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Cell-Free Massive MIMO for Wireless Federated Learning

Tung T. Vu, Duy T. Ngo, Nguyen H. Tran, Hien Quoc Ngo, Minh N. Dao, Richard H. Middleton

arXiv:1909.12567v5eess.SPcs.IT

TL;DR

Wireless federated learning must handle changing channels while limiting communication and computation costs. This paper proposes a CFmMIMO scheme that schedules optimization and FL iterations across large-scale coherence times, then jointly optimizes key training parameters. The design reduces training time by up to 55% versus baselines, and CFmMIMO has the lowest training time among the compared network architectures.

  • Problem

    Wireless FL designs can rely on channel state information remaining unchanged throughout the process, although channel variation can make optimized rates and power controls obsolete before training ends.

  • Method

    The paper develops a CFmMIMO scheme in which FL optimization and process intervals use large-scale coherence times, with joint design of local accuracy, transmit power, data rate, and UE processing frequency.

  • Results

    55%: the presented joint design reduces training time by up to 55% over baseline schemes, while CFmMIMO requires the lowest training time compared with cell-free TDMA massive MIMO and collocated massive MIMO.

  • Takeaways & Limitations

    CFmMIMO provides a general wireless-network scheme for supporting FL frameworks while accommodating channel stability at the large-scale coherence-time timescale.

Abstract

from arXiv · show

This paper proposes a novel scheme for cell-free massive multiple-input multiple-output (CFmMIMO) networks to support any federated learning (FL) framework. This scheme allows each instead of all the iterations of the FL framework to happen in a large-scale coherence time to guarantee a stable operation of an FL process. To show how to optimize the FL performance using this proposed scheme, we consider an existing FL framework as an example and target FL training time minimization for this framework. An optimization problem is then formulated to jointly optimize the local accuracy, transmit power, data rate, and users' processing frequency. This mixed-timescale stochastic nonconvex problem captures the complex interactions among the training time, and transmission and computation of training updates of one FL process. By employing the online successive convex approximation approach, we develop a new algorithm to solve the formulated problem with proven convergence to the neighbourhood of its stationary points. Our numerical results confirm that the presented joint design reduces the training time by up to $55\%$ over baseline approaches. They also show that CFmMIMO here requires the lowest training time for FL processes compared with cell-free time-division multiple access massive MIMO and collocated massive MIMO.

I. INTRODUCTION

Federated learning preserves data privacy by exchanging local model updates, but wireless implementations must address changing channels and costly update transmission. The paper proposes a CFmMIMO scheme with mixed-timescale optimization and reports substantial training-time reductions.

  • Motivation: Federated learning keeps local training data at user equipment while exchanging updates with a central server until the target accuracy is reached.This avoids transmitting raw data but uses users’ computational resources.
  • Problem: Existing wireless FL designs assume unchanged channel state information throughout training, making optimized rates and power controls obsolete as channels vary.The paper also questions the efficiency of orthogonal multiplexing for many users.
  • Proposed network: CFmMIMO uses a central processing unit and distributed access points to serve user equipment simultaneously, with channel hardening supporting stable operation during large-scale coherence times.Its coverage characteristics reduce susceptibility to unfavorable user links.
  • Proposed scheme: The proposed scheme lets any FL framework optimize performance before execution, with each iteration occurring within one large-scale coherence time and access points relaying training updates.Beamforming or filtering can be applied at the access points to improve update transmission.
  • Optimization: The example design minimizes FL training time by jointly optimizing local accuracy, transmit power, data rate, and users’ processing frequency in a mixed-timescale stochastic nonconvex problem.The problem models interactions among training, communication, and computation.
  • Results: 55%: the proposed solution reduces training time by up to 55% versus baseline schemes, while CFmMIMO achieves the lowest training time against cell-free TDMA and collocated massive MIMO.The proposed algorithm is reported to converge to at least a neighbourhood of stationary points.

B. Proposed Cell-Free Massive MIMO Network Structure to Support the General FL Framework

The proposed CFmMIMO framework separates FL-performance optimization from FL execution across large-scale coherence intervals, while adapting update transmission within each small-scale coherence time. It supports a general FL framework through CPU–AP–UE coordination and a four-step iterative process.

  • CFmMIMO connects a CPU to APs over backhaul links, with APs simultaneously serving participating UEs over shared wireless resources.
  • The framework divides stable large-scale-fading periods into one optimization interval followed by multiple FL-process intervals.
  • Only one optimization-algorithm iteration occurs within each large-scale coherence time, while short-term and long-term parameters are optimized on different timescales.
  • During FL execution, short-term parameters are refreshed before each iteration, and both this optimization block and one FL iteration fit within one large-scale coherence time.
  • Each FL iteration comprises downlink transmission, local UE computation, uplink transmission, and a CPU update, with channel estimation and update transmission arranged within one small-scale coherence time.
  • The scheme uses APs as relays and permits beamforming or filtering designs for training-update transmission, but focuses on FL support rather than simultaneous data transmission.

IV. DETAILED SYSTEM MODEL TO SUPPORT FL

This section introduces the CFmMIMO system model used to transmit and compute training updates during each FL iteration.

  • The system model covers training-update transmission and computation for each iteration of the FL process.

A. Steps (S1) and (S3) in Each Iteration of the FL Process: Model of Training Update Transmission

The model describes uplink pilot-based channel estimation and conjugate-beamforming transmission of global updates from the CPU through APs to UEs.

  • UL channel estimation: All UEs simultaneously transmit uplink pilots to the APs, which estimate UE–AP channels using MMSE estimation.
  • UL channel estimation: Channel hardening and the assumed indoor parameters make uplink channel-estimation time negligible within a small-scale coherence time.
  • Step (S1): The CPU sends each global downlink update to all APs over backhaul links before wireless transmission to the intended UE.
  • Step (S1): APs use conjugate beamforming with power-control coefficients to precode downlink update signals under per-AP power constraints.
  • Step (S1): The achievable downlink rate is constrained by the beamforming-dependent channel expression, determining the wireless delivery latency to each UE.

3) Step (S3) in each iteration of the FL process:

The uplink stage models UE transmission of local training updates to APs and CPU aggregation using channel-based reception and achievable uplink rates.

  • Each UE encodes its global uplink training update and transmits it with an amplitude controlled by its uplink power coefficient.
  • The UE-to-AP upload latency depends on the update size and the achievable uplink data rate.
  • APs receive the uplink signals, apply channel-based combining, and forward the processed information to the CPU.
  • The CPU detects the transmitted update symbols, while each UE’s achievable uplink rate is determined by its power-control configuration.

B. Step (S2) in Each Iteration of the FL Process: Model of Local Training Update Computation at UEs

The formulation models FL training time through UE computation, wireless transmission, and mixed-timescale optimization. It also accounts for energy and processing constraints while defining an ergodic iteration-time metric and effective training time.

  • Step (S2) computation: UE local-update latency depends on local iterations, dataset size, processing cycles per sample, and processing frequency.The CPU’s global-update aggregation latency is ignored because its computational resources are much more abundant than the UEs’.
  • UE energy consumption: Energy modeling includes uplink training-update transmission and local-update computation, while UL channel-estimation energy is neglected.The latter omission follows from the assumption that UL channel estimation takes negligible time relative to one FL training interval.
  • Iteration timing: Each FL iteration completes downlink transmission, local computation, uplink transmission, and related UE steps for all UEs before proceeding.Iteration time is determined by the relevant maximum wireless and computation delays, with CPU global-update time ignored.
  • Mixed-timescale structure: Local accuracy and short-term variables must be optimized on different timescales because changing local accuracy changes the number of FL iterations.Short-term variables include processing frequencies, downlink rates, and uplink rates.
  • Training-time objective: The ergodic time of one FL iteration averages iteration time over large-scale fading realizations, and effective training time measures one complete FL process.These definitions support the training-time minimization problem, which is nonconvex, stochastic, and tightly coupled while limiting UE energy and processing frequency.

VI. FL TRAINING TIME MINIMIZATION: PROPOSED ALGORITHM

The proposed solution decomposes the mixed-timescale problem into short-term optimization and convex-approximation steps. Successive inner approximations transform difficult constraints into tractable convex subproblems solved iteratively.

  • Algorithm design: Online successive convex approximation is tailored to solve the two-stage stochastic nonconvex training-time problem.The approach addresses the problem’s tight variable coupling rather than applying the general framework without modification.
  • Problem decomposition: For fixed local accuracy and large-scale fading, the formulation separates a short-term subproblem from a long-term master problem.The short-term problem optimizes variables such as transmission and computation controls, while the master problem handles long-term decisions.
  • Short-term approximation: The short-term problem is rewritten in epigraph form and approximated through convex lower and upper bounds for its nonconvex constraints.The resulting convex feasible set includes the original resource and timing constraints together with the introduced auxiliary variables.
  • Nonconvex constraints: The remaining nonconvexity arises from constraints involving auxiliary transmission and timing variables, motivating further inner approximation.The formulation identifies constraints (35f), (35g), and (35h) as challenging before their convex approximations are constructed.
  • Short-term algorithm: Algorithm 3 repeatedly solves the convex approximation, uses its solution as the next initial point, and stops when the target accuracy ε is reached.The inner approximations satisfy the properties required by the convergence framework.

B. Solving the Long-term Master Problem (31)

The long-term master problem is solved by updating a stochastic surrogate across large-scale coherence times. The surrogate combines historical information with a current approximation and is then convexified for optimization.

  • Surrogate update: At each large-scale coherence time, the stochastic cost is replaced by a sample surrogate built from the previous surrogate and the current approximate objective.The weighting parameter φ(n+1) controls the update between the two terms.
  • Objective approximation: The previous surrogate is linearized around the current local-accuracy iterate, while the current objective approximation adds a quadratic regularization term.The resulting expression provides a tractable approximation of the long-term objective.
  • Master-problem solution: The updated surrogate is used to form a convex approximation of the long-term master problem.This completes the long-term step of the online successive convex approximation procedure.

C. Solving the Overall Problem (29)

The overall algorithm alternates short-term optimization within each coherence time with long-term local-accuracy updates. After convergence, the resulting accuracy and adaptive transmission-computation variables are used during FL execution.

  • Overall algorithm: Algorithm 4 solves one short-term subproblem for each realized large-scale fading coefficient and uses its KKT solution to construct the long-term master problem.The master problem then produces an optimal local-accuracy candidate for the next update.
  • Long-term update: The local-accuracy iterate is updated by averaging its previous value with the master-problem solution using weighting parameter π(n+1).The weighting sequences are selected to satisfy the stated stochastic-approximation conditions.
  • FL execution: After convergence, FL uses the optimized local accuracy while short-term transmission and processing variables are updated within each short-term time block.The overall algorithm is rerun when large-scale fading statistics change.
  • Convergence: Convergence to a stationary point is guaranteed when the number of short-term iterations I(n) and the number of large-scale coherence times N both tend to infinity.The proof relies on the stated approximation properties and permits the Fritz John condition in place of the KKT condition for stationarity.

VII. CELL-FREE TDMA MASSIVE MIMO AND COLLOCATED MASSIVE MIMO FOR WIRELESS FEDERATED LEARNING

The paper formulates wireless FL training-time minimization for cell-free TDMA massive MIMO and collocated massive MIMO, adapting the proposed solution approach to each network. These baselines differ from CFmMIMO in their transmission structure, pilot requirements, and achievable-rate models.

  • Cell-free TDMA massive MIMO and collocated massive MIMO are introduced as comparison approaches for supporting wireless FL.Their problem formulations and solution algorithms are developed for comparison with CFmMIMO.
  • Cell-free TDMA needs pilot sequences of length 1, whereas CFmMIMO requires τt ≥ K for orthogonal pilots across K UEs.The channel-estimation model is otherwise equivalent when CFmMIMO pilots are pairwise orthogonal.
  • Cell-free TDMA transmits training updates sequentially in K equal orthogonal time slots, imposing a 1/K factor on downlink and uplink rates.Its effective FL training time is therefore formulated using the sequential transmission model.
  • The cell-free TDMA training-time problem has the same mathematical structure as problem (29), so a slightly modified Algorithm 4 solves it.The collocated massive MIMO formulation is likewise solved by a slightly modified Algorithm 4.
  • Collocated massive MIMO is modeled as a special CFmMIMO case in which all access points are collocated.Its downlink and uplink rates and power constraints are then specified for the resulting channel model.

VIII. NUMERICAL EXAMPLES

The numerical study evaluates Algorithm 4 under specified CFmMIMO simulation settings, then compares its convergence and effective training time with baseline designs. Algorithm 4 converges in fewer than 100 iterations and achieves substantial training-time reductions through joint optimization, while more UEs increase training time because interference and pilot contamination become stronger.

  • A. Parameters and Setup: The simulations use an existing FL framework as an example rather than evaluating a new FL framework on real datasets.The paper focuses its numerical results on Algorithm 4 and FL training-time minimization because real-dataset effectiveness was reported elsewhere.
  • 1) Effectiveness of the proposed algorithm:: Algorithm 4 converges in fewer than 100 iterations in the convergence experiment.Each iteration solves simple convex programs, so the paper expects low computational complexity.
  • 1) Effectiveness of the proposed algorithm:: 55% training-time reduction is achieved by Algorithm 4 over BL1 with M = 50 and K = 8.BL2 and BL3 also reduce time relative to BL1, but Algorithm 4 achieves further reductions of up to 49% and 43% over those baselines in reported settings.
  • 1) Effectiveness of the proposed algorithm:: Joint optimization of transmit power and local accuracy provides substantial training-time reductions beyond optimizing either design aspect separately.The comparisons include BL2, which optimizes local accuracy, and BL3, which optimizes downlink and uplink powers.
  • 1) Effectiveness of the proposed algorithm:: Increasing the number of APs decreases effective training time because array gain increases UE data rates.Increasing the number of UEs causes a dramatic training-time increase because mutual interference and pilot contamination become stronger.

2) Impact of key system parameters on the effective training time:

The effective training time is sensitive to local accuracy, processing frequency, energy limits, and UL-pilot length. CFmMIMO also outperforms cell-free TDMA massive MIMO and collocated massive MIMO in training-time comparisons.

  • Local accuracy: A 33% increase occurs with θmax = −40 dB versus θmax = −10, because lower local accuracy thresholds require more local-training iterations and reduce processing frequencies under Emax.The lower processing frequencies increase the time needed to compute local training updates.
  • UE processing frequency: A 19% increase occurs with fmax = 1.5 × 10^9 cycles/s versus fmax = 3 × 10^9 cycles/s because lower processing frequencies lengthen local-update computation.
  • UE energy consumption limit: A 9.4% increase occurs with Emax = 2 J versus Emax = 15 J when D = 1 km, as tighter energy limits can prevent the effective time from reaching the optimization optimum.Deep fading can make the energy constraint infeasible, although distributed antennas make simultaneous deep fading across all links unlikely.
  • UL-pilot length: UL-pilot lengths of τt = 1 and τt = 13 increase effective time by up to 10% and 1%, respectively, versus τt = 7.Large τt reduces data rates through Tc, whereas small τt increases pilot contamination and lowers data rates.
  • System comparisons: CFmMIMO reduces training time by up to 94% versus cell-free TDMA massive MIMO and up to 57% versus collocated massive MIMO.Distributed antennas reduce exposure to unfavorable UE links, while TDMA sequentially transmits updates and imposes a 1/K data-rate factor.
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