rho)) sqrt(m) b with contraction factor rho = 1 - 2 / (sqrt(kappa) + 1) + 2 nu / (sqrt(kappa) + 1), achieving O(sqrt(kappa) log(1 / varepsilon)) communication complexity even when each local proximal subproblem is solved only approximately via e = O(sqrt(kappa)) local gradient steps.

supports Source: FedSplit: An algorithmic framework for fast federated optimization · Locator: Section 3 Main Results (Theorems 1, 2, and 3, Corollary 1, Eqs. 21-28) & Appendix B

Limitation #r410-convexity-and-persistent-client-state-limitation

FedSplit's operator-splitting fixed-point and contraction guarantees require convex (or strongly convex) local objectives so that the reflected resolvent operators are non-expansive or contractive, and each client j must maintain a persistent local auxiliary variable z_j^{(t)} across communication rounds.

supports Source: FedSplit: An algorithmic framework for fast federated optimization · Locator: Section 2.2-2.3 (Algorithm 1) & Section 3.1-3.2 (Theorems 1-3)

Application #r410-ill-conditioned-heterogeneous-least-squares-and-logistic-regression-application

Applied to heterogeneous federated least-squares and binary logistic regression problems with high condition numbers kappa in {10, 10^4}, where FedSplit converges linearly to machine precision while FedGD and FedProx stagnate at non-vanishing optimality gaps.

supports Source: FedSplit: An algorithmic framework for fast federated optimization · Locator: Section 2.1 (Figure 1) & Section 4 Numerical Experiments (Figures 2-3)

⚠️ Curated Limitations & Negative Results

Theoretical contraction guarantees require convex or strongly convex local objectives.
Clients must maintain persistent auxiliary state vectors z_j^{(t)} across rounds.

Artificial Intelligence Ontology (AIO) Grounding

Mapped AIO Classes

AIO Target Class	Mapping Relation	Mapping Rationale
`AIO:FederatedLearning` Federated Learning	broad_match	FedSplit is a Peaceman-Rachford operator-splitting optimization method within Federated Learning.

Evidence Sources & Primary Literature

3 Sources

Source ID	Publication / Document	Type / Access	Reference & Checksum
`r410-primary-pdf`	FedSplit: An algorithmic framework for fast federated optimization Reese Pathak, Martin J. Wainwright	primary paper full_text	ID: https://proceedings.neurips.cc/paper/2020/hash/4ebd440d99504722d80de606ea8507da-Abstract.html sha256:250eb4e731c6…
`r410-arxiv-abs`	FedSplit: An algorithmic framework for fast federated optimization (arXiv abstract) Reese Pathak, Martin J. Wainwright	primary paper abstract_only	ID: https://arxiv.org/abs/2005.05238 sha256:7f5534466363…
`r410-metadata-cr`	FedSplit: An algorithmic framework for fast federated optimization (Crossref metadata) Reese Pathak, Martin J. Wainwright	primary paper abstract_only	ID: https://proceedings.neurips.cc/paper/2020/hash/4ebd440d99504722d80de606ea8507da-Abstract.html sha256:0486d4c37eed…

Ontology & Grounding Topology

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Curation Provenance

Curator: Antigravity

Method: ai_assisted

Model: Gemini

Created: 2026-10-19T08:13:00Z

Updated: 2026-10-19T08:13:00Z

Source File: data/curated/FedSplitOperatorSplittingForFastFederatedOptimization.yaml · Schema: curated_concept.yaml

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