Generalized differentiation in Wasserstein space and application to multiagent control problem

Abstract: Several concepts of generalized differentiation in Wasserstein space have been proposed in order to deal with the intrinsic nonsmoothness arising in the context of optimization problems in Wasserstein spaces. In this paper we introduce a concept of admissible variation encompassing some of the most popular definitions as special cases, and using it to derive a comparison principle for viscosity solutions of an Hamilton Jacobi Bellman equation following from an optimal control of a multiagent systems.

• The paper presents a novel admissible variation concept that unifies various differentiation approaches in Wasserstein space.
• It demonstrates the stability of these variations under perturbations and their equivalence to traditional derivative frameworks.
• The framework is applied to multiagent control problems, providing a robust solution via Hamilton-Jacobi-Bellman equations and viscosity methods.

Generalized Differentiation in Wasserstein Space

The paper "Generalized differentiation in Wasserstein space and application to multiagent control problem" (2511.15455) investigates a novel framework of differentiation within the Wasserstein space $\mathscr{P}(\mathbb{T}^d)$. It also applies this theoretical advancement to a multiagent control problem. The paper introduces an original notion, termed admissible variation, which is broad enough to encompass existing notions of derivatives within the Wasserstein space, thereby unifying several disparate approaches under a single cohesive framework.

Introduction to Wasserstein Space

The differentiation of real-valued functions on the Wasserstein space involves tackling the inherent nonsmoothness in such functional spaces. Traditional derivatives are not directly applicable due to the lack of linear structure, as $\mathscr{P}(\mathbb{T}^d)$, the set of probability measures on the $d$-dimensional torus, does not inherit the vector space properties necessary for classical differentiability concepts. Instead, the authors propose a general concept of variation, which serves to define differentiability within the field of probability measures.

Admissible Variations in Wasserstein Space

The notion of admissible variation introduced in this paper acts as a generalized derivative, capturing the essential features of several prior definitions while offering a unified approach to differentiation in Wasserstein space. Importantly, these variations allow for a consistent derivative framework under smoothness assumptions. This framework is sufficiently flexible to include the popular concepts of metric slopes and metric derivatives, offering a structure analogous to linear maps acting on tangent vectors.

Stability and Composition

The admissible variations are shown to be stable under perturbations by small plans, thereby permitting consistent application across a diverse range of problems. The paper emphasizes the robustness of these variations, illustrating their compositional properties and their alignment with other types of variations, such as transport map variations and Eulerian variations.

Differentiation Framework

Within this new framework, the paper explores several types of variations, applying the concept of admissible variations to define differential structures. The authors propose a methodology to compute derivatives along specific variations and prove an equivalence result, demonstrating that under appropriate smoothness conditions, existing notions of differentiability coincide with their proposed concept.

Comparison with Existing Notions

The proposed derivative is compared with previous frameworks, such as those by Cardaliaguet et al., with the paper demonstrating that the new concept encompasses existing definitions while offering additional flexibility and generality.

Application to Multiagent Control Problems

The theoretical advancements in differentiation are applied to a multiagent control problem, where the dynamics of agents are described through probabilistic measures. The paper characterizes the value function associated with a mean field control problem, particularly in scenarios involving a Bolza-type cost functional. It addresses the mixed dynamics of a continuum of agents and provides a formal solution framework through the derived Hamilton-Jacobi-Bellman (HJB) equation.

HJB Equation and Viscosity Solutions

The concept of sub- and super-differentials is employed to establish a comparison principle for viscosity solutions of the HJB equation, further substantiating the robustness and applicability of the new differentiation framework in multiagent control problems.

Conclusion

This paper provides a significant contribution to the field by introducing a generalized notion of differentiation in the Wasserstein space, offering a unified framework that enhances theoretical clarity and practical applicability in multiagent control contexts. The implications of this research extend to future developments in the handling of nonsmooth optimization in probability measures, potentially influencing how complex systems involving numerous interacting agents are modeled and controlled.