. In this paper, we propose and study an inverse boundary problem for the mean ﬁeld games (MFGs) governed by the ﬁrst-order master equation in a bounded domain. We establish a novel unique identiﬁability result by showing that the running cost function within any given proper (state) space-time subdomain is uniquely determined by the MFG boundary data of this subdomain. There are several salient features of our study. The MFG system couples two hyperbolic PDEs with one forward in time and the other one backward in time, and moreover there is a probability measure constraint. These make the inverse problem study new to the literature and highly challenging. We develop an eﬀective and eﬃcient scheme in tackling the inverse problem including the high-order variation in the probability space around a nontrivial distribution, and the construction of a novel class of CGO solutions to serve as the “probing modes”. Our study opens up a new ﬁeld of research on inverse problems for MFGs.

On an inverse boundary problem for mean field games

The theory of mean field games (MFGs) studies the limiting behaviors of large systems where the agents interact with each other in a certain symmetric way. The running and terminal costs are critical for the agents to decide the strategies. However, in practice they are often partially known or totally unknown for the agents, while the total cost is known at the end of the game. To address this challenging issue, we propose and study several inverse problems for MFGs. When the Lagrangian is a kinetic energy, we first establish unique identifiability results, showing that one can recover either the running cost or the terminal cost from knowledge of the total cost. If the running cost is limited to the time-independent class, we can further prove that one can simultaneously recover both the running and the terminal costs. Finally, we extend the results to the setup with general Lagrangians.

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Inverse problems for mean field games

In this work, we consider a novel inverse problem in mean-field games (MFGs). We aim to recover the MFG model parameters that govern the underlying interactions among the population based on a limited set of noisy partial observations of the population dynamics under the limited aperture. Due to its severe ill-posedness, obtaining a good quality reconstruction is very difficult. Nonetheless, it is vital to recover the model parameters stably and efficiently in order to uncover the underlying causes for population dynamics for practical needs. Our work focuses on the simultaneous recovery of running cost and interaction energy in the MFG equations from a finite number of boundary measurements of population profile and boundary movement. To achieve this goal, we formalize the inverse problem as a constrained optimization problem of a least squares residual functional under suitable norms. We then develop a fast and robust operator splitting algorithm to solve the optimization using techniques including harmonic extensions, three-operator splitting scheme, and primal-dual hybrid gradient method. Numerical experiments illustrate the effectiveness and robustness of the algorithm.

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A numerical algorithm for inverse problem from partial boundary measurement arising from mean field game problem

We introduce an optimal transport-based model for learning a metric tensor from cross-sectional samples of evolving probability measures on a common Riemannian manifold. We neurally parametrize the metric as a spatially-varying matrix field and efficiently optimize our model's objective using a simple alternating scheme. Using this learned metric, we can nonlinearly interpolate between probability measures and compute geodesics on the manifold. We show that metrics learned using our method improve the quality of trajectory inference on scRNA and bird migration data at the cost of little additional cross-sectional data.

Riemannian Metric Learning via Optimal Transport

A Coefficient Inverse Problem (CIP) of the determination of a coefficient of the Mean Field Games System (MFGS) of the second order is considered. The input data are generated by a single measurement event. Lateral Cauchy data, i.e. Dirichlet and Neumann boundary data are given for solutions of both equations forming the MFGS. In addition, solutions of both these equations are given at a fixed moment of time. H\"older stability estimates are obtained for both complete and incomplete lateral Cauchy data. These estimates imply uniqueness of our CIP. The apparatus of Carleman estimates is the main mathematical tool of this paper.

A Coefficient Inverse Problem for the Mean Field Games System

Mean-field games arise in various fields including economics, engineering, and machine learning. They study strategic decision making in large populations where the individuals interact via certain mean-field quantities. The ground metrics and running costs of the games are of essential importance but are often unknown or only partially known. In this paper, we propose mean-field game inverse-problem models to reconstruct the ground metrics and interaction kernels in the running costs. The observations are the macro motions, to be specific, the density distribution, and the velocity field of the agents. They can be corrupted by noise to some extent. Our models are PDE constrained optimization problems, which are solvable by first-order primal-dual methods. Besides, we apply Bregman iterations to find the optimal model parameters. We numerically demonstrate that our model is both efficient and robust to noise.

A mean field game inverse problem.

Decentralized Stochastic Gradient Descent (SGD) is an emerging neural network training approach that enables multiple agents to train a model collaboratively and simultaneously. Rather than using a central parameter server to collect gradients from all the agents, each agent keeps a copy of the model parameters and communicates with a small number of other agents to exchange model updates. Their communication, governed by the communication topology and gossip weight matrices, facilitates the exchange of model updates. The state-of-the-art approach uses the dynamic one-peer exponential-2 topology, achieving faster training times and improved scalability than the ring, grid, torus, and hypercube topologies. However, this approach requires a power-of-2 number of agents, which is impractical at scale. In this paper, we remove this restriction and propose \underline{D}ecentralized \underline{SGD} with \underline{C}ommunication-optimal \underline{E}xact \underline{C}onsensus \underline{A}lgorithm (DSGD-CECA), which works for any number of agents while still achieving state-of-the-art properties. In particular, DSGD-CECA incurs a unit per-iteration communication overhead and an $\tilde{O}(n^3)$ transient iteration complexity. Our proof is based on newly discovered properties of gossip weight matrices and a novel approach to combine them with DSGD's convergence analysis. Numerical experiments show the efficiency of DSGD-CECA.

Lisang Ding

Papers

A mean field game inverse problem.

DSGD-CECA: Decentralized SGD with Communication-Optimal Exact Consensus Algorithm