Decentralized Communication in MA-MDP

• The paper demonstrates that optimal coordination in MA-MDPs is achievable by decomposing global decision-making into tractable local subproblems with scheduled communications.
• It introduces the Dec-SMDP-Com framework and the MSBPI algorithm to efficiently balance communication costs against coordination quality.
• Empirical results from production control and grid rendezvous scenarios validate the approach’s effectiveness in reducing complexity while maintaining near-optimal performance.

A decentralized communication mechanism in the context of multi-agent Markov decision processes (MA-MDPs) refers to architectures, models, and algorithms enabling distributed agents to coordinate their actions and share information without continuous or unfettered global state exchange, often under explicit costs or constraints. Due to the computational intractability of optimal joint policy computation in general decentralized MDPs, recent theoretical and algorithmic work formalizes decomposition techniques that partition the global control problem into local subproblems with judiciously scheduled communication, leveraging periodic or event-based information exchange to re-synchronize decentralized agents while maintaining coordination efficacy and computational tractability.

1. Communication-Based Decomposition: Mechanism and Modeling

The communication-based decomposition mechanism ("CDM") defines a mapping from the global state $S$ to a tuple of agent-specific policies or "options" that operate independently between communication events. Specifically, CDM is expressed as

$$\mathrm{CDM} : S \rightarrow \left(\mathrm{Opt}_1, \mathrm{Opt}_2 \right),$$

where $\mathrm{Opt}_i$ denotes a set of temporally abstracted, single-agent options for agent $i$. Each option is constructed to terminate with a communication action. This decomposition allows the global MA-MDP to be partitioned into single-agent subproblems that operate autonomously, with periodic coordination via explicit (and potentially costly) communication. The decomposition unifies the benefits of local autonomy and global synchrony: agents execute pre-planned local strategies until certain events or schedules trigger communication, at which point the full joint state is restored and the process iterates.

Within the framework, each agent executes until its option terminates in a communication action $e \in E$, upon which new local options are reassigned based on the freshly synchronized state. The local process for agent $i$ is formally characterized by a tuple

$$\operatorname{opt}_i = \langle T_i : S_i \times X \rightarrow A\cup E, \ \beta_i, \ I_i \rangle,$$

where the termination condition $\beta_i$ is defined so that each option must end with a communication event.

2. Decentralized Semi-Markov Decision Process with Communication (Dec-SMDP-Com)

The mechanism is formally captured by the Decentralized Semi-Markov Decision Process with Direct Communication ("Dec-SMDP-Com"), defined as $(M, \mathrm{Opt}_1, \mathrm{Opt}_2, P^N, R^N)$:

• $M$ contains the states, action sets, transition dynamics, observations, and explicit communication costs of the original Dec-MDP.
• $\mathrm{Opt}_i$ encapsulates allowable options for agent $i$, each designed to force communication at completion.
• $$P^N\left(s', t+N \mid s, t, \mathrm{opt}_1, \mathrm{opt}_2 \right)$$ is the probability of arriving at joint state $s'$ at time $t+N$ after independently executing $\mathrm{opt}_1, \mathrm{opt}_2$ from $s$.
• $R^N(s, t, \mathrm{opt}_1, \mathrm{opt}_2, s', t+N)$ returns the expected (discounted and communication-inclusive) reward.

Agents execute local options to decouple their operation for as long as possible before synchronization via communication. Such events reset observational uncertainty and re-enable optimal reassignment of options with global awareness, which is critical when global reward functions couple otherwise independent agents.

3. Joint Optimization and Policy Iteration for Mechanism Synthesis

The search for the optimal communication-based decomposition is equivalent to solving the associated Dec-SMDP-Com optimally. This is accomplished via a heuristic multi-step backup policy-iteration algorithm (MSBPI) that searches over joint policy trees for pairs of options. Every policy tree encodes sequences of local actions terminating at leaves enforced to be communication acts. The backup operation for value function updates,

$$V(s, t, T) = \max_{\mathrm{opt}_1, \mathrm{opt}_2 \in \mathrm{OPT}_b} \sum_{k=1}^{\min(b, T-t)} P^N(s', t+k | s, t, \mathrm{opt}_1, \mathrm{opt}_2) \left[ R^N(s, t, \mathrm{opt}_1, \mathrm{opt}_2, s', t+k) + V(s', t+k, T) \right],$$

recurses over possible durations $k$ of temporally extended local options and globally synchronizes at each joint leaf. The algorithm uses branch-and-bound (pruning) to limit exploration of the double-exponential policy space. It is established that repeated policy improvements using this heuristic are guaranteed to converge to the Dec-SMDP-Com optimal policy.

4. Complexity Control via Restriction to Local Goal-Oriented Behaviors

Full option-space searches are computationally intractable. Restricting options to "goal-oriented" behaviors—where each agent is pre-assigned a local goal and follows an MDP-optimal policy to that goal—enables polynomial-time planning. In this reduced regime, the assignment of new local goals and resynchronization times are planned at communication events. The resulting LGO-MSBPI algorithm, operating over a restricted set of local goals $|G|$, has complexity $O(T^2 |S||G|)$, where $T$ is time horizon and $|S|$ is state space size. This trade-off between representational completeness and computational efficiency enables the practical deployment of decentralized communication planning in domains where optimal global coupling is prohibitive.

5. Integration of Human Knowledge and Myopic-Greedy Timing

When domain-specific local policies are known a priori—derived from human expertise or precomputed domain solutions—the decomposition process further simplifies. In this myopic-greedy timing approach, agents execute fixed local policies and only optimize over the timing of communication. The expected cost-to-go with and without communication is computed for each state: $$O_c(s_0, \pi_1^*, \pi_2^*) = \sum_{s} P(s | s_0, \pi_1^*, \pi_2^*) [R(s_0, \pi_1^*(s), \pi_2^*(s), s) + O_c(s, \pi_1^*, \pi_2^*)],$$

$$O_c(...) + C_z \quad \text{ if a communication act is taken.}$$

The optimal communication point is identified by a dynamic-programming recursion, selecting synchronization instants that yield lower expected overall cost (including communication) than proceeding without synchronization. This approach transforms the original problem into polynomial-time computation and lends itself readily to practical scheduling, especially in systems where agent behaviors can be pre-specified.

6. Empirical Results and Application Scenarios

The communication-based decomposition mechanism was empirically validated in:

• Production Control: A two-machine pairing scenario where agents must synchronize to accomplish joint tasks. The mechanism achieves targeted output with minor increases in costs due to communication, performance closely tracking a hypothetical setting with costless communication.
• Meeting under Uncertainty: Two agents attempting to rendezvous in a stochastic grid. Policies derived by the myopic-greedy approach and the communication-based mechanisms attain expected joint utilities near that of the free-communication optimum, but with substantially reduced communication frequency compared to suboptimal heuristics. This demonstrates the mechanism’s capacity to balance communication savings with high-quality coordination, across varying levels of uncertainty and communication cost.

7. Significance and Theoretical Impact

The communication-based decomposition mechanism establishes that in decentralized decision processes with costly communication:

• An optimal decomposition is not only computable via Dec-SMDP-Com but is amenable to algorithmic refinement, convergence guarantees, and tractable implementation through judicious restriction to local options or the use of human-provided local strategies.
• By periodic synchronization via optimal or near-optimal communication scheduling, agents overcome the challenge of globally coupled reward while minimizing communication overhead.
• The approach provides a foundation for extending decentralized MDP planning to practical domains such as distributed manufacturing, multi-robot navigation, and information gathering, offering systematic trade-offs between coordination quality, computational complexity, and communication efficiency.

A plausible implication is that in applied systems where communication costs or delays prohibit continuous synchronization, communication-based decomposition delivers a means to approach global optima without incurring the full complexity of joint policy planning or the performance loss of strictly local control.

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In essence, communication-based decomposition mechanisms exploit intermittent, strategically scheduled communication events to coordinate decentralized agents in MA-MDPs, yielding approximately optimal joint behavior with significant reductions in computational complexity and communication requirements, as formalized by the Dec-SMDP-Com model and supporting algorithms (Goldman et al., 2011).