Alexander Davydov

Alexander Davydov, Saber Jafarpour, Matthew Abate et al.

We use interval reachability analysis to obtain robustness guarantees for implicit neural networks (INNs). INNs are a class of implicit learning models that use implicit equations as layers and have been shown to exhibit several notable benefits over traditional deep neural networks. We first establish that tight inclusion functions of neural networks, which provide the tightest rectangular over-approximation of an input-output map, lead to sharper robustness guarantees than the well-studied robustness measures of local Lipschitz constants. Like Lipschitz constants, tight inclusions functions are computationally challenging to obtain, and we thus propose using mixed monotonicity and contraction theory to obtain computationally efficient estimates of tight inclusion functions for INNs. We show that our approach performs at least as well as, and generally better than, applying state-of-the-art interval bound propagation methods to INNs. We design a novel optimization problem for training robust INNs and we provide empirical evidence that suitably-trained INNs can be more robust than comparably-trained feedforward networks.

Saber Jafarpour, Alexander Davydov, Matthew Abate et al.

This paper proposes a theoretical and computational framework for training and robustness verification of implicit neural networks based upon non-Euclidean contraction theory. The basic idea is to cast the robustness analysis of a neural network as a reachability problem and use (i) the $\ell_{\infty}$-norm input-output Lipschitz constant and (ii) the tight inclusion function of the network to over-approximate its reachable sets. First, for a given implicit neural network, we use $\ell_{\infty}$-matrix measures to propose sufficient conditions for its well-posedness, design an iterative algorithm to compute its fixed points, and provide upper bounds for its $\ell_\infty$-norm input-output Lipschitz constant. Second, we introduce a related embedded network and show that the embedded network can be used to provide an $\ell_\infty$-norm box over-approximation of the reachable sets of the original network. Moreover, we use the embedded network to design an iterative algorithm for computing the upper bounds of the original system's tight inclusion function. Third, we use the upper bounds of the Lipschitz constants and the upper bounds of the tight inclusion functions to design two algorithms for the training and robustness verification of implicit neural networks. Finally, we apply our algorithms to train implicit neural networks on the MNIST dataset and compare the robustness of our models with the models trained via existing approaches in the literature.

Akash Harapanahalli, Samuel Coogan, Alexander Davydov

We present a novel framework that jointly trains a neural network controller and a neural Riemannian metric with rigorous closed-loop contraction guarantees using formal bound propagation. Directly bounding the symmetric Riemannian contraction linear matrix inequality causes unnecessary overconservativeness due to poor dependency management. Instead, we analyze an asymmetric matrix function $G$, where $2^n$ GPU-parallelized corner checks of its interval hull verify that an entire interval subset $X$ is a contraction region in a single shot. This eliminates the sample complexity problems encountered with previous Lipschitz-based guarantees. Additionally, for control-affine systems under a Killing field assumption, our method produces an explicit tracking controller capable of exponentially stabilizing any dynamically feasible trajectory using just two forward inferences of the learned policy. Using JAX and $\texttt{immrax}$ for linear bound propagation, we apply this approach to a full 10-state quadrotor model. In under 10 minutes of post-JIT training, we simultaneously learn a control policy $π$, a neural contraction metric $Θ$, and a verified 10-dimensional contraction region $X$.

William Retnaraj, Simone Betteti, Alexander Davydov et al.

Linear-threshold networks (LTNs) capture the mesoscale behavior of interacting populations of neurons and are of particular interest to control theorists due to their dynamical richness and relative ease of analysis. The aim of this paper is to advance the study of global asymptotic stability in LTNs with asymmetric neural interactions and heterogeneous dissipation under the structural Lyapunov diagonal stability (LDS) condition. To this end, we introduce a one-parameter family of LTNs that preserves the LDS condition and has a parameter-independent equilibrium set. In the fast limit, this family converges to a projected dynamical system (PDS), while in the slow limit, it converges to a discontinuous hard-selector system (HSS). Under LDS, we prove that the fast PDS limit is globally exponentially stable and that the HSS limit is globally asymptotically stable. This alignment suggests that the limiting systems capture essential mechanisms governing stability across the entire LTN family. Together with numerical evidence, these findings indicate that resolving stability at the fast and slow endpoints provides a promising and structurally grounded path toward establishing global stability for LTNs with biologically plausible recurrence and diagonal dissipation.

Saber Jafarpour, Alexander Davydov, Anton V. Proskurnikov et al.

Implicit neural networks, a.k.a., deep equilibrium networks, are a class of implicit-depth learning models where function evaluation is performed by solving a fixed point equation. They generalize classic feedforward models and are equivalent to infinite-depth weight-tied feedforward networks. While implicit models show improved accuracy and significant reduction in memory consumption, they can suffer from ill-posedness and convergence instability. This paper provides a new framework, which we call Non-Euclidean Monotone Operator Network (NEMON), to design well-posed and robust implicit neural networks based upon contraction theory for the non-Euclidean norm $\ell_{\infty}$. Our framework includes (i) a novel condition for well-posedness based on one-sided Lipschitz constants, (ii) an average iteration for computing fixed-points, and (iii) explicit estimates on input-output Lipschitz constants. Additionally, we design a training problem with the well-posedness condition and the average iteration as constraints and, to achieve robust models, with the input-output Lipschitz constant as a regularizer. Our $\ell_{\infty}$ well-posedness condition leads to a larger polytopic training search space than existing conditions and our average iteration enjoys accelerated convergence. Finally, we evaluate our framework in image classification through the MNIST and the CIFAR-10 datasets. Our numerical results demonstrate improved accuracy and robustness of the implicit models with smaller input-output Lipschitz bounds. Code is available at https://github.com/davydovalexander/Non-Euclidean_Mon_Op_Net.

Sean Jaffe, Alexander Davydov, Deniz Lapsekili et al.

Global stability and robustness guarantees in learned dynamical systems are essential to ensure well-behavedness of the systems in the face of uncertainty. We present Extended Linearized Contracting Dynamics (ELCD), the first neural network-based dynamical system with global contractivity guarantees in arbitrary metrics. The key feature of ELCD is a parametrization of the extended linearization of the nonlinear vector field. In its most basic form, ELCD is guaranteed to be (i) globally exponentially stable, (ii) equilibrium contracting, and (iii) globally contracting with respect to some metric. To allow for contraction with respect to more general metrics in the data space, we train diffeomorphisms between the data space and a latent space and enforce contractivity in the latent space, which ensures global contractivity in the data space. We demonstrate the performance of ELCD on the high dimensional LASA, multi-link pendulum, and Rosenbrock datasets.

Alexander Davydov, Franck Djeumou, Marcus Greiff et al.

Combining data-driven models that adapt online and model predictive control (MPC) has enabled effective control of nonlinear systems. However, when deployed on unstable systems, online adaptation may not be fast enough to ensure reliable simultaneous learning and control. For example, a controller on a vehicle executing highly dynamic maneuvers--such as drifting to avoid an obstacle--may push the vehicle's tires to their friction limits, destabilizing the vehicle and allowing modeling errors to quickly compound and cause a loss of control. To address this challenge, we present an active information gathering framework for identifying vehicle dynamics as quickly as possible. We propose an expressive vehicle dynamics model that leverages Bayesian last-layer meta-learning to enable rapid online adaptation. The model's uncertainty estimates are used to guide informative data collection and quickly improve the model prior to deployment. Dynamic drifting experiments on a Toyota Supra show that (i) the framework enables reliable control of a vehicle at the edge of stability, (ii) online adaptation alone may not suffice for zero-shot control and can lead to undesirable transient errors or spin-outs, and (iii) active data collection helps achieve reliable performance.

Alexander Davydov

We address the problem of verifying closed-loop contraction in nonlinear control systems whose controller and contraction metric are both parameterized by neural networks. By leveraging interval analysis and interval bound propagation, we derive a tractable and scalable sufficient condition for closed-loop contractivity that reduces to checking that the dominant eigenvalue of a symmetric Metzler matrix is nonpositive. We combine this sufficient condition with a domain partitioning strategy to integrate this sufficient condition into training. The proposed approach is validated on an inverted pendulum system, demonstrating the ability to learn neural network controllers and contraction metrics that provably satisfy the contraction condition.

Saber Jafarpour, Matthew Abate, Alexander Davydov et al.

Implicit neural networks are a general class of learning models that replace the layers in traditional feedforward models with implicit algebraic equations. Compared to traditional learning models, implicit networks offer competitive performance and reduced memory consumption. However, they can remain brittle with respect to input adversarial perturbations. This paper proposes a theoretical and computational framework for robustness verification of implicit neural networks; our framework blends together mixed monotone systems theory and contraction theory. First, given an implicit neural network, we introduce a related embedded network and show that, given an $\ell_\infty$-norm box constraint on the input, the embedded network provides an $\ell_\infty$-norm box overapproximation for the output of the given network. Second, using $\ell_{\infty}$-matrix measures, we propose sufficient conditions for well-posedness of both the original and embedded system and design an iterative algorithm to compute the $\ell_{\infty}$-norm box robustness margins for reachability and classification problems. Third, of independent value, we propose a novel relative classifier variable that leads to tighter bounds on the certified adversarial robustness in classification problems. Finally, we perform numerical simulations on a Non-Euclidean Monotone Operator Network (NEMON) trained on the MNIST dataset. In these simulations, we compare the accuracy and run time of our mixed monotone contractive approach with the existing robustness verification approaches in the literature for estimating the certified adversarial robustness.