Incentives in Federated Learning with Heterogeneous Agents

TL;DR

This work studies incentive design in federated learning with heterogeneous agents under a PAC accuracy objective, where an agent’s utility depends on who provides each sample. It shows that uncoordinated play can be highly inefficient and may lack pure equilibria, with the best equilibrium far from the social optimum. A central planner with full information can compute a near-optimal data-contribution plan via a polynomial-time LP relaxation that achieves a logarithmic approximation, and the authors design a Pay-What-You-Contribute mechanism that is strategyproof and unique among contribution-based transfers. The results offer a rigorous mechanism-design perspective for FL with heterogeneous data, bridging combinatorial optimization, learning theory, and public-good incentives, with implications for coordinating data-sharing across diverse domains.

Abstract

Federated learning promises significant sample-efficiency gains by pooling data across multiple agents, yet incentive misalignment is an obstacle: each update is costly to the contributor but boosts every participant. We introduce a game-theoretic framework that captures heterogeneous data: an agent's utility depends on who supplies each sample, not just how many. Agents aim to meet a PAC-style accuracy threshold at minimal personal cost. We show that uncoordinated play yields pathologies: pure equilibria may not exist, and the best equilibrium can be arbitrarily more costly than cooperation. To steer collaboration, we analyze the cost-minimizing contribution vector, prove that computing it is NP-hard, and derive a polynomial-time linear program that achieves a logarithmic approximation. Finally, pairing the LP with a simple pay what you contribute rule, where each agent receives a payment equal to its sample cost, yields a mechanism that is strategy-proof and, within the class of contribution-based transfers, is unique.

Figures (4)

• Figure 1: An agent’s expected loss falls as a larger share of a fixed training set comes from their own distribution. With a budget of $m$ data points, we sample $\lambda m$ from Agent 1 and $(1-\lambda)m$ from Agent 2 on FEMNIST LEAF, train a classifier, and repeat $100$ times. For each $m$, Agent 1’s loss decreases monotonically in $\lambda$, confirming that data are not exchangeable—utilities depend on who contributes. Full details in \ref{['sec:data-composition-performance']}.
• Figure 2: Samples from selected character classes for two FEMNIST agents, illustrating distinct handwriting styles.
• Figure 3: Agent 1's test accuracy as a function of the fraction of samples contributed from its own distribution (Agent 1), compared against random convex combinations from other agents (Agent 2). Each curve represents different total dataset sizes ($m$). The monotonic trend supports the weak–monotonicity \ref{['assumption:monotone']}.
• Figure 4: Visualization of the Set Cover reduction used in \ref{['thm:nphard']}. Left: Incidence matrix of the original Set Cover instance. Each row is a subset $E_j$ and each column an element $u_i$; a square means $u_i\in E_j$. Right: The corresponding bipartite graph construction: each node $x_j$ corresponds to subset $E_j$ and is connected to the hypotheses $h_i$ for which $u_i \in E_j$. A size-$m$ set cover on the left corresponds to an $m$-sample training set that forces ERM to output $h^\star$ on the right.

Theorems & Definitions (32)

• Theorem 1: PoS is unbounded
• Theorem 2
• Theorem 3: Approximation via Linear Programming
• Definition 1: Locally Oblivious Approximations
• Lemma 1
• Definition 2: Contribution-Based Mechanisms
• Theorem 4
• Corollary 1
• Theorem 5: Existence of Nash Equilibrium
• proof