Contractive Dynamical Imitation Policies for Efficient Out-of-Sample Recovery
Authors: Amin Abyaneh
Mahrokh Ghoddousi Boroujeni
Hsiu-Chin Lin
Giancarlo Ferrari-Trecate
Venue: ICLR 2025 Poster (2025)
Keywords: Learning from demonstration
Safe imitation learning
Robotics
Dynamical system
Contraction theory
Primary area: applications to robotics, autonomy, planning
TL;DR: We introduce a class of contractive imitation policies with theoretical guarantees and out-of-sample error bounds for robot learning.

Abstract: Imitation learning is a data-driven approach to learning policies from expert behavior, but it is prone to unreliable outcomes in out-of-sample (OOS) regions. While previous research relying on stable dynamical systems guarantees convergence to a desired state, it often overlooks transient behavior. We propose a framework for learning policies modeled by contractive dynamical systems, ensuring that all policy rollouts converge regardless of perturbations, and in turn, enable efficient OOS recovery. By leveraging recurrent equilibrium networks and coupling layers, the policy structure guarantees contractivity for any parameter choice, which facilitates unconstrained optimization. We also provide theoretical upper bounds for worst-case and expected loss to rigorously establish the reliability of our method in deployment. Empirically, we demonstrate substantial OOS performance improvements for simulated robotic manipulation and navigation tasks. See [sites.google.com/view/contractive-dynamical-policies](https://sites.google.com/view/contractive-dynamical-policies) for our codebase and highlight of the results.

Published as a conference paper at ICLR 2025
CONTRACTIVE DYNAMICAL IMITATION POLICIES FOR EFFICIENT OUT-OF-SAMPIN RECOVERY
Amin Abyaneh'; Mahrokh G. Boroujeni"; Hsiu-Chin Lin', Giancarlo Cerrari-Trecate
McGill University," 2 École Polytechnique Fédérale de Lausanne (EPFL)
ABSTRACT
Imitation learning is a data-driven approach to learning policies from expert be- havior, but it prone unreliable outcomes in out-of-sample (OOS) regions. While previous research relying on stable dynamical systems guarantees con- vergence to a desired state, it often overlooks transient behavior. We propose framework for learning policies modeled by contractive dynamical systems, en- suring that all policy rollouts converge regardless of perturbations, and in turn, enable efficient OOS recovery. By leveraging recurrent equilibrium networks and coupling layers, the policy structure guarantees contractivity for any parameter choice, which facilitates unconstrained optimization. We also provide theoretical upper bounds for worst-case and expected loss to rigorously establish the reli- ability of our method in deployment. Empirically, we demonstrate substantial OOS performance improvements for simulated robotic manipulation and navi- gation tasks. See codebase and highlight of the results.
wwt.//iiwwconttrattvveeccccmceee - dynamicalll policiiss for
INTRODUCTION
Imitation learning provides an intuitive policy optimization framework for executing complex robotic tasks by learning from expert demonstrations (Hussein al.) (017). However, naively repli- cating expert behavior can raise safety concerns during deployment, as the robot's 's trajectory may become unreliable when facing out-of-sample (OOS) states. For instance, the robot begins from an unknown initial state encounters environmental perturbations, it could fail to maintain a con- sistent and reliable behavior. To address this issue, the policy can be modeled using a dynamical system (DS) with rigorous reliability certificates (Ravichandar et al. 2020 Dawson al. 2023).
One such certificate is asymptotic stability (Devaney 2021]. which ensures that all trajectories eventually converge to the same equilibrium state, regardless of
the initial condition perturbations (KhansariZZadeh &
Expert
Billard 2011 Rana et al., 2020 Abbyaneh et al.] (002)).
While asymptotic stability guarantees convergence, it overlooks the transient behavior (Tsukamoto et al.) (202T).
Contractive
poliey
This phenomenon is illustrated in Fig. I for an arbitrary OOS initial state, where the trajectory induced by the sta- ble policy converges the target but fails to imitate the expert. In contrast, contractive DS policies ensure that the resulting trajectories exponentially converge to each other, and ultimately to the equilibrium state (Ravichan- dar et al. 2017: Mohammadi et al. 2024). We leverage the stronger notion of contractivity (Lohmiller & Slotine 1998) to ensure reliability in the transient phase, and as result, enhance the imitation quality.
Stable paliks
Out-of-sample state
Target
Figure 1: Policy rollouts generated by contractive and stable policies. While both policies eventually reach the tar- get, the contractive policy closely mim- ics the expert in the transient phase.
There are two main approaches to learning contractive DSs. The most common method relies constrained
"The first two authors collaborated equally. Corresponding author; ami abyaneh@mail mnngi 11.ca

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optimization to train the DS while satisfying learned or fixed contractivity certificates (Ravichandar ctt al., 2017 Blocher et al., 2017 Sindhwani et al. 2018). However, constrained optimization is computationally challenging for complex expert behaviors or high-dimensional state spaces. Addi- tionally, most constrained approaches trade off imitation accuracy for constraint satisfaction. The second approach is based on parameterized models with built-in contractivity guarantees for any choice of parameters (Mohammadi et al., 2024) Dawson et al. (202)). As a result, the model re- mains contractive even if the learning process is imperfect or disrupted.
In this work, we build on the foundation of two such formulations: (i) parameterization of con- tractive continuous- time DSs using recurrent equilibrium networks (RENs) (Martinelli al. 2023), and (ii) the coupling layer architecture (Papamakarios et al. (2021). The former ensures that the DS contractive with an adjustable rate, while the latter provides trainable bijective transformation that preserves contraction properties. Our experiments demonstrate that the coupling layers enhance
the model's representation power, leading to improved imitation accuracy and faster training, while maintaining the contractive nature of the DS.
Related work. Contractive imitation learning approaches are less common the literature, pri- marily due to the increased difficulty of enforcing contractivity compared to asymptotic stability. The majority existing methods train the DS under contractivity constraints (Blocher et al. 2017] Ravichandar et al. 2017: Siindhaani et al. 2018), resulting in computationally demanding op- timization problem. These methods assume specific representations of the DS or the contraction certificate mitigate the intensive computation. For example, [Ravichandar al. (2017) employ sum-of-squares programming with polynomials, while BBocher et al. (2017) and Ravichandar et al. (2017) use Gaussian mixture techniques. Additionally, [Sindhwani et al. (2018) adopt reproducing kernel Hilbert spaces. By focusing on these particular representations, these methods reduce the computational cost, but the cost of limiting the model's 's representation power.
More recently, Mohammadi et al. (2024) introduce a promising unconstrained neural architecture with inherent contractivity guarantees. The algorithm optimizes a parameterized negative-definite Jacobian in a lower-dimensional latent space and integrates it to derive the DS. Despite better perfor- mance in some high-dimensional environments, method also requires solving second-order differential equation, which is more complex than the standard first-order equations typically used (Rana et al. 2020 Sochopoulos et al. (024). Further discussion of the literature, including non-contractive methods, provided in App. B
mnder-performs in lower-dimensional ones. The
Contributions. We propose a State-only framework for learning Contractive Dynamical System policies (SCDS), which offers several distinct advantages. First, SCDS learns the policy solely from state measurements, eliminating the need to measure the expert's velocity. Hence, we address the cumulative error problem observed in previous methods that replicate expert velocity, where small errors can accumulate and lead to significant deviations in state space trajectories (Ravichandar al. 2020 Abyaneh al. 2024). Additionally, SCDS enables learning in latent space, with its dimension adapting to the problem at hand. It facilitates learning in high-dimensional state spaces by mapping to diwer-dimensional latent representations, while also providing flexibility in low- dimensional spaces through nigher-dimensional embeddings.
From a theoretical standpoint, we provide rigorous solution for OOS recovery by establishing an upper bound on the worst-case deviation from expert trajectories. SCDS also offers control over the transient behavior through an adjustable and learnable contraction rate. Training SCDS is highly efficient by relying on unconstrained optimization and neural ordinary differential equations (ODE) fixed-point differentiation for gradient computation (Chen et al. 2018). An overview of SCDS is outlined in Fig. 2
2 CONTRACTIVE POLICY FORMULATION
2.1 PROBLEM SETUP
Dataset. Consider a robot with a state y() E y at time E Rzo operating within state space y Ny,, The state y is typically considered to be the robot's joint task space configuration.
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Contractive policy (Sections 2.2, 2.3)
Contractive REN
Bijection block
12(0,7) 7)
Pe
Linear projection
90.1
90.K
Expert data (Section 2.1)
policy
update
Rollouts
Neural ODE
θ
initial states
(Section 3.1)
contractive solutions
Yo m
expert
Loss (Section 3.2)
rollouts
Target
y m
DTW
MSF.
y
Figure 2: Overview of the SCDS training scheme. The policy structure (top box) consists of REN, a linear projection, and a bijection block, ensuring contractivity for any choice of parameters, 0. In the forward pass (-), initial states are passed through a differentiable ODE solver to generate state rollouts. The loss function penalizes the discrepancy between the generated and expert trajectories, updating the policy via backpropagation (+)).
Within V, we have dataset D of M N expert demonstrations, defined as:
D y (y
-I),.-1) m=1²
where the m-th demonstration, y has Nm samples. We assume that all expert trajectories converge to common target state, - y for all m, but allow for different initial states and different sample sizes per trajectory. Note that our approach only requires state measurements, unlike most previous work, which rely both the state and its time derivative.
Objective. We aim to learn a policy that precisely replicates the expert trajectories in the dataset D while ensuring robustness against perturbations during deployment. In other words, the policy should be able to recover safely the robot encounters OOS states, either as a result of perturbations different initial states from those in D. We provide such guarantees by formulating the policy as a parameterized DS and leveraging tools from the contraction theory laid out in the next section.
2.2 BACKGROUND: CONTRACTIVE LATENT DYNAMICS
Consider an autonomous DS in continuous time 2(t) fz(())) where R. is the state and f: .R + R.N, specifies the dynamics. Contractivity is fundamental characteristic of f, ensuring that the solution trajectories of incrementally converge single trajectory, even when starting from different initial conditions or facing perturbations along the way (Lohmiller & Slotine [998] Tsukamoto et al. (2021). Therefore, contractive DS provides stability and predictability as the state evolves over time. A formal definition outlined in the following.
Definition 2.1. (Lohmiller Slotine 1998) The dynamical system à(() ((((() is contracting with the rate 7 RT,, for any two initial conditions, (0), z²(0) NR,, and some a IR,, the corresponding trajectories z" and z generated by the dynamics f satisfy:
[12)(1) - =2)( S ae -0 (2) 20)
4L 2
where || .1 denotes an L, norm.
Definition 2.1
implies that contractive DS guarantees robustness by exponentially forgetting per- turbations over time. The contraction rate controls the speed this process, with larger values indicating faster contraction. We define contractivity using the Lag (Fuclidean) norm.

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The key question is: How learn expressive and contractive DS with desired contraction rate? We start by adopting the unconstrained parameterization of continuous- time contracting DSs introduced by Martinelli al. (2023). In this approach, the dynamics f is modeled as a recurrent equilibrium network (REN), expressed as:
z(t) o [o(x(())
= -( v(t)
(3)
where e E No is an internal variable initialized zero, 7 is the contraction rate, and 0 E No is the model parameters. The nonlinear function o RR + R is applied element-wise and must be piecewise differentiable with first derivatives restricted
the interval [0, 1]. The mapping Q : RN x R4 + R No+ x (N+ N.) ensures that v(t) has a unique value for every 2 Q and that z satisfies the contractivity criteria in Eq. 2 for every 0 E R. The contraction rate
be trained or adjusted, as discussed in(App. C.I and influences the resulting model through S2. Note that RENs represent arbitrarily deep neural models by selecting the structure of the matrix (2(0, 7), thereby offering a high expressive power (Revay et al. 2023). Further background on contracting RENs and related proofs are provided in App. D
2.3 EXPRESSIVE CONTRACTIVE POLICIES
We formulate the policy de using a parameterized autonomous DS continuous time, given by:
(t) = go (z()))
(output transformation)
de
z(0) he (yo)
(initial condition)
(4)
i(t) fo (z()))
(latent dynamics)
where O E No collects all parameters.
Given an initial state yo. the DS Φ evolves through latent state z(() E R.N,, initialized through he RRN, -> R.N, and governed by the latent dynamics So IR.. R.V The latent state z(l) is mapped by Ge : TR= + R to (() E R²a,, which is the state planned by the policy. A low-level controller then converts this planned state into joint torque or velocity commands for the robot. Together, the DS de and the low-level controller form our imitation policy, which maps states to actions. For simplicity, we refer the DS as the policy. In the sequel, we describe different components Eq. 4
Latent dynamics. We model the latent dynamics with a contractive REN described in Eq. 3 Em- ploying a high-dimensional latent state, N= > Ny, increases the flexibility of our representation, as we empirically show in App. E.3 On the other hand, a low-dimensional latent state, N₂ < Ny, facilitates learning in some high-dimensional settings (Mohammadi al. 2024).
Output transformation. Owing to the architecture of fe. all solutions in the latent space converge to single one, regardless of the initial condition. Yet, it is crucial to maintain the same contraction property for the robot's s state trajectories, y. In essence, go is required to preserve contractivity, so that ÿ satisfies Eq. 2] for some constant and y. The same behavior is not guaranteed in the original REN formulation (Revay et al. 2023; Martinelli et al. 2023).
To address this, first change the dimension from the latent space the state space by applying learnable linear projection, Po z, where Po E N,N, Next, this transformation is followed by K coupling layers, 90,1
go K. which trainable bijective mappings from N to NN (Tabak
& Turner 2013 Rana et al. 2020). The resulting output transformation is described by:
(() go (z(()) - ge. (... .... (Po z(1)),
(5)
which preserves contractivity according to the next proposition.
Proposition 2.1. If the latent state satisfies the contractivity condition in Eq. with La norm, then any state trajectory ý obtained by the output map in Eg. Slalo satisfies this condition for every parameter No, Moreover, the contraction rate yy is invariant under the output transformation.
While Fq. 3 not immediately in the form (tt) f (z(()). it implicitly specifies the relation between x and since uniquely determined from
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The proof can be found in App. A.1 This proposition ensures that the mapping go from the latent space to the state space preserves the desired contractivity property.
Initial condition. The last element in Eq. to specify is the initial latent state z(0), which must be set to place the initial state ý(0) as close as possible to the desired yo- Using Eq. 5 this requirement translated into yo R 5(0) = go (z))). Bearing in mind that the bijective maps are invertible, we can express this relationship as Pe z()) R Jø.k (.. 90.1 (yo)). Then, a plausible solution is
determined by Pa's pseudoinverse:
z = ho (y)) P 90. k (-.. (yo)).
(6)
When Pe has full column rank, the pseudoinverse coincides with the left inverse, resulting in ýr(0) Yo- Otherwise, Eq. provides the -quares approximation to yo (Penrose 1956).
Policy architecture. At this stage, integrate the essential modules discussed earlier to construct
the final policy, by substituting Eq. Eq. and
Eq. 6 into the generic policy in Eq. 4
(()) = 90.1 (... (go.k (Poz(())
(outpua map)
z(0) = Pogo. H0
(initial condition)
$
[((t]] (v())
z(t)
= 2(0,7)
(latent dynamics)
(v(())
The policy is determined by its parameters and the contraction rate "y. In particular, for any desired y R4, the policy Og remains contractive regardless of the choice of 0 E RN, Therefore, can be optimized without imposing any constraints through computationally efficient automatic differentiation tools. In the section, we learn da effectively imitate the expert and build on the scalability of our method to learn intricate expert behaviors in high-dimensional spaces.
3 LEARNING CONTRACTIVE POLICIES THROUGH IMITATION
Each expert trajectory can be intuitively considered as the solution to an unknown initial value problem (IVP). We seek optimize the policy parameters 0 so that the policy do accurately models this IVP. To achieve this, we first explore solving the IVP in a differentiable way, then introduce loss function to measure the discrepancy between generated and expert trajectories, and finally optimize the policy to minimize this deviation through backpropagation.
3.1 DIFFERENTIABLE ODE SOLUTIONS
Consider the IVP in Eq. with an initial condition ýo y, which we refer to as icp(0e. ýo)- The generated solution, denoted by y. depends on 0. Therefore, the gradient information for the entire trajectory must be kept along the way to be leveraged by automatic differentiation to optimize ée. Luckily, the Neural ODE framework is capable of solving an IVP while efficiently storing the gradient information through implicit differentiation (Chen et al. 2018).
The Neural ODE framework solves the IVP in Eq. for selected horizon H, which determines the number integrations. Larger H enables the policy to imitate expert behavior with finer granular- ity at higher computational cost (see App. C.2). The IVP solutions, which synonymously call policy rollouts, are formed as differentiable trajectories: ý (yo, ý1, - We further improve the efficiency using batched multiple shooting, which generates array of rollouts corre- sponding to different initial conditions in parallel. Next, these policy rollouts are compared to expert demonstrations to train the policy 6e.
3.2 TRAJECTORY SPACE LOSS
We compare the policy rollout, ÿ. with each expert demonstration, y" using a discrepancy mea- sure ( y x yN + R20. where H and Nr, are the sizes of ý and ym, respectively. In the simplest case, where the trajectories of the same size, H Nn for all m, the function & can be defined as the mean squared error (MSE). Otherwise, a better alternative is dynamic time warp- ing (DTW) (Keogh & Ratanamahatana
2005 which can handle trajectories of different lengths.
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Indeed, unlike MSE, DTW is not sensitive time discrepancies: DTW is zero if two trajectories follow the same path but at different speeds. This behavior is desirable because spatial accuracy in robotic tasks is often the primary concern, rather than the execution velocity (Rana et al.) 2020
Sochopoulos et al. 2024). We employ the differentiable soft-D77W loss (Cuturi & Blondel, 2017) instead of the original formulation, which is tailored for gradient-based optimization (App. C.3)
Next, we introduce the loss function for multiple demonstrations. An intuitive approach to take convex combination of the discrepancies associated with each expert trajectory, given by:
M Am ())) y"),, s.t. ÿ = ivp(óo: Yo):
L(yo: 0)
A
(8)
where Am (Fn) E [0, 1] is the weight assigned to the difference from the m-th demonstration, and these weights satisfy the condition ΣM Am (fo) 1. naive choice for the combi- nation weights to set them all equal. Am (Vo) Mr. which results in an average loss. However, this approach can lead to a counterintuitive situation, as illustrated in Fig. 3] where the loss is nonzero even if the policy perfectly replicates one of the expert trajectories, i.e., Lyy"" 0) of when y = ym Instead, we propose selecting the combination weights inversely proportional to the squared distance between the given initial condition and the initial condition of the m-th demonstration,
Exper data.
Ralbart
Targel
Figure 3: Uniform weighting averages the deviation from both demonstrations,
(150- yo 12)
Am E1 (1 yo 113)
hence, L(vū; 0) * 0. In contrast, Eq. results in L(ys; 0) = O. For yo- the weights in Eg.. 9 assign higher importance to the closer yO's demonstration, which is intuitive.
The condition MM (5o) can easily be verified for the above weights. this case, the loss is zero when y = y", since only Am is nonzero, which is the expected behavior.
In real-world scenarios, there is often uncertainty about the initial robot state, Vo- A common approach to modeling this uncertainty is to assume that the initial state lies within a set Vo C y and has the distribution Po- Then, the true and empirical losses, denoted by L and É, are given by:
Su Su
L(po: 0)
A E Sam-P
LiVo;
unbiased
L(So: 0)
L 0)
(10)
estimation
where So { |So
s=1 is sampled set of initial states in Vo- The true loss, L. offers robust robust criterion for assessing 60 through averaging out the uncertainty in Yo but is impractical to compute. The empirical loss, É, is a computationally efficient approximation of L and can be minimized to train the policy. In the simple case where So consists of the initial conditions in D, the empirical loss boils M
down to C( [yv) m=mi 0) 37 EM E(pTT, y where ym is the solution to iup(óe, yū"). We employ this loss
mm
experiments.
3.3 OPTIMIZATION PROBLEM
The optimal parameters, 0", is learned by minimizing the empirical loss 2 as follows:
0" 2 arg min Me
E(1x0)) M J==1 0),
(11)
resulting in the optimal policy óø Since the policy parameterization in Subsec. 2.3 is contracting for all parameter choices, can solve the optimization problem over R No without additional con-
straints. The pseudocode for our method is presented in Alg. and its implementation details are discussed in App.
Remark 3.1. The contraction rate y in Alg. 1 can be either set manually or learned as model pa- rameter. Augmented Lagrangian methods can be used to learn higher contraction rates App. C.

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Algorithm State-only contractive policy learning
1: Require: Expert data D, contraction rate y, learning rate 17, horizon H
2: Initialize 0
Policy parameter initialization
3: while converged do
4: Sample yo E
5: Initialize z(0) + P 90 (fo)
b Latent initial state
Eq.
= [ 1H-1 +- Neural (DE( x(0) = (yr) H- t- go(z)
Differentiable ODE OoEver Subsec.
3.
Output map L4.
each y in D
9:
Find Am, f(y; y")) Compute L Lm Amlly, y
E4.
10
Eq.
Empirical Subsec. 3.2
Update 0 7V4
Gradient descent
return
OUT-OF-SAMPLA PERFORMANCE GUARANTEES
The policy Φg- is trained to replicate the trajectories in D. Thanks to its contractive behavior, even the robot starts from an unseen state ýo E Vo at deployment time, the generated trajectory will converge to those originating within D, likely leading to successful task completion. Still, evaluating the incurred loss L(yo: 0), as defined in Eq. 8 is crucial for ensuring reliability real-world scenarios. As precisely computing L for every Yo is computationally expensive due to the need to solve ipp( de. Yo), we instead provide an efficient upper bound. To this end, we first introduce an assumption regarding the locality of the initial state.
Assumption 4.1. The initial state yo lies within multi-focal ellipse region (Vineze 7982) with M focal points at the initial conditions in the dataset D, defined as:
M
Vo = -
Σ
150
- yo 1 R..
(12)
m=1
where R. constant scaling the region and M the number of expert demonstrations.
Assumption 4. requiirs the initial state to lie within bounded region relative to those in D, which realistic restriction. Under this assumption, we upper- bound the loss for given initial state.
Theorem 4.1. Consider a contractive policy the with contraction rate " and constant a as specified in Definition 2.7] Assume the loss function is the MSE and that all expert demonstrations and policy rollouts have the same length H. For ÿo E Yo satisfying Assumption 4.1 it holds that:
a R² (e 27 -
L(ýo: 0)
Am Am 1(5m
2 H 1)
where yth is the m-th demonstration, yrh m is the solution to ivp(óe. and L is defined in Eq. 8
The proof is outlined in App. A.2 Theorem 4.1 validates the intuition that contractive policy per- forms effectively in deployment through upper- bounding the incurred loss. The bound consists of two terms. The first term is a weighted sum of MSE between rollouts starting from the initial states in D and their corresponding demonstrations. These MSE values are precomputed, leaving only the coefficients Am [u] to be calculated for each ýo. The second term (ii) accounts for un- certainty in Yo. It decreases when R is smaller, reflecting less uncertainty, and when 6e is more contractive- either through smaller or a higher y-as as the influence of yo fades more quickly. Finally, we present an upper bound on the true loss.
Corollary 4.1.1. Under the same conditions Theorem 4.1 the true loss L defined in Eg. 10 upper-bounded by:
a² R22 22
C(po; 0) 5
max me (1,",,
1(Sm) +
M 7 -

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Target
Expert
00S
In-sample
Figure 4: In-sample and OOS policy rollouts for selected 2D tasks in the LASA dataset. The training process promotes higher contraction rates App. E.2). resulting in effective OOS recovery.
Target
Espert
In-sample
SCDS
SDS-EF
SNDS
Figure 5: Comparing SCDS the selected baselines on in-sample and OOS rollouts the 2D task space. BC results in diverging trajectories due to the lack of stability guarantees. While SNDS and SDS-EF ensure global stability and reach the target, they display large deviations from expert data. In contrast, SCDS efficiently recovers from OOS states through its contracting transient behavior.
where po is any distribution over the set Vo satisfying Assumption 4.1
The proof is provided App. A.3 Similar to Theorem 4.1 the bound tightens when the policy replicates the demonstrations more closely and the uncertainty level is lower, either due to smaller R and a or a larger 7
5 EXPERIMENTS
We conduct empirical studies the LASA dataset (Khansari-Zadch & Billard 2011), well the more complex Robomimic dataset (Mandleaar et al. (201)). These datasets are detailed in App. We deploy our policies in the Isaac Lab (Orbit) simulator (Mittal al. 2023) for the Franka Panda manipulator arm and the Clearpath Jackal wheeled robot. A highlight of the results is discussed here, while further experiments, computation time comparisons, and ablation studies the impact of the contraction rate, the latent space, and the coupling layers are provided in App. E
5.1 SETUP AND BASELINES
Baselines. We compare the accuracy of our approach to the following baselines: Stable Neural Dynamical System (SNDS) (Abyaneh et al. 2024)) Stable Dynamical System Learning using Fu- clideanizing Flows (SDS-EF) (Rana et al. (020). and Behavioral Cloning (BC) (Pomerleau) 1988] Mandlekar et al. 2020). Although BC does not guarantee stability, we include in our empirical evaluation to provide further ground for comparison. Other baselines trained using both states and their time derivatives, while SCDS relies solely on state measurements for training. The imple- mentation details and hyperparameters are discussed in App. H2
Evaluation. After training each policy, we generate two sets of rollouts: one with the initial state drawn from the dataset D (in-sample rollouts), and another with the initial state not in D (OOS
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Targer
Expert
In-sample
Can
Lif
LiB
nedem:
a. denss 30
n.dems:
n. deme:::: 50
Figure 6: In-sample and OOS policy rollouts for the 6D Can and Lift tasks in the Robomimic dataset, using both and 50 demonstrations. SCDS generates contractive trajectories even in higher- dimensional spaces, where expert trajectories exhibit non-converging patterns.
rollouts). To generate an OOS initial state, we first sample one of the demonstrations uniformly random and then sample uniformly from a hyper-sphere around it. The hyper-sphere around yo has radius of 0.1 y 1| The loss from in-sample rollouts corresponds to the empirical loss defined in Subsee. 3.2] while the OOS loss approximates the true loss given by Eq. 10] A low in-sample loss indicates the policy is expressive, whereas low OOS loss signifies it generalizes well. We use MSE and soft-DTW as the metrics é to capture the difference between rollouts and expert trajectories.
5.2 RESULTS AND DISCUSSION
LASA dataset. Fig. 4 illustrates OOS rollouts for different expert handwriting demonstrations in the LASA-2D (position) and the LASA-4D (position and velocity) datasets. For all random initial states, the induced trajectories reliably contract towards the expert demonstrations and accurately imitate them with high precision. Next, in Fig. 5 we compare the performance of SCDS with the introduced baselines for one of the motions. As shown, some trajectories generated by BC diverge from the target. which is expected since this baseline lacks stability guarantees. While the stable SDS-EF and SNDS baselines ensure convergence to the target, they lack efficient OOS recovery and deviate from expert data. Unlike the stable baselines, SCDS generates smoothly contracting trajectories that converge to the target, offering more precise replication of expert demonstrations.
Subsequently, we extend our comparison to all motions in the LASA dataset and summarize the results for in-sample and OOS rollouts in [Tb. 1 We also consider 4-dimensional version of the LASA dataset by replicating both the expert's 's position and velocity. As shown in the table, SCDS outperforms produces highly competitive results across all scenarios.
Robomimic dataset. We utilize the Robomimic dataset, specifically 6D (task space position and orientation) and 14D (joint position and velocity) expert demonstrations, to highlight the applica- bility of our method in high-dimensional spaces. Expert trajectories do not necessarily end at the same target, in contrast to the LASA dataset. |Fig. 6) shows OOS rollouts for policies trained single and multiple expert trajectories. When multiple trajectories are used, the contractive policy learns contract an average behavior to accomplish the task. Tab. presents results averaged across four tasks: Lift, Can, Square, and Transport. As shown, previous methods tend to overfit the dataset, resulting in high errors on OOS data. In contrast, SCDS demonstrates strong generalization, reducing the best baseline result by up to factor of 2.5.
Upper bound. We compute the upper bound on the true loss when using the MSE metric following Corollary 4.1.1 and report it in Tab. The constant a that appears in the bound is approximated empirically using a Monte Carlo approach. As expected, the MSE upper bound is consistently larger yet it remains relatively tight.
than the observed loss in Tab.
Robot deployment. Finally, we deploy the trained policies two sample cases: i) the Robomimic- 14D Lift manipulation task using the Franka Panda arm, and ii) a navigation task with the Clearpath Jackal wheeled robot following the Sine motion in the LASA-4D dataset. In both cases, the rollouts are tested in the Isaac Lab (Orbit) simulator (Mittal et al. 2023). The average OOS soft-DTW errors
Since this bounded region, an encapsulating multi-focal ellipse exists for

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Table 1: Evaluating in-sample and out-of-sample rollouts error on the LASA and the Robomimic datasets. The lowest (best) values for each metric and dataset are highlighted. The expressive power of SCDS enables it to achieve near state-of-the-an performance on in-sample data. More notably, consistently outperforms other baselines in OOS recovery, and conforms with the theoretical upper bound for MSE loss, CMSE as derived Sec.
Export
LASA-2D
LASA-4D
Robomimis-6D
Robomimie-14D
Metric
MSE
MSE
SNDS
0.02
1 0.01
0.72
0.14
0.03
1: 0.001
1.04
0.19
0.65 1: 0.55
1.26
0.70
2.42
1.37
4.15
0.92
BC
0.04
1 0.02
0.88
0.12
0.05
0.03
1.48
0:166
0.56
1: 0.32
1.88
0.20
1.75
0.22
4,86
0.58
SDS-EF
+ 0.01
0.85
0.13
0.02
1.10
0.10
0.50
0.28
1.01 0:0.6
3.30
0:75
5.65
0.58
SCDS
0.02
0.01
0.65
1. 0.05
0.03
0.01
0.72
0.1.1
0.56
0.22
1.05
0.37
1.68
0.45
4.10
0.40
0.6.
3.27
BC
8.63
+ 4.05
16.25
+ 5.27
19.25
7.3.4
27.48 + 6.83
11.05
4 7.41
19.94
+ 9.57
44.98
15.11
37.82
14.13
2.67
SCDS
0.32
0.15
1.72
4 0.54
1.09
0.211
2.58
0.30
0.89
0.26
1.71
0.53
2,68
0.65
6.27
0.48
CMSE
0.49 = 0.07
1.33 # 0.14
1.43 = 0.22
2,91 + 0.45
Tirgel
Expen data
Rebomimic Lift task: 141) joint space
SPace
Target
Figure 7: Simulation rollouts with Franka arm and Jackal mobile robots in Isaac Lab. Both robots start from random OOS initial state and swiftly converge expert data.
measured over 50 different initial states are 3.68 + 0.33 and 1.94 + 0.37, respectively. Snapshots of random rollouts are illustrated in Fig.
CONCLUSION, LIMITATIONS, AND FUTURE WORK
We introduced SCDS: neural contractive imitation policy that ensures all trajectories contract over time and ultimately converge the target. SCDS offers robustness OOS states throughout the transient phase, yielding stronger reliability than asymptotic stability. By parameterizing only con- tractive policies, SCDS learns the policy through efficient unconstrained optimization. Performance is further enhanced by learning in a high- nigh-dimensional latent space and employing coupling layers. Finally, we provide generalization guarantees when robot starts from unseen initial conditions.
Limitations. Learning from rapidly changing expert data demands longer horizon, and thus more computation. such, there exists trade-off between computational complexity and imitation accuracy for intricate expert behavior (App. C.2).
Future work. Subsequent research may extend the current method to long-horizon expert trajec- tories while maintaining reasonable computational load and the contractive nature of the policy. Additionally, RENs can be replaced by other parameterized contraction models. The applicability of SCDS to other robotie tasks, such quadrupeds and drones, is straightforward, and SCDS can imitate intricate expert behavior in these scenarios. Finally, applications contraction theory to stochastic imitation policies (Fu al. 2018) can be explored.
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ACKNOWLEDGMENTS
This work was supported a part of NCCR Automation, National Centre of Competence in Research, funded by the Swiss National Science Foundation (grant number 225155)2251555 and the NECON project (grant number 200021219431).
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Appendix
Table of Contents
Mathematical proofs
15
A.1 Proposition 2.1
15
A.2 Theorem
15
A.3
Corollary|4.11
16
Extended related work
B.1
Stable dynamical policies with restricted representations
17
B.2
Stable neural dynamical policies
17
B.3
Imitation policies with stability guarantees
17
B.
Recurrent equilibrium networks (RENs)
18
Design considerations
19
Contraction rate
Horizon
Soft DTW
19
C.4 Implicit depth and invertible layers
19
D
Continuous REN properties
20
D.1 Detailed structure
20
Universal contraction
Free parameterization constructability
21
E
Additional experiments
22
E.1
Further experiments the LASA datasct
22
E,2
Learning the contraction rate
22
E.3
Latent state dimension)
24
Ablation study; bijective layers
24
B.5
Ablation study: replacing REN with non-contractive model
25
Computation time
25
Longer horizon motions
26
F
Implementation details
28
F.1
Key components of SCDS
28
F2
Baselines.
29
F3
Computational resources
30
Simulation Isaac Lab
F5
Low-level controller
31
G
Datasets
32
LASA dataset
32
Robomimic dataset
32
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A MATHEMATICAL PROOFS
A.1 PROPOSITION
We first state and prove classic linear algebra lemma. Lemma A.1. Let P E Rmxx be an arbitrary matrix and m (P) denote its smallest singular value. For every vector v E R", it holds that:
P
Gmin (
|01
Proof. We start by expanding the right-hand side, as ||Pl|p/|ll = vf Pu/lell By the Courant-Fischer Theoren we obtain Amin ( P) S v² _ p Po/ Finally, since the singular values of P are the square roots of the eigenvalues of p" P, has Amin (P" p"p P) miin ( This completes the proof.
Below, we present the proof of Proposition 2.1
Proof. Let Vi and yo E R.V, represent two arbitrary initial states, and denote the state and latent state trajectories obtained by solving as y", y" and z", z",2", respectively. The proof proceeds in two steps. First, show that the linear projection preserves the contraction behavior at the same rate. We arrive at the following inequality through the following algebraic operations:
P Po (t) < Pe
2 (t) (t)
(norm definition)
< a Po
71 "||2" (0) (0))
(z satisfies Eq. 2
a Pell
Po 2o)z((0) Po 2. (0)]|
(Lemma A.
(13)
(Po)
a||Po| o (Po) as the new
where main (Po) denotes the smallest singular value of Pa. Hence, with a' constant, it follows from Eq. 13 that P& satisfies the condition in Eq.
Furthermore, established Manchester Slotine 2015 that bijective maps preserve con- tractivity. It can also be concluded from the proof in (Manchester & Slotine 2015) that y remains unchanged. Combined with the first step, this proves the proposition.
A.2 THEOREM 4.
We first state and prove a basic inequality.
ΣM M
Lemma A.2. For every I1. TM R4, holds that SM
1/2m
Proof of Lemma
A.2
We begin by rewriting the lemma as follows:
M
ΣM m=
Im
M
M
15
I'm
Σ
M
M
2m
m==1
m=1
m==1
Em'
(14)
Zmm Since Im R4 for all f1.... M), apply the inequality of arithmetic and geometric
means to derive:
M
M
M
M
Lmm'
III
II
Emm
1/M = 1.
(15)
M
Im
m=l
m=
For every square matrix A g with smallest eigenvalue (A) and every vector o E RR", it holds
Amin (A) S "²" Av/lmn//
"This inequality states that the arithmetic mean of list of non-negative real numbers is greater than equal the geometric mean of the same list.
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This inequality satisfies
14 thereby completing the proof.
Proof of Theorem 4.1
Using Eq. 8
and the triangle inequality for MSE, we have:
M Am (Fo) E(8, y"))
L(So: 0)
m=l
M
M
Am (Fo) ((5,, 8rd +
Σ m=1
Am (8o) Ely ym",
(16)
m=1
where y is the solution to vop(áo,So) for an arbitrary Yo E Vo- We expand the first loss term, assuming MSE loss is used:
fly, 8Th
1
Σ
yr
yi
1=
H-1
1
Σ
a²e 27// 11330 - y0 11
H 1=0
a (c 27 1)
1100 80
(17)
H (e
where the inequality follows from the contractivity definition in Eq. 2 evaluated at time 8/ H. We then simplify the first term in Eq. 16lusing 17 and the formula for An given by Eq. 9 keeping in mind that yo yo
M
a² (2 27 1) Am (8o) Ely, )r" z (c-2
M
(18)
H (e-27/ -
ΣM (1)80 - yo 12)
-1
Next, by applying Lemma we obtain:
M
F0 10
EM
11 - -xo0
R2
(19)
M 2
M
M
M
No
Yo
where the last inequality follows from the conic initial set, Go Vo. Eq.
Assumption 4.1
Combining
Eq. 16
18
and Eq. 19 completes the proof:
L(yo: 0) S a² (6 1)
R
M
Σ m
Am (8o) ((ym,,"
(20)
H He 1)
M
A.3 COROLLARY
Proof. First, we use the fact that the convex combination of some values is smaller than their maxi-
mum to obtain from Theorem 4.1 that:
a R (e 14-2
L(yo: 0) S
2(ymh " mE(1,"..M
max
+
H M (c 2//H -1)
for all yo E Vo. Next, notice that a bound holds for every y(0] E Vo. also holds when taking the expectation with respect any distribution over (00], i.c.:
e(ym y + a² R2 (e (c
C(po: 0
E So--P
L(So:
0)
<
max (L,-.,M M)
HM (e 21/H -1
(21)
Since the left-hand side is the definition of the true loss, L(po; 0), the proof is completed.
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B EXTENDED RELATED WORK
We introduced contractive imitation policies in Sec. 1 and an overview of these methods is pro- vided in Tab. 2 In this section, we extend the discussion to imitation policies in a broader context. Specifically, we review both stable and unstable imitation policies.
Table 2: Comparing contractive DS methods for imitation learning. Except for NCDS and SCDS, the rest attempt to solve a computationally challenging constrained optimization problem.
Unconstrained optimization
Adjustable contraction rate
State-only
Learn in latent space
C-GMR (Blocher et 2017
CDSP Ravichandir 2017
CVF
Smdiwani a 2018
NCDS
Motammad
only lower diwerdiimanioonal
SCDS
(ours)
B.1 STABLE DYNAMICAL POLICIES WITH RESTRICTED REPRESENTATIONS
common approach to ensuring global stability is through Lyapunov theory (Devaney. 202T). In this framework, Lyapunov candidate function must be defined each step of policy optimization, ensuring that the policy satisfies Lyapunov conditions with respect to this candidate. To increase the flexibility of these methods, the Lyapunov candidate is typically learned rather than fixed priori.
However, jointly learning both the policy and the Lyapunov candidate is computationally intensive. To alleviate this, prior efforts have often limited the hypothesis class of the policy the Lyapunov candidate, which in turn reduces the expressiveness and accuracy of imitation. In this context, Chansari-Zadel & Billard (2011) employs fixed quadratic Lyapunov candidate with dynamical policy represented by Gaussian mixture model. Nevertheless, trajectories with non-decreasing distance from the equilibrium are not consistent with this Lyapunov function and hence, cannot be achieved.
To address this limitation, Khansari-Zader & Billard (2014) and Abyaneh & Lin (2023) introduce trainable parameterized Lyapunov candidates, while Neumann & Steil (2055) uses diffeonorphisms to achieve similar goals. Despite these improvements, the expressive power remains limited. Ad- ditionally, scalability to higher-dimensional spaces remains challenge, primarily due to the com- plexity the underlying non-convex optimization.
B.2 STABLE NEURAL MYCAL POLICIES
Recent advancements in globally stable imitation learning have explored using neural networks to model policies, enhancing their expressive power. To ensure stability, line of research has focused on learning a simple stable DS and mapping it to a more complex one through trainable diffeomorphism (Rana al. 2020 Zhang al. (022). A diffeomorphism an invertible map that preserves stability and enables the generation of highly nonlinear policies. While these techniques have shown potential even for complex motion tasks, they have several limitations: they require large number of demonstrations during training, tend to produce policies that are quasi-stable and, most importantly, struggle to learn in high-dimensional state spaces due the need to preserve the invertibility of the diffeomorphic transformation.
Another class methods (Kolter & Manek 2019) Sochopoulos et al. 2024 Abyaneh et al. 2024) employs projection at each training step to enforce stability constraints. These projection-bases approaches have the advantage of requiring fewer demonstrations. However, projection does not inherently guarantee the smoothness of the resulting policy. As result, these policies can be difficult to train, particularly in high-dimensions spaces with rapidly changing dynamics.
B.3 IMITATION POLICIES WITH NO STABILITY GUARANTEES
A parallel research direction focuses on learning highly expressive policies without stability guar- antees. In this context, recent advancements in apprenticeship learning (Abbeel & Ng 2004) and
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inverse reinforcement learning (Ziebart al. 2008) can be leveraged for learning imitation policies. For instance, GAIL (Ho & Ermon 2016), AIRL (Fu et al. 2018), and VAIL (Peng et al. 2019) introduce adversarial IRL framework, with AIRL focusing
robustness to dynamic changes and VAIL. improving training stability. DART (Laskey et al. 2017) adds noise to the training data to mitigate compounding errors in "off-policy" methods. Dagger-style approaches (Ross & Bagnell 2010 Menda et al. 2019) iteratively aggregate new data from both expert and novice policies, of- fering some level safety assurance. However, the absence of stability guarantees in these methods means they offer no out-of-sample guarantees, which limits their applicability in real-world robotic environments
B.4 RECURRENT EQUILIBRIUM NETWORKS (RENS)
In this work, employed RENs to model the dynamics of our policy. RENs were introduced in the discrete-time setup by Revay et al. (2023) as a broad class of nonlinear dynamical models with built-in stability and reliability guarantees. Subsequently, Martinelli et al. (2023) extended this idea to continuous-time models, proving similar characteristics. In addition to being contraction by design, RENs highly flexible. For instance, they can represent all stable linear systems, all previously known sets of contracting recurrent neural networks, and all deep feedforward neural networks (Revay al.) 2023). RENs have been employed in various applications, such as system identification (Revay et al. 2023 Martinelli et al. 2023), control systems (Boroujeni et al. 2024). and optimization (Martin & Furier 2024).
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C DESIGN CONSIDERATIONS
SCDS involves design choices that balance accuracy with computational efficiency. In this section, we highlight the key design parameters that impact this trade-off and offer guidance how to select them for achieving optimal performance.
C.I CONTRACTION RATE
As highlighted in Sec. the contraction rate y can either be fixed learned alongside the model parameters. Higher values of y enforce more restrictive behavior, causing trajectories to contract exponentially faster, according
the definition of contraction theory Delimitonn 2.1 This intro- duces the first key trade-off: while higher contraction rates lead to faster out-of- recovery and improved robustness to unseen data, they also reduce the model's expressiveness.
The bijection block in the output map, presented in Eq. 5] helps mitigate this reduction in expressiv- ity. In our experiments, we achieve high contraction rates with bijection blocks of reasonable sizes, maintaining balance between contraction and policy expressivity. The specific hyperparameters used for the LASA and Robomimic datasets are detailed in Tab.
C.2 HORIZON
Extending the horizon length, denoted H, necessitates additional integrations by the ODE solver, which increases the computational complexity of the method. This is exemplified by a differentiable Euler integration step, expressed as:
Ze+1 Zz (t Vt E O,
(22)
The need to retain gradient information across all time steps further adds to memory and compu- tational demands. However, extending the horizon may be imperative to capture long-horizon or fast-evolving expert behaviors. In cases where long horizons are crucial-though not required for the Robomimic and LASA datasets- using more advanced integrators and reducing solver toler- ance can help manage computational costs. The horizon lengths for the LASA and the Robomimic datasets given in Tab.
C.3 SOFT DTW
The computation of Dynamic Time Warping (DTW) using its original formulation (Keogh & Ratanamahatana 2005) is known be resource-intensive. To address this, Soft-DTW (Cuturi & Blondel (017) presents a more computationally efficient alternative while still yielding acceptable precision. The soft-DTW algorithm assumes a matrix of admissible paths A, and defines the loss function as follows:
soft DT."DWW"""""
min 8 ym)
Σ d(ūs. )²
(23)
(R.))T
where min 9 denotes the soft-min operator with a smoothing factor 3, and d represents a distance metric, such the Euclidean distance. The soft mean operator defined by min (aj.... an) -Blog Σ, e aa./3 As + 0+, the soft-DTW loss converges to the original DTW loss. Impor- lantly, unlike standard DTW, the soft-DTW variant is differentiable across its entire domain, making it suitable for use as loss function in model training backpropagation.
C.4 IMPLICIT DEPTH AND INVERTIBLE LAYERS
The expressiveness of the REN architecture can be increased by increasing the dimension of v in Eq. 3 which deepens the implicit layer (Martinelli et al., 2023). However, our empirical obser- vations revealed that this approach substantially increases training time. Consequently, we chose to keep this parameter fixed and instead enhance representation power through the bijection block. Particularly, we show that increasing the number of invertible (bijective) layers enhances the rep- resentation power in App. E.4 The efficient implementation of these layers, covered in App. F.I reduces computational overhead, making the additional computations feasible.
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D CONTINUOUS REN PROPERTIES
We provide the corresponding theory, stating that continuous RENs are contractive for any choice of parameters. We encourage referring the original paper (Martinelli al. [023 Revay et al., 2023) to fully understand the mechanism behind the REN's contractive structure. However, highlights of core theorems are briefly glossed over in this section.
D.1 DETAILED STRUCTURE
We show an abstract format of the original REN formulation in Eq. given by,
A
B
B2
[x(()
bx by by
(tt)
C
D11
D12
w(t)
(24)
(t)
C2
D21
D22_
Lu(t)
Ā
w(t) = (v(()),
(25)
where sx() E R", ut() E Rm and y(() E RP are respectively the state (z in our paper), the input, and output (y in paper) time The function (-)) is entry-wise nonlinear activation function. The input and output of (-) are vtt), w(t) E IR9, respectively.
First, we restate the Eq. used in the main to describe RENs:
z(t)
z(() a (v((t))
2(0,7) v(t)
We make the following key alterations to lqq. 24 to derive Eq. 3 for use in the policy structure:
Remove the output equation, y(t). This involves setting D21 and D22 to zero while replac-
ing the linear transformation C2 with our contractivity-presevee output transformation in
Eq. 5
Eliminate the bias terms, bz, by, bu, as they are not essential for learning the autonomous
policies target.
Abstract the implicit relationship between u and w, eliminating its explicit dependency the latent state.
D.2 UNIVERSAL CONTRACTION
The first theorem in the paper guarantees contraction at a specific rate, contingent upon the existence of two matrix structures, as formally stated below.
Theorem D.1. (Martinelli et al. (223): Theorem 1) The REN architecture Eq. 3 contracting with the rate if there exists matrix P > O and a diagonal matrix A > 0 such that:
A P PA
-CT
PB W
x 0,
where,
W. = 2A - ADn - T.A.A
While some notations in this theorem are abstracted for simplicity in our paper, the key takeaway is that the existence of a positive-definite matrix P, along with diagonal and positive-deffi definnte matrix A, is sufficient to ensure the contractive nature of the architecture described in Eq. A detailed proof can be found the corresponding paper, and extensive experimental results further validate this theory in practice.
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D.3 FREE PARAMETERIZATION AND CONSTRUCTABILITY
Following [heorem D.I it is necessary to prove that there exists a way to construct the model parameter matrices, including A and P, making the entire architecture contractive by design and with the specified contraction rate. The following theorem essentially serves that purpose. It outlines procedure construct all the necessary matrices build contractive-by-desigge REN.
Theorem D.2. (Martinelli et al. (2023): Theorem 3) For any 0 WN,, and for any E, 0, there exist matrices (A,b) I and the matrices (Y, W, Z, P) 0, and way to construct these matrices for any choice of parameters, such that:
(-XT-Y-
U -- - W
P P= tep..
Note that 0 for autonomous models. Then, the latent state, z, in the continuous REN architec- ture defined by Eq. 3
E((()
z(t) ] [o(v(t))
= (2 v(()
contractive with the rate 7 for any choice of 0. Further, matrices (Y, P) and (A,b) can be computed unambiguously described in the proof.
The full proof for this theorem may he found in Martinelli et al. 2023 8. All the matrices
required in the theorem can be efficiently designed and computed using standard automatic differ- entiation tools, such as Py Torch, so that the resulting model is trainable, just like normal recurrent neural network.
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E ADDITIONAL EXPERIMENTS
We present set of additional experiments that highlight various features of SCDS, alongside abla- tion studies its key components: contractive RENs and the output map.
E.1 FURTHER EXPERIMENTS ON THE LASA DATASET
We illustrated in Sec. how the state trajectories contract towards each other. As explained in Subsec. 2.3) the latent state trajectories, also exhibit contraction. To illustrate this, trained a policy with different contraction rates for a sample motion from the LASA dataset and plotted the first element of the latent state when starting from various initial conditions in Fig. 8 In this experiment, the latent state has a dimension of 10. As expected, the latent states contract toward each other, with the contraction speed increasing for larger values of the contraction rate %
Rollouts
State
Contraction
Contaction
Contraction
Conraction 12.0
Time
Figure 8: Contractive behavior of the latent state. The figure illustrates the rollouts over time for varying dimensions of the latent state, under different contraction rates. Higher contraction rates accelerate convergence in both the latent state space and the robot's state space. The dimension of the latent state here is set 10.
In Fig. we complement our experiments Sec. on the LASA dataset by illustrating more expert motions. For all motions, SCDS produces trajectories that contract average of expert demonstrations, resulting in a high imitation accuracy.
Next, analyze the effect of the distance of OOS initial state from those in D in Fig. 10 and Tab. 3 As can be seen in the figure, SCDS demonstrates effective OOS recovery, even when the initial states very far from the in-sample ones.
Table 3: Evaluating out-of-sample rollouts eπror the LASA dataset. Compared to Table I the OOS initial states are sampled from wider distribution with radius of 0.25]| || for the LASA dataset. The lowest (best) values for each metric and dataset are highlighted. These results confirm that the recovery behavior observed in Fig. perssssss across all the other tasks in the LASA dataset.
Expert
LASA-2D
LASA-4D
Metric
MSE
soft-DTW
MSE
soft-DTW
SNDS
4.64 :: 2.50
11.06
2.35
6.57
3,56
14.78
1.08
BC
13.81
+ 6.08
29.25
+ 8:45
32,73
11.45
41.22 +: 11.51
SDS-EF
2.67
0.55
12.12
2.29
4.19
0.78
16.34
3.28
SCDS
0.23
2.12
1 0.70
1.44
0.35
3.40
0.56
E.2 LEARNING THE CONTRACTION RATE
As discussed in App. D and Eq. 3 we can either set the contraction rate to desired value (see Tab. 7) or learn as a model parameter. For learning y, the Lagrange multipliers method can be used which replaces the loss L. with:
M Lang ( m 0,7)) C((y)) m 0) (1(77) c).
(26)
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Targel
Expert
00S
In-sample
Figure 9: SCDS is tested over additional motions in the LASA dataset. In all cases, the induced trajectories accurately imitate the expert, even for out-of-sample initial states.
Start
Targei
Expert
00S
In-sample
R
R 0.2
R-044 0.4
R 0.8
R- 0.1
R- 0.4
Figure 10: Out-of-sample recovery for initial states sampled from regions with different volumes according to the definition of in in/Assumption 4.1 Notably, SCDS recovers even from more distant initial states.
where f..)) can be defined to encourage larger and the constants and are regularization terms. natural choice h can h(()) [7-70]2 with To being baseline value.
Fig. II illustrates the difference out-of-sample recovery between two cases: setting a low con- traction rate (top row) versus promoting higher contraction rates through Eq. 26 (bottom row). As shown, learning leads to a higher contraction rate, which results in trajectories that contract toward each other more quickly.
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Start
Targ::
DOO
In-sampli
Policies trained to have low (first row) and high (second row) contraction rates. SCDS can promote higher contraction rates during training achieve more robustness to out-of-sample initial states and perturbations.
Figure 11:
E.3 LATENT STATE DIMENSION
One of the key advantages of SCDS its ability learn latent space. As discussed Sec. I increasing the latent space dimension enhances the expressive power of the policy. We explore this aspect Fig. 12 for motion in the LASA dataset. As shown, a low-dimensional latent space restricts representation capacity, leading to poor imitation. However, as the latent dimension increases, imitation accuracy improves.
It is important to highlight that the policy remains contractive by design, regardless of the latent state dimension. This ensures that even the poorly trained policy the left-most plot of Fig. 12 remains globally stable and contracting.
Target
Expert data
00S
In-sample
Latent dimension:
Latent dimension: 16
Latent dimension: 64
Latent dimension: 128
Figure 12: Training results for a sample motion in the LASA dataset are plotted for various latent space dimensions. Training in higher dimensional latent space is more effective, especially when facing complex expert behavior that involves multiple demonstrations. Note that the time horizon is fixed here for all the rollouts.
E.4 ABLATION STUDY: BUECTIVE LAYERS
As discussed in Subsec. 2.3 bijective layers efficiently enhance the expressive power of the policy. To empirically validate this, we compare the in-sample loss, which is an indicator of the policy's expressiveness, in two cases: first, training policy with bijection layers (referred to SCDS) and second, training a policy without bijection layers (referred to as SCDS-NB) The REN struc- ture remains identical across both SCDS and SCDS-NB. The results for the LASA and Robomimic
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datasets presented in the first two of Tab. 4 As expected, SCDS consistently and signifi- cantly outperforms SCDS-NB.
However, directly comparing SCDS and SCDS-NB is not entirely fair, SCDS includes more parameters and, therefore, is expected to have greater flexibility. To address this, we introduce third policy, SCDS-NB* which lacks bijection layers but compensates by having more complex REN structure, achieved by increasing the size the implicit layer, No (see (seefSubsec. 2.2p. We show Tab. 4 that SCDS and SCDS-NB* achieve comparable performance. However, note that training SCDS is significantly faster than training SCDS-NB*, This finding demonstrates that using bijection layers more efficient alternative complicating the REN structure when seeking to increase flexibility.
important to
Table 4: Ablation study on invertible layers to improve the policy's expressive power. SCDS the standard policy formulation presented in Eq. 7 SCDS-NB uses the same REN structure but does not have bijection layers. SCDS-NB* makes SCDS-NB more complex by increasing the complex- ity of the implicit layer in the REN architecture. Still SCDS-NB uses no invertible layers. The best results are displayed for 50 trials on in-sample initial states.
Expert
LASA-2D
LASA-4D
Robomimic-D
Robominie-14D
Metric
MSE
soft-DTW
MSE
soft-DTW
MSE
soft-DTW
MSE
soft-DTW
SCDS
0.02 0.01
0.65 = 0.05
0,03 # 0.01
0.72 # 0.12
0,56 0.22
1.05 = 0.37
1.68 # 0.45
4.10 0.40
SCDS-NB
0.43 0.18
2.41 1: 0.75
1.45 0.30
3,38
1: 0.42
1.24 0.35
2.23 1: 0.74
3.49
1 0.85
8,24
1: 0.64
SCDS-NB'
0.01 0.00
0.06
0.04
0.02
0.13
0.45 0.15
0.26
1.81
0.24
4.87 0.32
E.5 ABLATION STUDY: REPLACING REN WITH NON-CONTRACTIVE MODEL
In this section, we analyze the impact the contractivity of REN on the resulting policy. To this end, we replace the REN in Eq. 7 with recurrent neural network (RNN), subclass of REN that is not contractive. Although the RNN architecture captures temporal dependencies, OOS trajectories not guaranteed to converge toward expert data, the resulting policy is no longer contractive.
We illustrate this behavior in Fig. 13 for both the LASA and the Robomimic datasets. Furthermore, Fig. 14 plots the average distance between trajectories for SCDS and the con-contractive baseline over time. Both figures highlight that some OOS trajectories diverge when using the RNN, while all trajectories consistently converge the target when using the contractive REN. This analysis reinforces the intuition behind employing contractive RENs as the foundation of the policy.
Start
Target
Expert data
00S
In-sample
No comiruction
Conitaction
No contraction
Contraction
Figure 13:
The effect of replacing the contractive REN with a non-contractive model, specifically recurrent neural network, is investigated across two candidate tasks. Some of the generated trajec- tories diverge in the no contraction case, which lacks guarantees on out-of-sample recovery.
E.6 COMPUTATION TIME
We compare the training time of SCDS with the employed baselines in Tab. using the same number of epochs and data points. The hyperparameters employed in this comparison are optimized for the best accuracy for each baseline, as detailed in Tab. 6 and Tab. 7 Despite the longer training
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NCON S0o4E
Distane
2.0
0.
1.5
02
5.9
5.0
2..0
Figure 14: Distance between induced trajectories for the policies using a contractive REN (left) and a non-contractive RNN (right), with a uniformly distributed set of initial states around the true initial conditions from the LASA dataset. Since the policy in the right figure is contractive or stable, the trajectories begin to diverge, causing the distance between neighboring trajectories to increase.
times, approach ensures contraction, offering stronger stability guarantee compared to the weaker stability guarantees provided by the stable baseline methods, or no guarantces at all in the case of BC.
is important to note that, since our method relies solely on state data rather than state time deriva- tives, the computational cost be significantly higher in lower-dimensional settings. However, this disparity decreases in higher dimensions, particularly when utilizing parallel computation for the generated trajectories. Additionally, switching discrete REN architecture notably reduce computation time, though at the expense of increased memory consumption.
Table 5: Evaluating the training time on the LASA and the Robomimic datasets in seconds. SCDS using either discrete or continuous REN architecture appears to be less computationally efficient in lower dimensions, but will shorten the gap in efficiency in larger datasets. Note that even though the same number of epochs are used here, the continuous SCDS method typically requires less than half of the number of epochs to achieve the same performance. Although SCDS has longer training times, it offers the stronger contractivity guarantee compared to the stability guarantees of the stable baselines the lack of any guarantees in the case of BC.
Expert SNDS
LASA-2D
LASA-4D
Robomimic-6D
Robomimic- 14D
156.8
14.1
372.5
22.8
435.2 1: 17.9
882.2 1 45.7
BC
57.2
10.1
62.9 + 12.5
91.5 + 17.4
103.0 36.5
SDS-FF
417.1
18.7
658.8 + 27.2
825.7 + 59.9
1150.2 + 94.5
SCDS-Continuous
1794.7
92.8
1767.2
94.6
2190.2
43.2
2711.1 173.5
SCDS-Discrete
32.6 381.9
18.2 509.0
610.5 1 36.3
857.4
E.7 LONGER HORIZON MOTIONS
In this section, we use the Snake expert demonstrations employed for experiments in LPV- DS (Figueroa & Billard 2018) and SNDS (Abyaneh al.) 2024). The Snake motion provides a stronger challenge than the typical tasks in the LASA dataset. Fig. 15 compares SCDS results against the best result mentioned in SNDS. Numerically, the average DTW for SNDS is reported as roughly 0. 058. However, SCDS results in a better imitation accuracy, with an average DTW error of 0.074 with low contraction rate, and 0.097 with high contraction rates as depicted in Fig. 15
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Policy
Reproduced
2222
Expert Demonstrations Start
SCDS (ours) trained Snake data
SNDS results for Snake motion (directly from the paper)
Figure 15: SCDS results for the Snake dataset demonstrate its ability to handle a more complex motion compared to the tasks in the LASA dataset. SCDS is applied starting from closer (left plot) and further (middle plot) initial conditions and is compared against SNDS (right plot). The complexity of the Snake task arises from the Snake dataset's longer time horizon and the need for rapid adaptation in DS learning.
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F IMPLEMENTATION DETAILS
Isaac simulator
Optimized contractive policy
Robotic system
Robot
Yr current,
Expert data
current
Noisy IC
Target
Franka
current state
Y
low-level
current state
lonque
Inverse dynamics
Dext Y++1
Low-level controller
state
Y +++
optimized
g
M(a) nla 4)=
Figure 16: Simulation and real-world deployment pipelines. The trained policy can be deployed to both real and simulated robots with minimal effort. Global stability and highly contractive behavior of induced trajectories ensure safety and precision during these zero-shot transfers.
This section is focused on key components of SCDS's efficient implementation, hyperparameters, and further details about our selected baselines and their architecture. Note that the entirety of SCDS's codebase, including the following modules, are efficiently implemented in PyTorch (Paszke et al. 2019).
F.1
KEY COMPONENTS OF SCDS
As discussed in Sec. 2 our approach benefits from three major design choices to enable (1) contrac- tivity by design and with an adjustable contraction rate, (2) additional expressivity, and (3) efficiently imitating the expert's 's behavior.
Tanps
Sinple sraeeoory
Nonliness DRLeCTOY
Trajctury (50 samples)
Trijettor: (20sse))
Figure 17: Left: Invertible coupling layer structure implemented as part of RealNVP. The affine transformation provides increased nonlinearity while maintaining bijection, and as a result, contrac- tion properties. Right: DTW still effectively compares trajectories with different lengths. This is particularly useful when learning long-horizon trajectories, or dealing with higher sampling rates.
(1) Continuous and discrete RENs network architecture: the Martinelli et al. 2023 continuous time approach, and the original
There are two implementations of the recurrent equilibrium
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Julia implementation of Revay et al. (2023)*s 's discrete-time method f formulation. Our repository supports both discrete and continuous REN architectures, though most of our experiments focus on continuous RENs. The discrete-time approach can be computationally more efficient, particularly over long horizons. However, it is less memory efficient, as the gradient tree expands with the horizon length.
In contrast, continuous RENs are compatible with Neural ODEs and still induce well-posed and contractive dynamical systems. Additionally, their compatibility with various integration schemes provides a degree of freedom in balancing accuracy and computational cost, as well as the ability to evaluate at arbitrary time points without the need for uniform time sampling (Martinelli al. 2023). Hence, the policy learned by using continuous RENs outputs smoother generated trajectories with adjustable granularity.
(2) Invertible coupling layers The RealNVP implementation of coupling layers (Dinh et al, 2017) is key mechanism for constructing invertible transformations while maintaining computa- tional efficiency. These layers split the input data into two parts, where one part remains unchanged,
and the other is transformed using function parameterized by the unchanged portion. Specifically, the transformed part is updated via an affine transformation, where the scale and shift parameters depend on the static part. This ensures that the transformation remains invertible, the unchanged part and the inverse of the affine transformation can be easily recovered. The structure of coupling layers allows for efficient computation of both the forward pass and the log-determinant of the Jacobian, which is crucial for likelihood estimation in normalizing flows.
(3) Soft and differentiable dynamic time warping Dynamic Time Warping (DTW) (Koogh & Ratanamahatana 2005) and its faster variant, soft-DTW (Cuturi & Blondel 2017). are often used to measure similarity between curves by aligning sequences that may vary in speed, timing, or sam- pling rate. However, DTW can computationally expensive, which is where soft-DTW becomes useful. Soft-DTW offers more efficient approximation with adjustable reduced complexity while maintaining relatively high accuracy.
DTW methods outperform MSE in scenarios where temporal variations differences in sampling rate are key to understanding the similarity between expert behavior and generated rollouts.
F.2 BASELINES
This section contains information about the baselines, especially their codebase and parameters, summarized in Tab. 6) Note that we were unable to compare our method against more recent con- tractive methods, especially Molammadi et al. (2024), there is no official implementation released with the paper.
SDS-EF encodes expert demonstrations as dynamical system trajectories on a Riemannian mani- fold, then transforms them into simpler straight-line paths latent Euclidean space using diffeo- morphisms (Rana et al. 2020p. The gradient flow dynamical system is designed so that trajectories in the latent space are linear and converge towards stable equilibrium. By ensuring that the dif- feomorphic transformation used map the original Riemannian manifold this Euclidean space smooth and invertible, the stability of the trajectories is maintained when mapping back to the original space with theoretical guarantees. Hence, SDS-EF can learn and reproduce complex expert data while providing asymptotic stability.
SNDS guarantees global asymptotic stability by jointly training neural networks for both the pol- icy and a trainable Lyapunov candidate (Abyaneh et al. 2024).. Therefore, ensures trajectories converge predictably even under perturbations or out-of-sample initial conditions. SNDS is tested on high-dimensional robotic tasks, and its differentiable loss aligns policy rollouts with expert demon- strations. The method shows strong performance in accuracy and stability across complex environ- ments, such as robotic arm control tasks, but suffers from non smoothness with rapidly changing expert behavior.
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BC is the standard approach to learning from expert demonstrations without additional safety stability guarantees. BC-RNN (Mandlekar et al. 2020 is a variation of the original behavioral cloning, which uses recurrent structures to capture emerging patterns over time. Nonetheless, still employ a vanilla BC implementation, as the experiments are not focused on long-horizon tasks with substantial recurring patterns.
Table 6: Baseline implementation, architecture, and hyperparameters
Bascline implementations
Method
Codebase
Stability certificate
SNDS (Abyanch al. 2024
rithub httm://minnbynnhisstahaa www.eeestaaee--miiiionnnn imitarion policyy
1yppunov, exponential, asyuptetic Lyapunov, arymptotic
SDS-EF (Kana et al 2020
eithub com/mranat ene licaamzzing Nows
(Pomerical 1988 Maidiekar et
2020
ctuub.cmm comt albBCO
None
Baseline parameters
SNDS
SDS-EF
BC
policy network: 256, 256,
bijection blocks:
network: 256, 256, 256, type: ResNer or FeedPorward activation: Re.ReLU regularization:
ICNV aetwork: [2, 64, 64, 1]
hidden layers: 100
activation: zn. Soffool
activation: un.ELU
0.01
coupling layer: 'rfan" le
0.01
Table 7: Hyperparameters of SCDS and their optimal values or ranges are shown for every key model parameter.
Param Deseription
Optimal value(s)
Y
Lower bound on the contraction rate of the policy
[1.0, 18.6). learnable
K
Number of coupling layer to increase the output nonlincarity
4 to 10
N2
Dimension of the latent state space
32 to 64
II
Forward simulation horizon for the generated trajectory Number of invertible layers in the outpul map (also used for hidden blocks)
20 to 50
K
4, 8
B
Complexity-accuree trade-off off parameter for soft-DTW (y in some references)
0.1
F.3 COMPUTATIONAL RESOURCES
For our experiments, we relied on computational server running on the Linux operating system, specifically Ubuntu 24.04.1 LTT. The server was equipped with high-performance NVIDIA RTX 4090 GPU with CUDA drivers version 12.6, which are more efficient when it comes to optimizing loss on several rollouts generated in parallel. Morcover, an array of CPUs, Intel Core i9-9900K, fea- turing 8 cores and 16 threads each, and 64 GB of DDR4 RAM (2x32 GB), were used in conjunction with the single GPU pipeline.
To conduct simulations with Isaac Lab for robotics experiments, required 4-6 GB of VRAM and 4-8 GB of system RAM, depending on the simulation task. We ran the simulations remotely on the server cluster. To interact with these simulations, used the Omniverse Streaming Client, allowing us to connect to the server and receive real-time broadcasts of the experimental scenes. This setup ensured that could handle the demanding requirements of our simulations while maintaining low latency and high-quality visualization during the remote experiments. Note that Isaac Lab enables parallel training and testing across randomized environments, as depicted in Fig. 18
F.4 SIMULATION IN ISAAC LAB
Isaac Lab is a cutting- outting-edge framework built on top of NVIDIA's Isaac Sim designed for creating and simulating complex robotics systems. It leverages the high-peeformance computational power of NVIDIA GPUs to efficiently simulate and train robots in a realistic virtual environment. By
https://isaccssmm www.:aannniiiiiiii..ii.aa.. http://develppernvvva....m//aaaa
github.thhhgt io/ saacc aw/indcx..hiin
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integrating Isaac Sim's physics-basee simulation with Isaac Lab's scalable infrastructure, run multiple parallel environments.
As shown in Fig. 18 this level of parallelism greatly reduces both training and testing time, the latter being particularly critical for our application. SCDS integrated into robotic systems using the workflow described in Fig. 16 The low-level controller translates contractive policies in the joint and task space of the robot into low-level joint torque or velocity commands, depending on the specific application. is important to note that low-level controller is necessary for joint position control, as the policy directly produces the next state. For task space position and orientation, only differential kinematics approach is needed to convert the end-effector pose into joint space.
Start
Target
Start
Parget
Out-of-sample recovery for 128 parallel environment
Figure 18: Efficient parallel testing of aut-of-sample behavior the Isaac Lab robotics simulator. Each robot is given a random initial condition drawn from a uniform distribution around true initial states. The parallel testing 128 sampled initial conditions yields the results in Tab. I
F.5 LOW-LEVEL CONTROLLER
Throughout this paper, we have assumed that there exists a low-level controller that converts the state planned by the DS de into motor commands (c.g., force or torque). This a standard assumption in learning high-level actions in imitation learning [Khansari-Zadeh & Billard 2011 Torabi et al.
2018 Chi et al. 2023), reinforcement learning (Bahl et al. 2020), and optimal control (Boroujeni al. 2021) In this section, we discuss designing this low-level controller.
In our policy formulation, the high-level part of the policy takes the current state the robot to the next desired state. This state representation experiments with the Franka Panda arm in Fig. 7 or in the robot's task space as in the experiments with Clearpath Jackal in Fig. 7
be defined either in the joint space as in
Once the high-level policy determines the next desired state, this information is relayed to a low- level controller responsible for generating low-level torque velocity commands. Depending on the specific requirements of the task and the robot's configuration, we utilize one of three types of controllers: a joint position controller, an inverse dynamics controller, or an inverse kinematics controller. The joint position controller directly commands the robot's 's joints to move to the specified positions. The inverse dynamics controller calculates the necessary torques or forces each joint to achieve the desired motion. The inverse kinematics controller computes the joint configurations needed to attain a specific end-effector position and orientation. This hierarchical control strategy enables precise and efficient motion control by leveraging the strengths of both high-level planning and low-level execution mechanisms.
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G DATASETS
We employ two well-known datasets evaluate our method real-world expert trajectories. The LASA dataset Khansari-Zadeh & Billard (01T) has long been the primary benchmark for learning DS- -S-based policies, we begin experiments with this standard dataset. However, do not stop after obtaining promising results, as SCDS demonstrates capabilities beyond the 2-dimensional and 4-dimensional expert demonstrations in the LASA dataset. To further challenge method, utilize the Rohomimic dataset Mandlekar et al. (2021) for the first time in the literature on learning policies modeled by DSs, allowing evaluate SCDS in more complex 6-dimensional and 14- dimensional spaces.
G.1 LASA DATASET
The LASA Handwriting Dataset is comprehensive collection of 2D handwriting motions recorded from human expert drawing on a tablet. It is particularly useful for visualization, and learning semi-complex dynamics. The dataset contains 30 distinct handwriting motions. Each motion is represented by 7 demonstrations, with the user drawing the desired pattern starting from different initial positions and ending the same final point, typically with some intersection between the demonstrations. The handwriting motions are defined in 2D Cartesian space, with the final target set at coordinates (0, 0) for all patterns, without loss of generality. The dataset provides consistent 1000 data points demonstration, interpolated to handle variations in timing between different demonstrations. The data structure provides insights into both the spatial and temporal dynamics of handwriting patterns, including the position, velocity, and acceleration of each motion.
Despite having 1000 samples per trajectory, the expert behavior may be replicated with very high accuracy using& limited horizon H of 50 to 100. Since SCDS employs the soft-DTW loss, which does not require the trajectories to have the same length, we can efficiently learn every motion in the LASA dataset while keeping the computational complexity minimum.
trajectory
Angle
CShape
CoublcBcodesLiinc
GShops
Leat
MShap
Stege
Sn
Srate
M
S
Spore
Traneznit
Vithag
Velecity profie
V
S
Figure 19: Expert demonstrations in the LASA dataset.
G.2 ROBOMIMIC DATASET
The Robomimic dataset is complete benchmark for robot learning from offline human demonstra- tions, designed to cover a wide range robot manipulation tasks. The dataset includes data gathered from experts performing variety of tasks through teleoperation, and also artificial data generated
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by policies which optimized using well-established reinforcement learning techniques. We fo- cus on the data generated by human experts for a set of designated tasks, namely, "Lift", "Can", "Square", and "Transport". The Robomimic dataset provides detailed measurements of the obser- vation space. Most notably, the dataset features are the robot's joint position (7D), joint velocity (7D), end-effector position (3D), end-effector orientation (3D), etc. We combine these features to get higher dimensional augmented state spaces (14D in the joint space, 6D in task space) to better evaluate SCDS and the selected baselines described in App. F.2
The expert demonstrations for these tasks are depicted in Fig. 20 The Lift and Can tasks are rela- tively easier to perform than the more complicated Square and Transport tasks. Notably, the average number of samples task is 48 + 6 (Lift), 116 = 14 (Can), 151 20 (Square), and 469 + 54 (Trans- port). Generally, we observe that the more complicated task this dataset, the more samples are required to describe it. There are between 200 and 300 demonstrations per task, but we typically restrict the learning to a dominant subset (between 80-90 percent) of these demonstrations train SCDS and leave the rest to the evaluation stage.
Lift
Can
Square
Transport
Position
Orientation
Figure 20: Position and orientation data for different tasks in the Robomimic dataset.
Lift
Can
Square
Trannsoott
Figure 21: Data collection setup for Lift, Can, Square, and transport tasks (Mandlekar al. 2021 We employ this collected data, which available in the Robomimic repository.
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Reviews:
[{"review_id": "8dYke9HEXw", "reviewer": "Reviewer_NYvK", "summary": "The paper addresses limitations in traditional imitation learning (IL) by introducing a framework based on contractive dynamical systems. The goal is to create imitation policies that reliably generalize to out-of-sample (OOS) states and remain stable under environmental perturbations. The approach leverages recurrent equilibrium networks (RENs) and coupling layers to ensure contractivity across all parameter values. By formulating the imitation problem as a differentiable ODE, the authors achieve efficient training with unconstrained optimization. The framework is demonstrated on robotics tasks, with improved OOS performance compared to existing methods.", "questions": "- The paper assumes that there is a given final state and that the initial states may be perturbed. However, how to deal with the cases where the target state is not fixed? For example, in the MuJoCo environments, the only requirement is to have the robot work successfully for 1000 steps and there is no constraint on final state as such. How to deal with this case?\n\n- The above also brings me to the question that if for every expert trajectory in the dataset, the final state changes. Will your method work in this case? If not, how would you address that? Currently, I see that the number of steps in each trajectory can be different but the final state is still the same.\n\n- How much is the algorithm sensitive to the quality of expert demonstration? If the data comes from a suboptimal policy, what should one expect would happen in this framework?\n\n- Maybe I missed this but does the method do interaction with the environment during learning? If yes, then some comparison of number of interactions made by different algorithms is needed.\n\n- The model’s performance is sensitive to the contraction rate, as mentioned by the author, I am curious how critical this is when we have to run the algorithm on some new task.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- The paper introduces a novel approach to imitation learning by employing contractive dynamical systems, going beyond traditional stability guarantees to enhance policy reliability in out-of-sample (OOS) regions and under perturbations.\n\n- The authors provide theoretical bounds for loss term.\n\n- The idea of integrating recurrent equilibrium networks (RENs) and coupling layers to ensure that the policy is contractive regardless of parameter choices seems interesting. This idea enables efficient training with unconstrained optimization and simplifying policy deployment.\n\n- Experimental results on LASA and Robomimic datasets demonstrate improvements in OOS recovery and imitation accuracy over baselines.\n\n- I appreciate that the authors provided code base for reproducibility. Also, the paper is well-written.", "weaknesses": "- No comparison of computational complexity along with wall clock time against the baselines is provided.\n\n- In the policy representation, the usage of RENs and coupling layers adds complexity. I am expecting a discussion around this as well as the complexity of the baselines.\n\n- While I appreciate that the authors provided bounds on the loss term, the result is too simple, I was expecting some sort of sample complexity result to actually see the sample requirements for their method.\n\n- SNDS paper also includes discussion on high-dimensional Snake demonstrations. I don't see experiments on such data in this work. \n\n- I have some questions that I would like to get answers to in the Questions section.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "UCIJKdRJ2s", "reviewer": "Reviewer_cFmX", "summary": "This paper proposes a method for learning an autonomous dynamical system model from data. Specifically, the data is assumed to be in the form of recorded state trajectories from a dynamical system controlled by an \"expert\" policy. The proposed model is \"contractive\" by construction; that is, all solutions of initial value problems (IVPs) using this model exponentially converge towards each other.\n\nThe paper claims that using a contractive model improves robustness of the learned model to out-of-sample (OOS) states. Specifically, if the model and expert-controlled systems are initialized at a state in a bounded region near those in the expert-demonstration data, then the error between the model and expert-controlled rolled-out state trajectories differs from the empirical loss by at most a constant. This claim is accompanied by theoretical and simulated experiment results.", "questions": "- Please clarify whether the work does or does not learn an implementable controller / policy for the system (e.g., robot). In the case of the latter, are there \"real-world scenarios\" (page 6, line 293) where this might be useful, particularly for the robot learning community? Describing these would be crucial to better establishing the proposed contribution.\n\n- Can you remedy the issue of non-injective $P_\\theta$ in Equation (7)? Since there are multiple possible $z_0 \\in \\mathbb{R}^{N_z}$ corresponding to a given $\\hat{y}_0 \\in \\mathbb{R}^{N_y}$ when $N_z > N_y$, is there a way to make a \"best\" choice?\n\n- Please clarify where the submitted experiment code actually uses the weights $ \\lambda_m(\\hat{y}_0) $. Do they make a difference in the results when compared to just using the standard mean?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "The paper shows a good effort put into communicating concepts from contraction theory and dynamical systems in an imitation learning context. The simulation experiments encompass a variety of systems that make them potentially compelling, in particular to the robot learning community. The choice of loss term weights in Equation (9) is, to the best of my knowledge, novel; moreover, their use in establishing the bound in Theorem 4.1 is neat.", "weaknesses": "My primary issue with this paper is its misuse of the term \"policy\", and its presentation as a paper that supposedly does policy learning. As mentioned in my summary of this paper, the proposed method learns a model of an _autonomous dynamical system_, i.e., one that is uncontrolled or has a fixed controller. In other words, given a system described by the ODE $\\dot{x} = f(x)$, or $\\dot{x} = f(x, u) = f(x, \\pi(x)) \\eqqcolon f_\\pi(x)$ with fixed controller $u = \\pi(x)$, the method in this paper learns an approximate model $\\hat{f}$ for $f$ or $f_\\pi$. This is not policy learning, which would instead attempt to explicitly learn a controller $\\pi$. Specifically, in an imitation learning context, policy learning would involve trying to learn whatever \"expert policy\" $\\pi$ gives rise to the data in the expert demonstration set.\n\nThis is an important distinction to make, particularly for the robot learning community that this paper seems to target. Learning an imitating policy would allow a user to then deploy the policy on the actual robot to autonomously recreate expert demonstrations (within some error). However, just learning a model for the closed-loop \"dynamical system + expert policy\" does not enable this. This diminishes the proposed contribution.\n\nThere are also a couple of mathematical issues (although they may be remediable):\n- In the linear projection $P_\\theta z$ with $P_\\theta \\in \\mathbb{R}^{N_y \\times N_z}$, from the paper and the accompanying code, the authors seem to use a much larger latent dimension $N_z$ than robot state dimension $N_y$. However, $P_\\theta$ then cannot be injective (i.e., have full column rank), and so the left inverse $P_\\theta^\\dagger$ does not exist in general (even if it may still be computable without errors on a computer using floating point arithmetic).\n- Each reciprocal in the _harmonic_ mean (not geometric mean as incorrectly mentioned in the appendix on page 16) in Equation (9) is only defined when the initial condition does not match any of the expert trajectories (otherwise you get 1 / 0). However, having the same reciprocal in the numerator may make the limit tend to 1 (since $\\lim_{a \\to 0^+} (1/a) / (1/a + 1/b) = \\lim_{a \\to 0^+} 1 / (1 + a/b) = 1)$, although this should be clarified in the paper. More importantly, the submitted code does not seem to use these weights at all and instead just uses the regular mean (see `loss = criterion(out, expert_trajectory).mean()` on line 83 in `ren_trainer.py`).", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "H4vPsYXIYf", "reviewer": "Reviewer_AXT4", "summary": "The paper proposes a method (SCDS) for imitating trajectories by learning a contractive latent dynamical systems. Contractivity means, that the resulting (latent) trajectories of any two initial states exponentially converge to the same trajectory with contractivity rate $\\gamma$. By enforcing contractivity, the paper aims to put more focus on approaching the goal in the same way as the demonstrations.\n\nFor modeling the latent dynamical system, SCDS uses a recurrent equilibrium network (REN, Martinelli et al., 2023), which already enforces contractivity by design. The latent state is mapped to the observed state by linearly projecting it to the desired dimension and then applying a learned bijection. The parameters of the REN, the projection and the bijection are learned end-to-end by minimizing a weighted MSE between the generated trajectory and the different demonstrations, where the trajectories are generated by first obtaining the latent initial state (by applying the inversed bijection of the observed initial state followed by a pseudo-inverse of the linear projection), and then integrating the latent dynamical system using a neural ODE. For computing the MSE between trajectories of potentially different length, SCDS applies dynamic time warping; the weights for the weighted MLE are computed such that the closest demonstrations obtain the largest weights.\n\nApart from presenting this architecture, the paper shows a lower bound on this weighted MLE for any initial states within a certain region of the demonstrated initial states, and compares the method with baselines on simulated robot environments.", "questions": "- Were the metrics in Table 1 computed by weighting the demonstrations using Eq. 9? Wouldn't this bias the results in favor of the proposed method?\n\n- Why isn't the contractivity in conflict with multimodal trajectory distributions?", "limitations": "", "rating": 5, "confidence": 3, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "The presentation of the method is very clear. Figure 1 illustrates the problem setting quite well, and Figure 2 gives a good overview of the pipeline.\n\nThe experimental results are good, and show that the reproduced trajectories behave similar to the demonstration, for initial states that are close to the demonstrated initial states.\n\nThe proposed pipeline is novel.\n\nThe claims seem to be correct, although the discussion of the experimental results need to be toned down, as the benefits are often not significant (e.g. in Table 1).", "weaknesses": "__Relevance__\n\nThe proposed method has several limitations:\n- First of all, it is to note that the problem setting only involves imitating state trajectories, without accounting for the robot dynamics and therefore the feasibility of the planned motions.\n- The methods assumes that all demonstrations end in the same state, which is rarely the case for practical problems.\n- By applying DTW, the method is not able to imitate velocity profiles.\n\n__Soundness__\n\n- The weighting for the MLE loss (Eq. 9) is not derived from first-order principles.\n\n- Assumption 4.1 seems to be overly strong, as it assumes that *all* (rather than any) demonstrated initial states are close to $\\hat{\\mathbf{y}}\\_0$.\n\n- The mapping from latent state to observed state is not invertible and therefore requires the use of a pseudo-inverse. As a consequence, the generated trajectory might not start at the correct initial state.\n\n- Although the experiments involve settings with multi-modal initial state distributions, it seems like such settings are not compatible with the aim of learning a dynamical system where all trajectories converge to the same one. I assume that approaching a goal from very directions would only be possible with loose contraction rates, which would defeat the main motivation of the paper. \n\n__Experiments__\n\n- The presentation of the experiments in Table 1 highlight the best mean MSEs, although the confidence intervals are quite large, which is a bit misleading.\n\n- The \"OOD\" initial states are still rather close to the initial states of the demonstrations. It would be interesting to evaluate the different methods for more distant initial states. It would also be interesting to evaluate the demonstrations at states that appeared during the end of a demonstration.\n\n- Qualitatively, the generated demonstrations tend to collapse to the same trajectory, which is not surprising given the objective. However, oftentimes, we are also interested in capturing the variance of the demonstration, which can be important for variable stiffness control. In that respect, the baseline SDS-EF yields arguably a better imitation.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "JAfKbVWbr0", "reviewer": "Reviewer_hPP6", "summary": "This paper studies how to learn a policy with contractive guarantee for out-of-sample recovery on a dynamical system. Remarkably, The paper 1) instantiates their DS policy through contractive REN, bijective transformation and neural ODE solver, with careful considerations of their each feature and applicability; and 2) provides thorough mathematical analysis on contractivity of the  proposed methods and an upper-bound of the loss when initial states locate in a multi-focal ellipse region. Finally, the authors provide empirical results on 2 datasets and a simulated environment, demonstrating the favorable performance of their methods.", "questions": "for concern 1,\n\n1. could you provide papers/references to showcase how your problem formulation could lead to  methods with successful real-world deployments/applications on robotic system? \n2. could you provide analysis on the realizability of the trajectory generated by your methods? i.e. does it always generate trajectory that is executable for the underlying robotic system? if not, could you provide analysis or experiments on the performance of your method under this situation? \n\nfor concern 2,\n\n1. In your experiments, could you set the initial states to be in some regions where state visitation density is low and show its performance? \n2. could you try training a diffusion model based policy (with treating the problem as future state generation conditioned on current state and train it with a data augmentation that adds the same level of ellipse noise to conditioned input state) and evaluating it in your experiments? I suspect this would already generate a decent performance on the experiments you conducted.", "limitations": "", "rating": 6, "confidence": 2, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "## originality:\nThe paper creatively integrates several interesting techniques, i.e. contractive REN, neural ODE and coupling layers into their methods, resulting in a successful and analyzable performance. To my knowledge, the method is both novel and effective, with a clean and elegant design.\n\n## quality:\nSome mathematical analysis are performed on the contractivity of the proposed method, SCDS. Notably, an upper-bound of the true loss $\\mathcal{L}(\\rho_0; \\theta)$ is derived and provides much insights.\n\n## clarity:\nThe paper is well written with clean words and appealing figures. It is easy to read and comprehend.\n\n## significance:\nUnder the condition considered (the initial state locates in a multi-focal ellipse of in-data samples), the method proposed offers rigorous guarantee of contractivity and favorable  performance, which is illustrated by numerical and visual results in experiment section.", "weaknesses": "I have 2 major concerns of this work:\n\n1.  the lack of consideration of  control input $u$, and the system dynamics with it, $\\dot{s} = f(s, u)$. Together with the ignorance of first-derivative of the system state. The author's  problem formulation reduces to how to generate state-only trajectory to meet the desired system requirement. Unfortunately, in my opinion, this approach of system modeling and problem formulation doesn't tackle the difficulty we usually meet in Robotics: 1) the real-world robotic system always has dynamics capacity and not every planned trajectory is equally executable hence the real-world impact of SCDS remains unknown; 2) Without the consideration of system dynamics in terms of control input, it's hard to assess the significance of the proposed method. Actually, under this simplified situation, the theoretical derivations and contractivity guarantees seem to be an over-kill.  One could simply generate trajectory interpolating towards the nearest in-sample point then exactly follow its trajectory.\n\n2. the overly-simplified experiment settings. Besides the simplification in problem formulation, the experiment settings are also too narrow. There are 2 settings of the initial states in the experiments: 1) the initial states are sampled directly from the dataset; and 2) the initial states are sampled from a multi-focal ellipse of the in-dataset samples, with a radius of 0.1 of its norm. I think this experiment setting is neither challenging nor fitting for the real-world cases where OOS recovery is in great need.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)