Regret Lower Bounds for Decentralized Multi-Agent Stochastic Shortest Path Problems

Multi-agent systems (MAS) are central to applications such as swarm robotics and traffic routing, where agents must coordinate in a decentralized manner to achieve a common objective. Stochastic Shortest Path (SSP) problems provide a natural framework for modeling decentralized control in such settings. While the problem of learning in SSP has been extensively studied in single-agent settings, the decentralized multi-agent variant remains largely unexplored. In this work, we take a step towards addressing that gap. We study decentralized multi-agent SSPs (Dec-MASSPs) under linear function approximation, where the transition dynamics and costs are represented using linear models. Applying novel symmetry-based arguments, we identify the structure of optimal policies. Our main contribution is the first regret lower bound for this setting based on the construction of hard-to-learn instances for any number of agents, $n$. Our regret lower bound of $Ω(\sqrt{K})$, over $K$ episodes, highlights the inherent learning difficulty in Dec-MASSPs. These insights clarify the learning complexity of decentralized control and can further guide the design of efficient learning algorithms in multi-agent systems.

💡 Research Summary

This paper establishes the first regret lower bound for decentralized multi-agent learning in Stochastic Shortest Path problems under linear function approximation, pinpointing the fundamental difficulty of coordination in unknown environments.

Background and Problem Formulation: The Decentralized Multi-Agent Stochastic Shortest Path problem models scenarios where multiple agents (e.g., robots, vehicles) must cooperatively reach a common goal in a stochastic environment. Each agent acts based on its local observations without sharing actions or private costs, though they may share learned parameters. To handle large state spaces, the transition dynamics and cost functions are assumed to have a linear structure: they are inner products between a known feature vector and an unknown parameter. The performance of any learning algorithm is measured by its cumulative regret compared to the optimal policy.

Core Challenges and Novel Techniques: Extending lower bound analyses from single-agent to multi-agent settings presents significant hurdles: an exponential state space, coupled dynamics and costs across agents, intractable value functions, and complex KL divergence terms. The authors overcome these with innovative methods:

1. Novel Feature Design: They construct a family of hard-to-learn two-node MASSP instances with specifically designed linear features that ensure valid probability distributions while enabling tractable analysis.
2. State Aggregation and Monotonicity: Instead of dealing with the exponential number of states directly, they partition the state space based on the number of agents at each node. They prove a monotonicity property of the optimal value function within these partitions, which is sufficient for deriving the lower bound without needing closed-form expressions.
3. Symmetry-based KL Divergence Bounding: Leveraging the symmetric structure of their constructed problem instances, they bound the KL divergence between different instance distributions efficiently, a key step in information-theoretic lower bounds.

Main Results: The primary contribution is a regret lower bound of Ω(√K) that holds for any number of agents n, where K is the number of learning episodes. This bound is derived by constructing a family of hard MASSP instances and analyzing the performance limit of any possible decentralized learning algorithm on them.

Significance and Implications:

• Fundamental Limit: This result characterizes the inherent statistical complexity of learning in decentralized linear MASSPs. It shows that a √K dependence is unavoidable, regardless of the algorithm’s design.
• Optimality of Existing Algorithm: The lower bound matches (up to logarithmic and constant factors) the upper bound Õ(√K) provided by a prior algorithm