Monotone, Bi-Lipschitz, and Polyak-Lojasiewicz Networks

This paper presents a new \emph{bi-Lipschitz} invertible neural network, the BiLipNet, which has the ability to control both its \emph{Lipschitzness} (output sensitivity to input perturbations) and \emph{inverse Lipschitzness} (input distinguishability from different outputs). The main contribution is a novel invertible residual layer with certified strong monotonicity and Lipschitzness, which we compose with orthogonal layers to build bi-Lipschitz networks. The certification is based on incremental quadratic constraints, which achieves much tighter bounds compared to spectral normalization. Moreover, we formulate the model inverse calculation as a three-operator splitting problem, for which fast algorithms are known. Based on the proposed bi-Lipschitz network, we introduce a new scalar-output network, the PLNet, which satisfies the Polyak-\L{}ojasiewicz condition. It can be applied to learn non-convex surrogate losses with favourable properties, e.g., a unique and efficiently-computable global minimum.

RobustNeuralNetworks.jl: a Package for Machine Learning and Data-Driven Control with Certified Robustness

Neural networks are typically sensitive to small input perturbations, leading to unexpected or brittle behaviour. We present RobustNeuralNetworks.jl: a Julia package for neural network models that are constructed to naturally satisfy a set of user-defined robustness constraints. The package is based on the recently proposed Recurrent Equilibrium Network (REN) and Lipschitz-Bounded Deep Network (LBDN) model classes, and is designed to interface directly with Julia's most widely-used machine learning package, Flux.jl. We discuss the theory behind our model parameterization, give an overview of the package, and provide a tutorial demonstrating its use in image classification, reinforcement learning, and nonlinear state-observer design.

Learning Over All Contracting and Lipschitz Closed-Loops for Partially-Observed Nonlinear Systems

This paper presents a policy parameterization for learning-based control on nonlinear, partially-observed dynamical systems. The parameterization is based on a nonlinear version of the Youla parameterization and the recently proposed Recurrent Equilibrium Network (REN) class of models. We prove that the resulting Youla-REN parameterization automatically satisfies stability (contraction) and user-tunable robustness (Lipschitz) conditions on the closed-loop system. This means it can be used for safe learning-based control with no additional constraints or projections required to enforce stability or robustness. We test the new policy class in simulation on two reinforcement learning tasks: 1) magnetic suspension, and 2) inverting a rotary-arm pendulum. We find that the Youla-REN performs similarly to existing learning-based and optimal control methods while also ensuring stability and exhibiting improved robustness to adversarial disturbances.

Learning Stable and Robust Linear Parameter-Varying State-Space Models

This paper presents two direct parameterizations of stable and robust linear parameter-varying state-space (LPV-SS) models. The model parametrizations guarantee a priori that for all parameter values during training, the allowed models are stable in the contraction sense or have their Lipschitz constant bounded by a user-defined value $\gamma$. Furthermore, since the parametrizations are direct, the models can be trained using unconstrained optimization. The fact that the trained models are of the LPV-SS class makes them useful for, e.g., further convex analysis or controller design. The effectiveness of the approach is demonstrated on an LPV identification problem.

Lipschitz-bounded 1D convolutional neural networks using the Cayley transform and the controllability Gramian

We establish a layer-wise parameterization for 1D convolutional neural networks (CNNs) with built-in end-to-end robustness guarantees. Herein, we use the Lipschitz constant of the input-output mapping characterized by a CNN as a robustness measure. We base our parameterization on the Cayley transform that parameterizes orthogonal matrices and the controllability Gramian for the state space representation of the convolutional layers. The proposed parameterization by design fulfills linear matrix inequalities that are sufficient for Lipschitz continuity of the CNN, which further enables unconstrained training of Lipschitz-bounded 1D CNNs. Finally, we train Lipschitz-bounded 1D CNNs for the classification of heart arrythmia data and show their improved robustness.

Direct Parameterization of Lipschitz-Bounded Deep Networks

This paper introduces a new parameterization of deep neural networks (both fully-connected and convolutional) with guaranteed Lipschitz bounds, i.e. limited sensitivity to perturbations. The Lipschitz guarantees are equivalent to the tightest-known bounds based on certification via a semidefinite program (SDP), which does not scale to large models. In contrast to the SDP approach, we provide a ``direct'' parameterization, i.e. a smooth mapping from $\mathbb R^N$ onto the set of weights of Lipschitz-bounded networks. This enables training via standard gradient methods, without any computationally intensive projections or barrier terms. The new parameterization can equivalently be thought of as either a new layer type (the \textit{sandwich layer}), or a novel parameterization of standard feedforward networks with parameter sharing between neighbouring layers. We illustrate the method with some applications in image classification (MNIST and CIFAR-10).

Learning over All Stabilizing Nonlinear Controllers for a Partially-Observed Linear System

We propose a parameterization of nonlinear output feedback controllers for linear dynamical systems based on a recently developed class of neural network called the recurrent equilibrium network (REN), and a nonlinear version of the Youla parameterization. Our approach guarantees the closed-loop stability of partially observable linear dynamical systems without requiring any constraints to be satisfied. This significantly simplifies model fitting as any unconstrained optimization procedure can be applied whilst still maintaining stability. We demonstrate our method on reinforcement learning tasks with both exact and approximate gradient methods. Simulation studies show that our method is significantly more scalable and significantly outperforms other approaches in the same problem setting.

Youla-REN: Learning Nonlinear Feedback Policies with Robust Stability Guarantees

This paper presents a parameterization of nonlinear controllers for uncertain systems building on a recently developed neural network architecture, called the recurrent equilibrium network (REN), and a nonlinear version of the Youla parameterization. The proposed framework has "built-in" guarantees of stability, i.e., all policies in the search space result in a contracting (globally exponentially stable) closed-loop system. Thus, it requires very mild assumptions on the choice of cost function and the stability property can be generalized to unseen data. Another useful feature of this approach is that policies are parameterized directly without any constraints, which simplifies learning by a broad range of policy-learning methods based on unconstrained optimization (e.g. stochastic gradient descent). We illustrate the proposed approach with a variety of simulation examples.

Recurrent Equilibrium Networks: Unconstrained Learning of Stable and Robust Dynamical Models

This paper introduces recurrent equilibrium networks (RENs), a new class of nonlinear dynamical models for applications in machine learning and system identification. The new model class has "built in" guarantees of stability and robustness: all models in the class are contracting -- a strong form of nonlinear stability -- and models can have prescribed Lipschitz bounds. RENs are otherwise very flexible: they can represent all stable linear systems, all previously-known sets of contracting recurrent neural networks, all deep feedforward neural networks, and all stable Wiener/Hammerstein models. RENs are parameterized directly by a vector in R^N, i.e. stability and robustness are ensured without parameter constraints, which simplifies learning since generic methods for unconstrained optimization can be used. The performance of the robustness of the new model set is evaluated on benchmark nonlinear system identification problems.

Lipschitz Bounded Equilibrium Networks

This paper introduces new parameterizations of equilibrium neural networks, i.e. networks defined by implicit equations. This model class includes standard multilayer and residual networks as special cases. The new parameterization admits a Lipschitz bound during training via unconstrained optimization: no projections or barrier functions are required. Lipschitz bounds are a common proxy for robustness and appear in many generalization bounds. Furthermore, compared to previous works we show well-posedness (existence of solutions) under less restrictive conditions on the network weights and more natural assumptions on the activation functions: that they are monotone and slope restricted. These results are proved by establishing novel connections with convex optimization, operator splitting on non-Euclidean spaces, and contracting neural ODEs. In image classification experiments we show that the Lipschitz bounds are very accurate and improve robustness to adversarial attacks.