Learning Neural Contracting Dynamics: Extended Linearization and Global Guarantees

It is unclear why the joint training of the encoder and the model is possible for ELCD but not in NCDS.

It is possible to jointly train the encoder and the model for NCDS as well. The authors of NCDS treat the encoder as a Variational Autoencoder (VAE) and train it by minimizing the evidence lower bound. The authors claim that the VAE training helps deal with the complexity of high dimensional systems. We argue, however, that the encoder and model must be trained jointly. If the encoder is trained independently of the model, then it is not necessarily true that the dynamics will be $l_2$-contracting in the learned transformed space. We confirmed this lack of contractivity experimentally, finding that VAE training of the encoder followed by training of the model did not produce reportable results. These poor results explain why we used joint training for NCDS for our baseline results. Since the NCDS code is not available, we have reproduced it to the best of our abilities.

The paper lacks a clear discussion of the specific types of problems where ELCD would be most applicable...

We envision applications in imitation learning where the resulting learned model has global guarantees of contraction, even away from the training data. Additionally, we imagine learning control systems of the form $\dot{x} = f(x) + Bu$, where $f$ is contracting. We could then leverage recent results on optimal control of contracting systems to design effective controllers for these systems [7]. We will update the introduction, motivation, and discussion about future research to highlight these points.

Is the ELCD model expressive enough to represent all contracting dynamical systems?

Thank you for your question.

It is known that any smooth nonlinear system with a hyperbolically stable fixed point can be exactly linearized inside its basin of attraction via a suitable diffeomorphism [8].

In essence, our extended-linear parameterization is tasked with learning the stable linear part, while the coupling layers aim to learn and approximate this suitable diffeomorphism.

So your question can be transcribed into the following: are coupling layers capable of approximating any diffeomorphisms?

Some results in this direction were obtained by [9,10]. The survey [11] confirms that those universality conditions hold for our choice in coupling layer [12].

However, the universality results in [9,10] apply to learning transformations between cumulative distribution functions. At this time, we cannot directly answer the universality question about diffeomorphisms.

Can the authors clarify... how the choice of diffeomorphism affects the contractivity of the learned dynamics?

You raise a good point. If we have a contracting dynamical systems, then any diffeomorphism applied to the system ensures that the resulting systems remains contracting (in a different Riemannian metric, however). In L291 we were commenting on the specific parametrization of NCDS [1]. Since the NCDS parametrization ensures contraction in the $l_2$ metric, it is possible that an arbitrary diffeomorphism applied to data coming from some contracting dynamics is not contracting in the $l_2$ metric (although it would still be contracting in some other metric). We realize this point was not fully clear in the first version of our paper and the revision will include a clearer more detailed explanation.

Complexity of approximating the contraction metric...

This is a good question. In our opinion, ELCD scales well independently of the complexity of approximating the contraction metric. This is because we do not compute the contraction metric at any step during training or evaluation. The complexity of approximating the metric would only arise if the end user needed to have an expression for it; something we do not need in our tasks.

First, we would like to thank you for finding our work to have a strong theoretical basis, well-written, and easy-to-read with reasonable experiments. Regarding larger-scale experiments, we have been unable to perform them at this time. We agree that they would add value and we envision executing them as future work.

The novelty of the method lies in learning a linear map...

Thank you for your time in reviewing our paper. Respectfully, we would like to clarify that we do not learn a linear map, but rather a nonlinear matrix-valued map $x \mapsto A(x,x^*)$ via the parametrization (11) to allow for nonlinear contracting dynamics in the latent space.

The main weakness of the paper is the experimental part...

We do focus on the ``usual suspects" in stable dynamics learning since they are common benchmarks studied in other works. These examples allow us to showcase that ELCD outperforms other state-of-the-art methods for learning stable dynamics in established benchmarks or well-known systems. Beyond LASA and the multi-link pendulum, we argue that the Rosenbrock dataset is novel and valuable (while the Rosenbrock dataset has not been featured in other works on learning dynamics, the Rosenbrock function is a classic function in optimization and ML). The Rosenbrock dynamics are novel in that they are stiff and evolve at different timescales. In each case we are able to showcase the superiority of ELCD compared to prior alternative methods.

The approach itself is by no means ``groundbreaking" but rather incremental...

We would argue that the approach is more than incremental in nature as we allow for learning a larger class of contracting dynamics compared to NCDS by our novel parametrization which captures asymmetric Jacobian matrices. The benefits of our parameterization are clearest in our updated results on the multi-link pendulum using continuous-time training. The pendulum has inherently oscillatory dynamics, whose Jacobian has complex eigenvalues. No learning model with symmetric Jacobian can represent this system, regardless of what diffeomorphism is used. Our parameterization is key towards learning this system. The commonality of the multi-link pendulum (and more geneally oscillatory dynamics) in the dynamical systems literature demonstrates the value of our contribution. Moreover, by directly parametrizing the extended linearization, we avoid the expensive line integral necessary in NCDS. Finally, we feel that the recent improvement from discretized-time to continous-time training is also valuable. As a testament to benefits of these changes, we can see the notably improved performance of ELCD compared to existing benchmarks in our numerical experiments.

The caption in Table 1 misses which metric is represented here...

You are correct that we were not precise enough in our caption of Table 1. To clarify, we are reporting the dynamic time warping distance (DTWD) between the trajectories of the learned model and the trajectories of the ground truth dataset. The $\pm$ corresponds to one standard deviation from the mean. We will clarify these details before the final submission.

Introduction

Thank you for clarifying your thought process. We will make the following changes to the introduction to make things clearer:

We will replace line 17 onward in the first paragraph with:

"Beyond approximating the vector field f, it is desirable to ensure that the learned vector field is well-behaved. In many robotic tasks like grasping and navigation, a well-behaved system should always reach a fixed endpoint. Ideally, a learned system will still stably reach the desired endpoint even when pushed away from demonstration trajectories. Additionally, the learned system should robustly reach the desired endpoint in the face of uncertainty. In other tasks, such as manufacturing, animation, and human-robot interaction, the learned system must smoothly follow a specific trajectory to its target. Following a trajectory is necessary to ensure function, as is the case with a manufacturing robot dynamically interacting with its environment, and safety, as is necessary with a robot occupying the same space as humans."

We will additionally replace lines 27-28 in paragraph three with:

"To ensure robustness and to allow for smooth trajectory following, there has been increased interest in learning contracting dynamics. A dynamical system is said to be contracting if any two trajectories converge to one another exponentially quickly with respect to some metric [25]. If a learned contracting system that admits a demonstration trajectory is pushed off that trajectory, it will follow a new trajectory that exponentially converges to the demonstration trajectory. "

To address those specific concerns:

One might actually be learning from data that comes from a dynamical system with multiple stable fixed points

Thank you for bringing this point up. Indeed, contracting dynamics can only have one stable fixed point. In this work, we are assuming that the underlying dynamics are globally contracting (hence with a unique fixed point) as this assumption ensures strong convergence and robustness guarantees for the learned dynamics, even away from training data.

As a possible extension to systems with multiple fixed points, one could imaging learning multiple ELCD models, one for the neighborhood of each fixed point and then (i) either devising a strategy to blend them for global problems, or (ii) depending on the initial condition, leveraging the specific relevant model.

Why does the stability of the learned dynamical system matter?

Like you mention, we envision applications in imitation learning where the resulting learned model has global guarantees of contraction, even away from the training data. We will update the introduction to be more specific as to why we want to learn globally contracting dynamics (and mention extensions to multiple equilibria and multiple local ELCD models).

In the partial contraction analysis...

Of course you are completely correct. We will be more precise and state that $x(t)$ is meant to be a specific solution trajectory of the original system $\dot{x}(t) = A(x(t),x^*)(x(t) - x^*)$.

Words about the controlled case...

Thank you for your curiosity on this topic. Our method can be naturally extended to learning control systems $\dot{x} = f(x) + Bu$ which are contracting in $x$ at fixed $u$. These systems automatically inherit an input-to-state stability property and an entrainment property, i.e., a periodic input $u$ results in a stable periodic solution $x$, see [6]. Moreover, control systems with a contracting drift term can be amenable to computationally efficient tools for optimal control, see [7]. We are actively studying the extension of ELCD to closed-loop contractivity guarantees, i.e., learning both $f$ and a closed-loop controller $u$ that offers global contraction guarantees (not just at the training data as is frequently studied in existing works).