Mean-field Problem Research Articles

Zero-Sum Games between Mean-Field Teams: Reachability-Based Analysis under Mean-Field Sharing

This work studies the behaviors of two large-population teams competing in a discrete environment. The team-level interactions are modeled as a zero-sum game while the agent dynamics within each team is formulated as a collaborative mean-field team problem. Drawing inspiration from the mean-field literature, we first approximate the large-population team game with its infinite-population limit. Subsequently, we construct a fictitious centralized system and transform the infinite-population game to an equivalent zero-sum game between two coordinators. Via a novel reachability analysis, we study the optimality of coordination strategies, which induce decentralized strategies under the original information structure. The optimality of the resulting strategies is established in the original finite-population game, and the theoretical guarantees are verified by numerical examples.

Mean-field neural networks: Learning mappings on Wasserstein space

We study the machine learning task for models with operators mapping between the Wasserstein space of probability measures and a space of functions, like e.g. in mean-field games/control problems. Two classes of neural networks based on bin density and on cylindrical approximation, are proposed to learn these so-called mean-field functions, and are theoretically supported by universal approximation theorems. We perform several numerical experiments for training these two mean-field neural networks, and show their accuracy and efficiency in the generalization error with various test distributions. Finally, we present different algorithms relying on mean-field neural networks for solving time-dependent mean-field problems, and illustrate our results with numerical tests for the example of a semi-linear partial differential equation in the Wasserstein space of probability measures.

A fast proximal gradient method and convergence analysis for dynamic mean field planning

In this paper, we propose an efficient and flexible algorithm to solve dynamic mean-field planning problems based on an accelerated proximal gradient method. Besides an easy-to-implement gradient descent step in this algorithm, a crucial projection step becomes solving an elliptic equation whose solution can be obtained by conventional methods efficiently. By induction on iterations used in the algorithm, we theoretically show that the proposed discrete solution converges to the underlying continuous solution as the grid becomes finer. Furthermore, we generalize our algorithm to mean-field game problems and accelerate it using multilevel and multigrid strategies. We conduct comprehensive numerical experiments to confirm the convergence analysis of the proposed algorithm, to show its efficiency and mass preservation property by comparing it with state-of-the-art methods, and to illustrate its flexibility for handling various mean-field variational problems.

Mean field games with absorption and common noise with a model of bank run

We consider a mean field game describing the limit of a stochastic differential game of N-players whose state dynamics are subject to idiosyncratic and common noise and that can be absorbed when they hit a prescribed region of the state space. We provide a general result for the existence of weak mean field equilibria which, due to the absorption and the common noise, are given by random flow of sub-probabilities. We first use a fixed point argument to find solutions to the mean field problem in a reduced setting resulting from a discretization procedure and then we prove convergence of such equilibria to the desired solution. We exploit these ideas also to construct ɛ-Nash equilibria for the N-player game. Since the approximation is two-fold, one given by the mean field limit and one given by the discretization, some suitable convergence results are needed. We also introduce and discuss a novel model of bank run that can be studied within this framework.

Mean Field Markov Decision Processes

We consider mean-field control problems in discrete time with discounted reward, infinite time horizon and compact state and action space. The existence of optimal policies is shown and the limiting mean-field problem is derived when the number of individuals tends to infinity. Moreover, we consider the average reward problem and show that the optimal policy in this mean-field limit is varepsilon -optimal for the discounted problem if the number of individuals is large and the discount factor close to one. This result is very helpful, because it turns out that in the special case when the reward does only depend on the distribution of the individuals, we obtain a very interesting subclass of problems where an average reward optimal policy can be obtained by first computing an optimal measure from a static optimization problem and then achieving it with Markov Chain Monte Carlo methods. We give two applications: Avoiding congestion an a graph and optimal positioning on a market place which we solve explicitly.

Polyak–Łojasiewicz inequality on the space of measures and convergence of mean-field birth-death processes

The Polyak–Łojasiewicz inequality (PŁI) in mathbb {R}^d is a natural condition for proving convergence of gradient descent algorithms (Karimi et al. in: Frasconi et al. (eds) Machine learning and knowledge discovery in databases, Springer International Publishing, Cham, pp 795–811, 2016). In the present paper, we study an analogue of PŁI on the space of probability measures mathcal {P}(mathbb {R}^d) and show that it is a natural condition for showing exponential convergence of a class of birth-death processes related to certain mean-field optimization problems. We verify PŁI for a broad class of such problems for energy functions regularised by the KL-divergence.

Nash equilibria for relative investors via no-arbitrage arguments

Within a common arbitrage-free semimartingale financial market we consider the problem of determining all Nash equilibrium investment strategies for n agents who try to maximize the expected utility of their relative wealth. The utility function can be rather general here. Exploiting the linearity of the stochastic integral and making use of the classical pricing theory we are able to express all Nash equilibrium investment strategies in terms of the optimal strategies for the classical one agent expected utility problems. The corresponding mean field problem is solved in the same way. We give four applications of specific financial markets and compare our results with those given in the literature.

Mean field control problems for vaccine distribution.

With the invention of the COVID-19 vaccine, shipping and distributing are crucial in controlling the pandemic. In this paper, we build a mean-field variational problem in a spatial domain, which controls the propagation of pandemics by the optimal transportation strategy of vaccine distribution. Here, we integrate the vaccine distribution into the mean-field SIR model designed in Lee W, Liu S, Tembine H, Li W, Osher S (2020) Controlling propagation of epidemics via mean-field games. arXiv preprint arXiv:2006.01249. Numerical examples demonstrate that the proposed model provides practical strategies for vaccine distribution in a spatial domain.

Estimation in tensor Ising models

AbstractThe $p$-tensor Ising model is a one-parameter discrete exponential family for modeling dependent binary data, where the sufficient statistic is a multi-linear form of degree $p \geqslant 2$. This is a natural generalization of the matrix Ising model that provides a convenient mathematical framework for capturing, not just pairwise, but higher-order dependencies in complex relational data. In this paper, we consider the problem of estimating the natural parameter of the $p$-tensor Ising model given a single sample from the distribution on $N$ nodes. Our estimate is based on the maximum pseudolikelihood (MPL) method, which provides a computationally efficient algorithm for estimating the parameter that avoids computing the intractable partition function. We derive general conditions under which the MPL estimate is $\sqrt N$-consistent, that is, it converges to the true parameter at rate $1/\sqrt N$. Our conditions are robust enough to handle a variety of commonly used tensor Ising models, including spin glass models with random interactions and models where the rate of estimation undergoes a phase transition. In particular, this includes results on $\sqrt N$-consistency of the MPL estimate in the well-known $p$-spin Sherrington–Kirkpatrick model, spin systems on general $p$-uniform hypergraphs and Ising models on the hypergraph stochastic block model (HSBM). In fact, for the HSBM we pin down the exact location of the phase transition threshold, which is determined by the positivity of a certain mean-field variational problem, such that above this threshold the MPL estimate is $\sqrt N$-consistent, whereas below the threshold no estimator is consistent. Finally, we derive the precise fluctuations of the MPL estimate in the special case of the $p$-tensor Curie–Weiss model, which is the Ising model on the complete $p$-uniform hypergraph. An interesting consequence of our results is that the MPL estimate in the Curie–Weiss model saturates the Cramer–Rao lower bound at all points above the estimation threshold, that is, the MPL estimate incurs no loss in asymptotic statistical efficiency in the estimability regime, even though it is obtained by minimizing only an approximation of the true likelihood function for computational tractability.

Extended mean-field control problem with partial observation

We study an extended mean-field control problem with partial observation, where the dynamic of the state is given by a forward-backward stochastic differential equation of McKean-Vlasov type. The cost functional, the state and the observation all depend on the joint distribution of the state and the control process. Our problem is motivated by the recent popular subject of mean-field games and related control problems of McKean-Vlasov type. We first establish a necessary condition in the form of Pontryagin’s maximum principle for optimality. Then a verification theorem is obtained for optimal control under some convex conditions of the Hamiltonian function. The results are also applied to studying linear-quadratic mean-filed control problem in the type of scalar interaction.

On the global bifurcation diagram of the Gelfand problem

For domains of first kind [7,13] we describe the qualitative behavior of the global bifurcation diagram of the unbounded branch of solutions of the Gel'fand problem crossing the origin. At least to our knowledge this is the first result about the exact monotonicity of the branch of non-minimal solutions which is not just concerned with radial solutions [28] and/or with symmetric domains [23]. Toward our goal we parametrize the branch not by the $L^{\infty}(\Omega)$-norm of the solutions but by the energy of the associated mean field problem. The proof relies on a carefully modified spectral analysis of mean field type equations.

Quantum computation for predicting electron and phonon properties of solids

Quantum chemistry is one of the most promising near-term applications of quantum computers. Quantum algorithms such as variational quantum eigen solver (VQE) and variational quantum deflation (VQD) algorithms have been mainly applied for molecular systems and there is a need to implement such methods for periodic solids. Using Wannier tight-binding Hamiltonian (WTBH) approaches, we demonstrate the application of VQE and VQD to accurately predict both electronic and phonon bandstructure properties of several elemental as well as multi-component solid-state materials. We apply VQE–VQD calculations for 307 spin–orbit coupling based electronic WTBHs and 933 finite-difference based phonon WTBHs. Also, we discuss a workflow for using VQD with lattice Green’s function that can be used for solving dynamical mean-field theory problems. The WTBH model solvers can be used for testing other quantum algorithms and models also.

Numerical Solution of Mean Field Problem with Limited Management Resource

Numerical Solution of Mean Field Problem with Limited Management Resource

Deep Learning and Mean-Field Games: A Stochastic Optimal Control Perspective

We provide a rigorous mathematical formulation of Deep Learning (DL) methodologies through an in-depth analysis of the learning procedures characterizing Neural Network (NN) models within the theoretical frameworks of Stochastic Optimal Control (SOC) and Mean-Field Games (MFGs). In particular, we show how the supervised learning approach can be translated in terms of a (stochastic) mean-field optimal control problem by applying the Hamilton–Jacobi–Bellman (HJB) approach and the mean-field Pontryagin maximum principle. Our contribution sheds new light on a possible theoretical connection between mean-field problems and DL, melting heterogeneous approaches and reporting the state-of-the-art within such fields to show how the latter different perspectives can be indeed fruitfully unified.

Improved stability for 2D attractive Bose gases

We study the ground-state energy of N attractive bosons in the plane. The interaction is scaled for the gas to be dilute so that the corresponding mean-field problem is a local non-linear Schrödinger (NLS) equation. We improve the conditions under which one can prove that the many-body problem is stable (of the second kind). This implies, using previous results, that the many-body ground states and dynamics converge to the NLS ones for an extended range of diluteness parameters.

Nonadiabatic Quantum Molecular Dynamics in Dense Manifolds of Electronic States.

Most nonadiabatic molecular dynamics methods require the determination of a basis of adiabatic or diabatic electronic states at every time step, but in dense manifolds of electronic states, such approaches become intractable. A notable exception is Ehrenfest molecular dynamics, which can be implemented without explicit determination of such a basis but suffers from unphysical behavior when propagation on a mean-field potential energy surface (PES) does not accurately reflect the true dynamics on multiple electronic states. Here we introduce the multiple cloning for dense manifolds of states (MCDMS) method, a systematically improvable approximation to the multiple cloning method. MCDMS avoids both the mean-field PES problem and the need to compute the full electronic spectrum. This is achieved by reformulating multiple cloning to use a subspace of approximate eigenstates constructed from the time-dependent Ehrenfest electronic wave function. By application to model systems, we show that this approach allows a substantial reduction in the size of the required electronic basis without significant loss in accuracy.

Quantum mean-field asymptotics and multiscale analysis

We study, via multiscale analysis, some defect of compactness phenomena which occur in bosonic and fermionic quantum mean-field problems. The approach relies on a combination of mean-field asymptotics and second microlocalized semiclassical measures. The phase space geometric description is illustrated by various examples.

Exchange and magnetic order in bulk and nanostructured Fe5Si3

The Curie temperature of bulk and nanostructured Fe5Si3 is investigated using experiments, density-functional simulations, and many-body model calculations. The bulk intermetallic, which crystallizes in the hexagonal D88 structure, exhibits several intriguing features: it does not exist as a room-temperature equilibrium phase, is close to the onset of ferromagnetism, and exhibits two crystallographically very different Fe sites. The samples, produced by rapid quenching (bulk) and cluster deposition (nanoparticulate thin films), have Curie temperatures of about 400 K. Interatomic exchange constants are calculated using the Kohn-Korringa-Rostoker (KKR) method and used to solve the multisublattice mean-field problem for the system. The Vienna ab initio simulation package (VASP) is employed to study the dependence of the Fe moment on the thermally induced spin misalignment, and a model calculation yields an estimate for quantum-spin-liquid corrections. The theory includes Heisenberg exchange but overestimates the Curie temperature, and a discussion is given regarding additional approaches to handle weakly ferromagnetic multisublattice intermetallic compounds.

Finite-size effects in the dynamics of few bosons in a ring potential

We study the temporal evolution of a small number N of ultra-cold bosonic atoms confined in a ring potential. Assuming that initially the system is in a solitary-wave solution of the corresponding mean-field problem, we identify significant differences in the time evolution of the density distribution of the atoms when it instead is evaluated with the many-body Schrödinger equation. Three characteristic timescales are derived: the first is the period of rotation of the wave around the ring, the second is associated with a ‘decay’ of the density variation, and the third is associated with periodic ‘collapses’ and ‘revivals’ of the density variations, with a factor of separating each of them. The last two timescales tend to infinity in the appropriate limit of large N, in agreement with the mean-field approximation. These findings are based on the assumption of the initial state being a mean-field state. We confirm this behavior by comparison to the exact solutions for a few-body system stirred by an external potential. We find that the exact solutions of the driven system exhibit similar dynamical features.

Compactness, existence and multiplicity for the singular mean field problem with sign-changing potentials

In this paper we consider a mean field problem on a compact surface without boundary in presence of conical singularities. The corresponding equation, named after Liouville, appears in the Gaussian curvature prescription problem in Geometry, and also in the Electroweak Theory and in the abelian Chern–Simons–Higgs model in Physics. Our contribution focuses on the case of sign-changing potentials, and gives results on compactness, existence and multiplicity of solutions.