Robust System Identification: Finite-sample Guarantees and Connection to Regularization
Authors: Hyuk Park
Grani A. Hanasusanto
Yingying Li
Venue: ICLR 2025 Poster (2025)
Keywords: dynamical system
time series
system identification
optimization
Primary area: learning on time series and dynamical systems

Abstract: We consider the problem of learning nonlinear dynamical systems from a single sample trajectory. While the least squares estimate (LSE) is commonly used for this task, it suffers from poor identification errors when the sample size is small or the model fails to capture the system's true dynamics. To overcome these limitations, we propose a robust LSE framework, which incorporates robust optimization techniques, and prove that it is equivalent to regularizing LSE using general Schatten $p$-norms. We provide non-asymptotic performance guarantees for linear systems, achieving an error rate of $\widetilde{\mathcal{O}}(1/\sqrt{T})$, and show that it avoids the curse of dimensionality, unlike state-of-the-art Wasserstein robust optimization models. Empirical results demonstrate substantial improvements in real-world system identification and online control tasks, outperforming existing methods.

Published as a conference paper at ICLR 2025
ROBUST SYSTEM IDENTIFICATION: NON-ASYMPTOTIC GUARANTEES AND CONNECTION TO REGULARIZATION
Hyuk Park, Grani A. Hanasusanto, Yingying Li University of Illinois Srrbana-Chappaie (hyukp2, gah, 0_101)@illinoiseed
ABSTRACT
We consider the problem of learning nonlinear dynamical systems from a single sample trajectory. While the least squares estimate (LSE) is commonly used for this task, it suffers from poor identification errors when the sample size is small or the model fails to capture the system's 's true dynamies. To overcome these limitations, we propose a robust LSE framework, which incorporates robust optimization techniques, and prove that it is equivalent regularizing LSE using general Schatten p-norms. We provide non-asymptotic performance guarantees for linear systems, achieving an error rate of ((1/VT), and show that it avoids the curse of dimensionality, unlike state-of-thh the-trr Wasserstein robust optimization models. Empirical results demonstrate substantial improvements in real-world system identification and online control tasks, outperforming existing methods.
INTRODUCTION
Many real-world problems require learning unknown dynamical systems from data. Examples can be from identifying the dynamics of mechanical systems like autonomous driving predicting time-series data such as climate patterns and financial trends (Ng et al. 2006 Hong et al.) 2008 Louka et al. 2008 Brunton al. 2016 Alaskar 2019). In the control community, the problem of estimating the parameters of a dynamical system is referred to as system identification. System identification is crucial since accurate estimation of underlying systems is integral developing safe and reliable control systems.
In this work, we propose a robust system identification method for a certain class of nonlinear dynamical systems, assuming only single trajectory of data is available. Specifically, the system expressed as a linear combination of known nonlinear functions of the state and control inputs. Such system models have been widely applied since they accommodate broad range of dynamic behaviors. One of the simplest system identification algorithms the least squares method or the least squares estimate (LSE), which minimizes the squared prediction errors of the given samples. Due to the stochastic nature of the data, the performance LSE cannot be deterministically guaranteed. Moreover, since the data comprises single trajectory of states resulting from the evolution of the dynamical system, the samples are non-i.i.d. Recent works (Simchowitz et al. [0018] Jedra & Proutiere 2020) provide non-asymptotic analyses of LSE, specifically addressing system identification errors with respect to a finite number of non-i.i.d. samples. They show that the error decays as fast as the optimal rate (LIVTT) where T' denotes the number of samples.
Although these theoretical results are promising, the empirical performance of LSE may suffer, particularly when only few samples are available or the model is misspecified. This limitation is critical in applications where data collection is inherently restricted, or the true dynamics are highly complex. To address these issues, we propose robust approach that combines robust optimization with LSE by formulating a min-max optimization problem, referred to as the robust LSE problem. We show that the robust LSE problem can be cast as convex semidefinite program (SDP), making tractable to solve. Additionally, we provide a non-asymptotic analysis for our approach and demonstrate that robust LSE achieves a near-optimal enror rate Õ /T). Interestingly. we show that our robust LSE problem is equivalent to the LSE problem with additional regularization term

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based on the general Schatten p-norm. While a few special cases of Schatten p-norms have been used to regularize LSE problems (Abbasi- Yadkori & Szepesvári 2011 Sun et al. (022), these methods do not guarantee asymptotic convergence to the true system parameters under single trajectory. To our knowledge, our work is the first to provide a non-asymptotic analysis for LSE with general Schatten p-norm regularization under single trajectory.
An alternative data-driven robust approach to ours is the state-of-the-an Wasserstein robust optimiza- tion (Mohajerin Esfahani & Kuhn 2018), which has gained considerable attention in the machine learning community for its promising performance (Shafieezadeh Abadeh et al. (0118) Liu et al. 2022
Nietert et al., [024 Bai et al. 2024). However, this approach suffers from the curse of dimensionality: for systems with a high-dimensiona state space, its error rate decays slowly.
The contributions of this paper are summarized as follows:
i. We introduce a novel system identification algorithm that combines robust optimization with the LSE framework. Additionally, we establish the equivalence between our robust LSE problem and the LSE problem regularized by the Schatten p-norm. This is significant for the regularization framework because, as noted in Abu-Mostafa et al. (2012). "most of the regularization methody used successfully in practice are heuristic methods. Thhe equivalence, however, enables regularization to borrow a probabilistic interpretation from robust optimization, suggesting that the regularization term should be data-dependent to ensure good sampee (i.e., test) performance.
ii. We provide a theoretical performance guarantee for our robust method, achieving an error rate of (1)/VT). This result is notable not only as the first performance guarantee for regularized LSE under the single trajectory setting but also because it shows that our robust LSE circumvents the curse of dimensionality, unlike the emerging data-driven Wasserstein robust optimization models-hence. offering new insights into the robust regression literature.
ii.. We conduct numerical experiments that demonstrate substantial performance improvements in real-world system identification tasks, such as short-term wind speed prediction. We also conduct extensive experiments with synthetic dynamical systems to validate our approach. Additionally, we showcase its effectiveness in online control tasks by integrating our robust LSE with existing online linear quadratic control algorithms, demonstrating consistently better performance compared to existing methods.
FUTHER LITERATURE REVIEW
There has been recent emergence of interest in deriving non-asympiotic systems identification
errors. Most works focus on analyzing performance of the standard LSE (Simchowitz et al. 2018
Faradonbeh et al., 2018 Sarkar & Rakhlin 2019 Mania et al. 2019; Foster al. 2020, Dean et al.
2020;Jedrr & Proutiere 2020 Sattar et al. [202]] Kowshik et al., 2021 Sattar & Oymak 2022 Mania et al.) 2022 Li et al. 2023]] One advantage of analyzing the standard LSE is that the system identification error term, which is the main interest of the analysis, can be analytically obtained using the solution the LSE problem. This term can then be broken down in various ways, enabling different approaches to address the resulting components.
While theoretical guarantees for LSE appear promising, the empirical performance degrades in real-world applications where available data is scarce, resulting in subpar estimates (Sun et al.) 2022). We employ robust optimization techniques (Ben-Tal al. 2009) to enhance the resilience of LSE. The key idea of robust optimization is to find solutions that perform optimally against the worst-case realizations of uncertain data. Dean et al. (2020) assume i.i.d. samples and utilize the standard LSE for system identification. They construct an uncertainty set of system parameters around the resulting estimate. They then solve a min-max problem, referred to as the robust LQQR problem, to determine the best control input against the worst-case system parameter in the uncertainty set. In contrast, our approach directly formulates a min-max problem for system identification, seeking the best system parameter against the worst-case realizations of the data. To the best of our knowledge, no prior work has proposed a robust LSE problem formulation presented in this paper.
As stated in contributions, the non-asymptotic analysis proposed in this paper is not limited to our robust LSE. It can be extended to the regularized LSE problem. where the regularization term is defined as the Schatten p-norm of a quadratic function of system parameters multiplied by
2

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user-defined (scalar) tuning parameter- henceforth referred to the regularization parameter. Special cases of the Schatten p- p-norm regularization are proposed in the literature (Abbasi- Yadkoni & Szepesvári 2011: Sun et al. 2022). In Abbasi-- Yadkori & Szepesvári (201T), they introduce the squared Frobenius norm of system parameters with the regularization parameter set to some small fixed value, hence the convergence of their estimate the true system parameter is not guaranteed. In Sun et al. (2022), they consider systems with limited state observations, i.c., states cannot be directly observed. They introduce the nuclear norm of the Hankel matrix to their LSE problem and derive the non-asymptotic impulse response estimation errors only for the MISO (multi- input single-output) system under the assumption that i.i.d. samples are available. These papers do not provide guidance on how to adjust the regularization parameter as function of the number of available samples.
NOTATION
Bold lower-case letter and upper-case letter X represent vector and matrix, respectively, while regular font æ indicates a scalar. For any square matrix X E Rexn, the trace operation is denoted tr( (r( The spectral radius denoted ((X) ) is the largest absolute value of the eigenvalues of X. An n X n dimensional identity matrix is denoted as 1n. For any n X m matrix X E Ruxm | lp represents the Schatten p-norm of the matrix, which is defined as | (tr(X') X )3 For several special cases of the Schatten p-norm, we may interchangeably use the following notations: nuclear norm || ||. = || II.. Frobenius norm || || = || ll2. and operator norm || || 11x In the entire paper, we will not use matrix norms induced by vector norms to prevent any confusion. We may use | 2l2 for the Euclidean norm (i.e., e norm). In this case, the notation should still be clear since the norm is taken on a vector (i.c., bold lower-case). To denote an n X n positive (semi definite matrix X, we may interchangeably use X + (X E SQ) and X 0 (X 0). We use the standard (-)) notation to describe the complexity of function. In addition, (() is used to suppress multiplicative terms with logarithmic dependence.
PROBLEM STATEMENT
Consider the following class of nonlinear dynamical system models
1 0 p(1,, u.) + Wt. t = 0
(I)
Let E RR, E R" and u. E R represent the state, control input, and noise at time I, respectively. The feature map Φ T x R. - RR is an arbitrary, known nonlinear function. To simplify notation, we the feature map interchangeably as Φ(2,, uz) and o(zz), where øt [x. u, E | is the augmented vector the state and control input.
Our goal is to recover the unknown parameters 0* R2xp from a single trajectory of data. The model (T) is versatile enough to capture a broad range of real-world applications, from mechanical systems like autonomous helicopters and bipedal robots to time-series models commonly used in
financial markets, weather prediction, and epidemiology for modeling disease spread (Ljung 1998
Ng al.
2006 Hong al. 2008 Louka al. 2008 Brunton al. 2016 Alaskar 2019).
2.1
LEAST SQUARES ESTIMATE
The least squares estimate (LSE) is widely used for system identification. Given a single trajectory (T- 3r), the LSE denoted as Or minimizes the sum of the squares of the residuals:
1 + T
or E arg min
Σ
|
t=0
Let us refer to the minimization in (2) as the LSE problem. Note that the objective function in is quadratic in 0. Therefore, we can rewrite the LSE problem as
min
æt-1 Ō(zz)
I, T
0
Ø141
min tr (G())
(3)
T 1=
0
0
(zz))
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T T
where G
n
O0 and 22
1
x++1 ((zi)
x++1 ((z))
(4)
We can express the true LSE problem by substituting DD in with its expectation, namely, S E E( Then, obtain the system parameter by solving the true LSE problem:
0" E arg min ir (G(0)))))
(5)
From (5) clear that obtaining the true system parameter requires knowledge of Ui-- while the empirical estimate 52y inherently contains estimation errors that depend on the available data. In other words, poor estimate f27 may lead to inferior performance, which is commonly the case when the sample size is small or, in our context, short trajectory of data.
ROBUST LEAST SQUARES ESTIMATE
As mentioned earlier, the estimate 6yy based on T non-i.i.d. samples may fail to accurately capture Oz when T is small. Even with a sufficiently large T, the standard LSE 0r (2) may still perform poorly in practical upplications where the model (I) does not adequately reflect the true behavior of the dynamical system. To address this issue, we formulate a robust version the LSE problem to obtain a robust estimate, denoted as O
Or E arg min max tr (G(0) where - nEU
= { E s : 1|2 - Erllp sc]-
(6)
The proposed approach (6) first constructs the uncertainty UP which contains all positive semidefinite matrices Ω that are within a distance 2 from the estimate 2n in the Schatten p-norm. Then, seeks a minimizer 0 that performs best under the worst-case matrix Z in U.f However, the min- problem in (6) is difficult to solve directly since the objective function involves a maximization problem. In the following, we introduce an equivalent semidefinite program (SDP) for the robust LSE problem. To knowledge, the proposed formulation has not been derived in the literature of robust regression problems.
Theorem 1. For any given uncertainty sel parameters p 2 (as in the Schalten p-norm) and the robust LSE problem in (6) can be equivalently reformulated as the SDP
min S.t.
tr(
c
T
0
R
(n+m)
r
TES2atm
H $s4 S
I
0
r
(7)
0
H
Lm
TT
> 0,
where || lly is the dual Schatten norm of [|p that q such that ! y +
1.
The proof of this theorem is in the supplementary material A
Note that the Schatten p-norm defined in (6) corresponds to the Schatten g-norm in the objective function in (7) For any 2 1, the reformulation 77) is convex SDP. In particular, for several choices of q such as = 1, 2, oo, it readily solvable by off-the-shelf commercial solvers. Moreover, the computational complexity of our approach is invariant to the number of samples T, i.e., the size of the SDP ( remains the same regardless of T since the model only requires the matrix Q7.
Interestingly, the robust LSE problem admits an equivalence to the regularized LSE problem as shown in the following corollary.
Corollary 1. For any given uncertainty set parameters 2 1 and e 2 O, the robust LSE problem 6 equivalent to the LSE problem with the Schatten q-norm regularization term as follows:
min (G()) + +el|G(0)|l+
(8)
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The proof of this corollary is in the supplementary material
few remarks are in order. If the nuclear norm (i.c.. 9 = 1) is used in 8). then we have elI e|| = ell0 + en. Thus, the regularization term simplifies to squared Frobenius norm regularization on 0, and the resulting problem constitutes a tractable quadratic program. Corollary [1 further draws an interesting connection between the robust LSE and the regularized LSE in (Abbasi- Yadkori & Szepesvári (2011). In that work, the regularization parameter is set to a small fixed value. However, regulareee
lacks a clear explanation of how the regularization impacts the performance of the LSE since introduced merely to ensure the invertibility of the matrix Σt= Pt+1 (known as the Gram matrix as discussed in the following section). In this case, a convergence rate on the system identification eιror cannot be established. Our result, therefore, not only provides a justification for the use of squared Frobenius norm regularization but also guidance on how to control the parameter the sample size T increases. The recent work Sun et al. (2022) uses a Hankel nuclear norm regularization to identify low-order linear systems using i.idd. trajectories. They recognize that the regularization term yields better performance than the unregularized LSE when the number of samples is small. However, to the best of our knowledge, there is prior non-asymptotic analysis for the LSE with general Schatten norm regularization under the single trajectory assumption. In Section 4 we provide non-asymptotic analyses for the robust LSE problem, which ultimately results in system identification errors.
4 PERFORMANCE GUARANTEES
Among the simplest examples of (I) is the linear system, where the feature map is defined as o(æ,, u.) [a u, ] E 1a+m In this case, the system evolves according .z+11 A' B'u, + Wt, where the unknown parameters are 0 [A' B*] E (=x(n=m), In the literature, much of the statistical analysis concerning LSE performance focuses on understanding the sum of outer products, _T-1 I1411 known as the Gram matrix. Suppose that the noise is Gaussian, e.g.. we ~ NO, 2, and the sequence of control inputs is generated by a Gaussian distribution, e.g., U ~ WN~~ (O, I)) fort = o..... T 1. Then, the expected Gram matrix EE 1(+10141) 1]. which corresponds the first diagonal block of 127 in (5), can be nicely represented as matrix- valued function of the unknown system parameter 0*:
T-1
T-1 t Σ T = - t=0
E
Σ
C++ 26
1+1
E.)(AA)
I=o am0
Of course, the expected Gram matrix not accessible to us since requires 0*.
In this section, we discuss the aon-asymptotic guarantees of our robust approach for linear systems under the single trajectory assumption. Specifically, our goal is show that the system identification errors of our robust method matches the near-optimal rate O(1/VT). This suggests that introducing robustness incurs only negligible costs in terms of T, while providing significant empirical improve- ments over the unregularized LSE in (2) (henceforth referred to as the standard LSE), as discussed in the following section.
As noted in Tsiamis et al. (2023), despite its apparent simplicity, the linear system remains challenging to analyze. The majority of research in statistical learning for system identification has focused on linear systems, as this setting allows for more tractable theoretical analysis. While a few papers have analyzed certain classes of nonlinear systems, these often sidestep the core challenges by focusing on
systems that exhibit near-linear behavior (Foster et al. 2020 Sattar et al. 2021 Kowshik et al. 202T Mania al. 2022).
Sattar& & Oymak 2022
For the standard LSE, many works have established optimal rates of convergence. A key advantage of analyzing the standard LSE is that the system identification error can be directly derived from the analytical solution to the LSE problem in 2 Specifically, the error is given by Or 0"
(E 2 ) T-1 =x= -1 This expression allows the error term to be decomposed in various ways to enable different types of analysis (Simchowitz et al.) [018 Sarrarr& & Rakhlin (019)Jedr Proutiere| 2020). However, these decomposition techniques do not apply to our robust LSE, as the eтror term for the robust estimator, i.e., 0r 0*, no longer has a convenient analytical form. Therefore, different approach is required for analysis.
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4.1 ASSUMPTIONS
We formally state our assumptions for the analysis in this section.
A1. We consider a strictly stable system, i.e., ((*)) <
A2. The data, i.e., observed states of the system [ collected in single trajectory of length T +
denoted as (aze)= E (R*(T+1) with the initial state 3t
A3. Let (Fih20 be a filtration and (æ, 120 be stochastic process such that at is F measurable.
A4. The noise w, I a martingale difference sequence with respect F, with F. and E/ [i-1] Σ.. > 0.
AS. Furthermore, we assume that wy is a o.. re.-conditionally nub-Gaussian random vector with respect to Fe. i..., for any unit vector inner-produce v w; follows the inequality E[exp(v w;)| S exp(Iiu]]o0 /2)).
I" there exists some constant o² 0 such that the
A6. The control input u, E Rh is an i.i.d. -2-sub-Gaussian random vector with E/u,] = 0 and Elu,u, - In other words, we inject sub-Gaussian exploration noise into the system to identify the system parameter 0*.
These are standard assumptions the literature. particular, A3.
A5. enable us to utilize tools from the self-normalized process (Abbasi- Yadkori et al. 201T). The main challenge in our analysis arises from the single trajectory assumption in A222, as this trajectory consists of non-i.i.d. samples. Due to this difficulty, some previous works rely ii.d. trajectories are available, taking only the last state from each trajectory to ensure that those T samples i.i.d. Our main theoretical contribution lies in deriving non-asymptotic guarantees of the proposed robust LSE method (6), using non-i.i.d. samples.
more stringent assumption that T multiple
We first provide the non-asymptotic coverage guarantee of our uncertainty set in (6). which is eventually used as the main ingredient for our system identification error analysis.
Proposition 1. (Non-asymptotic coverage guarantee). For any significance level & E (0, 1]. we have
P [22 E ((,(5) 2 1
(10)
Here, (6) is the uncertainty set for the robust LSE problem (6) defined as follows:
(6( = (2 E mm ||2 - 112-r S <(6) and <(6) Õ(LVT))
(11)
The proof of this proposition is in the supplementary material
Before discussing the main results, we make several comments about Proposition Although (37 provides more explicit form for the upper bound e(ō) in (II) by identifying the universal constants, would be an overly conservative estimate and thus lack practical usage by itself. Instead of relying on such conservative priori bound, it is common practice to calibrate the regularization parameter, using the cross- vross-validation procedure (Arlot & Celisse 2010 Mohajerin Esfahani & Kuhn 2018 Shatieezadeh Abadeh al. 2018 Bach 2024). In the following section, we will use cross-validation to select the initial regularization parameter. In this context, P I becomes practically useful, as the rate O(I/VT) offers guidance on how to scale the regularization parameter of the robust LSE as the sample size increases. We believe that these results are insightful from a broad range of perspectives, including machine learning, system identification, and robust optimization.
the system identification literature, regularization is often applied merely to ensure strong convexity, with the regularization parameter typically set heuristically to a small value (Abbasi- Yadkori & Szepesvári 2011; Sun et al. 2022). Our results, however, suggest that the regularization parameter should be detendependent to achieve good out-of-sample performance guarantees. To the best of our knowledge, there are no existing works that provide theoretical analysis of the regularized LSE under single trajectory.
In the literature on robust optimization, Mohajerin Esfahani & Kuhn (0188) introduces the state- of-the-art Wasserstein robust optimization model under the i.i.d. data setting. While their model

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could, in principle, be alternative to ours, unfortunately, their analysis reveals that the Wasserstein model suffers from the curse of dimensionality, i.e., their enror rate (1//T²) becomes slower as the dimension of the state space, n, increases. Although a more recent work Gao (202)) addresses this challenge by deriving dimension-free non-asymptotic guarantees under the Lipschitz continuity assumption with respect to 0, it may not easily generalize and, more importantly, it is not applicable
to our problem. While the Wasserstein model has garnered significant attention in the machine
learning community for its promising performance across various applications (Shafieezadeh Abadeh al. 2018; Liu et al. 2022] Bai et al.) 2024, Nietert et al., 2024), there is still an open question about whether the error rate can be improved. In this regard, our results provide new insights into the open question by avoiding the curse of dimensionality, even under the more stringent assumption of non-i..dd data from single trajectory.
we now turn to the analysis of the non-asymptotic system identification errors. Specifically, the following theorem applies to both the robust LSE and the regularized LSE, due to their equivalence.
Building upon the insights from Proposition [
Theorem 2. (System identification errors). Suppose that e(0) is the upper bound in 11 Then, for any significance level 8 (0, 11, as long as
T T())= 400
200
+ logdet (TTT:-1)
log
+
2(n + m) log
8
3
where
r
(0))
ou and Ti
+
E 1 7
0
8
we have the following system identification errors
0 - -Orl c(8) m/mm/nnfi (n, m)
P
(2 + 2| + | [el
21-6,
(12)
ā
where ā
24 20
min o2)
The proof of this theorem is in the supplementary material
D
5 NUMERICAL EXPERIMENTS
In this section, we present numerical experiments to evaluate the performance of our proposed method. Both the proposed approach and benchmark models are implemented in Python 3.7. Specifically, neural network model is implemented using TensorFlow (Abadi et al. (2055). while the optimization problem (7) is modeled with the CVXPY (Diamond B (016) interface and solved using the commercial solver MOSEK ApS) 2024). All experiments were conducted on a laptop equipped with a 6-core, 2.3 GHz Intel Core 17 CPU and 16 GB of RAM. The SDP formulations for the examples
in this section can be solved in under 0.1 seconds. In the supplementary material
we provide the E
mean computational times for several example systems.
We compare our robust LSE with the standard LSE for the wind speed prediction problem (Tas cikaraoglu & Uzunoglu 2014) and learning synthetic dynamical systems. Additionally, we consider an online control task where combine robust LSE with the existing online linear quadratic (LQ) control algorithms. We then compare the regret of different algorithms to demonstrate how improved performance in system identification be translated into more reliable control systems.
As commented earlier, while the theoretical error rate ((1/VT) derived in Proposition| | is still useful, choosing the regularization parameter directly from the theoretical upper bound c(ō) leads to a too conservative estimate 0r. In fact, similar argument is made in Dean et al. 2020). Instead of adopting the theoretical guarantee, the authors use the standard bootstrap method to obtain an empirical upper bound (6) on the system identification errors of the standard LSF,, i.e., |0 0rl S z(0). For the robust LSE, we use a 3-fold cross- cross-validation procedure to determine an initial value of the regularization parameter, as follows. We split the samples into three equal-sized subsets where two of the three subsets are put together learn the robust estimate. The resulting estimate is then tested on the remaining set for all (a 10*)/vT where {1, 31 5, 7, 9} and b {-3....,3) This process is repeated three times for different partitions of the samples to choose the that performs best overall.
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5.1 WIND SPEED PREDICTION
We address the wind speed prediction problem as an example of learning an underlying nonlinear time-series model. Accurate wind speed prediction essential for the safe integration wind energy into power grids. However, the nonlinear nature of wind speed makes this task particularly challenging and an active research area, with various approaches being proposed, including physics- based models and neural network- based designs (Louka et al., 2008; Liu et al. 2018; Chen et al. 2021 Cai et al. 2021] Theuer et al. 2021] Hazarika et al. 2022). A recent work Chen et al. (2024) decomposes the wind speed data into simpler nonlinear components known intrinsic mode functions (IMFs) using the Hilbert Hilbert-Huang transform (HHT) based on complementary ensemble empirical mode decomposition. Neural network models employed learn the IMFs, and the standard LSE is used to determine the optimal weights for these learned nonlinear functions.
Note that the nonlinear function of() is not given explicitly in this experiment but learned via the neural network model, making the problem more challenging than our problem setup. We chose the wind speed prediction problem because successful implementation would demonstrate that complex nonlinear systems can be effectively learned by combining machine learning and optimization methods without extensive domain knowledge. Using the wind speed data from fedesoriano (2022). we implemented the optimized HHTTNAR method from Chen et al. (2024) (here simply referred to as LSE), along with our robust version, and evaluated the prediction accuracy the next daily wind speeds. Figure shows prediction results for both methods using single trajectory of 30 sample points (i.c., T 30).
Wind power (in mis)
True daily wind speed
25
LSE
20
Robust LSE
15
10
50
Day
Figure 1: Daily wind speed predictions for 50- period. The training sample size is 30 for both LSE (red) and robust LSE (green). The predictions for the next 50 days are compared to the actual wind speed (solid gray line).
We replicated the experiments across 20 different datasets with varying training data sizes (i.e., increasing 7)) and recorded the root mean squared errors (RMSE) as measure of system identification (i.e., prediction) errors. shows that our approach achieves significant improvement over standard LSE.
200
Mean RMSE (in log)
LSE
Mean improvement tin %6)
Robust LSE
Robust LSE
10
150
100
3
50
Figure 2: Mean wind speed prediction errors over 20 test datasets: mean RMSE (solid lines) on log scale, with the 10th and 90th percentiles represented by filled areas (left) and mean percentage improvement of the robust LSE over the standard LSE (right)
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5.2 LEARNING DYNAMICAL SYSTEMS
Instead of focusing particular dynamical system examples (in the upcoming online control experiments, we will consider standard examples from the literature), we randomly generated five sets of 500 synthetic systems 0 [A* B"] Rx each set having the same spectral radius p(A*) ranging from 0.1 to 1.0. We compared the system identification errors of the robust LSE and the standard LSE as we collected more samples (i.e., increasing T) over time. We observed that the smaller pl A is, the greater performance improvement the robust LSE achieves over the standard LSE. Figure
shows the mean system identification errors the operator norm when pl (A*) = 1.0.
%6
30
Robust LSE
Mean norm errors
3.0
Robust LSE
25
2.5
Mean Improvement
20
2.0
15
1.5
1.0
0.5
40
T
Figure 3: Mean system identification errors over 500 synthetic systems with ((A)) = 1.0: mean errors (solid lines) the operator norm, with the 10th and 90th percentiles represented by filled areas (left) and mean percentage improvement of the robust LSE over the standard LSE (right)
we tested the stability of the synthesized controller based on the estimated systems in a marginally stable system. We also conducted additional experiments, including one on high-dimennn digeenimenii system, and another comparing our method with the Wasserstein model.
In the supplementary material F
5.3 ONLINE LINEAR QUADRATIC CONTROL
To showcase how our robust LSE can used to design reliable control systems, we performed online LQ control tasks using standard examples in the literature: i) the longitudinal flight control of Boeing 747 from Ishihara al. (1992), ii) a marginally unstable Laplacian system from Dean et al. (2020). and iii) UAV in 2D plane from Zhao et al. (021). We considered several online L.Q algorithms proposed recent years: 1) OFULQ from Abbasi- Yadkori & Szepesvári (201 2) STABL BL from (2022). Broadly speaking, these algorithms conduct two main tasks: identifying the system and deriving the best control input. In particular, OFULQ and STABL utilize the standard LSE for their system identification task. Hence, we can replace the standard LSE with the robust LSE which we referred to as 4) R-OFULQ and 5) R-STABL.
Lale et al. 2022). 3) ARBMLE from Mete
For each of the algorithms 1)-5), we conducted 500 simulations over a time horizon of T 1000. recording the mean regrets. Due to space limitations, we present only the results for the Boeing 747 example in Figure see the plots for other examples included in the supplementary [ In Figure we show only the robust algorithms 4) and 5) for the plot. As mentioned, however, our online control algorithms can be adjusted by selecting different
in the SDP 7 to maintain clarity of
values of the parameter corresponding the Schatten q-norm in the SDP (7) A detailed discussion on choosing the parameter q is provided in the supplementary material
Every algorithm presented in experiments requires several parameters. We adopted the parameter setups suggested by the corresponding papers. However, we acknowledge that their setups are not identical to each other. For example, some papers recording regret after = 50, while others assume tight upper bound on | 0zl is available at each time step f. Irrespective of the choice of the Schatten norm parameter q, our algorithms offer significant advantages over other benchmark algorithms. The results demonstrate not only that the robust LSE can be utilized for various online control algorithms, but also that optimizing the regularization parameter in real-time (i.e., with respect T) for both the robust LSE and the regularized LSE is indeed advantageous.

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Published as a conference paper at ICLR 2025
10
Mean regret
108
10
10c
10a
200
800
1000
T
ARBMLE
OFU
...
STABL
R-OFU
R-STABL
Figure 4: Mean regret over 500 replications: i) Boeing 747
CONCLUDING REMARKS
We present robust framework for system identification by leveraging robust optimization to im- munize standard LSE against small-sample estimation errors and model misspecifications. We derive non-asympootic guarantees system identification errors by analyzing the concentration of single-sample trajectory. Notably, robustifying the estimation achieves near-optimal error rate and demonstrates substantial empirical improvements. While our analysis is based on a single trajectory, our framework can be straightforwardly applied to simpler setting where multiple trajectories are available.
Our proposed formulation constitutes a simple semidefinite program, which can be efficiently implemented using standard off-the-shelf solvers. In the special case where the co-norm is used in the uncertainty set, the formulation reduces to an efficiently solvable quadratic program. The experimental results on the real-world wind prediction problem highlight the significant advantages of our robust model, achieving unprecedented performance. When deployed in online LQQ control algorithms, the robust system estimates yield substantially lower regret than standard LSE, further demonstrating the practical benefits of our approach.
Our current work focuses on fully observable systems, which presents a key limitation. Future research will aim to develop robust optimization framework for identifying partially observable systems with performance guarantees. Additionally, while the complexity of our SDP formulation in (8) is independent of the number of samples, it becomes impractical for high-dimensional systems, as solving SDPs generally scales poorly with dimensionality. For large-scale system identification, recommend setting q 1 in (8) reduce the SDP quadratic program, though this approach sacrifices the flexibility of selecting the Schatten norm parameter.
ACKNOWLEDGMENTS
This work is supported by the National Science Foundation grants 2343869 and 2404413.
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SUPPLEMENTARY MATERIAL
A PROOF OF THEOREM
Proof. Dualizing the inner maximization problem with the constraint ||s2 STllp S e given by our
uncertainty set, we have
max min tr(G((0)2) + - 2z0 X20
A/122 12T112
max min tr(G((0)1)) + de max tr [ (2 -(tr S
(13)
X20
(r (Nr 52)
max min tr(G(())2) + de
min
(14)
0=0
X20
=
min ir ( Ae X20,
max 100
(r((G(0) r)22
(15)
S
tr (râr) + de s.t. r =
I 0
= min X
0
T
(16)
tr (Târ) Ac s.t. r
I 0
I
min X20.
H ž 0.
(17)
0
H
0
SA, 0=0
In the first equality 13). we use the definition of the dual norm for 2||k ā2r lp. As in the second
equality (14), we can convert the maximization to minimization since max f()) = min - -())- The third equality (13) exploits strong duality by following the standard results of the convex analysis (see Theorem 1, Chapter in Luenberger (1997), ). The feasible set of (A, T) defined in
(14 is a convex set, and the objective function of the inner minimization problem is convex in (A, r). Furthermore, we can show the existence of an interior point the feasible set, that is, there always exists some such that the following strict inequality holds: |[l < À for any À 0. Hence, strong duality holds. Then, the maximization over Ω in (15) leads to restriction of the feasible set which is given by the constraint in (16). In other words, (G - r) in (5)) needs be negative semidefinite. In the last equality, we linearize the quadratic term 0 T 0 by following Lemma 4 in Mittal et al. (2020). Then, we can combine the minimization in (17) with the minimization over 0 in (7). Finally, reversing the
epigraphic reformulation ||T]| < À in the equality (17) yields the problem formulation (7). which is a semidefinite program.
B PROOF OF COROLLARY
Proof. By reversing the epigraphic reformulation | S À in (6).
we have
min tr (râr)) + eellD s.t. r 1
I,
0
oTo T
(18)
=G())
Suppose that A, B, O and A z B. Then, the following is true: tr(AC) 2 Ir(BC). Recall that positive semidefinite inequality 2 implies ordering on matrices known as Loewner's ordering. One property of the Loewner's 's ordering is that A > B + (A) 2 (B( for all i where a,(-) denotes the i-th singular value of the corresponding matrix (note that the converse not necessarily true). Also, by definition, the Schatten q-norm is equivalent to the &d no-m of the vector of singular values, i.e., ||||| l [[(r((A,,
(A) || = (EE lo, ()) Using these facts, we can
conclude that r G()) holds when I and 0 are the minimizer of the problem (88) Hence, the problem (8) is equivalent to 8)
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C PROOF OF PROPOSITION 1
Proof. Proving that the guarantee 10) holds amounts to showing that the distance between S2 and ft is small w.h.p.: ||2 f2rlp I e(0) w.p. least 1 Ó. Here, we derive the upper bound e(0) for p oo, i.c. the case where the norm in (TT) defined by the Schatten oc-norm (equivalently, operator norm || = || .. Due the equivalence of norms, is easy to show similar bounds for any 2 with different dimensional factors.
Note that S2y can be explicitly expressed follows:
T
a++1 3t
E++1
3t
(19)
T
t=
11+11
(A'x; B²u + wi) x! (A'x, + B w)s u
3t (A*x,
B² + wi)
x u,x,
Ut x; + B un + wi)
u
(20)
Similarly, 24 is expectation following inequalities:
20
i.e., 22 = E[Str]. Hence, using 200. we can establish the
T-1
[2
1rl
S
2
R
L
Σ
T
(=0
1=0
1
B Σ W
T
w
+2 ||
B
E
U
U
T
T
L=
1=0
1=U
(0)
T-1
T-1
T-1
+ +1++ B
T
Σ
U
Σ
i
T
+2
Wi
U
W
(d)
(c)
Our goal is to bound each of the terms (a)-(e), and then combine the results to complete the proof.
(a):
Notice that we analyze the difference between the Gram matrix and its expectation with factor (1/T). Similar results are discussed in Jedra & Prouticre (2020). First, we introduce the preparatory result.
Suppose ({A) for a matrix A _x
Consider block Toeplitz matrix
I
0
A
1n
0
0
H,
R nnxnt
(21)
0
A
1
I,
Then, for any t t2 1, there exists finite constant J(A) > 0 that only depends A such that
H. & J(A) = Σ ||
(22)
==0
where J(A) is specifically the limit of a matrix power series Σ' |||
Jedra & Proutiere (2020) analyze the sample complexity of the unregularized LSE where unknown system is uncontrolled. (i.e., identifying only A+). We can derive a similar result to Lemma 2 in their work.
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Under i..d. sub- Gaussian exploration noise, our dynamic system can be written 2++1 T. where 7. is a zero mean noise with a covariance matrix En = Binon. Then, can define vectorized states of the system up time T::
A'
+
Σ
x1
7
0
0
C/ E RT where Cy = IE
:
0
0
E
++
SHT
TT
7/--1
7T--1
0
0
Σn
Eo
and f
E "R" is isotropic, i.e., EE ]
lhr
(23)
Fr
To simplify the notation, us define the reciprocal of the square root matrix of the expected Gram matrix as follows:
1122
T-1 ΣΣ
-1/2
Σ
M
1=
Then, we can establish the following equalities:
T-1
T-1 M
I|M
Σ a, M-
I,
sup
In u M
(24)
||
1=0
T
sup
Mu 1|
E || x. Mu 27
(25)
|
1=0
sup
E Hy CT/C/// - E
Σ Mu H7 H7 C//22/111
(26)
||
sup
ΣΣ Mu HrC,// - |Σ Hr C
(27)
[|
Mu u
0
0
where EMu
E RT in (26) is block diagonal matrix.
0
0
0
Mu
The first equality 24 1 is the variational form of the operator norm. In the last equality 27 we
f for an isotropic vector &. The objective use the fact that D = tr(D" DEBE }) |II
function in (27) be written _ WE |B Well where (E W indexed by
of matrices W is referred to as chaos process.
We omit the remaining steps since they are identical the proof of Lemma in Jedra & Proutiere (2020) once we recognize that (27) is the supremum of a chaos process. The main idea for the remain- ing steps that the Hanson- Wright inequality (Hanson Wright 1971) provides the concentration bound on (27) when u is fixed. Then, we can use the c-net argument, i.e., discretizing the feasible region -uu ||al|2 < 1} and combining the bounds for all E (()) where U((e) an e-net of U. Following this idea, for 8 E (0, 1]. we have
Pr T
Σ
x
S (()(())
> 1-6 5, where
|| ||M
Hr
||C,ll (log (2) c2n)
[rr| ||C,,|l (log (3)
c2n)
E(c) (6) = uu max
(28)
V
ei T
Note that H in 28 can be further bounded by some finite constant J (A*) due to the preparatory
result (22) However, we have not made the explicit dependence of (() in terms of T' yet as
||M in (28) grows with T. We defer the discussion to where the bounds on (b) and (c) are established since the same issue arises.
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(b) and (c):
The same technique is applied to (b) and (c). Hence, we only show the derivation for (b). Note that since the noise term w is independent of æt, the expectation in (b) is zero matrix. Hence, we only T- wrar ||. Assuming Σi Ty is invertible (at the moment), need to analyze (1/T) -
can break (b) into the product of two terms as follows:
1122
T
1/2
Σ w
Σ
Σ
Σ
T
- 1/2
1/2
$ T
w w
(29)
self-normalized martingule
persistent excitation term
As denoted, the stochastic process in 29 is referred the self self-normalized martingale whose non-asymptotic bounds are already analyzed in Abbasi- Yadkori & Szepesvári (2011). Hence, we can invoke the following results to obtain the bound on the self- self-normalized term.
Suppose that V = -t= a. + V where V e [T/2] T (0*) is positive definite matrix with a universal constant 0, ensuring the invertibility of Vz. Then, for 5 E (0, 1]. we have
1/2
(Vr) 5" EDwll log (Vace(())
P
Σ
w
Σ
S 541
1
(30)
det (0") det T2 ((
long T log
log
(31)
Note that Vi in (30) is the only term that depends on 7 and it increases at most logarithmically as T grows. We make few comments before proceeding: i) the bound (30) has to be probabilistic since the invertibility (i.e., positive definiteness) of Et- x. cannot be guaranteed deterministically; ii) the lower bound on 7 in (31). i.e., the minimum number of samples that ensures the invertibility of T--1 Ty w.h.p., called the burn-in time. Here, we use the big- O notation for the burn-in in time only because we want to streamline the exposition. We make the quantity explicit in the proof of Theorem 2 under sufficient conditions.
Subsequently, we derive an upper bound on the persistent excitation term in similar to one in 288) Hence, can establish the following inequalities:
Note that the term
1/2
1/2
E
T
Σ
xr T
nd
E
C
S n
(T))()) (6)
t=
(32)
w.p. at least 6.
In the first inequality, we the following fact: | for any A, B St. The second inequality follows from (28). By the reverse triangle inequality, we can further derive the following upper bound on the persistent excitation term:
///2|
<
A
B n A- B
1/2
Σ x t=0
<
E
Σ 1=0
+ n v (T)
(33)
Recall that have not addressed the term |M 'Il in ((a) (6). In fact, the term | x x 1 in 333 is equivalent to ||M as denoted above. Using the definition of the expected Gram matrix
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Published as a conference paper at ICLR 2025
p
obtain the following inequalities:
1/2
T-1
1/2
1/2
E
E
Σ
Σ r (0))
1/2
Σ (A')* (o +Ew) (A
)
1/2
T
(A') (o. +ww) (A'T)²
+00
SVT||
Emil 1
(A*)"
B'' + Ew
1/2 ((A))
(34)
-O(VT).
The first equality holds since the expected Gram matrix is positive semidefinite and 34 follows from the preparatory result (2 Here, emphasize |M- grows at the rate of O(v TT). Hence, combining 30 and (3) with the factor 1/T yields that (b) upper-bounder by O(1/VT). Moreover,
since M = OvVT), we can claim that E(a) (6) in 28 is at most (L/VTT).
(d) and (e):
They can be addressed by the standard concentration inequality for a covariance matrix (see Theorem 6.5 in Wainwright (2019).. For (d), under i.i.d. sub-Gaussian exploration noise, we can claim that there exist universal constants C1, C2; C3 > O such that
1= u.u.
ΣT u.u.
2 1 6,
(35)
lo5(())
los( Tea
log( +))
where (6) og ca (v
O(//V)) For (e), we can
Tes
apply the same concentration inequality by defining an augmented random vector v. [u w. ]T since
T
1
IE
u
w.u,
E
Σ
Valn
Σ
T
T
1=1
1=1
Therefore, there exists universal constants E C2 03
such that
T
T
Σ
wiu.
Σ
wis!
S (o)
2 21-
(36)
where (6) max (02,
(v
mmm T
max (o o2t)
(v
log ? Tea
log( "t Tes
O(1/TT).
Finally, combining (a)-(e), we have
<(0) = ci |Vlog (3) + onn
(log (2) c2n)
5td
T
+
T
C3 VT
V
log(T) + log
g
n + m
m T
log
log ($²
+ c1
+
(37)
T
T
Here, C1, C22 C3. C4, and Cs are universal constants. Removing the dependence of 37 on variables other than 7' establishes the guarantee (10).
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D PROOF OF THEOREM 2
Proof. The guarantce (10) in Proposition Timplies that the following holds:
min tr (G(O)E)
S min tr (G(O))r)+ ( + 2 2 8.
0
0
=((0)
=g(0)
Let f(0) and g be the objective function of the true and robust LSE problems, respectively. First, we show that g(0) is an a-strongly convex function with high probability (w.h.p.). Following the definition of strong convexity, showing strong convexity amounts to showing that g can be rewritten as g(0) (() a where g((@) is a convex function and a > 0. Note that (r( contains the convex quadratic function, i.e., tr(1/TEin " 0). As shown in Tsiamis et al. (2023), under i.i.d. exploration noise, the stochastic process of 2₁ [a]] u!' satisfies the block martingale small ball (BMSB) condition with parameters (A, I [A/2) 3/20) where parameter is can be set a positive integer and
[k/2] (0*)
0
k//2
2 l
the covariance matrix of 2|k/2
can be shown that the BMSB condition can guarantee the persistent excitation w.h.p. (see Proposi- tion 2.5 in (Simchowitz et al. (2018)) Therefore, by setting & = 2, we can establish the following persistent excitation the stochastic process 2, for 2 (6) (defined earlier):
T
P
Σ
أ 2 (ll(n+m) 2 1-6, where ā
min fou o2)
(38)
16
20
Hence, we can claim that for any significance level 5 E (0, 1], 9(0) is ā-strongly convex with probability (w.p.) at least 1 1-6 when T is sufficiently large. Suppose g(0) indeed an â-strongly convex function. Then, we can upper-bound the system identification errors as follows:
0 - S 2nd IVVg (0)) 2//min(n, m) V eeg (0*)).
(39)
The first inequality follows from the properties strong convexity. The second inequality holds due to the equivalence of norms. To ease the notation, we define the following block matrix notations for S2 and Ωt
Q W
W E+
W
Ω2-
and ΩT
(40)
W
T
E
Then, we can write the gradient in Subsequently, we can establish the following inequalities:
399
Veg (0*)
2W
20 ËT +
Veg (0 2W + 20' ËT + +e(6)Vol
| - 1-(WW-AW))
+ 20 (E* -B"= + +c())oll(((0)))
AQ AW AW AE (c(()
(41)
sup AQ AW AW AE (c(()
|2A 20' (1)
(42)
sup
2 A| + |20
+
[[W AE
AW
Sc(8)
2c( + [|c(5) + +
= (2) + 02
+ Vo ((0"))l
(43)
The first inequality 4TT) holds due to our guarantee Proposition 1. the next equality 422 we
cancel out the terms W and E" using the optimality condition for the true LSE problem, namely. Vol (0*) = - W 0* E Combining (38 and (39 [[VVeg(0* (39h replaced by (43)
using the union bound yields the claim.
19

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Published as a conference paper at ICLR 2025
E
COMPUTATIONAL TIME
Boeing 747
ii) Laplacian
iii) UAV
100
6.09H-02
5.58E-02
6.61E-02
400
6.30E-02
5.10E-02
5.20E-02
1000
4.90E-02
6.09E-02
5.08E-02
Table Mean computational time (in seconds) over 100 replications for solving the example systems in the SDP formulation; as shown here, the computational time is invariant to the sample size T.
F FURTHER EXPERIMENTS
Mean norm errors
Frequency stabiliity
1.0
0.9
10a
0.7
Time, T
Time, T
LSE
Robust LSE
Figure 5: We consider marginally stable system (i.e., p(Ar) 1). In the second plot, we compute the optimal LQ controller K using the estimated system 0r [Ār Br] and show the frequency
that the synthesized controller is stable, i.e., p(A* B KT) < 1.
Mean run time (sec)
Mean norm errors
0.20
0.15
0.05
Time,
Time, T
LSE
Wasserstein LSE
Robust LSE
Figure Comparison between our robust LSE model (green) and the Wasserstein model (blue), with both models fine-tuned using cross-validation. The Robust model demonstrates superior performance terms mean norm errors. As shown the right subplot, the run time for our model remains constant regardless of sample size while the run time for the Wasserstein model increases with sample size.
20

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Published as a conference paper at ICLR 2025
Mean norm errors
Mean improvement (%)
80
40
10-
25
180
140
200
Time,
Time,
LSE
Robust LSE
Figure 7: Performance comparison a large-scale system: we compare our proposed method (green) with the standard LSE (red) in a large-scale linear system where 0 500 100, Robust LSE shows superior performance, similar to other experiments in this paper.
long-range memory and under-actuated system
Mean norm errors
10
Mean improvement (%)
60
50
40
30
4-
10
40
60
80
40
60
80
Time, T
Time, T
Figure 8: Performance comparison with long-range memory and unter-actuated system: the system has p()*) = 0.995 and A' E R 20x20 and B E 20x2 The results show the superior performance
of ours (green) over LSE (red).
21

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Published as a conference paper at ICLR 2025
G
MORE ONLINE LQ CONTROL RESULTS
10c
105
Mean regret
10
103
102
200
400
600
800
1000
T
ARBMLE
OFU
STABL
R-OFU
R-STABL
Figure 9: Mean regret over 500 replications: ii) marginally unstable Laplacian system:
105
Mean regret
104
103
200
400
600
800
1000
T
ARBMLE
OFU
...
STABL
R-OFU
R-STABL
Figure 10: Mean regret over 500 replications: iii) UAV 2D plane
22

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Published as a conference paper at ICLR 2025
H CHOICE OF THE SCHATTEN NORM PARAMETER !
Since the regularization term in 7 is defined using the Schatten norm, the parameter q-or equiva- lently p in 6 be chosen based the distribution of the eigenvalues of the underlying system. To illustrate this relationship, we conducted stylized experiments, as shown in Figure
Uneven eigenvalues
Even eigenvalues
Mean norm errors
1.2
1.7
1.1
1.6
1.5
1.4
1.3
0.8
1.2
0.7
11
1.0
10-2
10
10a
10a
10a
10-2
10-1
100
10: 10a
Radius parameter, &
Radius parameter, E
==
Figure 11: Choosing the optimal Schatten norm parameter q in 7 We generate two linear systems for the experiment. The system in the left subplot has uneven eigenvalues (λı 0.9 and
A2 = 0 = 0.1), while the system the right subplot has even eigenvalues (A1 = X2 = A3 = 0.9). We conduct 300 simulations across different choices of the Schatten and the radius parameter (on the x-axis), plotting the mean norm error (on the y-axis) to identify the optimal radius for each value of q. As shown, q performs the best for the system with uneven eigenvalues, while oo performs the best for the system with even eigenvalues.
parameter g E {1, oc)
Furthermore, we validated our findings through adaptive control tasks, as demonstrated in Figure 12
Boeing 747 (uneven eigenvalues)
UAV (even eigenvalues)
Mean regret
10'
10a
10'
10
10'
200
600
800
1000
200
400
600
800
1000
Time, T
Time,
T
R-STABL (q = 1)
ARBMLE OFU
STABL
R-STABL (q 00)
Figure 12: Adaptive control tasks with different choices of i) Boeing 747 represents system with uneven eigenvalues, whereas iii) UAV corresponds to system with even eigenvalues. We conducted 500 simulations to evaluate adaptive control tasks using different values of in Robust STABL (TABL..... The results align with our interpretation choice q as described in Figure II
23


Reviews:
[{"review_id": "BJQGmokR4Q", "reviewer": "Reviewer_q1qq", "summary": "Building on recent work by authors such as Simchowitz, Jedra and others, this paper\nstudies system identification for linear systems in the one trajectory setup,\nshowing that a regularized form of least-squares estimation (LSE)---with the\nregularization being a Schatten $p$-norm---has favorable nonasymptotic convergence,\nand (empirically) outperforms unregularized LSE on some standard benchmarks.", "questions": "My main question is regarding the extent to which there is novelty in the proof of the main theorem (Theorem 2).", "limitations": "", "rating": 5, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The problem is a significant one, and the theoretical results presented by the authors are nicely executed and nicely presented.", "weaknesses": "Clearly it's appropriate to regularize least-squares estimators in settings in\nwhich there isn't much data, and for matrix estimands a standard matrix norm such\nas the Schatten $p$-norm is a natural choice.  This would have seemed to be a\nreasonable starting place for the paper.  But, instead of just starting with this\n(statistical) perspective, the current paper starts with a robust optimization\n(minimax) perspective.  In the authors' words, \"seeking the best system parameter\nagainst the worst-case realizations of the data.\"  But the latter motivation is\nrather different from the motivation of small data sets, and it's a bit odd to\nstart with robustness if the focus is small samples.\n\nThe authors do show that the minimax perspective yields the Schatten $p$-norm\nregularization.  Perhaps this is just a matter of taste, but this sequence of\nlogic doesn't make much conceptual sense to me.\n\nA robustness perspective would seem to make sense if the issue is one of misspecification,\nand indeed the authors state that \"it [the LSE] suffers from poor identification\nerrors when the sample size is small or the model fails to capture the system's true\ndynamics.\"  It's the latter statement that seems well aligned with the robustness\nmotivation.  But the paper doesn't seem to follow up on this issue, either theoretically\nor empirically (e.g., by considering misspecification).\n\nWith respect to the theoretical results, they are indeed very nicely executed and presented,\nbut I'm not sure how much novelty there is.  My sense is that this is a redeployment of tools\nfrom the papers of Simchowitz, Jedra and others; I'd be happy to stand corrected if that's not the case.", "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": "LOf3vPdim9", "reviewer": "Reviewer_Wdgh", "summary": "In this paper, the authors propose a novel estimator for discrete-time, single-trajectory system-identification via Schatten-norm regularization. The authors claim that the proposed estimator combines the desiderata of rate-optimal guarantees of the standard least-squares estimator and robustness to misspecification a la Wasserstein robustness. The authors support their claims on some numerical experiments, demonstrating that in the presence of misspecification (e.g. from nonlinear dynamics), the proposed estimator has better convergence properties than the standard LSE.", "questions": "Please see the above Weaknesses section for areas to clarify.\n\nAlso, it would be good to expand on the comparison to the DRO literature. Notably, why is (Gao, 2023) not applicable to even the iid version of this least-squares problem? For example, if we instead pretend to draw each point of the trajectory from its marginal distribution and apply a beta-mixing blocking argument, would (Gao, 2023) be applicable to the independent process?\n\nRui Gao, \"Finite-sample guarantees for wasserstein distributionally robust optimization: Breaking the curse of dimensionality\", 2023.\n\nKuznetsov and Mohri, \"Generalization bounds for non-stationary mixing processes\", 2016.", "limitations": "", "rating": 8, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The paper is in general well-written and straightforward to understand. The method derivation is quite clean, demonstrating that min-max robust LSE can be turned into an semi-definite program, which can then be reduced further to an unconstrained Schatten-norm-regularized least-squares problem. Something of note is that the proposed SDP only interacts with the trajectory through the empirical Gram matrix, and thus does not need to scale with the length of the trajectory. The theoretical guarantees of the estimator also demonstrate the rate-optimality of the procedure, which is otherwise not true for out-of-the-box guarantees of Wasserstein robust estimators. Numerical experiments demonstrate that not only is the estimator performant compared to LSE, it can be slotted into various online control frameworks to great effect.\n\nI think the paper is good enough for acceptance; I have bumped my score up after the author's response.", "weaknesses": "Since the proposed method is a regularized version of the least-squares estimator, it inherently has an additional moving part in the regularization parameter. In theory, for the robustness set to cover the population Gramian with high probability, the regularization parameter is determined by the convergence rate of the empirical Gramian to the population one, and thus can only be as good as the (typically conservative) upper bounds. Therefore, the authors propose as a heuristic to determine the best scaling of the regularization parameter via cross-validation, and demonstrate that this tuned method outperforms LSE by a large relative amount. However, this is somewhat unenlightening, as this is comparing a fine-tuned algorithm to *the* base estimation algorithm (which is technically subsumed by the proposed class of estimators). It would be more informative if the comparison were to a different class of (fine-tuned) robust estimators.\n\nThe paper also assumes strict stability. While this is likely for technical reasons in the proof, as the authors follow Jedra and Proutiere's (2020) Hanson-Wright approach, something that is left wanting in this paper is whether proposed estimator also performs better in the face of (near) marginal stability, since the authors point out in Section 5.2 that the estimator actually performs relatively better in the very-stable regime. In the same vein, it would be more enlightening to see stabilization rate of synthesized controllers from the estimated matrices $\\hat A, \\hat B$ as a function of data, e.g. via certainty-equivalent LQR, as this is a more practical notion of estimator robustness.\n\nThe paper also contains a few confusing parts. Most notably, the uncertainty level $\\epsilon(\\delta)$ in (11) is set with minimal details on the parameters therein. Since this quantity shows up in the error bound in Theorem 2, it would be good to at least show the valid upper bound on $\\epsilon(\\delta)$ derived in the appendix, since the main paper suggests $\\epsilon(\\delta) = \\tilde{\\mathcal O}(1/\\sqrt{T})$ is the quantity up to universal constants, while in reality it contains some (rather important) system-dependent quantities.\n\n**Minor comments**\n- Should provide mathematical definition of subgaussianity in the assumptions.\n\n- Should define what $\\|\\nabla_\\theta \\|\\mathbf A(\\theta^\\star)\\|_q\\|$ is in (12). Also, this seems to differ from the quantity in the appendix proof (42).\n\nJedra and Proutiere, \"Finite-time Identification of Stable Linear Systems Optimality of the Least-Squares Estimator\", 2020.", "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": "VIGwqSgdaQ", "reviewer": "Reviewer_JZsQ", "summary": "The authors study the problem of identifying non-linear system dynamics from a single trajectory. They introduces a robust framework for system identification and reveal the connection to regularized empirical risk minimization. They provide non-asymptotic guarantees for linear systems and demonstrate an improved error rate of $\\tilde{\\mathcal{O}}(1/\\sqrt{T})$, which the authors claim avoids the curse of dimensionality faced by other robust methods such as Wasserstein robust optimization. Empirical results validate the effectiveness of robust LSE in system identification and online control tasks.", "questions": "1) The authors mention the use of different Schatten norms but does not provide insight into the practical or theoretical consequences of varying $p$. Could the authors expand more on this?\n\n2) Could the authors provide more intuitions on the key factors for avoiding the course of dimensionality in their results? The way the error bounds is presented in Theorem 2 it is not clear the benefit of the regularization term.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "**Clarity of exposition:** The paper is well-written and identifies the limitations of the standard LSE in system identification under limited data or misspecified models, providing motivation for the proposed robust LSE.\n    \n**Theoretical contribution:** The paper presents non-asymptotic error bounds that establish the effectiveness of Schatten $p$-norm regularization, positioning the method as a theoretically grounded alternative to Wasserstein robust optimization.\n\n**Empirical Validation:** Several numerical experiments are conducted, including tasks in wind speed prediction and online control, with substantial improvements over classical LSE.", "weaknesses": "**Discussion of the theoretical results:** The authors very briefly discuss their main results and the implications of the regularization term in their bounds. In particular, it is important to note that $T$ still scales with the system dimensions, i.e., $n+m$ in Theorem 2. It is important to make clear the difference between the curse of dimensionality and the unavailable dependence of the system dimension in the burn-in.", "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": "nVUqwf4gO9", "reviewer": "Reviewer_X6xt", "summary": "1. An equivalent SDP formulation for robust LSE with Schatten p-norm regularization \n2. finite-sample guarantees for system identification with single trajectory", "questions": "1. Due to the setup (perfect state measurement and known model $\\phi$), Thm. 1 holds with no significant difference for linear and nonlinear $\\phi$. What if the case where only $y_t=x_t+v_t$ is available or $\\phi$ has some uncertainty? For simplicity, we can assume that $\\phi$ is linear.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1. very nice and solid theory\n2. provide theoretical guarantees for a hard system identification problem", "weaknesses": "1. Is single trajectory a common case in real applications, although it might be an interesting theoretic topic? Could you please provide more motivate examples? Online control of linear system might be one, however the existing works most focus on learning a controller directly rather than following the conventional pipeline (i.e. system identification and then model-based control). What is the advantage of conventional pipeline in the low data region?\n2. The authors claim that unlike the SOTA Wasserstein robust optimization, their approach can avoid the curse of dimensionality. However, no comparison study w.r.t. the SOTA method is provided in the paper. And the examples are of very low dimension. The largest linear model is in Sec. 5.2 with $\\theta \\in \\mathbb{R}^{5\\times 10}$.\n3. Very strong theory but very weak experiments. Most of them are either synthetic or low-dimension datasets. \n4. The results in Sec.4 rely on strong assumptions. For example, if the application is online control or wind prediction, then A2 and A6 do not hold.", "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": "dvcQorP3ik", "reviewer": "Reviewer_pX9E", "summary": "This paper introduces a robust least squares estimation (LSE) framework that combines robust optimization with the LSE approach to learn nonlinear dynamical systems from a single sample trajectory, demonstrating its equivalence to regularizing LSE with a general Schatten \np-norm. For the linear case, the authors provide non-asymptotic performance guarantees of the framework under certain conditions and show that the error rate is invariant to system dimensionality.", "questions": "1. In line 162, the authors state that they can express the true LSE problem by substituting $\\hat\\Omega_T$ in equation 3 with its expectation. Is there any literature to support this? What do you mean by \"the true LSE problem\"? In my opinion, it is more reasonable to obtain the true LSE problem by substituting $x$ and $\\phi$ in equation 2 with their expectation. \n2. How do the authors prove that the optimal solution to equation 5 is $\\theta^*$?\n3. The proposed robust LSE framework is closely related to the Schatten p-norm. Hence, what is the value of p in your experiments? In particular, it is necessary to show the performance of the robust LSE across different norms.\n4. In Sections 5.1 and 5.2, the authors only compare the proposed method with the LSE. Comparisons with other classical methods are highly recommended, such as LSE regularization with special cases of the Schatten p-norms mentioned in line 54 and the Wasserstein robust optimization discussed in line 59.\n5. It seems that the definition of $\\Sigma_\\omega$  in equation 9 is missing.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "The authors propose a robust LSE framework to learn nonlinear dynamical systems and demonstrate its equivalence to the LSE problem regularized by the Schatten p-norm. They also provide theoretical performance guarantees for the framework applied to linear dynamical systems under certain conditions。", "weaknesses": "While the paper considers nonlinear dynamical systems, the theoretical results on convergence are limited to linear cases. Additionally, the paper lacks comparisons with other classical methods, such as LSE regularization with special cases of the Schatten \np-norms mentioned in line 54 and the Wasserstein robust optimization discussed in line 59.", "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)