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FedSplit: An algorithmic framework for fast federated optimization

Reese Pathak, Martin J. Wainwright

arXiv:2005.05238v1cs.LGmath.OCstat.ML

TL;DR

Existing federated optimization procedures can have fixed points that do not correspond to stationary points, even for deterministic convex updates. The paper introduces FedSplit, an operator-splitting family for distributed convex minimization, and proves correct fixed points with convergence guarantees across convex settings. Its scope is primarily finite convex losses with Lipschitz continuous gradients, while convergence claims also depend on conditions such as suitable stepsizes.

  • Problem

    Existing federated procedures can fail to preserve stationary fixed points of the original optimization problem, even in deterministic convex settings.

  • Method

    FedSplit applies operator-splitting techniques to distributed convex minimization with additive client-assigned costs.

  • Results

    FedSplit has fixed points corresponding to optima of the original problem and comes with convergence guarantees for several convex minimization settings.

  • Takeaways & Limitations

    FedSplit provides a provably correct framework for federated convex optimization and is supported by experiments confirming theoretical predictions and favorable conditioning scaling.

  • Takeaways & Limitations

    The analysis focuses on finite convex losses with Lipschitz continuous gradients and does not immediately cover constrained, nonsmooth, or nonconvex problems.

Abstract

from arXiv · show

Motivated by federated learning, we consider the hub-and-spoke model of distributed optimization in which a central authority coordinates the computation of a solution among many agents while limiting communication. We first study some past procedures for federated optimization, and show that their fixed points need not correspond to stationary points of the original optimization problem, even in simple convex settings with deterministic updates. In order to remedy these issues, we introduce FedSplit, a class of algorithms based on operator splitting procedures for solving distributed convex minimization with additive structure. We prove that these procedures have the correct fixed points, corresponding to optima of the original optimization problem, and we characterize their convergence rates under different settings. Our theory shows that these methods are provably robust to inexact computation of intermediate local quantities. We complement our theory with some simple experiments that demonstrate the benefits of our methods in practice.

1 Introduction

Federated optimization brings distributed optimization to large, heterogeneous networks, but existing procedures can lack convergence theory and correct fixed points. The paper introduces FedSplit, an operator-splitting approach for distributed convex minimization, with convergence analysis and experiments.

  • Federated learning distributes estimation and learning across remote clients facing heterogeneous resources and data, unreliable communication, massive scale, and privacy constraints.
  • Existing federated procedures include stochastic-gradient and proximal methods such as FedSGD and FedAvg, adapted to settings where clients may fail.
  • Deterministic analogues of these methods can lack convergence theory, and their fixed points need not be stationary points of the original problem even with full local datasets.
  • FedSplit applies operator splitting to distributed convex minimization with client-assigned finite convex costs having Lipschitz continuous gradients.
  • The paper builds on operator-splitting methods for distributed optimization, specifically applying Peaceman-Rachford splitting to the distributed convex problem.
  • FedSplit retains fixed points corresponding to original optima and provides convergence guarantees, while experiments confirm theoretical predictions and favorable scaling with problem conditioning.

2 Existing algorithms and their fixed points

The paper finds that common federated gradient and proximal procedures can converge to fixed points that are not solutions of the original problem, especially with multiple local updates. Their behavior is correct only in restricted cases such as a single gradient step or shared local minimizers.

  • Existing procedures: FedSGD and FedProx are analyzed through deterministic full-data updates, representing idealized limits of stochastic local procedures.FedSGD performs multiple local gradient steps before averaging, while FedProx uses proximal updates.
  • FedSGD: When e = 1, deterministic federated gradient updates converge to a minimizer of the consensus problem whenever the sequence converges and local losses are convex.The single-step case preserves the correct fixed points under the stated convergence and convexity conditions.
  • FedSGD: When e > 1, federated gradient fixed points generally satisfy a different relation and need not be minimizers of the distributed or consensus problem.The discrepancy already appears with e = 2 and multiple devices.
  • Restricted correctness: The correct fixed points can be preserved for multi-step gradient or proximal procedures when all local cost functions share a common minimizer, but this assumption is unrealistic in federated learning.The paper notes that shared local optima would reduce the need to share data across devices.
  • FedProx: FedProx typically finds zeros of the summed gradients of Moreau envelopes rather than zeros of the summed gradients of the original losses.For m > 1, these fixed-point relations are generally different.
  • Illustrative simulation: In a simple least-squares experiment, FedProx and deterministic multi-step FedSGD fail to converge to the correct fixed point, while FedSGD with e = 1 is the exception.The figure shows nonzero optimality-gap error floors for the failing methods.

3 A splitting framework and convergence guarantees

FedSplit reformulates federated optimization as a monotone inclusion and applies Peaceman–Rachford splitting to obtain correct fixed points and convergence guarantees. Under convex smooth losses, the analysis covers geometric convergence, inexact proximal updates, and iteration complexity.

  • Operator-theoretic framework: FedSplit applies Peaceman–Rachford splitting to the consensus reformulation of distributed convex minimization.The formulation uses the sum of client objectives with equality constraints enforcing consensus.
  • Algorithm: The FedSplit procedure maintains local device vectors, performs local proximal and centering steps, and averages them at the central server.The algorithm uses proximal solvers that may be exact or approximate.
  • Correctness: FedSplit fixed points correspond to optimal solutions of the original distributed problem, unlike fixed points of several previously proposed procedures.This fixed-point guarantee does not itself establish that the iterates converge.
  • Convergence guarantees: Under strongly convex and smooth losses, FedSplit has geometric convergence to the optimum even when proximal updates are inexact.The stated stepsize uses the smallest strong convexity parameter and largest smoothness parameter across clients.
  • Inexact proximal updates: With gradient-based approximate proximal updates, e = 10 gradient steps suffice in experiments to track exact proximal updates to relatively high precision.The analysis gives an error bound whose dependence on the number of local gradient steps decays exponentially.
  • Scope and limitations: The guarantees focus on finite convex functions with Lipschitz continuous gradients and require stepsize choices involving smoothness and strong convexity parameters.The framework does not immediately cover constrained, nonsmooth, or nonconvex problems, and reliable proximal stepsize selection remains unresolved.

4 Experiments

The experiments evaluate FedSplit on least-squares and logistic-regression problems, comparing exact and inexact proximal updates with federated gradient methods. FedSplit preserves favorable convergence behavior under approximate local computation and has milder dependence on problem conditioning.

  • Experimental setup: The experiments cover least-squares and logistic-regression problems, using exact FedSplit, inexact FedSplit with e ∈ {1, 5, 10}, and FedGD for comparison.Least-squares instances include isotropic and spiked design-matrix ensembles; logistic-regression instances use synthetic Gaussian features and Bernoulli labels.
  • Inexact updates: Inexact FedSplit proximal updates introduce error floors, while increasing the number of local gradient steps reduces the resulting approximation error.Figure 2 attributes the floors to approximate proximal solves; Figure 3 reports floors for e ∈ {1, 5}.
  • Conditioning: As conditioning worsens, FedGD degrades more rapidly than FedSplit, while both procedures retain geometric convergence.Figure 2 similarly reports a more severe slowdown for federated gradient in the poorly conditioned spiked least-squares case.
  • Convergence: Both FedGD with e = 1 and FedSplit exhibit linear convergence on the logistic-regression experiments.The inexact FedSplit variants preserve linear convergence up to their error floors.
  • Inexact updates: 10 local updates per communication round allow inexact FedSplit to track the exact procedure below 10−6 accuracy on logistic regression.This result is reported for the e = 10 implementation.
  • Conditioning: At κ = 10000 and tolerance ε = 10−3, FedGD requires about 34000 iterations versus roughly 400 for FedSplit.The comparison illustrates the favorable conditioning dependence observed for FedSplit, at the expense of a more expensive local update.

5 Proofs

The proofs establish convergence guarantees for FedSplit and analyze how iteration complexity depends on problem conditioning. The theory predicts geometric convergence, with FedSplit’s conditioning dependence milder than FedGD’s.

  • Operator framework: The proof section introduces the proximal and reflected resolvent operators used in the FedSplit analysis.The reflected resolvent is also called the Cayley operator.
  • Conditioning dependence: For κ ∈ {100, 1000, 10000}, both FedGD and FedSplit show geometric convergence, but their rates decay as conditioning worsens.The FedGD rate degrades more rapidly as κ increases.
  • Iteration complexity: At ε = 10−3, FedGD has approximately linear iteration-complexity dependence on κ, whereas FedSplit has sublinear dependence.These empirical estimates are reported as consistent with the theoretical guarantees.

5.1 Proofs of guarantees for FedSplit

This section proves the main guarantees for FedSplit, including correct fixed points, convergence under strong and weak convexity, and convergence with approximate proximal updates.

  • Guarantees: The FedSplit guarantees include fixed-point correctness, convergence in strongly and weakly convex settings, and convergence with approximate proximal updates.These results are identified as Proposition 3, Theorems 1 and 2, and Corollary 1.

5.2 Proof of Proposition 3

The proofs show that FedSplit’s fixed points are minimizers of the distributed objective and establish contraction-based convergence, including robustness to approximate proximal updates. They also derive regularized weakly convex guarantees and corresponding iteration bounds.

  • Fixed-point correctness: At a FedSplit fixed point, summing the local first-order conditions yields the zero-gradient condition for the aggregate objective.Because the aggregate objective is convex, this condition implies minimization.
  • Operator structure: The separable distributed objective makes its proximal and reflected-resolvent operators block-separable across clients.The exact and approximate reflected resolvents therefore act blockwise on the product space.
  • Inexact convergence: Approximate proximal updates are analyzed as a perturbed operator, with residuals controlling the distance between practical iterates and the exact FedSplit dynamics.The proof combines contractivity with bounds on the perturbation induced by approximate updates.
  • Exact convergence: The idealized FedSplit operator is contractive under stepsize s = 1/√(ℓ∗L∗), with contraction coefficient ρ = 1 − 2/(√κ + 1).Contractivity gives a unique fixed point and supports the convergence proof.
  • Contraction proof: The convergence analysis uses contractivity of the reflected proximal operator for the separable objective and nonexpansiveness of the equality-subspace operator.These properties yield the contraction bound for the composite FedSplit operator.
  • Weak convexity: For weakly convex objectives, regularization produces a strongly convex smooth problem whose FedSplit iterates satisfy an explicit iteration bound for ε-optimality.The proof relates the regularized objective Fλ to the original objective F and chooses λ to control the approximation error.

5.3 Characterization of fixed points

This section characterizes fixed points for FedSGD and FedProx, using common client variables and, for FedProx, the proximal-operator identity.

  • The section presents two fixed-point results, one for FedSGD and one for FedProx.These results are stated as part of the analysis introduced in Section 3.1.
  • FedSGD limit points are represented as tuples of client variables satisfying a fixed-point relation.The characterization begins with a limit point (x⋆_1, . . . , x⋆_m) in (R^d)^m.
  • The FedSGD characterization uses the fact that the client variables are common at a fixed point.
  • FedProx fixed points are characterized using the proximal operator identity for convex functions.The identity is substituted into display (42) to obtain the stated characterization.

6 Discussion

The discussion summarizes FedSplit as an operator-splitting method that corrects fixed-point failures of earlier federated procedures and provides convergence guarantees for convex minimization. It also identifies open questions involving stochasticity, approximation accuracy, asynchrony, delays, and privacy.

  • FedSplit addresses the failure of some earlier methods to have fixed points corresponding to optima, even under convex deterministic updates.The paper frames this as a correctness problem in federated optimization.
  • FedSplit uses operator splitting for federated minimization and retains correct fixed points while providing convergence guarantees for various convex problems.The guarantees are identified in Theorems 1 and 2.
  • The analysis is limited to deterministic algorithms, leaving stochastic approximations to FedSplit's proximal updates for future work.The discussion highlights the unresolved tradeoff between approximation accuracy and convergence rates.
  • The current framework assumes synchronous client updates, leaving behavior under stragglers and computation delays for future study.
  • Privacy-preserving local updates remain an important federated-learning desideratum not addressed by the current discussion.The paper specifically points to suitable privacy guarantees for local data as an open concern.
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