Personalized Multi-Agent Average Reward TD-Learning via Joint Linear Approximation

We study personalized federated linear stochastic approximation (LSA), a framework which notably encompass personalized temporal difference learning. In this setting, heterogeneous agents collaborate to solve distinct linear fixed-point equations, each corresponding to an agent-specific learning problem. A central open question in personalized learning is whether a single method can adapt to an unknown level of heterogeneity by converging to each agent's personalized solution in all regimes while achieving a linear speedup in the number of agents when their learning problems are sufficiently similar. We answer this question affirmatively by introducing PF-LSA, a minimalist algorithm that mixes each agent's local stochastic update with the average update across agents, at no additional computational cost relative to standard federated methods. We prove that PF-LSA, achieves best-of-both-worlds guarantees without any prior knowledge on the level of heterogeneity. Our analysis is based on a sharp decomposition of the error into consensus and disagreement components. The consensus error decays rapidly, whereas the disagreement error decays more slowly but becomes negligible in low-heterogeneity regimes.

Despite the popularity of the actor-critic method and the practical needs of collaborative policy training, existing works typically either overlook environmental heterogeneity or give up personalization altogether by training a single shared policy across all agents. We consider a federated actor-critic framework in which agents share a common linear subspace representation while maintaining personalized local policy components, and agents iteratively estimate the common subspace, local critic heads, and local policies (i.e., actors). Under canonical single-timescale updates with Markovian sampling, we establish finite-time convergence via a novel joint linear approximation framework. Specifically, we show that the critic error converges to zero at the rate of $\tilde{\mathcal{O}}(1/((1-γ)^4\sqrt{TK}))$, and the policy gradient norm converges to zero at the rate of $\tilde{\mathcal{O}}(1/((1-γ)^6\sqrt{TK}))$, where $T$ is the number of rounds, $K$ is the number of agents, and $γ\in (0,1)$ is the discount factor. These results demonstrate linear speedup with respect to the number of agents $K$, despite heterogeneous Markovian trajectories under distinct transition kernels and coupled learning dynamics. To address these challenges, we develop a new perturbation analysis for the projected subspace updates and QR decomposition steps, together with conditional mixing arguments for heterogeneous Markovian noise. Furthermore, to handle the additional complications induced by policy updates and temporal dependence, we establish fine-grained characterizations of the discrepancies between function evaluations under Markovian sampling and under temporally frozen policies. Experiments instantiate the framework within PPO on federated Hopper-v5 action-map heterogeneity, showing gains over Single PPO and FedAvg PPO and downstream transfer from the learned shared trunk.

In this paper, we study cooperative multi-agent reinforcement learning (MARL) where the joint reward exhibits submodularity, which is a natural property capturing diminishing marginal returns when adding agents to a team. Unlike standard MARL with additive rewards, submodular rewards model realistic scenarios where agent contributions overlap (e.g., multi-drone surveillance, collaborative exploration). We provide the first formal framework for this setting and develop algorithms with provable guarantees on sample efficiency and regret bound. For known dynamics, our greedy policy optimization achieves a $1/2$-approximation with polynomial complexity in the number of agents $K$, overcoming the exponential curse of dimensionality inherent in joint policy optimization. For unknown dynamics, we propose a UCB-based learning algorithm achieving a $1/2$-regret of $O(H^2KS\sqrt{AT})$ over $T$ episodes.