Multi-dimensional Stackelberg Game-based Incentive Mechanism for Differential Private Federated Learning with Non-IID Data

Incentive mechanisms are essential for boosting client engagement in differential private federated learning (DP-FL). However, existing Stakelberg games-based incentive mechanisms typically assume that client decisions are one-dimension and that data is independent and identically distributed (IID) across clients. In reality, data distributions are often non-IID and clients have multi-dimensional resources decisions, including data quantity, privacy, and label distribution. Therefore, in this paper, we present a novel multi-dimensional Stackelberg game-based incentive mechanism for DP-FL with non-IID data, aiming to maximize the total utility of clients and server by seeking a balance between the clients' multi-dimensional decisions and the server's payment. Specifically, we first formulate the utility functions of both server and clients under multi-dimensional decisions and then model the interactions between server and clients as a single-leader-multiple-followers Stackelberg game. To derive the optimal decisions that maximize their utilities, we theoretically prove the existence of a Stackelberg equilibrium between server and clients. Due to the difficulty to directly calculate the Stackelberg equilibrium, we propose a bi-level multi-agent reinforment learning algorithm to learn the optimal decisions for both server and clients by trial and error. Extensive simulation results demonstrate that our proposed method outperforms the baselines in terms of total utility.