Johan Peralez

Johan Peralez, Aurèlien Delage, Jacopo Castellini et al.

The centralized training for decentralized execution paradigm emerged as the state-of-the-art approach to $ε$-optimally solving decentralized partially observable Markov decision processes. However, scalability remains a significant issue. This paper presents a novel and more scalable alternative, namely the sequential-move centralized training for decentralized execution. This paradigm further pushes the applicability of the Bellman's principle of optimality, raising three new properties. First, it allows a central planner to reason upon sufficient sequential-move statistics instead of prior simultaneous-move ones. Next, it proves that $ε$-optimal value functions are piecewise linear and convex in such sufficient sequential-move statistics. Finally, it drops the complexity of the backup operators from double exponential to polynomial at the expense of longer planning horizons. Besides, it makes it easy to use single-agent methods, e.g., SARSA algorithm enhanced with these findings, while still preserving convergence guarantees. Experiments on two- as well as many-agent domains from the literature against $ε$-optimal simultaneous-move solvers confirm the superiority of our novel approach. This paradigm opens the door for efficient planning and reinforcement learning methods for multi-agent systems.

Rafael F. Cunha, Jacopo Castellini, Johan Peralez et al.

Multi-agent planning and reinforcement learning can be challenging when agents cannot see the state of the world or communicate with each other due to communication costs, latency, or noise. Partially Observable Stochastic Games (POSGs) provide a mathematical framework for modelling such scenarios. This paper aims to improve the efficiency of planning and reinforcement learning algorithms for POSGs by identifying the underlying structure of optimal state-value functions. The approach involves reformulating the original game from the perspective of a trusted third party who plans on behalf of the agents simultaneously. From this viewpoint, the original POSGs can be viewed as Markov games where states are occupancy states, \ie posterior probability distributions over the hidden states of the world and the stream of actions and observations that agents have experienced so far. This study mainly proves that the optimal state-value function is a convex function of occupancy states expressed on an appropriate basis in all zero-sum, common-payoff, and Stackelberg POSGs.

Abderrahim Bendahi, Adrien Fradin, Johan Peralez et al.

Neural operators have emerged as a powerful, discretization-invariant framework for solving partial differential equations (PDEs). Although established approaches like the Deep Operator Network (DeepONet) have successfully achieved universal approximation for operators, and architectures such as Fourier Neural Operators (FNOs) have shown algebraic convergence rates, a precise theoretical connection between the continuous theory and its discrete numerical implementation remains a challenge. Specifically, the relationship between the continuous formulation and the discrete numerical stability has yet to be fully explored. In this paper, we address this gap by establishing theoretical guarantees for the discretization error and stability of neural operator approximation schemes. We prove analytical bounds that link solution regularity to input discretization, providing a formal quantification of neural operator accuracy under real-world numerical constraints. We derive these bounds to the specific cases of State Space Model-based Neural Operators (SS-NOs) and FNOs, thus providing a new discretization error theorem for these models. Additionally, through an input-to-state stability (ISS) analysis, we formally assess the impact of discretization on the stability of SS-NOs results obtained in the continuous domain. Our empirical experiments on 1D and 2D benchmarks validate our theoretical bounds and show the robustness of SS-NOs under varying resolutions.

Clara Lucía Galimberti, Johan Peralez, Daniele Astolfi et al.

The Kazantzis-Kravaris-Luenberger (KKL) observer provides a general framework for nonlinear state estimation by immersing the system dynamics into a stable linear or nonlinear latent dynamics. However, the performance of KKL observers relies heavily on the specific choice of these latent dynamics, which is often heuristic. This paper proposes a methodology to learn a KKL observer that combines global stability guarantees with local optimality. We derive a condition on the latent dynamics such that the observer locally mimics the behavior of a Minimum Energy Estimator (Mortensen observer). We then employ Deep Learning to approximate the KKL transformation and the latent dynamics, using neural network architectures that structurally enforce the contraction property. The proposed strategy is validated through numerical simulations on nonlinear benchmarks, demonstrating a good performance in the presence of state and measurement noise.

Johan Peralez, Aurélien Delage, Olivier Buffet et al.

A recent theory shows that a multi-player decentralized partially observable Markov decision process can be transformed into an equivalent single-player game, enabling the application of \citeauthor{bellman}'s principle of optimality to solve the single-player game by breaking it down into single-stage subgames. However, this approach entangles the decision variables of all players at each single-stage subgame, resulting in backups with a double-exponential complexity. This paper demonstrates how to disentangle these decision variables while maintaining optimality under hierarchical information sharing, a prominent management style in our society. To achieve this, we apply the principle of optimality to solve any single-stage subgame by breaking it down further into smaller subgames, enabling us to make single-player decisions at a time. Our approach reveals that extensive-form games always exist with solutions to a single-stage subgame, significantly reducing time complexity. Our experimental results show that the algorithms leveraging these findings can scale up to much larger multi-player games without compromising optimality.