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Wouter Jongeneel

Publications and source records attributed to Wouter Jongeneel.

11 recordsLinked to original sources
arXiv2025

On topological properties of closed attractors

The notion of an attractor has various definitions in the theory of dynamical systems. Under compactness assumptions, several of those definitions coincide and the theory is rather complete. However, without compactness, the picture becomes blurry. To improve our understanding, we characterize in this work when a closed, not necessarily compact, asymptotically stable attractor on a locally compact metric space is homotopy equivalent to its domain of attraction. This enables a further structural study of the corresponding feedback stabilization problem.

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arXiv2025

A generalized global Hartman-Grobman theorem for asymptotically stable semiflows

Recently, Kvalheim and Sontag provided a generalized global Hartman-Grobman theorem for equilibria under asymptotically stable continuous vector fields. By leveraging topological properties of Lyapunov functions, their theorem works without assuming hyperbolicity. We extend their theorem to a class of possibly discontinuous vector fields, in particular, to vector fields generating asymptotically stable semiflows.

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arXiv2025

Ordering and refining path-complete Lyapunov functions through composition lifts

A fruitful approach to study stability of switched systems is to look for multiple Lyapunov functions. However, in general, we do not yet understand the interplay between the desired stability certificate, the template of the Lyapunov functions and their mutual relationships to accommodate switching. In this work we elaborate on path-complete Lyapunov functions: a graphical framework that aims to elucidate this interplay. In particular, previously, several preorders were introduced to compare multiple Lyapunov functions. These preorders are initially algorithmically intractable due to the algebraic nature of Lyapunov inequalities, yet, lifting techniques were proposed to turn some preorders purely combinatorial and thereby eventually tractable. In this note we show that a conjecture in this area regarding the so-called composition lift, that was believed to be true, is false. This refutal, however, points us to a beneficial structural feature of the composition lift that we exploit to iteratively refine path-complete graphs, plus, it points us to a favourable adaptation of the composition lift.

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arXiv2024

A Large Deviations Perspective on Policy Gradient Algorithms

Motivated by policy gradient methods in the context of reinforcement learning, we identify a large deviation rate function for the iterates generated by stochastic gradient descent for possibly non-convex objectives satisfying a Polyak-Łojasiewicz condition. Leveraging the contraction principle from large deviations theory, we illustrate the potential of this result by showing how convergence properties of policy gradient with a softmax parametrization and an entropy regularized objective can be naturally extended to a wide spectrum of other policy parametrizations.

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arXiv2023

On continuation and convex Lyapunov functions

Suppose that the origin is globally asymptotically stable under a set of continuous vector fields on Euclidean space and suppose that all those vector fields come equipped with -- possibly different -- convex Lyapunov functions. We show that this implies there is a homotopy between any two of those vector fields such that the origin remains globally asymptotically stable along the homotopy. Relaxing the assumption on the origin to any compact convex set or relaxing convexity to geodesic convexity does not alter the conclusion. Imposing the same convexity assumptions on control Lyapunov functions leads to a Hautus-like stabilizability test. These results ought to be of interest in the context of learning stability certificates, policy gradient methods and switched systems.

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arXiv2022

Imaginary Zeroth-Order Optimization

Zeroth-order optimization methods are developed to overcome the practical hurdle of having knowledge of explicit derivatives. Instead, these schemes work with merely access to noisy functions evaluations. One of the predominant approaches is to mimic first-order methods by means of some gradient estimator. The theoretical limitations are well-understood, yet, as most of these methods rely on finite-differencing for shrinking differences, numerical cancellation can be catastrophic. The numerical community developed an efficient method to overcome this by passing to the complex domain. This approach has been recently adopted by the optimization community and in this work we analyze the practically relevant setting of dealing with computational noise. To exemplify the possibilities we focus on the strongly-convex optimization setting and provide a variety of non-asymptotic results, corroborated by numerical experiments, and end with local non-convex optimization.

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arXiv2022

Efficient Learning of a Linear Dynamical System with Stability Guarantees

We propose a principled method for projecting an arbitrary square matrix to the non-convex set of asymptotically stable matrices. Leveraging ideas from large deviations theory, we show that this projection is optimal in an information-theoretic sense and that it simply amounts to shifting the initial matrix by an optimal linear quadratic feedback gain, which can be computed exactly and highly efficiently by solving a standard linear quadratic regulator problem. The proposed approach allows us to learn the system matrix of a stable linear dynamical system from a single trajectory of correlated state observations. The resulting estimator is guaranteed to be stable and offers explicit statistical bounds on the estimation error.

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arXiv2021

Topological Linear System Identification via Moderate Deviations Theory

Two dynamical systems are topologically equivalent when their phase-portraits can be morphed into each other by a homeomorphic coordinate transformation on the state space. The induced equivalence classes capture qualitative properties such as stability or the oscillatory nature of the state trajectories, for example. In this paper we develop a method to learn the topological class of an unknown stable system from a single trajectory of finitely many state observations. Using a moderate deviations principle for the least squares estimator of the unknown system matrix $θ$, we prove that the probability of misclassification decays exponentially with the number of observations at a rate that is proportional to the square of the smallest singular value of $θ$.

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arXiv2021

Small errors in random zeroth-order optimization are imaginary

Most zeroth-order optimization algorithms mimic a first-order algorithm but replace the gradient of the objective function with some gradient estimator that can be computed from a small number of function evaluations. This estimator is constructed randomly, and its expectation matches the gradient of a smooth approximation of the objective function whose quality improves as the underlying smoothing parameter $\delta$ is reduced. Gradient estimators requiring a smaller number of function evaluations are preferable from a computational point of view. While estimators based on a single function evaluation can be obtained by use of the divergence theorem from vector calculus, their variance explodes as $\delta$ tends to $0$. Estimators based on multiple function evaluations, on the other hand, suffer from numerical cancellation when $\delta$ tends to $0$. To combat both effects simultaneously, we extend the objective function to the complex domain and construct a gradient estimator that evaluates the objective at a complex point whose coordinates have small imaginary parts of the order $\delta$. As this estimator requires only one function evaluation, it is immune to cancellation. In addition, its variance remains bounded as $\delta$ tends to $0$. We prove that zeroth-order algorithms that use our estimator offer the same theoretical convergence guarantees as the state-of-the-art methods. Numerical experiments suggest, however, that they often converge faster in practice.

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