Raphaël M. Jungers

Antoine Aspeel, Damien Dasnoy, Raphaël M. Jungers et al.

In radiation therapy, tumor tracking is a challenging task that allows a better dose delivery. One practice is to acquire X-ray images in real-time during treatment, that are used to estimate the tumor location. These informations are used to predict the close future tumor trajectory. Kalman prediction is a classical approach for this task. The main drawback of X-ray acquisition is that it irradiates the patient, including its healthy tissues. In the classical Kalman framework, X-ray measurements are taken regularly, i.e. at a constant rate. In this paper, we propose a new approach which relaxes this constraint in order to take measurements when they are the most useful. Our aim is for a given budget of measurements to optimize the tracking process. This idea naturally brings to an optimal intermittent Kalman predictor for which measurement times are selected to minimize the mean squared prediction error over the complete fraction. This optimization problem can be solved directly when the respiratory model has been identified and the optimal sampling times can be computed at once. These optimal measurement times are obtained by solving a combinatorial optimization problem using a genetic algorithm. We created a test benchmark on trajectories validated on one patient. This new prediction method is compared to the regular Kalman predictor and a relative improvement of 9:8% is observed on the root mean square position estimation error.

Adrien Banse, Licio Romao, Alessandro Abate et al.

We introduce an adaptive refinement procedure for smart, and scalable abstraction of dynamical systems. Our technique relies on partitioning the state space depending on the observation of future outputs. However, this knowledge is dynamically constructed in an adaptive, asymmetric way. In order to learn the optimal structure, we define a Kantorovich-inspired metric between Markov chains, and we use it as a loss function. Our technique is prone to data-driven frameworks, but not restricted to. We also study properties of the above mentioned metric between Markov chains, which we believe could be of application for wider purpose. We propose an algorithm to approximate it, and we show that our method yields a much better computational complexity than using classical linear programming techniques.

Matthew Philippe, Gilles Millerioux, Raphaël M. Jungers

We study computational questions related with the stability of discrete-time linear switching systems with switching sequences constrained by an automaton. We first present a decidable sufficient condition for their boundedness when the maximal exponential growth rate equals one. The condition generalizes the notion of the irreducibility of a matrix set, which is a well known sufficient condition for boundedness in the arbitrary switching (i.e. unconstrained) case. Second, we provide a polynomial time algorithm for deciding the dead-beat stability of a system, i.e. that all trajectories vanish to the origin in finite time. The algorithm generalizes one proposed by Gurvits for arbitrary switching systems, and is illustrated with a real-world case study.

Adrien Banse, Venkatraman Renganathan, Raphaël M. Jungers

We extend the notion of Cantor-Kantorovich distance between Markov chains introduced by (Banse et al., 2023) in the context of Markov Decision Processes (MDPs). The proposed metric is well-defined and can be efficiently approximated given a finite horizon. Then, we provide numerical evidences that the latter metric can lead to interesting applications in the field of reinforcement learning. In particular, we show that it could be used for forecasting the performance of transfer learning algorithms.

Adrien Banse, Alessandro Abate, Raphaël M. Jungers

Labeled Markov Chains (or LMCs for short) are useful mathematical objects to model complex probabilistic languages. A central challenge is to compare two LMCs, for example to assess the accuracy of an abstraction or to quantify the effect of model perturbations. In this work, we study the recently introduced Cantor-Kantorovich (or CK) distance. In particular we show that the latter can be framed as a discounted sum of finite-horizon Total Variation distances, making it an instance of discounted linear distance, but arising from the natural Cantor topology. Building on the latter observation, we analyze the properties of the CK distance along three dimensions: computational complexity, continuity properties and approximation. More precisely, we show that the exact computation of the CK distance is #P-hard. We also provide an upper bound on the CK distance as a function of the approximation relation between the two LMCs, and show that a bounded CK distance implies a bounded error between probabilities of finite-horizon traces. Finally, we provide a computable approximation scheme, and show that the latter is also #P-hard. Altogether, our results provide a rigorous theoretical foundation for the CK distance and clarify its relationship with existing distances.

Guillaume O. Berger, Raphaël M. Jungers

We consider the problem of repetitive scenario design where one has to solve repeatedly a scenario design problem and can adjust the sample size (number of scenarios) to obtain a desired level of risk (constraint violation probability). We propose an approach to learn on the fly the optimal sample size based on observed data consisting in previous scenario solutions and their risk level. Our approach consists in learning a function that represents the pdf (probability density function) of the risk as a function of the sample size. Once this function is known, retrieving the optimal sample size is straightforward. We prove the soundness and convergence of our approach to obtain the optimal sample size for the class of fixed-complexity scenario problems, which generalizes fully-supported convex scenario programs that have been studied extensively in the scenario optimization literature. We also demonstrate the practical efficiency of our approach on a series of challenging repetitive scenario design problems, including non-fixed-complexity problems, nonconvex constraints and time-varying distributions.

Guillaume O. Berger, Raphaël M. Jungers

We investigate the Probably Approximately Correct (PAC) property of scenario decision algorithms, which refers to their ability to produce decisions with an arbitrarily low risk of violating unknown safety constraints, provided a sufficient number of realizations of these constraints are sampled. While several PAC sufficient conditions for such algorithms exist in the literature -- such as the finiteness of the VC dimension of their associated classifiers, or the existence of a compression scheme -- it remains unclear whether these conditions are also necessary. In this work, we demonstrate through counterexamples that these conditions are not necessary in general. These findings stand in contrast to binary classification learning, where analogous conditions are both sufficient and necessary for a family of classifiers to be PAC. Furthermore, we extend our analysis to stable scenario decision algorithms, a broad class that includes practical methods like scenario optimization. Even under this additional assumption, we show that the aforementioned conditions remain unnecessary. Furthermore, we introduce a novel quantity, called the dVC dimension, which serves as an analogue to the VC dimension for scenario decision algorithms. We prove that the finiteness of this dimension is a PAC necessary condition for scenario decision algorithms. This allows to (i) guide algorithm users and designers to recognize algorithms that are not PAC, and (ii) contribute to a comprehensive characterization of PAC scenario decision algorithms.

Cláudio Gomes, Raphaël M. Jungers, Benoît Legat et al.

We propose an algorithm to restrict the switching signals of a constrained switched system in order to guarantee its stability, while at the same time attempting to keep the largest possible set of allowed switching signals. Our work is motivated by applications to (co-)simulation, where numerical stability is a hard constraint, but should be attained by restricting as little as possible the allowed behaviours of the simulators. We apply our results to certify the stability of an adaptive co-simulation orchestration algorithm, which selects the optimal switching signal at run-time, as a function of (varying) performance and accuracy requirements.

Pierre-Yves Chevalier, Julien M. Hendrickx, Raphaël M. Jungers

We consider the problem of determining the existence of a sequence of matrices driving a discrete-time consensus system to consensus. We transform this problem into one of the existence of a product of the transition (stochastic) matrices that has a positive column. We then generalize some results from automata theory to sets of stochastic matrices. We obtain as a main result a polynomial-time algorithm to decide the existence of a sequence of matrices achieving consensus.

Pierre-Yves Chevalier, Julien M. Hendrickx, Raphaël M. Jungers

We address the problem of determining if a discrete time switched consensus system converges for any switching sequence and that of determining if it converges for at least one switching sequence. For these two problems, we provide necessary and sufficient conditions that can be checked in singly exponential time. As a side result, we prove the existence of a polynomial time algorithm for the first problem when the system switches between only two subsystems whose corresponding graphs are undirected, a problem that had been suggested to be NP-hard by Blondel and Olshevsky.