When Should Agents Coordinate in Differentiable Sequential Decision Problems?

TL;DR

This work addresses when a team of agents should coordinate in differentiable, sequential decision problems by translating coordination into a timing problem over intervals. It introduces a Hessian-based second-order framework that distinguishes coordinated (jointly optimal) from uncoordinated (Nash) behavior and extends this to dynamic horizons, allowing the team to choose which time-steps to coordinate. An algorithm is developed to generate first-order solutions, classify their coordination intervals, and solve for an optimal coordination schedule that trades off coordination costs against the value of coordinated solutions. The approach is demonstrated on toy robot-separation and dynamic horizon scenarios, showing that longer horizons generally necessitate coordination across longer time spans to avoid high-cost uncoordinated outcomes. The framework provides a principled, computational method to balance communication costs with the benefits of coordination in differentiable motion-planning tasks.

Abstract

Multi-robot teams must coordinate to operate effectively. When a team operates in an uncoordinated manner, and agents choose actions that are only individually optimal, the team's outcome can suffer. However, in many domains, coordination requires costly communication. We explore the value of coordination in a broad class of differentiable motion-planning problems. In particular, we model coordinated behavior as a spectrum: at one extreme, agents jointly optimize a common team objective, and at the other, agents make unilaterally optimal decisions given their individual decision variables, i.e., they operate at Nash equilibria. We then demonstrate that reasoning about coordination in differentiable motion-planning problems reduces to reasoning about the second-order properties of agents' objectives, and we provide algorithms that use this second-order reasoning to determine at which times a team of agents should coordinate.

Figures (4)

• Figure 1: Uncoordinated behavior is suboptimal. Two agents minimize $l(x,y) = \tau\left(x^2 +y^2\right) + \gamma \exp\left(- (x - y/\rho\right)^2 )$. (a) The cost landscape shows two coordinated solutions ($\bigstar$) at local minima, and one uncoordinated solution ($\bigstar$) at a saddle point with higher cost. (b) At the saddle, each agent individually perceives a local minimum: the slice along the $x$-axis (with $y=0$ fixed) shows positive curvature, and vice versa. (c) Solutions are classified by their Hessian structure: coordinated solutions ($\bigstar$) have a positive definite full Hessian, while uncoordinated solutions ($\bigstar$) have only positive definite diagonal blocks.
• Figure 2: Coordinated behavior is valuable in dynamic settings. We plot $z_t$ for the three first-order solutions that we find by randomly initializing a root-finding algorithm. Agents one and two follow the blue and orange trajectories, respectively, and the green bars represent the set of intervals $S_\mathbf{u} = \{S ~|~ \mathbf{u} \in \mathsf{SOL}_S\}$, i.e., the intervals during which the agents are coordinated for each solution. We remove dominated sets of time intervals when plotting $S_\mathbf{u}$, for example, we only plot the interval $\{1,\ldots,6\}$ for the solution in (a), even though the solution is coordinated on any subset $S$. We also remark that we also found solutions where the trajectories of agents one and two were swapped, which we do not depict. We additionally annotate the cost $l$ of each trajectory. There are two coordinated solutions of differing costs shown in (a) and (b). There is an additional uncoordinated solution that the agents may reach if they do not coordinate on the correct time-steps. Thus, by coordinating, the team can reduce the chance of reaching a high-cost solution. Note that the solutions in (b) and (c) are different solutions, and this difference is most apparent at $T = 6$.
• Figure 3: The optimal coordinated solution finds the smallest interval that differentiates coordinated and uncoordinated behavior. We depict the optimal distribution $p^*$ for the problem in \ref{['eq:dyn_stack_coordination']} when $T=6$, where we color the heatmap by the value of $p^*$ for each interval. We observe that the optimal solution places probability mass on intervals that contain $\{4,5\}$, with higher mass being placed on shorter intervals. This solution reflects the fact that, in order to avoid high-cost uncoordinated solutions, the team only needs to be optimal in the decision variables corresponding to time-steps $4$ and $5$.
• Figure 4: Coordination is necessary in longer horizon settings to handle diverse uncoordinated solutions. In (a-d), we plot $z_t$ for four of the first-order solutions we generate in the case where the horizon $T$ is $10$. We additionally depict the optimal solution to the coordination problem \ref{['eq:dyn_stack_coordination']} in (e). We observe a diverse set of behaviors among the first-order solutions, and we observe that, in the solution to the coordination problem \ref{['eq:dyn_stack_coordination']} for $T = 10$, the team pays to place high probability mass on the interval $\{3,4,5,6,7,8\}$. We remark that the behavior where agents swap positions twice, as in (d), does not appear among the coordinated solutions we find.

Theorems & Definitions (8)

• definition thmcounterdefinition: Coordinated solution
• theorem thmcountertheorem: First-order necessary conditions nocedal2006numerical
• definition thmcounterdefinition: Uncoordinated solution
• lemma thmcounterlemma
• remark thmcounterremark
• definition thmcounterdefinition: Dynamic coordination intervals
• remark thmcounterremark
• remark thmcounterremark