Maximum likelihood inference for high-dimensional problems with multiaffine variable relations

Maximum Likelihood Estimation of continuous variable models can be very challenging in high dimensions, due to potentially complex probability distributions. The existence of multiple interdependencies among variables can make it very difficult to establish convergence guarantees. This leads to a wide use of brute-force methods, such as grid searching and Monte-Carlo sampling and, when applicable, complex and problem-specific algorithms. In this paper, we consider inference problems where the variables are related by multiaffine expressions. We propose a novel Alternating and Iteratively-Reweighted Least Squares (AIRLS) algorithm, and prove its convergence for problems with Generalized Normal Distributions. We also provide an efficient method to compute the variance of the estimates obtained using AIRLS. Finally, we show how the method can be applied to graphical statistical models. We perform numerical experiments on several inference problems, showing significantly better performance than state-of-the-art approaches in terms of scalability, robustness to noise, and convergence speed due to an empirically observed super-linear convergence rate.

Learning to Boost the Performance of Stable Nonlinear Systems

The growing scale and complexity of safety-critical control systems underscore the need to evolve current control architectures aiming for the unparalleled performances achievable through state-of-the-art optimization and machine learning algorithms. However, maintaining closed-loop stability while boosting the performance of nonlinear control systems using data-driven and deep-learning approaches stands as an important unsolved challenge. In this paper, we tackle the performance-boosting problem with closed-loop stability guarantees. Specifically, we establish a synergy between the Internal Model Control (IMC) principle for nonlinear systems and state-of-the-art unconstrained optimization approaches for learning stable dynamics. Our methods enable learning over arbitrarily deep neural network classes of performance-boosting controllers for stable nonlinear systems; crucially, we guarantee Lp closed-loop stability even if optimization is halted prematurely, and even when the ground-truth dynamics are unknown, with vanishing conservatism in the class of stabilizing policies as the model uncertainty is reduced to zero. We discuss the implementation details of the proposed control schemes, including distributed ones, along with the corresponding optimization procedures, demonstrating the potential of freely shaping the cost functions through several numerical experiments.

Unconstrained Parametrization of Dissipative and Contracting Neural Ordinary Differential Equations

In this work, we introduce and study a class of Deep Neural Networks (DNNs) in continuous-time. The proposed architecture stems from the combination of Neural Ordinary Differential Equations (Neural ODEs) with the model structure of recently introduced Recurrent Equilibrium Networks (RENs). We show how to endow our proposed NodeRENs with contractivity and dissipativity -- crucial properties for robust learning and control. Most importantly, as for RENs, we derive parametrizations of contractive and dissipative NodeRENs which are unconstrained, hence enabling their learning for a large number of parameters. We validate the properties of NodeRENs, including the possibility of handling irregularly sampled data, in a case study in nonlinear system identification.

Universal Approximation Property of Hamiltonian Deep Neural Networks

This paper investigates the universal approximation capabilities of Hamiltonian Deep Neural Networks (HDNNs) that arise from the discretization of Hamiltonian Neural Ordinary Differential Equations. Recently, it has been shown that HDNNs enjoy, by design, non-vanishing gradients, which provide numerical stability during training. However, although HDNNs have demonstrated state-of-the-art performance in several applications, a comprehensive study to quantify their expressivity is missing. In this regard, we provide a universal approximation theorem for HDNNs and prove that a portion of the flow of HDNNs can approximate arbitrary well any continuous function over a compact domain. This result provides a solid theoretical foundation for the practical use of HDNNs.

Follow the Clairvoyant: an Imitation Learning Approach to Optimal Control

We consider control of dynamical systems through the lens of competitive analysis. Most prior work in this area focuses on minimizing regret, that is, the loss relative to an ideal clairvoyant policy that has noncausal access to past, present, and future disturbances. Motivated by the observation that the optimal cost only provides coarse information about the ideal closed-loop behavior, we instead propose directly minimizing the tracking error relative to the optimal trajectories in hindsight, i.e., imitating the clairvoyant policy. By embracing a system level perspective, we present an efficient optimization-based approach for computing follow-the-clairvoyant (FTC) safe controllers. We prove that these attain minimal regret if no constraints are imposed on the noncausal benchmark. In addition, we present numerical experiments to show that our policy retains the hallmark of competitive algorithms of interpolating between classical $\mathcal{H}_2$ and $\mathcal{H}_\infty$ control laws - while consistently outperforming regret minimization methods in constrained scenarios thanks to the superior ability to chase the clairvoyant.

Robust online joint state/input/parameter estimation of linear systems

This paper presents a method for jointly estimating the state, input, and parameters of linear systems in an online fashion. The method is specially designed for measurements that are corrupted with non-Gaussian noise or outliers, which are commonly found in engineering applications. In particular, it combines recursive, alternating, and iteratively-reweighted least squares into a single, one-step algorithm, which solves the estimation problem online and benefits from the robustness of least-deviation regression methods. The convergence of the iterative method is formally guaranteed. Numerical experiments show the good performance of the estimation algorithm in presence of outliers and in comparison to state-of-the-art methods.

Neural System Level Synthesis: Learning over All Stabilizing Policies for Nonlinear Systems

We address the problem of designing stabilizing control policies for nonlinear systems in discrete-time, while minimizing an arbitrary cost function. When the system is linear and the cost is convex, the System Level Synthesis (SLS) approach offers an exact solution based on convex programming. Beyond this case, a globally optimal solution cannot be found in a tractable way, in general. In this paper, we develop a parametrization of all and only the control policies stabilizing a given time-varying nonlinear system in terms of the combined effect of 1) a strongly stabilizing base controller and 2) a stable SLS operator to be freely designed. Based on this result, we propose a Neural SLS (Neur-SLS) approach guaranteeing closed-loop stability during and after parameter optimization, without requiring any constraints to be satisfied. We exploit recent Deep Neural Network (DNN) models based on Recurrent Equilibrium Networks (RENs) to learn over a rich class of nonlinear stable operators, and demonstrate the effectiveness of the proposed approach in numerical examples.

On Robust Classification using Contractive Hamiltonian Neural ODEs

Deep neural networks can be fragile and sensitive to small input perturbations that might cause a significant change in the output. In this paper, we employ contraction theory to improve the robustness of neural ODEs (NODEs). A dynamical system is contractive if all solutions with different initial conditions converge to each other asymptotically. As a consequence, perturbations in initial conditions become less and less relevant over time. Since in NODEs, the input data corresponds to the initial condition of dynamical systems, we show contractivity can mitigate the effect of input perturbations. More precisely, inspired by NODEs with Hamiltonian dynamics, we propose a class of contractive Hamiltonian NODEs (CH-NODEs). By properly tuning a scalar parameter, CH-NODEs ensure contractivity by design and can be trained using standard backpropagation and gradient descent algorithms. Moreover, CH-NODEs enjoy built-in guarantees of non-exploding gradients, which ensures a well-posed training process. Finally, we demonstrate the robustness of CH-NODEs on the MNIST image classification problem with noisy test datasets.

Distributed neural network control with dependability guarantees: a compositional port-Hamiltonian approach

Large-scale cyber-physical systems require that control policies are distributed, that is, that they only rely on local real-time measurements and communication with neighboring agents. Optimal Distributed Control (ODC) problems are, however, highly intractable even in seemingly simple cases. Recent work has thus proposed training Neural Network (NN) distributed controllers. A main challenge of NN controllers is that they are not dependable during and after training, that is, the closed-loop system may be unstable, and the training may fail due to vanishing and exploding gradients. In this paper, we address these issues for networks of nonlinear port-Hamiltonian (pH) systems, whose modeling power ranges from energy systems to non-holonomic vehicles and chemical reactions. Specifically, we embrace the compositional properties of pH systems to characterize deep Hamiltonian control policies with built-in closed-loop stability guarantees, irrespective of the interconnection topology and the chosen NN parameters. Furthermore, our setup enables leveraging recent results on well-behaved neural ODEs to prevent the phenomenon of vanishing gradients by design. Numerical experiments corroborate the dependability of the proposed architecture, while matching the performance of general neural network policies.

Bayesian Error-in-Variables Models for the Identification of Power Networks

The increasing integration of intermittent renewable generation, especially at the distribution level,necessitates advanced planning and optimisation methodologies contingent on the knowledge of thegrid, specifically the admittance matrix capturing the topology and line parameters of an electricnetwork. However, a reliable estimate of the admittance matrix may either be missing or quicklybecome obsolete for temporally varying grids. In this work, we propose a data-driven identificationmethod utilising voltage and current measurements collected from micro-PMUs. More precisely,we first present a maximum likelihood approach and then move towards a Bayesian framework,leveraging the principles of maximum a posteriori estimation. In contrast with most existing con-tributions, our approach not only factors in measurement noise on both voltage and current data,but is also capable of exploiting available a priori information such as sparsity patterns and knownline parameters. Simulations conducted on benchmark cases demonstrate that, compared to otheralgorithms, our method can achieve significantly greater accuracy.