Kostas Margellos

Filiberto Fele, Kostas Margellos

We consider a multi-agent noncooperative game with agents' objective functions being affected by uncertainty. Following a data driven paradigm, we represent uncertainty by means of scenarios and seek a robust Nash equilibrium solution. We treat the Nash equilibrium computation problem within the realm of probably approximately correct (PAC) learning. Building upon recent developments in scenario-based optimization, we accompany the computed Nash equilibrium with a priori and a posteriori probabilistic robustness certificates, providing confidence that the computed equilibrium remains unaffected (in probabilistic terms) when a new uncertainty realization is encountered. For a wide class of games, we also show that the computation of the so called compression set - a key concept in scenario-based optimization - can be directly obtained as a byproduct of the proposed solution methodology. Finally, we illustrate how to overcome differentiability issues, arising due to the introduction of scenarios, and compute a Nash equilibrium solution in a decentralized manner. We demonstrate the efficacy of the proposed approach on an electric vehicle charging control problem.

Nikolaus Vertovec, Kostas Margellos, Maria Prandini

We consider a moving target that we seek to learn from samples. Our results extend randomized techniques developed in control and optimization for a constant target to the case where the target is changing. We derive a novel bound on the number of samples that are required to construct a probably approximately correct (PAC) estimate of the target. Furthermore, when the moving target is a convex polytope, we provide a constructive method of generating the PAC estimate using a mixed integer linear program (MILP). The proposed method is demonstrated on an application to autonomous emergency braking.

Nikolaus Vertovec, Sina Ober-Blöbaum, Kostas Margellos

Hamilton-Jacobi reachability methods for safety-critical control have been well studied, but the safety guarantees derived rely on the accuracy of the numerical computation. Thus, it is crucial to understand and account for any inaccuracies that occur due to uncertainty in the underlying dynamics and environment as well as the induced numerical errors. To this end, we propose a framework for modeling the error of the value function inherent in Hamilton-Jacobi reachability using a Gaussian process. The derived safety controller can be used in conjuncture with arbitrary controllers to provide a safe hybrid control law. The marginal likelihood of the Gaussian process then provides a confidence metric used to determine switches between a least restrictive controller and a safety controller. We test both the prediction as well as the correction capabilities of the presented method in a classical pursuit-evasion example.

Junaid Ahmed Memon, Allan Andre Do Nascimento, Kostas Margellos et al.

This paper presents a prototyping framework for distributed control of multi-robot systems, aimed at bridging theory and practical testing of distributed optimization algorithms. Using the Single Program, Multiple Data (SPMD) paradigm, the framework emulates distributed control on a single computer, with each core running the same algorithm using local states and neighbour-to-neighbour communication. We demonstrate the framework on a four-quadrotor position-swapping task using a non-cooperative game-theoretic distributed algorithm. Computational time and trajectory data are compared across the supported dynamics levels: a point-mass model, a high-fidelity quadrotor model, and an experimental hardware testbed using Crazyflie quadcopters. The results show that the framework provides a low-cost and accessible approach for validating distributed algorithms.

Niall O'Sullivan, Licio Romao, Kostas Margellos

Conformal prediction and scenario optimization constitute two important classes of statistical learning frameworks to certify decisions made using data. They have found numerous applications in control theory, machine learning and robotics. Despite intense research in both areas, and apparently similar results, a clear connection between these two frameworks has not been established. By focusing on the so-called vanilla conformal prediction, we show rigorously how to choose appropriate score functions and set predictor map to recover well-known bounds on the probability of constraint violation associated with scenario programs. We also show how to treat ranking of nonconformity scores as a one-dimensional scenario program with discarded constraints, and use such connection to recover vanilla conformal prediction guarantees on the validity of the set predictor. We also capitalize on the main developments of the scenario approach, and show how we could analyze calibration conditional conformal prediction under this lens. Our results establish a theoretical bridge between conformal prediction and scenario optimization.

André Bertolace, Konstatinos Gatsis, Kostas Margellos

Decision making and learning in the presence of uncertainty has attracted significant attention in view of the increasing need to achieve robust and reliable operations. In the case where uncertainty stems from the presence of adversarial attacks this need is becoming more prominent. In this paper we focus on linear and nonlinear classification problems and propose a novel adversarial training method for robust classifiers, inspired by Support Vector Machine (SVM) margins. We view robustness under a data driven lens, and derive finite sample complexity bounds for both linear and non-linear classifiers in binary and multi-class scenarios. Notably, our bounds match natural classifiers' complexity. Our algorithm minimizes a worst-case surrogate loss using Linear Programming (LP) and Second Order Cone Programming (SOCP) for linear and non-linear models. Numerical experiments on the benchmark MNIST and CIFAR10 datasets show our approach's comparable performance to state-of-the-art methods, without needing adversarial examples during training. Our work offers a comprehensive framework for enhancing binary linear and non-linear classifier robustness, embedding robustness in learning under the presence of adversaries.

Nikolaos Kariotoglou, Kostas Margellos, John Lygeros

We discuss the computational complexity and feasibility properties of scenario based techniques for uncertain optimization programs. We consider different solution alternatives ranging from the standard scenario approach to recursive variants, and compare feasibility as a function of the total computation burden. We identify trade-offs between the different methods depending on the problem structure and the desired probability of constraint satisfaction. Our motivation for this work stems from the applicability and complexity reduction when making decisions by means of recursive algorithms. We illustrate our results on an example from the area of approximate dynamic programming