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Fast and Scalable Multi-Agent Distribution Matching via Partitioned Optimal Transport
Kooktae Lee, Ruchika Singh
TL;DR
The paper addresses the computational cost of global discrete optimal transport for large-scale terminal distribution matching. It partitions agents and target samples into spatial blocks and combines local assignments with finite-horizon control. Simulations report faster assignment solves while maintaining matching performance comparable to global methods.
Problem
Global discrete optimal transport becomes computationally expensive as the numbers of agents and target samples grow, complicating large-scale terminal distribution matching.
Source: §I · p. 1Method
The framework partitions agents and target samples into spatially corresponding blocks and integrates local assignments with barycentric finite-horizon control.
Source: §I · p. 1, §IV · p. 4Results
At M = 300 and N = 1500, K = 5 achieves 0.49 s (11× speedup) and K = 10 achieves 0.21 s (26× speedup), versus 5.48 s for the centralized approach, while maintaining comparable matching performance.
Source: §VII · p. 8, §VIII · p. 8Takeaways & Limitations
Block mass balance preserves global feasibility and yields a Wasserstein upper bound, while alternating local assignment and control guarantees monotonic cycle-to-cycle descent of the transport surrogate.
Source: §VIII · p. 8
Abstract
from arXiv · showhide
This paper presents a scalable optimal-transport-based framework for terminal distribution matching in multi-agent systems. While optimal transport provides a natural way to measure distributional mismatch and assign agents to a desired spatial distribution, global discrete transport can become computationally expensive for large-scale systems. We address this bottleneck by partitioning agents and target samples into spatially corresponding blocks and solving smaller local transport problems. Under a mass-balance condition, the resulting restricted coupling remains feasible for the global problem and provides an upper bound on the Wasserstein cost. The local assignments generate target locations for finite-horizon agent control, applicable to both linear and nonlinear dynamics. By alternating local assignment and control, we establish a cycle-to-cycle descent guarantee for the resulting transport surrogate. The proposed framework therefore enables scalable terminal distribution matching while retaining a rigorous connection to the Wasserstein objective. The technical soundness of the proposed results is validated through simulations.
I. INTRODUCTION
This paper formulates multi-agent deployment as matching the agents’ terminal spatial distribution to a target distribution. It addresses the computational cost of global discrete optimal transport with a spatially partitioned assignment framework.
A. Terminal Distribution Matching
- Terminal distribution matching steers the agents’ configuration at a prescribed time toward a given target distribution.The target samples represent the desired spatial distribution.Source: §I · p. 1, A. Terminal Distribution Matching · p. 2
I. INTRODUCTION
- Optimal transport measures mismatch between current and desired distributions while assigning agents to target samples.The paper builds on the centralized terminal-matching formulation of.Source: §I · p. 1
- Discrete optimal transport becomes computationally expensive as the numbers of agents and target samples grow.The centralized formulation of sequentially couples M agents with N target samples.Source: §I · p. 1
- The paper partitions agents and target samples spatially, using block mass balance to preserve global feasibility and provide a Wasserstein upper bound.It also integrates localized assignments with barycentric control and gives a cycle-to-cycle descent guarantee for the transport surrogate.Source: §I · p. 1
B. Discrete Optimal Transport
The framework uses discrete optimal transport to assign target mass and form agent-level targets, then reduces assignment cost by partitioning the problem into corresponding spatial blocks. Equal normalized mass in each paired block supports feasible local transport problems.
B. Discrete Optimal Transport
- A feasible transport coupling provides an upper bound on the squared 2-Wasserstein distance, even when the coupling is not globally optimal.The transport plan assigns mass between agents and target samples under prescribed marginals.Source: B. Discrete Optimal Transport · p. 2
C. Terminal Distribution Matching: Sequential Assignment and Control
- Sequential assignment allocates target mass to agents, forms barycentric terminal references, and repeats after finite-horizon control using updated agent states.This converts distribution matching into agent-level control problems, but still sequentially couples M agents with N target samples.Source: C. Terminal Distribution Matching:… · p. 2
A. Spatial Partition and Mass Balance
- Spatial partitioning groups agents and target samples in corresponding regions, with each paired block required to have equal normalized empirical mass.The blocks need not contain equal numbers of agents and target samples; for example, 10 of 30 agents can pair with 200 of 600 target samples.Source: A. Spatial Partition and Mass Balance · pp. 2–3
- Block mass balance ensures each agent block can be transported entirely to its corresponding target block, allowing smaller local transport problems while preserving global feasibility.The analysis does not require a particular partitioning procedure.Source: C. Terminal Distribution Matching:… · p. 2, A. Spatial Partition and Mass Balance · p. 3
B. Block-Restricted Coupling and Feasibility
Under block mass balance, the block-restricted transport set is nonempty and contained in the unrestricted coupling set. Thus, its feasible couplings preserve global feasibility and upper-bound the squared Wasserstein distance.
B. Block-Restricted Coupling and Feasibility
- Under block mass balance, the block-restricted coupling set is nonempty and every such coupling is feasible for the original optimal transport problem.Local couplings can be assembled from corresponding blocks while preserving the prescribed marginals.Source: B. Block-Restricted Coupling and… · p. 3
- Because a block-restricted coupling is also an unrestricted coupling, its transport cost upper-bounds the squared Wasserstein distance.The Wasserstein cost minimizes over all global couplings.Source: B. Block-Restricted Coupling and… · p. 3
- Setting cross-block entries to zero yields a global coupling with the prescribed row and column marginals.The resulting matrix is nonnegative and satisfies the block restriction.Source: B. Block-Restricted Coupling and… · p. 3
C. Barycentric Targets
The transport plan gives each agent a barycentric target, which is a uniquely characterized point obtained from its assigned target samples.
C. Barycentric Targets
- The barycentric target for each agent is formed from its transport-plan weights, which define a probability vector.Each agent is assigned a single reachable point rather than a distribution of targets.Source: C. Barycentric Targets · p. 3
- For each agent, its barycentric target is the unique minimizer of the stated target-sample-weighted objective.The first-order condition identifies a unique stationary point, which is the unique global minimizer.Source: C. Barycentric Targets · p. 3
D. Barycentric Control Interpretation
Because the Wasserstein objective depends on a state-dependent optimal coupling, direct control analysis is difficult. Fixing a feasible coupling yields a transport-cost upper bound that decomposes into barycentric tracking and a coupling-dependent residual, supporting agent-level target-reaching control.
D. Barycentric Control Interpretation
- Fixing the coupling avoids directly analyzing the Wasserstein objective’s state-dependent minimization and turns the control step into agent-level finite-horizon target-reaching problems.The resulting barycentric targets can be computed through smaller local transport problems using the spatially partitioned coupling.Source: D. Barycentric Control Interpretation · pp. 3–4
- A feasible fixed coupling gives a transport-cost upper bound on the Wasserstein distance, so reducing that cost reduces a certified upper bound on the distribution-matching objective.The connection between this cost and barycentric tracking was established in prior work [7].Source: D. Barycentric Control Interpretation · p. 4
- For a fixed coupling, the transport cost decomposes into a barycentric tracking term and a coupling-dependent residual independent of the agent states.The residual is weighted dispersion among the target samples assigned to each agent; it is zero when each row sends all its mass to one target.Source: D. Barycentric Control Interpretation · p. 4
IV. FINITE-HORIZON BARYCENTRIC CONTROL
The framework converts assigned barycentric targets into finite-horizon control problems. Linear dynamics admit exact target reaching under a reachability condition, while nonlinear dynamics require a controller satisfying a terminal-contraction condition.
IV. FINITE-HORIZON BARYCENTRIC CONTROL
- The control step treats each agent’s assigned barycentric target as a finite-horizon target-reaching problem, with exact closed-form control for linear dynamics and a general contraction condition for nonlinear dynamics.The stated results are standard finite-horizon control facts, with derivations referred to.Source: §IV · p. 4
A. Linear Dynamics
- For linear dynamics, full row rank of the finite-horizon reachability matrix ensures arbitrary terminal states are achievable, and the minimum-norm controller reaches each barycentric target exactly in H steps.The exact control sequence is stated as a result carried over from Lemma 2 in.Source: A. Linear Dynamics · p. 4
B. Nonlinear Dynamics
- For nonlinear dynamics, the framework assumes a controller contracts each agent’s terminal distance to its assigned barycentric target by a factor ρ in [0, 1).A finite-horizon feedback controller or nonlinear MPC implementation can satisfy the condition; the LTI controller is the special case ρ = 0.Source: B. Nonlinear Dynamics · p. 4
V. CYCLE-TO-CYCLE DESCENT AND WASSERSTEIN GUARANTEES
The assignment–control procedure applies each control sequence over a full horizon before re-solving restricted optimal transport. Under persistent block membership and the stated control assumptions, the transport surrogate is nonincreasing across cycles, with strict descent when agents begin away from their targets.
A. Control Cycles and the Control-Step Descent
- Each cycle spans H steps with the coupling and barycentric targets fixed during control; the control-step descent follows because the fixed coupling’s residual is unchanged across the cycle.The cycle-level analysis separates descent from the control step and from subsequent reassignment.Source: A. Control Cycles and the Control-Step… · p. 5
B. Cycle-to-Cycle Surrogate Descent
- Persistent block membership keeps the previous cycle’s coupling feasible at the next cycle, allowing reoptimization over the same restricted feasible set to combine with control-step descent.The assumption is naturally satisfied with fixed agent assignments or horizons short enough to prevent boundary crossings.Source: A. Control Cycles and the Control-Step… · p. 5, B. Cycle-to-Cycle Surrogate Descent · p. 5
- The full assignment–control iteration has a monotonically nonincreasing surrogate under Assumptions 1, 3, and 4, and descent is strict when at least one agent starts a cycle away from its assigned barycentric target.The iteration applies the computed sequence over H steps before re-solving restricted optimal transport and updating targets.Source: B. Cycle-to-Cycle Surrogate Descent · p. 5, §V · p. 5
C. Wasserstein Upper Bound
A block-restricted coupling remains a feasible global coupling during a cycle when block membership persists. Its transport cost therefore upper-bounds the Wasserstein distance, but monotonicity is guaranteed for the surrogate, not necessarily for the true Wasserstein optimum.
C. Wasserstein Upper Bound
- The block-restricted surrogate is an upper bound on the true Wasserstein distance throughout a cycle, and its cycle-to-cycle descent provides certified bounds without recomputing global optimal transport at every step.The same coupling retains the required marginal masses as agent locations change during the cycle.Source: C. Wasserstein Upper Bound · p. 6
- Persistent block membership keeps the cycle’s coupling feasible for the global transport problem, which makes its evaluated transport cost a valid Wasserstein upper bound.The coupling remains feasible at each time in the cycle.Source: C. Wasserstein Upper Bound · p. 6
- The guarantee is monotonicity of the restricted surrogate, not necessarily of the true Wasserstein distance, whose optimal global coupling may differ.The upper-bound surrogate can decrease even when the Wasserstein optimum behaves differently across cycles.Source: C. Wasserstein Upper Bound · p. 6
D. The Linear Time-Invariant Case
In the LTI case, exact barycentric control gives an equality in the control-step decrease, and restricted OT reassignment adds further surrogate-cost reduction. Spatial partitioning also decomposes the assignment into smaller local problems, with an O(K^2) reduction in the equal-cardinality regime.
D. The Linear Time-Invariant Case
- In the LTI case, exact control gives an equality in the control-step decrease, and restricted reassignment further decreases the surrogate across cycles.The control-step inequality becomes an equality; the subsequent coupling update supplies an additional decrease.Source: D. The Linear Time-Invariant Case · p. 6
A. Local Assignment and Computational Complexity
- Partitioning decomposes the global restricted transport problem into K independent local problems that can run in parallel.The local couplings are assembled along the block diagonal.Source: A. Local Assignment and Computational… · pp. 6–7
- For balanced, comparable blocks, each local problem has approximately M/K agents and N/K target samples instead of M and N globally.This replaces one M × N problem with K problems of approximately (M/K) × (N/K) size.Source: A. Local Assignment and Computational… · p. 7
- In the equal-cardinality one-to-one regime, local Hungarian assignments yield an ideal O(K^2) computational reduction versus centralized assignment.This special case assumes M = N and has vanishing dispersion; each block has O((M/K)^3) worst-case complexity.Source: A. Local Assignment and Computational… · p. 7
B. Algorithm Summary
The algorithm fixes a spatial partition, solves local transport problems at each cycle, and uses the resulting barycentric targets for finite-horizon control. The cited theorem guarantees that the surrogate is nonincreasing across iterations.
B. Algorithm Summary
- The partition is constructed offline and kept fixed, establishing persistent agent-block memberships before execution.These memberships satisfy Assumption 4.Source: B. Algorithm Summary · p. 7
- At each cycle, independent blockwise OT problems produce a global coupling and barycentric targets, followed by finite-horizon control.The controller uses the stated linear-dynamics law or a controller satisfying the specified condition for nonlinear dynamics.Source: B. Algorithm Summary · p. 7
- Theorem 1 guarantees nonincreasing surrogate values during the iterative assignment-control procedure.Fig. 2 depicts the partitioned surrogate and Wasserstein distances over successive cycles.Source: B. Algorithm Summary · p. 7, Figure 2 · p. 7
- Fig. 1 contrasts centralized transport trajectories with partition-restricted transport using K = 5 blocks.The caption identifies the two methods but does not state a quantitative outcome.Source: Figure 1 · p. 7
VII. SIMULATIONS
Simulations compare partition-restricted and centralized transport, showing substantial assignment-time reductions alongside comparable distribution-matching behavior. They also illustrate monotone surrogate descent and the stated feasibility and Wasserstein-bound properties.
VII. SIMULATIONS
- Both methods disperse agents from the initial cluster across the four target modes, while the mass-balanced restricted coupling remains feasible for the original OT problem.The simulation uses M = 30 agents, N = 500 target samples, and K = 5 spatial blocks.Source: §VII · p. 7
- The partitioned surrogate decreases monotonically across cycles, and the cycle-boundary Wasserstein distance is bounded above by the next surrogate value.The paper presents these as simulation checks of the descent guarantee and the upper-bound relationship.Source: §VII · p. 8
- The partitioned method has a slightly higher Wasserstein distance than centralized OT, trading a comparable matching-performance sacrifice for substantial computational speedups.The paper attributes the difference to the centralized method's unrestricted transport flexibility.Source: §VII · p. 8
- At M = 300 and N = 1500, K = 5 solves in 0.49 s (11× speedup) and K = 10 in 0.21 s (26× speedup), versus 5.48 s centrally.At fixed M = 40, K = 10 is roughly 22× faster as N scales; speedup grows with problem size in both sweeps.Source: §VII · p. 8, Figure 3 · p. 8
VIII. CONCLUSION
- The simulations demonstrate the intended properties of mass-balanced feasibility, restricted-surrogate descent, and reduced assignment computation through spatial decomposition.The conclusion reports solve-time reductions of an order of magnitude or more with matching performance comparable to global methods.Source: §VII · p. 8, §VIII · p. 8
- The conclusion states that block mass balance preserves feasibility and that finite-horizon barycentric control yields monotonic cycle-to-cycle surrogate descent while the surrogate upper-bounds Wasserstein distance.These guarantees accompany the reported computational gains and comparable matching performance.Source: §VIII · p. 8