Joint Service Pricing and Cooperative Relay Communication for Federated Learning

I. INTRODUCTION

• Centralized training data raises security and privacy concerns, while direct model-update communication may be unavailable because of limited range or energy-inefficient because of high transmission power.The introduction also notes that device participation requires incentives rather than relying on sacrificial volunteers.Source: §I · p. 1
• Mobile devices provide learning services for a price and cooperatively relay model updates to the model owner.A relay combines its own update with received updates using an average operator, leaving update-file size unaffected and not significantly affecting energy consumption.Source: §I · p. 1
• Pricing and relay choices must be considered jointly because larger training-data sizes can motivate the model owner while lowering a device’s probability of receiving relay service.Devices set prices for their training data, and the model owner chooses the data size for each device.Source: §I · pp. 1–2
• The proposed design is described as supporting smooth handling of massive model-update volumes and reducing communication congestion for the model owner and mobile devices.Devices choose their own data prices for individual profit optimization while cooperating to transfer updates.Source: §I · p. 2

II. SYSTEM DESCRIPTION

• Each device trains locally on part of its data to generate model updates for the model owner's model.The model owner negotiates the training-data size for each device.Source: §II · p. 2
• The model owner's learning accuracy increases with the total training-data size, with diminishing marginal increases under the increasing concave accuracy function.The model owner’s utility incorporates this relationship and the per-unit training-data price.Source: §II · p. 2
• Devices can relay one another's updates, with routing constraints requiring a single next hop and a path to the model owner.The routing design also requires at least one device to connect to the owner's access point, and limits relay hops so each update reaches the owner.Source: §II · p. 3
• A relay must receive a device's update before finishing its own computation; device service time includes computation, relaying, and transmission.The relay-arrival constraint compares these periods with the relay device's computation time.Source: §II · p. 3
• Devices sharing a relay node generate mutual interference, and the transmission-rate model uses channel gains, distance-based path loss, and noise.The path-loss coefficient is specified as α ≥ 2.Source: §II · p. 3

III. STACKELBERG GAME FORMULATION

• Mobile devices act as leaders and choose per-unit data prices, relay nodes, and transmission powers; the model owner follows by choosing training-data sizes.The owner is the buyer of the learning service, and devices are its sellers.Source: §III · p. 4
• The lower-level subgame has the model owner choose training-data sizes given the devices' prices.The training-data-size vector is denoted sd.Source: §III · p. 4
• After the owner's demand is determined, the upper-level subgame has devices choose prices and cooperative transmission decisions.The model defines the transmission-power matrix P and the devices' profit vector Π.Source: §III · p. 4
• The game defines a Stackelberg equilibrium as the solution for the model owner and the mobile devices.The model specifies upper bounds for the devices' data prices and transmission powers.Source: §III · p. 4

IV. EQUILIBRIUM ANALYSIS

• Backward induction solves the lower-level owner's subgame first, then substitutes its Nash equilibrium into the upper-level device subgame for analysis by the exterior point method.The lower-level solution is shown to exist by concavity and to be unique by negative definiteness of the utility Hessian.Source: §IV · p. 4

A. Solution to Lower-level Subgame

• The model owner's lower-level solution is obtained from the first derivative of its utility with respect to each device's training-data size.The resulting best response is denoted sd*; the analysis adopts a Weibull model for trackable analysis.Source: A. Solution to Lower-level Subgame · p. 4
• Strict concavity of the accuracy function and linearity of the price term make the owner's utility concave in each training-data size, establishing existence of a solution.The stated analysis adopts the Weibull model as suggested in.Source: A. Solution to Lower-level Subgame · p. 4
• A negative-definite Hessian, with negative diagonal and zero off-diagonal elements, establishes uniqueness of the lower-level solution.Source: A. Solution to Lower-level Subgame · p. 4

B. Solutions to Upper-level Subgame

• The mobile-device profit functions are obtained by substituting the model owner’s optimal data demand into the device profit functions.Source: B. Solutions to Upper-level Subgame · p. 4
• The upper-level subgame is analyzed at a Nash equilibrium, where no player can improve its profit by changing strategy while others’ strategies stay fixed.Source: B. Solutions to Upper-level Subgame · p. 4
• The nonlinear constraints are handled with the exterior point method, which rewrites each device’s constrained profit objective as an unconstrained objective using a large positive penalty coefficient M.Source: B. Solutions to Upper-level Subgame · p. 5

V. PERFORMANCE EVALUATION

• The evaluation models nine mobile devices, with bandwidth and channel gain set to 10, noise variance to 1, and model-update size to 0.1.Source: §V · p. 5
• Device-to-device and device-to-access-point distances are uniformly distributed over a 10-by-10 plane, while wireless-power and processing costs are generated using Gaussian distributions.Source: §V · p. 5
• The relay fee is set to 0.0096, and the processing-rate, averaging-time, and function-coefficient vectors are specified for the nine devices.Source: §V · p. 5

A. Numerical Result

• Mobile device 3 routes through device 7, reducing its wireless energy consumption, while devices 5 and 6 share device 4 as relay despite mutual interference.The reported comparison says using device 4 is more energy efficient for devices 5 and 6 than other relay choices or direct transfer to the central cloud.Source: A. Numerical Result · p. 5, Figure 2 · p. 5, Table II · p. 5
• Device 7 has the highest unit training-data price because its received update combines information from devices 7 and 3; device 4’s price is lower than device 8’s, attributed to data quality and model-owner preference.Source: A. Numerical Result · pp. 5–6, Figure 3 · p. 6
• Device 4 receives the largest training-data allocation, and its slow processing rate makes it a relay for devices 5 and 6, whose allocations are much smaller.Devices 5 and 6 use device 4 to avoid direct access-point transfers, improving energy efficiency.Source: A. Numerical Result · p. 6, Figure 4 · p. 6

• The game jointly analyzes mobile devices’ transmission and training-data pricing strategies and the model owner’s learning-service subscription.Source: §VI · p. 6
• The model captures how device transmission strategies affect transmission rates and relay nodes, and how service subscription affects the owner’s utility.Source: §VI · p. 6
• The analysis considers interference costs in device profits and training-data size effects on both owner utility and device profits.Source: §VI · p. 6
• The owner’s optimal strategy is studied using best response, and device equilibrium strategies using the exterior point method; long-run interactions remain future work.Source: §VI · p. 6