A New Approach to Controlling Linear Dynamical Systems
Authors: Anand Paresh Brahmbhatt
Gon Buzaglo
Sofiia Druchyna
Elad Hazan
Venue: ICLR 2026 Poster (2026)
Keywords: Online Convex Optimization
Online Control
Linear Dynamical Systems
Primary area: learning theory
TL;DR: We give a new algorithm for online control of LDS based on spectral filtering with exponential improvement in running time.

Abstract: We propose a new method for controlling linear dynamical systems under adversarial disturbances and cost functions. Our algorithm achieves a running time that scales polylogarithmically with the inverse of the stability margin, improving upon prior methods with polynomial dependence maintaining the same regret guarantees. The technique, which may be of independent interest, is based on a novel convex relaxation that approximates linear control policies using spectral filters constructed from the eigenvectors of a specific Hankel matrix.

Under review as a conference paper at ICLR 2026
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A NEW APPROACH TO CONTROLLING LINEAR DYNAMICAL SYS-
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TEMS
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Anonymous authors
Paper under double-blind review
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ABSTRACT
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We propose new method for controlling linear dynamical systems under adversarial disturbances and cost functions. Our algorithm achieves a running time that scales polylogarithmically with the inverse of the stability margin, improving upon prior methods with polynomial dependence maintaining the same regret guarantees. The technique, which may be of independent interest, is based on novel convex relaxation that approximates linear control policies using spectral filters constructed from the eigenvectors of specific Hankel matrix.
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INTRODUCTION
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Controlling linear dynamical systems (LDS) under adversarial disturbances is a core challenge at the interface of control and online learning, with direct relevance to reinforcement learning, robotics, and sequential decision making. Formally, given control inputs u, E R" and disturbances w, E Rd, the state X, E Rd evolves as
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X++1 Ax; + Bu, + W.-
(1.1)
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At each step a convex cost ea (x,, Uz) is revealed, and the learner must choose controls u, to minimize cumulative loss.
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026 The classical Linear Quadratic Regulator (LQR) assumes quadratic costs and i.i.d. stochastic noise Kalman (960),
027 These assumptions rarely hold in modern applications, where disturbances and costs may be structured, correlated, or even adversarial. Consider flying drone the wild: the dynamics (A, B) follow physics, but disturbances such as wind or malfunction are unpredictable. The cost captures both the control objective and the environment, and our immediate actions must adapt over time. For instance, reaching a target location requires adjusting turns on the fly in response to other drones, birds, trees, buildings. Such scenarios call for a more general and robust model of control. We therefore study the fully adversarial setting, with known lime-nnvrriant dynamics, full state observation, and convex Lipschitz costs, commonly referred as online control Hazan & Singh (2025).
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Minimizing Regret in Control. In adversarial settings, minimizing cumulative cost is generally intractable, as the decision maker only observes the loss after choosing the control input. Instead, following the standard approach in online decision making, we aim to compete with the best stationary policy π E II in hindsight. This is captured by the notion of regret, defined as
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T Regrety (A, = Σ ce (x (xx(.u")))
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min Σ ce (x7 u,) TEll
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where xz is the state induced by the algorithm's controls uz, and xạ is the state under the fixed policy л.
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043 A natural choice for IT is the class of linear feedback policies u, = Kx,, standard in control. While optimal in certain settings, finding the best such controller is nonconvex problem. We therefore adopt improper learning. competing with the best linear policy using a broader class.
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046 To enable efficient learning and analysis, it is common impose structure the comparator class for instance, strong stability (Cohen et al. 2018) or diagonal strong stability (Agarwal et al.] (2019)). Following this line, we define in Definition 3.4 the class of diagonalizably stable policies, which preserves sufficient expressiveness for general LDS control.
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Marginally Stable Comparators: For linear policies u, = Kx,, stability is determined by the spectral radius p of While small p ensures rapid
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the closed-loop matrix A + BK, with state evolution x, = E' (A
(A+ (k))--l,-- decay of disturbances, many applications favor marginal stability, where p = 1 - for small 7. This regime preserves long-term memory and yields smoother, more energy-efficiem control, useful in settings like roboties, thermal systems, and satellite dynamics. See Section B for an example.
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1.1
OUR CONTRIBUTIONS
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Spectral Representation for Control: We introduce Online Spectral Control (OSC), which compresses past distur-
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bances into low-dimensional spectral representation. This viewpoint provides universal feature map for online control: reduces the problem to regression on compact spectral features, enabling a provably more efficient algorithm that achieves logarithmic dependence the stability margin while retaining optimal regret guarantees.
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Exponential Runtime Improvement: We prove that
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(OSC) & O(TT
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where y the stability margin. The runtime scales only polylogarihmmicaly in 1/7, improving on the polynomial dependence of GPC (Agarwal et al. 2019a) via fast online convolution (Agarwal et al. 2024a).
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Method
Regret
Time
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Costs
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LQR
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Fixed Known Quadratic
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Online LQ Cohen
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GPC (Agarwal et al. 2019a)
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Online Convex Lipschitz
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OSC (our Algorithm
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Online Convex Lipschitz
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Table I: Comparison of different control methods. The highlighted row corresponds to our proposed approach. In the
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regret bounds, we hide polylogarithmic factors by the notation (-). Our method is the only one to perform in the most general setting with the best running time.
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Empirical Evaluation: JectionSSpresent ablations showing that OSC matches or outperforms GPC across variety of settings, while using far fewer parameters and integrating naturally with nonlinear models.
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1.2
RELATED WORK
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Control of Dynamical Systems: Control theory, grounded in deep mathematical foundations and with a long history of practical applications, dates back to self-regulating feedback mechanisms in ancient Greece. The first formal mathematical treatment is attributed to James Clerk Maxwell (Maxwell) 1868). For a historical perspective, see Fernández Cara & Zuazua Iriondo (2003). The problem of stabilizing general dynamical systems has been shown to be NP-hard (Ahmadi 2016): see Blondel & Tsitsiklis (2000) for a comprehensive survey on the computational complexity of control.
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Linear Dynamical Systems: Even simple questions about general dynamical systems are intractable. In control, Lyapunov pioneered the use of linearization to analyze local stability of nonlinear systems (Lyapunov) 1992). The seminal work of Kalman (1960) introduced state-space methods and showed that any LDS can be controlled under stochastic assumptions and known quadratic costs.
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Online Stochastic Control: Early work in the ML. community on control focused on the online LQQ setting (Abbasi-
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Yadkori & Szepesvári 2011 1 Dean et al. 2018 Mania et al. 2019: Cohen et al. 2019). achieving vT regret with polynomial runtime. A parallel line of work (Cohen et al. (2018) studied online LQR with adversarially chosen quadratic losses, also achieving /T regret. In all of these works, regret measured against the best linear controller in hindsight.
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Online Nonstochastic Control: Recent methods for fully adversarial control are surveyed in Hazan & Singh (025).. These approaches typically learn a parameterized mapping from past disturbances control inputs, with the number of parameters- and hence the ountim--sallng polynomially with the inverse of the stability margin. Our work builds on the setting of Agarwal al. (2019a), proposing more efficient algorithm with similar regret guarantees.
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Subsequent work has refined the regret bounds under additional assumptions. AAgawal et al. (2019b) established logarithmic regret for strongly convex costs in the presence of stochastic or semi-adversarial noise, while Foster & Simchowizz (2020) derived similar guarantees for known quadratic costs under fully adversarial noise.
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Beyond the full-information setting, [Simchowitz al. (2020) extended the framework to partial observation, Minasyan et al. (022) analyzed adaptive regret in unknown and time-narying systems, and et al. (2023); Suggala et al. (2024) studied online control with bandit feedback.
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Online control has also seen applications in diverse domains, including mechanical ventilation (Suo et al. meta-optimization (Chen & Hazan 2023), and the regulation of population dynamies (Golowich et al. 2024
2022
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al.
2025).
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Online Convex Optimization: Our method reduces online control to online convex optimization. For background on regret minimization and online learning, see Cesa-Biaanhi & Lugosi 2006):
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Hazan
(0066)
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Learning in Linear Dynamical Systems: Hardt et al. (2016) showed that unknown linear dynamical systems (LDSs) can be learned by applying random Gaussian inputs and optimizing via gradient descent. This approach was later extended to highherdimennional and marginally stable systems by Sarrar & Rakhlin (2019). More recently, (Bashi al. (2023a) introduced tensor-based methods for learning LDSs, and Bakshi et al. (2023b) extended these techniques to handle mixtures of LDSs, following the formulation of Chen & Poor (002). While these tensor-based approaches avoid dependence on the system's 's condition number, their computational complexity still scales with the hidden dimension a limitation addressed by recent developments in spectral filtering.
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Spectral filtering was originally introduced for sequence prediction in online learning, as a means of bypassing the non-convexity inherent in learning linear dynamical systems (Hazan al.. 2017 Agarwal et al. 2024b).
Spectral Filtering:
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Hazan et al. (2018) extended the method to systems with non-symmetric dynamics, though their analysis incurred a dependence the hidden dimension. This limitation was subsequently addressed by Marsden & Hazan (025), who incorporated autoregressive structure and used Chebyshev polynomial approximations to obtain dimension- bounds.
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While spectral filtering has proven effective in prediction settings, its application to control remains limited. To our knowledge, the only prior use is by Arora et al. (2018), who applied spectral filtering as a black-box subroutine in the offline LQR setting with unknown dynamics. In contrast, we address the online control problem, and develop new spectral filtering approach applicable to any controllable system (Definition 3.D. competing with the best diagonalizably stable policy in hindsight (Definition 3.4). 3.44)
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The key technical difference is that, unlike in prediction settings where past responses are available, online control requires leveraging system stability and integrating over a different set of eigenvalues, resulting in distinct Hankel structure (see (2.1)
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ALGORITHM AND MAIN RESULT
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Our approach employs a convex relaxation of the linear control problem by computing universal spectral filters and then learning a linear combination of the multiplication of past disturbances with these filters, as described in detail in Algorithm| I The spectral filters are universal in the sense that they are independent of the specific linear dynamical system (LDS)) at hand, the initial state, the disturbances, and the cost functions. In fact, they correspond exactly to the top eigenvectors of the matrix H E Rmxx m for some memory m, whose entries are given by
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---)- Hy i
(2.1)
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where is an assumed bound on the system's ' instability margin, formally defined in Definition 3.4 the filters in Figure The number of required eigenvectors, denoted by h, also determines the number of learnable parameters.
We illustrate
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Spectral Filters (y=0.001)
Spectral Filters [y=0.00
Spectral Filters (y=0.05)
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Figure 1: Entries of the first six eigenvectors of Hl0o, plotted coordinate-wise.
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2.1 ALGORITHM
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In Algorithm D we learn a mapping from disturbances w, to controllers u, However, instead of learning a linear mapping of the disturbances, we learn a linear mapping of specific matrix product involving the disturbances and the
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Algorithm Online Spectral Control Algorithm
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Input: Horizon T, number of parameters h. memory m, step size 77, convex constraints K. 1xxxxd
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2: Compute {(o,, +5) the top h eigenpairs of H from Eq. (2.1)
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Initialize M Rxxxxd
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4: for / 0, : do
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Define W W-1ttmm = (W-11... Wi-m] E Rdxm
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Compute control u, Ei=1 of
1/4 / WW
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7: Observe the new state X and record We = X(+1 - Ax - Bu,
Set Mih 11x [M 1/ Me (1)
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9: end for
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10: return M
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filters. Notably, line 7 we explicitly leverage our knowledge of the system to compute the disturbances from the observed state and the chosen controller.
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Finally, Algorithm[[ [ simply an instance of Projected Online Gradient Descent (Zinkevich) 2003), applied to some
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convex set K. in which each element is sequence of h matrices from Inxd, and the loss functions are the specific memory-less loss functions l, as in Definition
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2.2
MAIN RESULT
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While we defer the formalization of our assumptions and notations to Section
3 present our main result here:
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Theorem 2.1 (Main Theorem). Let e₁ be any sequence of convex Lipschitz cost functions satisfying Assumption 3.3 Then, Algorithm achieves the following
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and let the LDS be controllable (Definition) 3.1) and satisfy Assumption 3.2 regret bound:
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CoCIVT 74
(GTdd 733
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Regrety (OSC,
log3
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where S is the class of linear policies defined in Definition
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This result holds under the following choice of imputs:
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(i) 7
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(i) h = log 7 log
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(iii) 7) C2V Tike'
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(iv) K is the set from Definition
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where the constants are defined as follows:
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Co S 103,
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Note that Algorithm maintains most h O (polylog (T/7)) parameters at each step Moreover, the matrix
multiplications between W.-rttm and Φ; (in line Algorithm 1) for each time step E [T], can be computed by
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zero-padding Φ; to be T-dimensional and performing online convolution with the disturbance stream {wz Weer.. Using
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the efficient online convolution technique introduced in Agarwal et al. (2024a) for this task, we obtain the following corollary:
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Corollary 2.2. The average running time of each the
the T executions of the inner loop in Algorithml Ais O (log' (T/r)).
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Proof Roadmap. The proof of Theorem 2. follows a two-step strategy. First, we show that the class of diagonalizably stable linear policies can be approximated by a family of spectral controllers with bounded parameters (Lemma 4.3)
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Second, we analyze the regret of Algorithm [ which performs projected online gradient descent over this spectral policy class. The analysis leverages convexity the loss functions and boundedness of the feasible set. A high-level overview of the analysis appears in Section and full proofs are provided in Sections A
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3
PRELIMINARIES
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3.1
NOTATION
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We use to denote states, u for control inputs, and w for disturbances. The dimensions of the state and control spaces are denoted by = dim(x) and n = dim(u), respectively. Matrices related to the system dynamics and control policy are denoted by capital letters A, B, K, M. For convenience, write w, 0 for all < 0.
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For any t2 t1. define Westt, (edx(t=-(t+t) as the matrix whose columns are W We. in that order. Additionally, given matrices M E RRadd we define Mi:h E RRxddx as their concatenation along the third dimension.
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Given a policy и, we denote the state and control at time t by (xf, u," when following я. If IT is parameterized by a set of parameters θ, and the context makes the inputs clear, use (x, u or (x,(O), (+)) to refer the same quantities. For simplicity, we use (x,, U₁) without any superscript or argument to refer to the state and control at time under Algorithm T
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3.2
SETTING
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We now present the key definitions and outline the assumptions used throughout the paper. Our setting considers adversarial noise and convex cost functions. We begin by defining controllability:
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Definition 3.1. An LDS as in (L.] controllable if the noiseless LDS given by X Ax; Bu, be steered to any largel state from any initial state.
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Since the disturbances are non stochastccccc we assume without loss of generality that X The following assumptions formalize the notions of bounded disturbances and Lipschitz continuity of cost functions over bounded domains.
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Assumption 3.2. The system matrix is bounded, i.e., [B]] S KB- The disturbance at each time step is also bounded, i.e.. ||well W.
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Assumption 3.3. The cost functions cx u) are convex. Moreover, as long as ||x||| ||u||| S D, the gradients are bounded:
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[x, u)ll, (x, u)ll GD.
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In Definition 3.1 Cohen et al. (2018), the notion of a (k, stronglyyy stable linear policy introduced. Since our spectral analysis focuses diagonalization, in Definition 3.4 we extend this to the notion of (ss diddionnallll stable policies.:
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Definition 3.4. A linear policy K is (r,, d)diiooonalizably stable if there exist matrices L, H satisfying A + BK HLH such that the following conditions hold:
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diagonal with nonnegative entries.
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The spectral norm of L is strictly less than one, i.e., || n.
S
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The controller and the transformation matrices are bounded, i.e., ||]]] H|| || H || S.K.
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We denote by S {K : K is (R. diddooonnalialb stable} the set of such policies, and, with slight abuse of notation, also use S to refer to the class of linear policies SXx, where S E S. Each policy in S is fully parameterized by the matrix E gaxd
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Definition 3.4 is similar to diagonal strong stability (Definition 2.3 of Agaawal et al. (2019h)), with the key difference that we require L to have real eigenvalues Note that by Ackermann's 's formula (Ackermann, 1972. Galiaskarov al. 2023), there always exists K E S that controls the noiseless system. Finally, we introduce the tollowing assumption for clarity of presentation, which we later relax in Section
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Assumption 3.5. The zero policy K 0 lies in S
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For simplicity, we assume that K. KB, W > 1 and γ s 2/3, without loss of generality.
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'The requirement of nonnegative eigenvalues can be relaxed by integrating over larger set; it is imposed here for ease of presentation.
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*Assumption 3.5 can be relaxed via a simple reduction, as outlined in Section Theorem 2.1 holds without this assumption, up to additional factor of x in C1.
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4
ANALYSIS OVERVIEW
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We present the proof of our main result here, while deferring the proofs of technical lemmas to the appendix. Before proceeding, we outline the key technical considerations involved in establishing our final regret bound.
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Algorithm D learns a convex relaxation the policy class S, referred to as the spectral policy class, defined as follows:
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Definition 4.1. [Spectral Controller] The class of Spectral Controllers with parameters, memory m and stability Y is
defined as:
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}
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IT R.m,y
SC "..m., (We-1:t-m) = Σ
1/1 M.W, 1:t mbb -i 0i x=1
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(1-))44-11
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where Φ; E RD, RR,,, E R. are the 1th top eigenvector and eigenvalue of H E Rmxm such that Hig" policy in this class is fully parameterized by the matrices M1sh E Rnxdxx
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Any
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To enable learning via online gradient descent, we require a bounded set of parameters:
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Definition 4.2. The set of bounded spectral parameters defined as
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K = Mi:h E Axxxx ||x M
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||4
S
3 MIMa S n² // VV
2h
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In Section A, we prove that the spectral policy class can approximate S up to arbitrary accuracy. Formally, we state this
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as:
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Lemma 4.3. For any linear policy K E S, there exists a SC policy with M E K. such that for any E (0, 1):
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Σ
|o((xM , u.")
cx(xx u S ET,
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(i)
1 log
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} and (ii) h 2 2 log T Plog
Do0CC d Eq
log T log 1/4 (2) log (#G3 )) where C1 is as defined
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in Theorem 2.7
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We further note that in Algorithm]] [ online gradient descent not performed on the actual cost function, but on a modified cost function, referred to as the memory-less loss function:
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Definition 4.4. We define the memory-less loss function at time t as
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(M = ct (x4 (Mas),, un (M1sh))
(4.1)
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Classical results in online gradient descent provide a regret bound with respect to loss functions (M,)))). However, our regret defined in terms of the actual costs e (xt, uz). Nevertheless, in Section B.3 well approximated by & (M(M) We formally state this result as:
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we prove that (x,,, u,) is
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Lemma 4.5. Algorithm is executed with v) = C2v Fim Then for every E [T].
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6C1 60i
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|x, un) - ((((WW) S
log log!/4
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T//
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where Or and C2 are defined in Theorem 2.7
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Proof of Theorem 2.1
For 1/VT, observe that choice of m and h satisfies the conditions in Lemma 4.3 (using 2 1 and 0 < < 1). Hence, using Lemma 4.3 and the Definition 4.4 we get:
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the fact that T, d, C
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min Σ (MM) CK
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min KES
cx u T
CIVT 7
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We now invoke the regret of the Online Gradient Descent, stated Theorem 3.1 in Hazan (2016). By Lemma B.T the set IC is convex, and by Lemma B.2 the memory-less loss functions convex functions. Furthermore, using the bound
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on lipschitz constant of the memory-less loss computed in Lemma B.3 and the bound of the diameter of K, for our choice 1/. we get a regret bound. For our choice of and m, the regret bound evaluates to:
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384CIVT
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Σ & (Mia)) - min
Σ 4 (MT1))
log²
(4.3)
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M.EK
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Next, 4.5 gives us the following bound on the difference between the memory-less loss & (M and the cost
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incurred by the algorithm (x Uz) for every & E [2']. Taking the sum over all & [2':
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T
r
24CIVT log
(000C.d)
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a (x,,u)) u.)
Σ
l (Min))
(4.4)
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7/
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Finally, we add equations (4.2)
4.3)
4.4)
together to obtain the result.
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EXPERIMENTS
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We compare our method to GPC. Unless otherwise noted, we control an LDS with state dimension d = 10 and a single control input = 1. Disturbances are Gaussian and the cost is a fixed quadratic function, randomly generated at = 0 and normalized by the dimensions. The system matrices are sampled as follows: A is diagonal with entries drawn uniformly from [0.5, 0.95]. and B has entries (o, (N, 1). Training uses Adam Adam Kingma & Ba (0144) with learning rate 10 3i = 0.9, B22 0.999, and projection onto the unit ball at each step. Performance is averaged over 20 runs, and report the mean of the last 100 observed costs each time step. We report loss (rather than regret), since the optimal comparator in hindsight is not directly observable for a realized disturbance sequence. Both methods use the last 50 disturbances input features; GPC therefore has 50 d parameters, while OSC uses only the top 15 Hankel cigenvectors, totaling 15 x d parameters.
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5.1
COMPARISON OF DIFFERENT MODELS AND SYSTEMS
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One way to view GPC is performing linear regression past disturbances. Analogously, Algorithm can be interpreted as linear regression on the convolution of disturbances with spectral filters. From this perspective, the method amounts to linear regression on a more expressive set of features: the spectral filters compress the information in the disturbances while preserving the structure most relevant for prediction. While we have a solid theoretical understanding the linear setting, modern deep learning methods often benefit from richer models. Motivated by this, we replace the linear model with a neural network operating on the same spectral features, and in this section we demonstrate the advantages that these features provide.
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359 We evaluate GPC and OSC on both linear and nonlinear systems, and with either simple linear regression model or two-layer ReLU neural network, with 100 hidden units. The nonlinear system (L.SS ReLU) follows
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Tt+1 = ReLU((A; + But) + We.
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Across settings, OSC uses fewer parameters yet achieves long-run performance comparable to GPC. Hidden layers improve both methods, showing that spectral features remain effective with nonlinear models and can serve as inputs to more powerful architectures (e.g., CNNs or Transformers), potentially amplifying their empirical benefits.
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5.2
ABLATING STATE DIMENSIONALITY
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We study the effect of the state dimension d with control input fixed at 1. Figure 3 reports results for d e {2, 5, 10, 20). with loss normalized by d. Performance degrades as d grows, consistent with the intrinsic difficulty of controlling larger state spaces through single input channel.
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5.3
ABLATING NUMBER OF PARAMETERS IN OSC
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OSC uses far fewer parameters than GPC: with memory 50, GPC requires 50d parameters, whereas OSC uses only hd, where h is the number of Hankel eigenvectors. Figure 4 shows that performance improves with i up to about h = 20, beyond which returns diminish. Even with 5, OSC is competitive with GPC, and by h = 20 their performance is nearly identical, highlighting the compactness of spectral features.
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(c) LDS ReLU: GPC vs OSC (no hidden units)
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Figure 2: Comparison of GPC and OSC under linear (LDS) and nonlinear (LDS ReLU) dynamics, using either linear head (no hidden units) or 100-unit MLP head.
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427 Figure 3: Effect of state dimensionality (loss normalized by d): higher dimensions make it harder control an
Figure 4: Effect of spectral parameters h: even small h gives strong performance, and by h 20 OSC matches GPC with far fewer parameters.
428
LDS with a single input.
430
431
8

---

Under review as a conference paper at ICLR 2026
432
5.4
ABLATING STABILITY
434
We now examine how system stability affects performance. vary the distribution of diagonal entries of A: previously drawn uniformly from [0.5, 0.95]. and now from [0.5, 0.99) and [0.5, 0.999). Larger eigenvalues correspond to a smaller stability margin, making the control problem harder. Figure Souure/sshor shows that while overall loss increases in the less stable regime, OSC and GPC maintain comparable performance.
435
436
437
438
439
GPC
440
70
0SC
90
441
00
442
60
443
70
444
50
445
446
447
448
30-
449
2000
4000
6000
8000
10000
2000
40.00
6000
8000
10000
450
Step
Step
451
Diagonal entries in [0.5, 0.99]
Diagonal entries in [0.5, 0.999]
452
453
Figure 5: Effect of system stability on performance. Smaller stability margins (larger eigenvalues of A) make the problem harder, but OSC remains competitive with GPC across regimes.
454
455
456
5.5
ABLATING DISTURBANCES
457
458
We next study the effect of the disturbance distribution. Beyond Gaussian noise, we consider Rademacher noise (i.i.. 11) and a deterministic sinusoid. Figure 6 shows that OSC and GPC behave similarly across disturbance types. When disturbances follow a simple nonlinear rule (e.g., sinusoid), spectral filtering is less natural than directly learning the rule, but OSC still remains competitive.
459
460
462
463
50
464
45
465
466
467
468
0 35
469
470
471
472
473
474
475
(a) Rademacher disturbances
(b) Sinusoidal disturbances
477
Figure Effect disturbance distribution on performance. OSC and GPC exhibit similar relative behavior under random (Rademacher) and structured (sinusoidal) disturbances.
478
479
480
481
6
CONCLUSION AND DISCUSSION
482
We introduced spectral-fittering approach for online control of linear dynamical systems with adversarial disturbances. By relaxing linear policies into Hanneel-based spectral features, we obtain an efficient convex formulation that preserves optimal regret while greatly reducing runtime and parameters. These spectral features provide compact representations of dynamics and open the door to integration with modern deep learning models for large-scale control.
484
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Under review as a conference paper at ICLR 2026
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487
488
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Naman Agarwal, Brian Bullins, Elad Hazan, Sham M. Kakade, and Karan Singh. Online control with adversarial disturbances, 2019a. URL htt.:/ttt://arxxt. wttp:///.orr//aa/112...03333
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Naman Agarwal, Xinyi Chen, Evan Dogariu, Vlad Feinberg, Daniel Suo, Peter Bartlett, and Elad Hazan. Futurefill: Fast generation from convolutional sequence models, 2024a. URL URL.htpp:://rrxivvorr/ http//bb//220000 23110.3766 03766
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Noah Golowich, Elad Hazan, Zhou Lu, Dhruv Rohatgi, and Y. Jennifer Sun. Online control in population dynamics, 2024.
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Moritz Hardt, Tengyu Ma, and Benjamin Recht. Gradient descent learns linear dynamical systems. (bss1609.00591.. 2016. htLhttt:////
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Elad Hazan. Introduction to online convex optimization, 2016. httppppa/aaaaai..ooo/ass////////
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Elad Hazan and Karan Singh. Introduction online control, 2025. URL htttps://aaxiv..rr/////////// 2221
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Elad Hazan, Karan Singh, and Cyril Zhang. Learning linear dynamical systems via spectral filtering, 2017. URL
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Elad Hazan, Holden Lee, Karan Singh, Cyril Zhang, and Yi Zhang. Spectral filtering for general linear dynamical systems, 2018. URL /tLttss:://aaxi org/ hatg//ab//eee abs///as../1///////.//////// 1802 03981
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Alcksandr Mikhailovich Lyapunov. The general problem of the stability of motion. International joturnationaaouunalll journal of control, 55 (3):331-534, 1992.
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Horia Mania, Stephen Tu, and Benjamin Recht. Certainty equivalence is efficient for linear quadratic control. In Advances in Neural Information Processing Systems, pp. 10154-10164, 2019.
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Annie Marsden and Elad Hazan. free regret for learning asymmetric linear dynamical systems, 2025. URL https://arxi>> org/
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James Clerk Maxwell. I. on governors. Proceedings of the Royal Society of London. 16:270-283, 1868. doi: 10. 10.1008//sl..1667. .0055. URL http:p/p///ooollocccoyyooo...hap 1b.10///////0000/ppppp/
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Edgar Minasyan, Paula Gradu, Max Simchowitz, and Elad Hazan. Online control of unknown time-varying dynamical systems, 2022. lttpss////aa org. wwwp/t/////.//oo.../t.../e..aeet 2202.222222 07890
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Tuhin Sarkar and Alexander Rakhlin. Near optimal finite time identification of arbitrary linear dynamical systems. In Kamalika Chaudhuri and Ruslan Salakhutdinov (eds.), Proceedings of the 36th International Conference Machine Learning, volume 97 of Proceedings of Machine Learning Research, pp. 5610-5618. PMLR, 09-15 Jun 2019. URL
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Max Simchowitz, Karan Singh, and Elad Hazan. Improper learning for non-stochastic control, 2020. URL https //arxiv.or//ea abbpgg/bb//0000 09255
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Arun Suggala, Y. Jennifer Sun, Praneeth Netrapalli, and Elad Hazan. Second order methods for bandit optimization and control, 2024. URL/attps://axxiv wrg. abtp//bb//2222 08929
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590 Y. Jennifer Sun, Stephen Newman, and Elad Hazan. Optimal rates for bandit nonstochastic con- trol. In A. Oh, T. Naumann, A. Globerson. K. Saenko, M. Hardt, and S. Levine (eds.).
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Advances in Neural Information Processing Systems, volume 36, pp. 21908-21919. Curran Asso- ciates, Inc., 2023. URL httt:ttttt/s//ootteeet ngg
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Under review as a conference paper at ICLR 2026
594
Daniel Suo, Naman Agarwal, Wenhan Xia. Xinyi Chen, Udaya Ghai, Alexander Yu, Paula Gradu, Karan Singh, Cyril Zhang, Edgar Minasyan, Julienne LaChance, Tom Zajdel, Manuel Schottdorf, Daniel Cohen, and Elad Hazan. Machine learning for mechanical ventilation control, 2022. URL
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06779
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Martin Zinkevich. Online convex programming and generalized infinitesimal gradient ascent. ICML'03, pp. 928-935. Pres,, 2003. ISBN 1577351894.
599
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602
A APPROXIMATION RESULTS
603
604 To prove Lemma 4.3 we shall show that the class of (k. didigoonall stable linear policies can be approximated using Open-Loop Optimal Controllers (Definition [A.1] in Lemma [A.2] We then show that the class of open-loop
605
optimal controllers can be approximated by the class of spectral controllers in Lemma A.3
We begin by defining the
606
class of open-loop optimal controllers as follows:
607
608
Definition A.1 (Open Loop Optimal Controller). The class of Open Loop Optimal Controllers of with memory m is
609
defined as:
610
m
611
TOOOCC_
OLOC #m.K (:t-1:tmm)
K
_ (A+
BK) w
612
613
Any policy in this class is fully parameterized by the matrix K E Rdxa and the memory y E Z.
614
615
Next, we state and prove Lemma A.2 which shows that any linear policy in S can be approximated up to arbitrary accuracy with an open-loop optimal controller of suitable memory.
616
617
618
Lemma A.2. Let a linear policy K E S. Then, for m 2 log
[BGk 67
and E E (0,
619
620
2x
621
Σ
K.m [x(x kmm .u k.m) - - u
S T
K.m x
K,m u1
S
622
623
624
Proof. We begin by bounding the difference in the state and the difference in the control inputs. The difference in cost is bounded using the fact that the cost functions are lipschitz in the control and state. We begin by unrolling the expressions for u and x in terms of we:
625
626
627
628
K U
K
E(A+
(A
x
K
A
t-i + Σ A Bu,
(A.1)
629
1=1
1=1
1=1
630
631
Notice that for all t 5 m, u
ut K. and hence x
Kim Xx
For t m,
632
m (A+ BK)" i- 'w---- K >
633
u Km
K
(A+ + BK) Wi-1
=
634
1=1
635
636
K Σ
(A + BK)
637
i=m+1
638
639
< k W
Σ (1-7)'-1
A ineq.. C-S] C-S]
640
#=m+1
641
3
642
(1 m
643
644
Using this together with (A.1) and the fact that uiK u. K.m for any S m, we similarly get:
645
646
t-m
xx K
Km
Σ 11-1B(ul" - (K
Σ
1 B(uk
K.m
647
12

---

Under review as a conference paper at ICLR 2026
648
and by using Assumption 3.5 we can write:
649
650
x x; mm
S KBR
K.m
16 K WnxW W
(1 mm
651
72
652
1=1
653
Using the fact that KBK² / > 1, we get the following uniform bound:
654
655
max { |
u, Kimll
||
11111 K.m |}
KB W
(1 m1
656
?2
657
Whenever m
2
1 log
nax²) which is indeed true from our choice of E and m, this implies that the
658
659
Km
k u K.
S x W/h. Using Lemma A.4 |xx]
1 < x w//,,, by triangle inequality
:
660
661
X
Kmmll
u
Kim
I 2x Wx/ w/y. Thus, the sum of costs is bounded using lipschitzness of ot as follows:
662
663
T Σ
[x(x
K )
ex (x, Kmm u, kmm)l
s Σ
2Gx²
T
K.m x X,
+
+|1 u - u K.m
664
665
W2
666
(1 - mm- T S 7 2
T
cchocce of ml
667
668
669
670
We shall now prove that every open-loop optimal controller can be approximated up to arbitrary accuracy with a spectral controller.
671
672
Lemma A.3. For every open loop optimal controller OLOC such that K E S and ||xKm|| ||, ||| there exists an spectral controller SCC
S 2x 2x/
Kmmm M.M.MMMM with M KC such that:
673
674
675
676
Σ
co(xx u cx(xx
K.m -u S TT.
k,
2
677
678
679
for any s E (0 and h 2 log I log
[000G " " /md log T log
093/2
1/1
(2
680
681
Proof. Since K e S. there exists a diagonal L Rax= in Definition 3.4
so that:
682
684
683
u
Km
K
BK)
(A
W. K
HD
H
w
685
1=1
686
i-1 Then write -i-1= = 2===1 a
e and obtain
687
688
d
d H W = K
689
ut = KKK
K.m
H
Σ
i-1 ezee] 0j
eze]
Σ
Heje T H -1 Σ o
w,
690
1=1
==1
1=1
691
Recall Wr Wit-- (w......, Wi-m] E Rdxm define tha [1, [1,a,.... m- E m and get
692
693
694
u, Kim
= KK Σ
H H We-1t-mHa
695
j=1
696
m
m
697
KK
Heje
H We-1d-m Σ 0 Ha;
Σ
T = -mm
698
j=1
699
700
K
Σ H H 1 :5 Ma;
Wi-la-ma.
701
13

---

we have Under review as a conference paper at ICLR 2026
702
Let S ". be the spectral controller with M = 0i - KH (Ei=1 0f No, e,e, H for all i [h]. Note that
703
704
| || S R3
of 1/4 (0j, Hai) 1A yisjsm.
705
max
[E(d)
706
707
and from the analysis of Lemma
C.4
we have that of 1/4 (()))((r))) S / Thus, || S k3 V7
7
2h
Then.
708
709
710
1
K.
K
H
H H o May cje}
711
1=++1
712
112
713
S W m
Φ May
W----mm S W v
714
1=
1=1
715
30k W m
2
716
log
1/4
exp
Lemma C44
V7
16 log T
717
i=h+1 1=1
718
30k W mmd
1/1 (2)
719
log
exp
dr
VT
16 log
720
721
50k W /md
1/1 2
2
722
log log
exp
16 log T
723
724
725
M x
x
K.m
A B M" -
Bu
726
727
728
S KBR Σ (1-
|
u
Assumption 3.5
729
730
50 W Vmd
log T log
1/4
(2) exp
72h h 16 log T
731
73/2
732
Using the fact that K kB
2x22/1 1/7 > 1, we get uniform bound:
733
734
max {|12 Km
x
M' 11.11.5.m K. u, |13 S
50mgk W /md
log T log
log
1/4
2
exp
x² h
73/2
16 log T
735
736
Whenever h 2 2oo Tlog 500ggn? /ind
log T Tlog (2) which is indeed the case for our choice o and h. this implies
737
Vi K.ml S x² W/7. Hence, by triangle inequality |xx" M
738
that x 15" - x
K.m|
u."""
u,
||,
|u M'
I Wk/W// Thus,
739
the sum of costs bounded as follows:
740
T
W T x 7
741
(x,
M"
co(x,
K.m
K.m
u,
K.m
X
M
X,
K,m
+
M u.
742
743
300G W W2 /md
log T Tlg
1/4
2
exp
.T
744
73/2
16 log T
745
x T.
|choice of hb]
746
2
747
748
749
We conclude the proof of Lemma 4.3 using Lemmas A.2 and A.3 as follows:
750
751
Proof of Lemma
4.3 K is a (x, 7)
diagonalizably stable linear policy and E (0, 1). Hence, by Lemma A.2 for the [ W22 we have that
752
choice of m
log
753
754
W
755
ce (x
K.m u K.m) ca (x K u
) <
E,,
xx
x Kim
ug
K.m
14

---

Under review as a conference paper at ICLR 2026
756
Since
xx M E K. such that:
.1
K.m [.K.ml)
<
W
for any h 2 2lo log
( W
/idd
log TegTlog' log" 1/4
(2) 5
there exists an
757
279/2
758
759
M ) cx(x, cx(xx"
Kim
T. u
2
760
761
In particular, since 2x 2 x<] for > 0.5 and since 2 log sC
log(8) 0.5, m S 2 log
[8G
762
exd
eq
763
Thus, for h 2 2 log T log triangle inequality, get:
gouGGenn" Wdd log T log 1/4 (2) log!
((
there exists such an M E K. Using
En
eqs
764
765
766
|e(xx u.") cx(x u s SE.
767
768
769
Finally, we derive the bound on the state and control obtained by following a linear policy from S, which we use in the proof of Lemma A.2
770
771
Lemma A.4. For any K E S, the corresponding states x and control inputs n are bounded by
772
773
x²2
W
x W
k
774
775
776
Proof. As in many other parts of this paper, we first write the states as linear transformation of the disturbances:
777
778
K
Σ (A
BK) We-t
x
779
780
781
7)² w
782
783
784
785
Then, since u, = Kx with ||]]| S
we obtain the result.
786
787
788
B LEARNING RESULTS
789
790
B.1 CONVEXITY OF LOSS FUNCTION AND FEASIBILITY SET
791
To conclude the analysis, we first show that the feasibility set K. is convex and the loss functions are convex with respect to the variables Mish- This follows since the states and the controls are linear transformations of the variables.
792
793
794
Lemma B.1. The set KC = { Mi1 E Rxxxd ||x
I M
<
3x"W
| S x3
2h is convex.
795
796
Since x,M u. are linear in Mish. from the convexity of the norm, the fact that the sublevel sets of a convex function convex and that the intersection of convex sets is convex, we are done.
Proof.
797
798
799
Lemma B.2. The loss (M))) is convex in Mi:h-
800
801
Proof. The loss function f, is given by (MMi)) = az (x (Maa), ), u, (M(M))) Since the cost e, is a convex function with respect to its arguments, we simply need to show that xM and u depend linearly on Mi The state is given by
802
803
804
1/4 M.W, 2:-- m- -.t M.W.-an---1e 10)) W.-1
805
M
Ax M
Bul + W
Ax 1
+B B
Σ
806
1=1
By induction, we can further simplify
807
808
809
M
A
Wi-i
A B
1/4 M, W. Wi-i-11-----
i=1
i=1
j=1
15

---

Under review as a conference paper at ICLR 2026
810
which is a linear function of the variables. Similarly, the control is given by
811
812
813
u = of M.W 1-
//M
814
815
Thus, we have shown that X₂ (M))) and ur (M..)) are linear transformations of Misk A composition of convex and linear functions is convex, which concludes our Lemma.
816
817
818
B.2
LIPSCHITZNESS OF (-)-)
819
820
The following lemma states and proves the explicit lipschitz constant of (c)).
821
Lemma B.3. For any Mishs Mi,.. E K. it holds that,
822
/mhh 6GNBR W
(2) || Mi
823
aM (((M)s))
log"
824
7/
825
826
Proof. Taking the difference in controls:
827
S W Vmllo" 1/4 (2) 4
828
[lu() (MMM)) Σ
// (M, - M.
Σ
|| Mill
829
7 j=1
830
By unrolling the recursion, we have:
831
832
h
833
xx(Mis))= 1 Wi-i+ Σ A B
Σ
)
1/4
M, W. W
834
5=1
835
Taking the difference.
836
837
h
(M1ih) (M( =
Σ
A 1 B
(M, M,) -
838
839
1=1
j=1
840
A [|--||| |||||||||
Σ
10
1/4 ||M - MMlll W
841
842
j=1
843
x² - 7)--1 W W /m 1/4 (2)
Σ
844
|| Mr - MIll
845
1=1
846
(2)
847
S W mog
Σ M,
-M]l
848
1=1
849
Using the fact that xBx² //y > 1. we get uniform bound:
850
851
KBR W vm log
(2)
h
max (Mxxa(Masn) X1 II
u. u,MM
1/4
Σ j=1
MM - MIll
852
7
853
854
Using the lipschizness of the cost function from Assumption the definition K. we have
855
(Mi.h) (M(i)s))) = (c.(xx((M.))) (M1s)) un (M - cx(x. (M..) u. (MM.))
856
W
857
(M) (Mi.)) + +1lu, (M()))
858
859
6GkBn Vm
log (2) Σ M M/
h
1/4
860
72
862
Finally, we upper bound each M, - M3|| by M - Mi to get the result.
863
16

---

Under review as a conference paper at ICLR 2026
864
B.3
LOss FUNCTIONS WITH MEMORY
865
866
The actual loss C₁ at time is not calculated on x, (MM.)) but rather on the true state x2, which in turn depends on different parameters M for various historical times Nevertheless, cx(X,, ut) is well approximated by & MMt)).
867
868
as stated in Lemma 4.5 and proven next.
869
Proof of Lemma 4.5 By the choice step size 7,, and by the computation the lipschitz constant of & w.r.t Maa
870
Lemma B.3
we have:
871
2k3
2h
872
"7
'T'
873
874
where L is the lipschitz constant of & w.r. Mis. computed in Lemma
B.3
Thus, for each E [k]],
875
876
877
M|M - M S ||M s
Σ
|M
- S in 2ix 2h
V
7T
878
s=t-i+1
879
Observe that u, un (MMM) ) We use the fact proved above to establish that x, and xx( are close. Observe that X₁ and X (MM...) can be written as
880
881
883
882
(M
Σ A'
W
Σ
1/4 Σ
oj M.W-------mOO
884
i=1
==1
885
h
886
X
1 W Σ
A f- B
1/4 1 W j
M
887
1=1
1=1
1=1
888
Evaluating the difference,
889
890
h Σ
||x, - X.MMMMl 5
1
-M
891
892
1=1
j=1
893
S 2x n W v v log" 1/4
(2)
2h
Σ i(1
((1-))*-1
894
7T
895
896
2x KB Rk²RB W Vmh
log
1/1
2
897
/T
898
By definition, ( (.) ce (xe(M (M 11)), (AM and by the definition of and the projection used in Algorithm
899
we have by Assumption 3.3
900
901
(MM -ce u:)l = (c us (M(M).))
ce (xt, u:)l
902
W
903
||x (x.(Min
xxll
904
905
6GKBN&W22 V2 mh
2 1/4 loga/a
log
906
7T/2
907
908
909
910
C
SPECTRAL TAIL BOUNDS
911
912
We use the following low-approxemate rank property of positive semidefinite Hankel matrices, from Beckermann & Townsend (001)):
913
914
Lemma C.1. [Corollary 5.4 in Beckermann Townsend (0166) Let Hn be PSD Hankel matrix of dimension n.
915
Then,
916
x²
-2k+2
917
())2((1n)) < 16 exp
(H)))
(log(8(n/2111
17

---

Under review as a conference paper at ICLR 2026
918
Define the matrix
919
1-y Hm = i Halta da.
920
921
922
923
where ja [1, a,....... .......,".... am=1I= E RR", and note that (Hm)is= (1-7))+--1 particular, this is a PSD Hankel matrix (+j-1
924
of dimension m. We prove the following additional properties related to it:
925
Lemma C.2. Let o be the jth top singular value of Hm Then, for all T 2 10, we have
926
927
n3j
928
jj Σ 156800 log
exp
4 4log T
- log
2
929
930
Proof. We begin by noting that for any J.
931
m
(1 71- 21-1
00
(1
7)
2 2-77 7
)
1 1
(2) 71
932
of S Tr(Hm))
Σ
S 7)
Σ
2i +1 (1 7)
log
log
933
2i
7
2
934
Now, since T 2 10 implies 8|T/2]/T > T, we have by Lemma
C.I that
935
936
2
π
02+2k S 012k < 8 log
exp
937
log
938
2
939
1120 log
exp
2 log T
940
Thus, we have that for all j.
941
942
2
(j - 2)
2
x²j
943
oj < 1120 log
exp
2 log T
156800 log
?
-exp
2 T
944
945
946
Lemma C.3. For all T E N and 0 s a S 15-1- -, we have:
947
|| 12 < 1/7,
948
949
2.
|| 1|
< 2//72
950
951
952
Proof. The first inequality can obtained by evaluating:
953
1-
2m am
954
||a = Σ 2i-2
1 (1 7)2 S 7
1 1- a
955
956
To obtain the second inequality, observe that in the summation form,
da || Emng(i (2i 2)-3 and hence it
957
958
monotonically increases with m. Thus, if the limit exists for
d du || as m - oo then the limit is the supremum.
959
Evaluating the derivative for the closed form expression of || and taking the supremum, we get:
960
961
d
|| 12
2ma 2m
1 (1 a²) - (1 a 2m (-2o)
20)
< sup meN
962
da
(1 (1-42)2
963
2a - 2ma 2m-1 + 2ma 2m+1 -2 2m+1
964
sup meN
(1 22)²
965
2a 2ma 2m-1 + 2ma 2m+1
-0 2m+1
966
lim
967
m-
(1 2²2²
968
2a
2(1 -7)
2
<
969
(1 (1-a") 2/2
(1 (1 2)22
72
970
971
18

---

Under review as a conference paper at ICLR 2026
972
Lemma C.4. Let } be the eigenvectors of Hm. Then for all E [T], a a [0, 10,1-7))
7.. -1),
and 7 S 2/33,
973
974
Ia 03 $ v 2 1/4
log"
1//2/4
1/4
(2)
2
exp ( 16 (-16rt)
m²j
975
vy
976
977
Consider the scalar function g(a) (wo Φ3 )a over the interval [0, 1 - 7]. First, notice that by definition of Φ) as the eigenvectors of Hm, and or as the corresponding eigenvalues, we have
Proof.
978
979
g(a)da=) (oodo=/ [f/-" (0] Ma) dd Sor² f--
980
03 Mall 0 0j ZnØz = -.-
981
982
Since || is 2/ 2/72-lipschitz in the interval [0, 7]. and g(ax) is its projection
g(a) is also 2/72-lipschitz. Note that g(a) is also a non-negative function and integrates to oj over the interval [0, |0,1- >/- Say is the maximum
983
984
value achieved by g(a) for all a E [0, >] then R S ||aall < 1/7. Subject to achieveing the maximum at R. the 7
985
non-ne non-negative 2/12-lips -ipschiit function over [0, 1 -1 with the smallest integral is given by:
986
2 a,0
987
(a) max R
988
989
for which fu'
A(a)da- R2n2/ whenever y 2/3. Thus, we get that R S 2o, and hence |nTØl1 7
990
Using the upper bound o, from lemma C.2 get the result.
991
992
993
994
D STABILIZED SPECTRAL POLICY
995
996
Assumption 3.5 restricts to competing only against systems for which the zero matrix (x, di-diagonalizably stable. This implies that there exists a decomposition:
997
998
A = HLH
999
where |H]], | S K. L is diagonal, and |||| < 1 - 7. However, our proofs only require the bound:
1000
1001
Vie N,
||All| x² (1-
1002
which holds even if L is not diagonal. In fact, suffices for the zero matrix be (x, (007)))sssso --ttonng stable, as defined in
1003
Cohen et al. (2018). For completeness, we recall the definition:
1004
Definition D.1 (Definition 3.1 in Cohen et al. (0018)) linear policy K is (n. -tttronll stable if there exist matrices
1005
L, H such that:
1006
A+ BK -HH
1007
and the following conditions hold:
1008
1009
The spectral norm of L is strictly smaller than unity, i.e., ||4|| 1-7
1010
The controller and the transformation matrices are bounded, i.e., |||]|, [|]]| ||--1|| '1 k.
1011
1012
However, the assumption of 0 being (R, -tsstongly stable linear controller can be further relaxed by using
1013
precomputed (x, sttroonly stable matrix Ko. This can be done using an SDP relaxation described in Cohen al.
1014
2018). Given access to (K, -sstrrngly stable Ko. we learn stabilized spectral policy using online gradient descent defieed as follows:
1015
1016
u == Kox M +
1/4 W.
1017
1018
Consider playing ú, = u, + Kox, instead of u, at each t E [T]. Observe that the system
1019
1020
X:+1
Ax; + Bü, Wi,
(D.1)
1021
when controlled by ú, behaves the same as the system
1022
X = (A + BK,)x, + Bu, + W.
(D.2)
1023
1024
when controlled by u, This means that the sequence of states in both the cases is the same. Thus, since the 0 matrix is (54,7)ssrongh) stable for system (D.2) the regret of our algorithm on system (D.2) is bounded by our result. By the
1025
structure of our proofs, for system D.2) each one of
19

---

Under review as a conference paper at ICLR 2026
1026
max
k x
X
K.m
/ 4t
K.m
1027
1028
(ii) max
K.m x A5
/ km
u A
11
1029
1030
(iii) max X (M1( - x, (A11 (lllaa/llllllMMaaa (1
u. (MM
1031
(iv) max { || - xx(Mii+))l Il I - (1l
1032
1033
remains bounded. Observe for any state and control if
Kox then:
1034
max {||x - x'11, || - 1'|)) & max {||x - x'I,, I1,, I|u - n'll +
||x x'
|]
S 2x max {||x x'1,, -'1l)
1035
Thus, replacing u, (M) with (M) in (iii) and (iv) yields the same bound with additional factor of 2k. Now, observe
1036
that u Kx Kox (K Ko)x i(k-Ko, Define
1037
1038
m
K.m = Kox + (K Ko) _ (A BK) u,
1039
Wi-fi
1040
1041
and choose
1042
M o. 1/ Ko)H Oo. Mar H-1
1043
Vi E [M]
1044
1045
K.m and u. with " in (i) and (ii), we get the same bounds with an additional factor of 2k. This allows us conclude that, when competing against the same policy class S, we get an upper bound the regret with the same order of growth with respect to T and 1/7.
Now, replacing u with ū, (K-K))) K.m with it
Ko) u1,
1046
1047
1048
1049
E ADVANTAGE OF SMALLER Y
1050
1051
Previous works require stability margin of y (()ppllloog()) to ensure an O(polylog(T)) running time. In contrast, this work shows that setting (2(1Thh for k (1, 1/12) maintains sublinear regret while still guaranteeing O(polylog(T)) running time. In this section, we construct example demonstrating that choosing 7 1/T for E (0, 1/12) results significantly lower aggregate loss compared to ; = 1/polylog(T). Consider a noiseless linear dynamical system with parameters b R. governed by the update equation:
1052
1053
1054
1055
1056
1057
It-1 = ax2 + buz
1058
The loss function at each time step is defined as:
1059
VI E [T), c(x, u) = max 11.
1060
1061
Since this is scalar system, for sufficiently large the class of (x, 7) diagoonnallll stable controllers reduces to:
1062
S(7) = {ko E E a bb < 1
1063
1064
If the initial state zu = 1, then:
1065
T
T
T (k)
T -1-
1 (1 7)
1066
min kES())
calar, u.) = min Σ
(xx) =
max RES(S) 2=1
Σ
1067
kE)())
tm1
1068
Using the fact that 0 1-1 2 " we can upper and lower bound this expression as:
1069
1070
1 1- Σ ca(a,, uz) 5
1071
min NES(T)
7
1072
1073
Using the lower bound,
1074
1075
min kES(1 RES(1/potylog((T) tm1
(x) 2 -polylog(T)
1076
1077
and using the upper bound,
1078
T
ca(x,, < u -T - 5 -T*
1079
min kES(1/Tk)
20

---

Under review as a conference puper at ICLR 2026
1080
Hence, the difference in the minimum costs of the two cases is lower bounded as:
1081
1082
T
T
1083
min kES(L/polglog(T)
Co (a2, ue)
min KEB(1/T
Σ ca Uz) 2 /2 polylog(T) (2(
1084
1085
Thus, choosing 7 1/7Th results in significantly lower for the best controller in the policy class S( (which compete against) compared to the case when = 1/polylog(T). In particular, the improvement is by polynomial
1086
1087
factor in T.
1088
1089
F USE OF LARGE LANGUAGE MODELS
1090
1091
We used large language model (ChatGPTT assist with editing and polishing the writing of this paper. Specifically, the model was used to improve clarity, conciseness, and readability of some sections. All technical content, proofs, algorithms, and experiments were developed entirely by the authors. The model did not contribute to research ideation, discovery, or experimental design.
1092
1093
1094
1095
1096
1097
1098
1099
1100
1101
1102
1103
1104
1105
1106
1107
1108
1109
1110
1111
1112
1113
1114
1115
1116
1117
1118
1119
1120
1121
1122
1123
1124
1125
1126
1127
1128
1129
1130
1131
1132
1133
21


Reviews:
[{"review_id": "SGmNecRANv", "reviewer": "Reviewer_71FK", "summary": "This paper considers the problem of controlling a known linear dynamical system under possibly adversarial disturbances and time-varying costs. The authors propose an approach motivated by \"spectral filters\", wherein the eigenvalues/eigenvectors of a particular Hankel matrix determined by a user-set stability margin (over)estimate are used to parameterize a control law that maps past disturbances to controls. Compared to prior work in \"non-stochastic control\", by using this particular parameterization, the proposed method attains regret bounds that improve the dependence on the stability margin, as well as improve the run-time dependence on the stability margin from polynomial to polylogarithmic.", "questions": "Please see under Weaknesses.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "Overall, this paper is well-written and the message is rather clear. The proposed method is a novel adaptation of prior literature in both non-stochastic control and spectral filtering that seems to improve existing regret and runtime guarantees for this problem under relatively relaxed theoretical conditions. The approximation techniques involving the Hankel matrix are interesting, and as far as I know rather novel in the space of (linear) control. Thus, I think this paper contains worthy and interesting content for the niche of learning on dynamical systems researchers.", "weaknesses": "Some immediate points that deserve some discussion are as follows:\n\n1. The results seem to rely on the dynamics matrices $A,B$ being known. How well do the proposed results extend to the unknown dynamics case?\n\n2. The proposed method seems rather tied to linear dynamics parameterization. How does one naturally apply this method to non-linear dynamics/policy parameterization? On that note, the discussion of applying GPC and OSC to nonlinear systems (with potentially nonlinear parameterization) in experiment Section 5.1 is rather terse, and it is not immediately clear how the proposed methods are being applied---the appendix doesn't seem to have further information on this front.\n\n3. The numerical experiments lack comparisons to even simple baselines. For example, considering some of the experiments concern control with Gaussian or zero-mean stochastic disturbances, it would be instructive to see the loss dynamics of, e.g. online LQR. Furthermore, in the known-linear-dynamics (potentially quadratic-loss case), there is quite a lot of literature on robust control, see e.g. classical mixed H2/H$\\infty$ control, competitive control [1], and adversarially robust synthesis [2]. I think it is both worth including some discussion about robust control perspectives, and a couple more simple baselines beyond disturbance-action filter approaches to get a better sense of the method's relative performance.\n\n4. As a minor comment, it should be noted the margins of the submission seem to have been different than the template.\n\n[1] Goel and Hassibi, \"Competitive Control\"\n\n[2] Lee et al. \"Performance-Robustness Tradeoffs in Adversarially Robust Linear-Quadratic Control\"", "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": "nPuKXwe5D9", "reviewer": "Reviewer_Enzf", "summary": "This paper proposes an online control algorithm for linear stochastic systems with adversarial disturbance. The proposed algorithm approximates a policy using spectral features derived from Hankel matrix eigenvectors, allowing efficient updates with regret comparable to existing approaches while significantly improving runtime. The author demonstrates that the proposed algorithm can match or surpass existing baselines empirically while using far fewer parameters.", "questions": "- Could the authors provide more intuition behind the motivation for using spectral filters? What advantages do spectral features offer compared to other feature representations in online control?\n- Could the authors comment on the actual runtime improvement over baseline methods in the experiments?", "limitations": "", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "- The use of spectral filtering in online control appears to be a novel idea that could have a broader impact on the control domain.\n- The theoretical analysis is thorough and clearly demonstrates improvement over prior approaches.", "weaknesses": "- The empirical evaluation remains relatively narrow in scope. Including results on higher-dimensional systems would strengthen the paper.\n- While the main advantage of the proposed algorithm lies in its runtime efficiency, this aspect is not evaluated in the experiments.", "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": "X28BjlhN3Q", "reviewer": "Reviewer_tXpd", "summary": "The paper proposes a new \\emph{spectral convex relaxation} for online (adversarial) control of linear dynamical systems. \nIt replaces explicit system-dependent features with a fixed, universal set of \\emph{Hankel eigenvector filters}, \nthereby transforming control into an online convex optimization problem. \nThe resulting algorithm achieves a regret bound of $\\widetilde{\\mathcal{O}}(\\gamma^{-4}\\sqrt{T})$, \nwhere $\\gamma$ denotes the \\emph{stability margin} of the comparator policies (i.e., the spectral radius bound of the closed-loop matrix). \nThis dependency in $\\gamma$ is better compared to existing approaches.\nThe method runs in polylogarithmic time per step and is supported by a detailed approximation analysis \nand experiments demonstrating a competitive runtime efficiency compared to existing baselines.", "questions": "1) What can be said when $A,B$ are unknown (cf. weakness section above)? \n    Do the results extend to estimated $(\\widehat A,\\widehat B)$, and how do identification errors \n    $\\|A-\\widehat A\\|$, $\\|B-\\widehat B\\|$ affect the feature construction, stability, and regret guarantees?\n\n2) When referring to a ``convex cost,'' do you mean convex jointly in $(x,u)$ for each $t$? \n    Please clarify the precise convexity and Lipschitz assumptions, including in which variables and norms they hold.\n\n3) In the first displayed equation of Section 1.1, the policy class in the regret definition is missing. \n    Please specify explicitly over which set of policies or feedback gains the infimum is taken.\n\n4) In Algorithm 1, the loss function $\\ell_t$ is referenced but not defined at that point. \n    It only appears later in Equation (4.1), which interrupts the reading flow. \n    Please either define $\\ell_t$ where Algorithm~1 appears or add a forward reference in the algorithm caption.\n\n5) Definition 3.1: can the stated condition occur at any time $t$? \n    I assume yes, please make the time quantification explicit.\n\n6) Definition 3.4: please be precise that $K$ denotes a feedback gain that induces a linear policy; \n    $K$ itself is not a policy. Consider rephrasing accordingly to avoid ambiguity.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "The paper is very well written and the overall presentation is clear and easy to follow. \nThe structure and flow of ideas make it straightforward to understand the motivation, \ntechnical setup, and implications of the proposed approach. \nBoth the algorithmic framework and the main theoretical result Theorem 2.1) are presented in a transparent way, \nwith all relevant quantities and assumptions clearly defined. \nIn particular, the paper explicitly spells out the concrete hyperparameter choices \n$m$, $h$, and $\\eta$ that lead to the stated regret bound, \nwhich greatly improves the reproducibility and interpretability of the theoretical results. \nA further strength is the improved dependence on the stability margin $\\gamma$ in the regret bound, \nwhich represents a significant advance over previous approaches whose computational or regret guarantees scale more unfavorably with $\\gamma$. \nFinally, the discussion of related work is comprehensive and well organized, \nsituating the contribution within the broader literature on online control and learning-based methods.", "weaknesses": "**Weaknesses and Suggestions.**\n\n 1) *Reliance on known $(A,B)$ is unrealistic in many applications.*\n  The entire pipeline assumes exact knowledge of the system matrices to reconstruct disturbances and build spectral features.\n  This limits applicability to model-mismatch or unknown-dynamics settings.\n  Suggestion: Add a discussion (or short appendix) on robustness to misspecified $(A,B)$ and outline a data-driven variant where\n  $(\\widehat A,\\widehat B)$ are estimated online (or pre-estimated) with explicit stability conditions under identification error,\n  e.g., bounds of the form $\\|A-\\widehat A\\|,\\|B-\\widehat B\\|\\le\\varepsilon$ and their impact on regret constants.\n\n  2) *Placement of Theorem 2.1 (many forward references).*\n  As written, Theorem 2.1 appears before the technical setup of Section 3, causing heavy forward referencing and making the statement hard to parse on first reading.\n  Suggestion: Move Theorem 2.1 to immediately \\emph{after} Section 3, once the comparator class, spectral filters, and surrogate loss are fully defined.\n  This will make the result self-contained at its first occurrence.\n\n  3) *Reorder Section 3 before the main results.*\n  Closely related to the previous point: placing the entire Section~3 (setup, assumptions, and construction of the spectral controller)\n  before the main theorem would improve readability and reduce back-and-forth navigation.\n  Suggestion: New order: (i) problem setup and assumptions, (ii) spectral feature construction \\& surrogate loss (current Section 3),\n  (iii) then algorithm and main regret theorem.\n\n  4) *No empirical verification of the $\\sqrt{T}$ regret scaling in Theorem 2.1.*\n  The current experiments do not test the $\\sqrt{T}$ dependence, so they cannot confirm the rate empirically.\n  Suggestion (add a minimal experiment):\n\n   4a) Fix a stable LTI instance and cost sequence within the paper's assumptions (convex Lipschitz costs), and fix a stability margin $\\gamma$ (e.g., by choosing a comparator with spectral radius $1-\\gamma$).\n    \n4b) Run the proposed method for horizons $T\\in\\{2^{10},2^{11},\\dots,2^{18}\\}$, keeping all other hyperparameters at the prescribed values in Theorem 2.1 (the stated $m,h,\\eta$ schedule).\n   \n4c) For each $T$, compute regret with respect to the comparator class used in the theory (or as close as feasible), average over multiple seeds, and report the mean $\\pm$ std.\n   \n4d) Plot $\\log(\\text{regret})$ vs.\\ $\\log T$ and report the fitted slope (expectation: slope $\\approx\\frac{1}{2}$ within confidence bands).\n    \\item (Optional) Repeat for a few $\\gamma$ values to visualize the $\\gamma$-dependence of constants in the regret (even if asymptotically the slope stays $\\approx\\frac{1}{2}$).", "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": "8RZFFq5e2z", "reviewer": "Reviewer_AjXG", "summary": "This paper proposes a new algorithm for online control of linear dynamical systems (LDS) under adversarial disturbances and convex cost functions. The central idea is to use a spectral representation of control policies, transforming the non-convex control problem into a convex online learning problem using Hankel-based spectral filters. This allows efficient computation while preserving strong theoretical guarantees.", "questions": "1. Let us consider an extreme case where System (1.1) does not have any external disturbance (i.e. $w_t \\equiv 0$). Then, under this setup, Line 7 of Algo. 1 will produce $\\tilde{W}=0$. Moreover, whatever $M^t$ is optimized, there is no effective control input, i.e. $u_t=0$. Do those theoretic claims still hold?\n\n2. Follow the previous question. If $w_t$ is small, how will the proposed algorithm behave? Will the loss decreases slower? Or the decay rate does not change?\n\n3. The examples consider simple diagonal $A$ and dense $B$. How about the controllable canonical form where both $A$ and $B$ has certain sparse property? If there is one control input $u\\in \\mathbb{R}^{1}$ but many states $x\\in \\mathbb{R}^n$ with large $n$, then the effect on $x(n)$ is delayed by $n$ steps. Will it causing some large oscillations in this loss curve?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "1. The spectral filters for marginally stable LDS can effectively compress the information of external disturbance. Thus, it allows efficient computation while preserving strong theoretical guarantees.\n\n2. The work sounds solid, although I didn't check the detail proofs.", "weaknesses": "1. **The criticism on LQR is not accurate**\n\nFor modern industrial applications, the top three control algorithms are likely to be PID, LQR and MPC, despite that those theoretical assumptions are not satisfied. Moreover, the example of drone flying control system is not a good motivate example for the proposed approach. A simple reason is that those scenarios mentioned in this paper are currently handled by the above three algorithms in many commercial products.  \n\nAlso, the statement about the benefits of marginal stability is misleading: \"...yields smoother, more energy-efficient control, useful in settings like robotics, thermal systems, and satellite dynamics\". It can lead to small control effort if the open-loop system is unstable or marginally stable. If open-loop is very stable, then one needs extra control effort to reduce the stability margin. Due to its small robustness margin, marginally stability is often not desired in applications related to robotics and satellites. A common setup in those applications is a strong stable tracking controller together with a smooth reference generator. It can yield smooth control input while remaining strong robustness against disturbances.\n\n2. **Some highly related works  are missing**\n\nIn Section 1.2, the first two paragraph dates back to the beginning of control theory, which is not necessary. On the other side, it missed some more recent and highly related works called **system level synthesis (SLS)** [1,2] in the control literature. From my understanding, the proposed approach is a special case of SLS, which belongs to a more general framework called **Youla parameterization**. Compared to SLS, the proposed approach has two new features: 1) in SLS setup, the cost function is often known while this work considers the online convex optimization setup; 2) SLS is for general stability while this paper has a tailed algorithm for marginally stable LDS. \n\n3. **Does the proposed approach guarantee stability?**\n\nIn this work, it only assumes that the cost function is convex and Lipschitz. Let us consider a linear cost on $x$ and $u$ and there is not constraint $\\mathcal{K}$. Then, Algorithm 1 will drive the state to infinity.  Maybe it needs some additional assumptions, e.g., $c(x,u)$ is bounded from below.\n\n4. **Lack of experiments on adversarial cost and disturbance**\n\nThis paper claims those benefits in the theory but no experimental illustration is provided. It only investigates simple disturbances including Gaussian,  Rademacher noise and deterministic sinusoidal. And the cost is a fixed quadratic.\n\n[1] Anderson et al. System level synthesis. Annu. Rev. Control, 2019.\n\n[2] Wang et al. A system level approach to controller synthesis. IEEE-TAC, 2019.", "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)