Ziad Kobeissi

Research

My research interests covers:

Partial Differential Equations
Modelling and Numerical Methods
Optimal Control
Mean Field Games

Machine Learning
Optimization
Reinforcement Learning
Deep Neural Networks

Probability and Statistics
Stochastic Differential Equations
Stochastic Algorithms
Markov Decision Processes

See my publications and preprints on my Google Scholar and arXiv pages, or in the following list:
*: the names of the authors are in alphabetic order.

The tragedy of the commons: A Mean-Field Game approach to the reversal of travelling
Ziad Kobeissi*, Idriss Mazari-Fouquet* and Domenec Ruiz-Balet*, 2023
Preprint
```
@article{kobeissi2023tragedy,
  title={The tragedy of the commons: A Mean-Field Game approach to the reversal of travelling waves},
  author={Kobeissi, Ziad and Mazari-Fouquet, Idriss and Ruiz-Balet, Domenec},
  journal={arXiv preprint arXiv:2303.01365},
  year={2023}
}
```
The goal of this paper is to investigate an instance of the tragedy of the commons in spatially distributed harvesting games. The model we choose is that of a fishes' population that is governed by a parabolic bistable equation and that fishermen harvest. We assume that, when no fisherman is present, the fishes' population is invading (mathematically, there is an invading travelling front). Is it possible that fishermen, when acting selfishly, each in his or her own best interest, might lead to a reversal of the travelling wave and, consequently, to an extinction of the global population? To answer this question, we model the behaviour of individual fishermen using a Mean Field Game approach, and we show that the answer is yes. We then show that, at least in some cases, if the fishermen coordinated instead of acting selfishly, each of them could make more benefit, while still guaranteeing the survival of the population. Our study is illustrated by several numerical simulations.
Temporal Difference Learning with Continuous Time and State in the Stochastic Setting
Ziad Kobeissi and Francis Bach, 2022
Preprint
```
@inproceedings{kobeissi2023temporal,   
title = {Temporal Difference Learning with Continuous Time and State in the Stochastic Setting},
author = {Kobeissi, Ziad and Bach, Francis},
booktitle={arXiv},
year = {2023},
}
```
We consider the problem of continuous-time policy evaluation. This consists in learning through observations the value function associated to an uncontrolled continuous-time stochastic dynamic and a reward function. We propose two original variants of the well-known TD(0) method using vanishing time steps. One is model-free and the other is model-based. For both methods, we prove theoretical convergence rates that we subsequently verify through numerical simulations. Alternatively, those methods can be interpreted as novel reinforcement learning approaches for approximating solutions of linear PDEs (partial differential equations) or linear BSDEs (backward stochastic differential equations).
A Non-asymptotic Analysis of Non-parametric Temporal-Difference Learning
Eloïse Berthier, Ziad Kobeissi and Francis Bach, 2022
Neurips spotlight 2022
```
@inproceedings{berthier2022non,
title={A Non-asymptotic Analysis of Non-parametric Temporal-Difference Learning},
author={Berthier, Elo{\"\i}se and Kobeissi, Ziad and Bach, Francis},
booktitle = {Advances in Neural Information Processing Systems},
year={2022}
}
```
Temporal-difference learning is a popular algorithm for policy evaluation. In this paper, we study the convergence of the regularized non-parametric TD(0) algorithm, in both the independent and Markovian observation settings. In particular, when TD is performed in a universal reproducing kernel Hilbert space (RKHS), we prove convergence of the averaged iterates to the optimal value function, even when it does not belong to the RKHS. We provide explicit convergence rates that depend on a source condition relating the regularity of the optimal value function to the RKHS. We illustrate this convergence numerically on a simple continuous-state Markov reward process.
Mean field games with monotonous interactions through the law of states and controls of the agents
Ziad Kobeissi, 2022
NoDEA: Nonlinear Differential Equations and Applications
```
@article{kobeissi2022mean,
title={Mean field games with monotonous interactions through the law of states and controls of the agents},  
author={Kobeissi, Ziad},  
journal={Nonlinear Differential Equations and Applications NoDEA},
volume={29},
number={5}, 
pages={52},
year={2022},
publisher={Springer}
}
```
We consider a class of Mean Field Games in which the agents may interact through the statistical distribution of their states and controls. It is supposed that the Hamiltonian behaves like a power of its arguments as they tend to infinity, with an exponent larger than one. A monotonicity assumption is also made. Existence and uniqueness are proved using a priori estimates which stem from the monotonicity assumptions and Leray–Schauder theorem. Applications of the results are given.
On classical solutions to the mean field game system of controls
Ziad Kobeissi, 2021
CPDE: Communications in Partial Differential Equations
```
@article{kobeissi2022classical,
title={On classical solutions to the mean field game system of controls},
author={Kobeissi, Ziad}, 
journal={Communications in Partial Differential Equations},
volume={47},
number={3}, 
pages={453--488},
year={2022},
publisher={Taylor \& Francis}
}
```
We consider a class of mean field games in which the optimal strat- egy of a representative agent depends on the statistical distribution of both the states and controls. We prove some existence results for the forward–backward system of PDEs in a regime never considered so far, where agents may somehow favor a velocity close to the average one. The main step of the proof consists of obtaining a priori estimates on the gradient of the value function by Bernstein’s method. Uniqueness is also proved under more restrictive assumptions. Finally, we discuss some examples to which the previously mentioned results apply.
Mean field games of controls: Finite difference approximations
Yves Achdou* and Ziad Kobeissi*, 2020
AIMS: Mathematics in Engineering; special issue dedicated to Italo Capuzzo Dolcetta
```
@Article{achdou2021mean,
title = {Mean field games of controls: Finite difference approximations},
journal = {Mathematics in Engineering},
volume = {3},
number = {3},
pages = {1-35},
year = {2021},
issn = {2640-3501},
doi = {10.3934/mine.2021024},
url = {https://www.aimspress.com/article/doi/10.3934/mine.2021024},
author = {Yves Achdou and Ziad Kobeissi},
keywords = {mean field games, interactions via controls, crowd motion, numerical simulations, finite difference method},
}
```
We consider a class of mean field games in which the agents interact through both their states and controls, and we focus on situations in which a generic agent tries to adjust her speed (control) to an average speed (the average is made in a neighborhood in the state space). In such cases, the monotonicity assumptions that are frequently made in the theory of mean field games do not hold, and uniqueness cannot be expected in general. Such model lead to systems of forward-backward nonlinear nonlocal parabolic equations; the latter are supplemented with various kinds of boundary conditions, in particular Neumann-like boundary conditions stemming from reflection conditions on the underlying controled stochastic processes. The present work deals with numerical approximations of the above megntioned systems. After describing the finite difference scheme, we propose an iterative method for solving the systems of nonlinear equations that arise in the discrete setting; it combines a continuation method, Newton iterations and inner loops of a bigradient like solver. The numerical method is used for simulating two examples. We also make experiments on the behaviour of the iterative algorithm when the parameters of the model vary.
On the implementation of a primal-dual algorithm for second order time-dependent mean field games with local couplings
L. Briceno-Arias*, D. Kalise*, Z. Kobeissi*, M. Laurière*, A. Mateos Gonzalez* and F. J. Silva*, 2019
ESAIM: Proceedings and Surveys
```
@article{briceno2019implementation,
title={On the implementation of a primal-dual algorithm for second order time-dependent mean field games with local couplings},
author={Briceno-Arias, Luis and Kalise, Dante and Kobeissi, Ziad and Lauriere, Mathieu and Gonz{\'a}lez, A Mateos and Silva, Francisco J},
journal={ESAIM: Proceedings and Surveys},
volume={65},
pages={330--348},
year={2019},
publisher={EDP Sciences}
}
```
We study a numerical approximation of a time-dependent Mean Field Game (MFG) system with local couplings. The discretization we consider stems from a variational approach described in [14] for the stationary problem and leads to the finite difference scheme introduced by Achdou and Capuzzo-Dolcetta in [3]. In order to solve the finite dimensional variational problems, in [14] the authors implement the primal-dual algorithm introduced by Chambolle and Pock in [20], whose core consists in iteratively solving linear systems and applying a proximity operator. We apply that method to time-dependent MFG and, for large viscosity parameters, we improve the linear system solution by replacing the direct approach used in [14] by suitable preconditioned iterative algorithms.
PhD Thesis: Some contributions to the theory of mean field games
Ziad Kobeissi
```
@phdthesis{kobeissi:tel-03274530,
  TITLE = {{Contributions to the theory of mean field games}},
  AUTHOR = {Kobeissi, Ziad},
  URL = {https://theses.hal.science/tel-03274530},
  NUMBER = {2020UNIP7150},
  SCHOOL = {{Universit{\'e} Paris Cit{\'e}}},
  YEAR = {2020},
  MONTH = Oct,
  KEYWORDS = {Mean Field Games ; System of partial differential equations ; Numerical simulations ; Jeux {\`a} champ moyen ; Syst{\`e}me d'{\'e}quations aux d{\'e}riv{\'e}es partielles ; Simulations num{\'e}riques},
  TYPE = {Theses},
  PDF = {https://theses.hal.science/tel-03274530/file/KOBEISSI_Ziad_va.pdf},
  HAL_ID = {tel-03274530},
  HAL_VERSION = {v1},
}
```
This thesis deals with the theory of mean field games (MFG for short). The main part is dedicated to a class of games in which agents may interact through their law of states and controls; we use the terminology mean field games of controls (MFGC for short) to refer to this class of games. First, we assume that the optimal dynamics depends upon the law of controls in a Lipschitz way, with a Lipchitz constant smaller than one. In this case, we give several existence results on the solutions of the MFGC system, and one uniqueness result under a short-time horizon assumption. Second, we introduce a scheme and make simulations for a model of crowd motion. Thrid, under a monotonicity assumption on the interactions through the law of controls, we prove existence and uniqueness of the solution of the MFGC system. Finally, we introduce an algorithm for solving MFG systems of variational type, we use a preconditioned strategy based on a multigrid method.

Ziad Kobeissi

Postdoctoral researcher, Inria Paris and ILB

Research