Learning Neural Contracting Dynamics: Extended Linearization and Global Guarantees

Summary

The paper proposes a novel neural network architecture, Extended Linearized Contracting Dynamics (ELCD), for learning dynamical systems with global contractivity guarantees. The authors leverage the concept of extended linearization to factorize the vector field and enforce negative definiteness of its symmetric part, ensuring global exponential stability and equilibrium contraction. To extend contractivity to more general metrics, they introduce a latent space representation and learn diffeomorphisms between the data and latent spaces. The method is evaluated on various datasets, demonstrating superior performance compared to existing models, including Neural Contractive Dynamical Systems (NCDS).

Strengths

The paper addresses the important problem of learning stable and robust dynamical systems, which has significant implications in various fields. The proposed ELCD model is theoretically sound, with guarantees of global contractivity, a property that ensures robustness to perturbations. The use of extended linearization and latent space representation is innovative and allows the model to capture a wider range of contracting dynamics. The experimental results are promising, showing that ELCD outperforms existing methods on several benchmark datasets. ELCD addresses some limitations of NCDS. Specifically, ELCD parameterizes the vector field directly, avoiding the computationally expensive Jacobian integration step in NCDS. ELCD also allows for asymmetric Jacobians, leading to increased model expressivity compared to NCDS, which is restricted to symmetric Jacobians. ELCD trains the encoder jointly with the dynamics model, allowing the encoder to learn representations specifically tailored for capturing contracting dynamics. This is in contrast to NCDS, which trains the encoder separately as part of a Variational Autoencoder (VAE) before training the dynamics model. The joint training in ELCD potentially leads to better performance by optimizing the encoder for the specific task of representing contracting dynamics.

Weaknesses

The assumption of contractivity in the underlying dynamics might limit the applicability of the model to real-world systems where this assumption may not hold. The current implementation is restricted to trajectories converging to a fixed point and dynamics in Euclidean space, which could be extended to more general settings. This limitation is particularly evident in the LASA handwriting dataset, where the model struggles to capture the translation invariance inherent in human handwriting. In reality, the end-point location of handwriting is not fixed, and the entire letter can be translated without affecting its validity. Also, handwriting can often exhibit overlapping trajectories, which cannot be modeled by the current implementation. Moreover, the learned model can only generate one handwriting. It is unclear why the joint training of the encoder and the model is possible for ELCD but not in NCDS. The authors could address this point to clarify the advantages of their approach. The paper lacks a clear discussion of the specific types of problems where ELCD would be most applicable and where it might not be suitable. A more detailed analysis of the model's strengths and weaknesses in different scenarios would be beneficial.

Questions

1. Is the ELCD model expressive enough to represent all contracting dynamical systems? If not, what are the limitations, and can they be formally characterized? 2. The paper states that "contraction is invariant under differential coordinate changes..." (L227) but also mentions that "If the data is not contracting in the transformed space..." (L291). These statements seem contradictory. Can the authors clarify this discrepancy and explain how the choice of diffeomorphism affects the contractivity of the learned dynamics? 3. The computational complexity of approximating the contraction metric is mentioned as a potential limitation. Can the authors elaborate on the scalability of the method to high-dimensional systems and suggest possible strategies to mitigate this issue? 4. The paper focuses on trajectories converging to a fixed point. How can the method be extended to handle more complex dynamical behaviors, such as limit cycles or chaotic attractors? 5. Why is joint training of the encoder and the model possible for ELCD but not in NCDS?

Rating

8

Confidence

4

Soundness

4

Presentation

3

Contribution

3

Limitations

The limitation sections address some of the limitations of the current approach, but is insufficient. The current implementation of ELCD is limited to trajectories converging to a fixed point and dynamics in Euclidean space. This restricts its applicability to problems where these assumptions hold. For example, while ELCD is applied to model handwriting in the LASA dataset, it struggles to capture the translation invariance inherent in human handwriting, where the end-point location is not fixed. It would be beneficial to discuss the specific types of problems where ELCD would be most applicable and where it might not be suitable.

Summary

The paper presents Extended Linearized Contracting Dynamics (ELCD), a dynamical system with neural network components that has global contraction guarantees. They demonstrate improved efficiency and performance on the trajectory fitting LASA dataset (upto 8 dim), pendulum dataset (upto 16 dim), Rosenbrock dataset (upto 16 dim).

Strengths

1. Strong theoretical basis for the ELCD model. 2. Well-written and easy-to-read paper. 3. Reasonably sized experiments for a paper with a primarily theoretical contribution.

Weaknesses

1. If the authors would like to present more of a empirical contribution, it would help to scale the experiments to something of the order of 100s of dimensions such as in common RL environments like mujoco-humanoid.

Questions

Please see above.

Rating

6

Confidence

2

Soundness

3

Presentation

3

Contribution

3

Limitations

NA

Summary

The paper presents a method for learning contracting representations from a dynamical system. The novelty of the method lies in learning a linear map and a coordinate transform map (Diffeomorphism), which extends the types of contracting systems that can be learned. The experiments show that the proposed methods performs better empirically than Neural Contractive Dynamical System and other approaches for learning contracting dynamics.

Strengths

- The paper is well structured and easy to follow - The problem motivation beyond the vast amount of methods on learning stable dynamics is well formulated - The relation to recent literature on this topic is also well done

Weaknesses

The main weakness of the paper is the experimental part. There are details missing and it feels "rushed", e.g., table caption unclear (see below). Additionally, he set of tasks are very limited and don't go beyond the "usual suspects" in stable dynamical system learning (e.g. pendulum chains). Besides the experiments, a point of limitation would be that the approach itself is by no means "groundbreaking" but rather incremental in the context of learning contracting dynamics (whether empirically it is truly groundbreaking cannot be assessed from the provided experiments)

Questions

The caption in Table 1 misses which metric is represented here and also what the +- means (this should not be implicitly assumed)

Rating

4

Confidence

2

Soundness

3

Presentation

3

Contribution

3

Limitations

The paper discusses some extensions to having multiple fixed points. However, the paper lacks a true discussion on limitations and scenarios where the approach may fail (not necessarily due to the approach but also including due to learning it)

Summary

This paper proposes a novel parameterization of the extended linearisation form of a dynamical system that guarantees global contractivity. Whilst the most basic form of this parameterization only ensures contractivity in some (implicitly defined) metric, a latent space version is also proposed that enables more flexible learning of dynamics that are contractive in more general metrics. Experiments are provided that show superiority over related methods.

Strengths

This paper is very well written; it provides a very good intro / recap of contractivity which was very useful to me at least, and a clear articulation of what was missing from previous works on the subject (e.g. NCDS is computationally expensive and has overly constrained (symmetric) Jacobians, no global contraction guarantees, etc). The idea of directly parameterising the extended linearization of the system is very neat, and so is the use of the converse contraction theorem of Giesl et al to prove the main contractivity result. Although the experiments are a bit thin, I find them convincing enough.

Weaknesses

The authors very quickly gloss over the motivations for learning contractive dynamics; I had to go read the NCDS paper to understand the real-world relevance of such dynamics. Let me write down below what occurred to me when reading the intro (which mostly betray my lack of experience in potential practical applications of this paper), and the authors may or may not want to modify their introduction to preempt this potential confusion. My initial thoughts when reading the first paragraph of the intro, in particular “it is desirable to ensure [...]”: - For one, one might actually be learning from data that comes from a dynamical system with multiple stable fixed points, so why is it desirable to enforce a single one in the learned model? - Relatedly, why does the stability of the _learned_ dynamical system matter? I would assume that what typically matters in robotics/control is that the actual physical system implemented (and in particular, the closed control loop) be stable and robust? What does that have to do with the learned model? Is model stability important because an unstable model (where the ground truth is stable) would extrapolate very badly outside the region where there is data? (I was thinking of a situation where deep learning is used to learn an internal model of a plant from observational data, e.g. for subsequent control purposes; in which case (i) autonomous dynamics are not necessarily very relevant and (ii) it's unclear why those dynamics would be contractive in general; now I understand that there are applications in e.g. imitation learning where it might be useful to ensure that the learned dynamics have such guarantees, but it was hard to guess from the current intro; the intro of the NCDS paper helped).

Questions

In the partial contraction analysis of Eq 13, it might be worth stating that x can be any function of time (as opposed to a constant). This seems critical to the eventual conclusion, where you say "we can pick y2(t) = x(t)", as this statement implicitly relies on a trajectory of the original system also being a valid trajectory of the virtual system. I'm guessing this is very standard but as someone not familiar with this type of partial contraction analysis I thought I'd mention it. A few typos - l.41 “enforce an exponentially decay the Lyapunov function” - l.52 “are constrainted” - l.53 “there are points [...]” → there may be points [...] - l.161 “prior words” → prior works - l.258: diffeopmorphisms - l.287 minimize the ELBO → maximize the ELBO - l.290 the data is corresponds - l.456: metrc - just a suggestion for your acronym: "ELCoD" would be easier to pronounce ("ell-cod") than "ELCD"

Rating

7

Confidence

3

Soundness

4

Presentation

4

Contribution

3

Limitations

Could the authors please comment on whether requiring contraction behaviour (and in particular, the existence of a unique stable equilibrium) might be too constraining in some applications? I wonder if the authors could also say a few words about the controlled case -- i.e. driven systems as opposed to autonomous systems. E.g. if the autonomous system $\dot{x} = f(x)$ obeys the guarantees provided by this work, what guarantees might be automatically inherited by the corresponding control affine system $\dot{x} = f(x) + B u$?