Optimal Control of Probability Distributions via Optimal Transport

Core Concepts

The solution of dynamic programming in probability spaces can be obtained by combining the solution of dynamic programming in the ground space and the solution of an optimal transport problem.

Abstract

The paper studies discrete-time finite-horizon optimal control problems in probability spaces, where the state of the system is a probability measure. The authors show that the solution of such problems can be obtained by combining two ingredients: (1) the solution of dynamic programming in the "ground space" (i.e., the space on which the probability measures live) and (2) the solution of an optimal transport problem.
The key insights are:

The cost-to-go in the probability space is equal to the multi-marginal optimal transport discrepancy, with the transportation cost being the cost-to-go in the ground space.

The optimal state-input distribution results from the combination of the optimal input in the ground space and the solution of the optimal transport problem.

This separation principle reveals that the "low-level control of the agents of the fleet" (how does one reach the destination?) and the "fleet-level control" (who goes where?) are decoupled. The authors provide several examples and counterexamples to illustrate their results and expose potential pitfalls.
The proofs rely on novel stability results for the (multi-marginal) optimal transport problem, which are of independent interest.

Stats

Many optimal control problems of stochastic or large-scale dynamical systems can be framed in the probability space, whereby the state is a probability measure.
The optimal control problem in the probability space can be written as:
inf_{u_k:X_k->U_k} G_N(μ_N, ρ_N) + Σ_{k=0}^{N-1} G_k(μ_k, u_k, ρ_k),
subject to the measure dynamics μ_{k+1} = f_k(·, u_k(·))#μ_k.

Quotes

"The solution of dynamic programming in probability spaces results from two ingredients: (i) the solution of dynamic programming in the "ground space" (i.e., the space on which the probability measures live) and (ii) the solution of an optimal transport problem."
"A separation principle holds: The "low-level control of the agents of the fleet" (how does one reach the destination?) and "fleet-level control" (who goes where?) are decoupled."

Key Insights Distilled From

Dynamic Programming in Probability Spaces via Optimal Transport

by Anto... at arxiv.org 04-09-2024

https://arxiv.org/pdf/2302.13550.pdf

Deeper Inquiries

How can the proposed approach be extended to handle constraints on the probability measures, such as covariance constraints or penalties?

In order to extend the proposed approach to handle constraints on the probability measures, such as covariance constraints or penalties, we can incorporate these constraints into the formulation of the optimal transport problem.
For covariance constraints, we can introduce constraints that enforce certain relationships or bounds on the covariance matrices of the probability measures. This can be achieved by adding terms to the cost function that penalize deviations from the desired covariance structure. By including these constraints in the optimization problem, we can ensure that the resulting probability measures satisfy the covariance constraints.
Similarly, for penalties on the probability measures, we can introduce penalty terms in the cost function that penalize deviations from certain properties or distributions. These penalties can be designed to encourage the probability measures to adhere to specific criteria or distributions while optimizing the overall control problem.
By incorporating these constraints and penalties into the optimization framework, we can extend the proposed approach to handle a wider range of constraints on the probability measures, providing a more flexible and robust solution for control problems in probability spaces.

What are the implications of the separation principle for the design of transportation costs in applications where the underlying metric is not known a priori?

The separation principle outlined in the context has significant implications for the design of transportation costs in applications where the underlying metric is not known a priori.
One key implication is that the separation principle allows for a modular approach to designing transportation costs. By decoupling the low-level control of individual agents from the fleet-level control, designers can focus on optimizing the control strategies for individual agents independently of the overall fleet coordination. This modular approach enables more flexibility in designing transportation costs tailored to specific requirements and constraints.
Additionally, the separation principle suggests that the transportation costs can be designed to capture different aspects of the control problem. For example, the costs can be structured to incentivize certain behaviors or trajectories at the individual agent level, while also considering the overall fleet coordination objectives. This flexibility in designing transportation costs can lead to more effective and efficient control strategies in complex applications where the underlying metric is not known in advance.
Overall, the separation principle provides a framework for designing transportation costs that are tailored to the specific needs and constraints of the application, allowing for a more nuanced and effective control strategy in scenarios where the underlying metric is uncertain.

Can the insights from this work be leveraged to develop efficient distributed or decentralized control strategies for large-scale multi-agent systems?

Yes, the insights from this work can be leveraged to develop efficient distributed or decentralized control strategies for large-scale multi-agent systems.
The separation principle outlined in the context provides a clear framework for decoupling the control of individual agents from the fleet-level coordination. This separation allows for the development of distributed control strategies where each agent can optimize its behavior independently based on local information and objectives, while still contributing to the overall fleet-level goals.
By leveraging the insights from this work, designers can develop distributed control algorithms that utilize the optimal transport framework to coordinate the actions of multiple agents in a decentralized manner. This approach can lead to more efficient and scalable control strategies for large-scale multi-agent systems, where centralized control may not be feasible or practical.
Furthermore, the modular nature of the separation principle enables the development of adaptive and robust control strategies that can dynamically adjust to changes in the environment or system conditions. This adaptability is crucial for large-scale multi-agent systems operating in complex and dynamic environments.
In conclusion, the insights from this work can serve as a foundation for the development of efficient distributed or decentralized control strategies for large-scale multi-agent systems, offering a framework for optimizing fleet-level coordination while maintaining individual agent autonomy and efficiency.