Improved rates for prediction and identification for partially observed linear dynamical systems

Identification of a linear time-invariant dynamical system from partial observa-1 tions is a fundamental problem in control theory . Particularly challenging are sys-2 tems exhibiting long-term memory . A natural question is how learn such systems3 with non-asymptotic statistical rates depending on the inherent dimensionality ( or-4 der ) 𝑑 of the system , rather than on the possibly much larger memory length . We5 propose an algorithm that given a single trajectory of length 𝑇 with gaussian ob-6 servation noise , learns the system with a near-optimal rate of ‹𝑂 ( ︁ » 𝑑𝑇 ) ︁ in ℋ2 er-7 ror , with only logarithmic , rather than polynomial dependence on memory length.8 We also give bounds under process noise and improved bounds for learning a9 realization of the system . Our algorithm is based on multi-scale low-rank approx-10 imation : SVD applied to Hankel matrices of geometrically increasing sizes . Our11 analysis relies on careful application of concentration bounds on the Fourier do-12 main – we give sharper concentration bounds for sample covariance of correlated13 inputs and forℋ∞ norm estimation , which may be of independent interest.14 1 Introduction15 We consider the problem of prediction and identification of an unknown partially-observed linear16 time-invariant ( LTI ) dynamical system with stochastic noise,17 𝑥 ( 𝑡 ) = 𝐴𝑥 ( 𝑡− 1 ) + 𝐵𝑢 ( 𝑡− 1 ) + 𝜉 ( 𝑡 ) ( 1 ) 𝑦 ( 𝑡 ) = 𝐶𝑥 ( 𝑡 ) + 𝐷𝑢 ( 𝑡 ) + 𝜂 ( 𝑡 ) , ( 2 ) with a single trajectory of length 𝑇 , given access only to input and output data . Here , 𝑢 ( 𝑡 ) ∈ R𝑑𝑢 are18 inputs , 𝑥 ( 𝑡 ) ∈ R𝑑 are the hidden states , 𝑦 ( 𝑡 ) ∈ R𝑑𝑦 are observations ( or outputs ) , 𝜉 ( 𝑡 ) ∼ 𝑁 ( 0 , Σ𝑥 ) 19 and 𝜂 ( 𝑡 ) ∼ 𝑁 ( 0 , Σ𝑦 ) are iid gaussian noise , and 𝐴 ∈ R𝑑×𝑑 , 𝐵 ∈ R𝑑×𝑑𝑢 , 𝐶 ∈ R𝑑𝑦×𝑑 , 𝐷 ∈ R𝑑𝑦×𝑑𝑢20 are matrices . Partial observability refers to the fact that we do not observe the state 𝑥 ( 𝑡 ) , but rather21 a noisy linear observation 𝑦 ( 𝑡 ) .22 As a simple and tractable family of dynamical systems , LTI systems are a central object of study for23 control theory and time series analysis . The problem of prediction and filtering for a known system24 dates back to [ Kal60 ] . However , in many machine learning applications , the system is unknown and25 must be learned from input and output data . Identification of an unknown system is often a necessary26 first step for robust control [ DMM+19 , BMR18 ] . In a long line of recent work , the interplay between27 machine learning and control theory has borne fruit in an improved understanding of the statistical28 and online learning guarantees for prediction , identification , and control for unknown systems . In29 machine learning , LTI systems also serve as a simple model problem for learning from correlated30 Submitted to 35th Conference on Neural Information Processing Systems ( NeurIPS 2021 ) . Do not distribute . data in stateful environments , and can give insight into understanding the successes of reinforcement31 learning [ Rec19 , TR19 ] and recurrent neural networks [ HMR18 ] .32 Partial observability poses a significant challenge to system identification : In the fully observed33 setting , given access to 𝑥 ( 𝑡 ) , there is no obstacle to learning the matrices directly through linear34 regression . However , in the partially observed setting , the most natural form of the optimization35 problem is non-convex.36 Systems exhibiting long-term memory are particularly challenging to learn . Restricting to marginally37 stable systems , this occurs when the spectral radius of 𝐴 , 𝜌 ( 𝐴 ) , is close to 1 , and it implies that the38 output at a particular time can not be accurately estimated without taking into account inputs over39 many previous time-steps—on the order of 𝑂 Ä 1 1−𝜌 ( 𝐴 ) ä times steps . Such systems often arise in40 practice . A particular class of such systems are those exhibiting multiscale behavior , with different41 state variables that change on vastly different timescales [ CR10 ] . For example , the body ’ s pH level is42 affected both by long-term changes on a timescale of days or weeks , as well as breathing rate which43 changes over a timescale of seconds . For such systems , it makes sense to discretize at the scale of the44 fastest changing variable , which leads to a long memory for the slowest-changing variable . With few45 exceptions , existing guarantees for learning partially observed LTI systems degrade as the memory46 length increases . However , counting the number of parameters in the model ( 1 ) – ( 2 ) suggests that the47 right measure of statistical complexity is the intrinsic dimensionality of the system , not the memory48 length . This leads to the following natural question.49 Question : How can we learn partially observed LTI systems with ( non-asymptotic ) statistical rates50 that depend on the intrinsic dimensionality of the system , rather than the memory length ? 51 Despite the simplicity of the question , little in the way of theoretical results are known . We focus52 on the particular problem of learning the impulse response ( IR ) function of the system—which fully53 determines its input-output behavior—in ℋ2 norm . This is a natural norm for prediction problems54 as it measures the expected prediction error under random input . Known guarantees for learning the55 IR depend on the memory length . One particularly undesirable consequence is that for a continuous56 system with time discretization ∆ going to 0 , the memory scales as 1/∆ ( while the system order57 stays constant ) , leading to suboptimal estimation by an arbitrarily large factor.58 Our key contribution is an algorithm and analysis that gives statistical rates that are optimal up to59 logarithmic factors . Unlike previous works , our rates depend on the system order 𝑑—the natural60 dimensionality of the problem—and only logarithmically on the memory length of the system . Our61 algorithm is based on taking a low-rank approximation ( SVD ) of the Hankel matrix , which is a62 widely used technique in system identification . We consider a multiscale version of this algorithm,63 where we repeat this process for a geometric sequence of sizes of the Hankel matrix . This is essential64 for obtaining a stronger theoretical guarantee . In the setting of zero process noise , we prove that our65 algorithm achieves near-optimal ‹𝑂 ( ︁ » 𝑑 ( 𝑑𝑢+𝑑𝑦 ) 𝑇 ) ︁ rates inℋ2 error for the learned system.66 Our analysis relies on careful application of concentration bounds on the Fourier domain to give67 sharper concentration bounds for sample covariance and ℋ∞ norm estimation , which may be of68 independent interest . While we consider our algorithm in a simple setting , we hope that this is a69 first step to understanding and improving more complex subspace identification algorithms . Indeed,70 SVD and related spectral methods are a standard step used in subspace identification algorithms71 such as N4SID ; our analysis suggests that SVD has an important “ de-noising effect ” .72 We also give improved bounds for system identification , that is , learning the matrices 𝐴 , 𝐵 , 𝐶 , 𝐷73 using the Ho-Kalman algorithm [ HK66 ] , with ‹𝑂 ( ︁ » 𝐿𝑑 ( 𝑑𝑢+𝑑𝑦 ) 𝑇 ) ︁ rates.74 1.1 Notation75 Norms . We use ‖·‖ to denote the 2-norm of a vector . For a matrix 𝐴 , let ‖𝐴‖ = ‖𝐴‖276 denote its operator norm , 𝜌 ( 𝐴 ) denote its spectral radius ( maximum absolute value of eigen-77 value ) , and ‖𝐴‖F denote its Frobenius norm . For a matrix-valued function 𝑀 ( 𝑡 ) ∈ C𝑑1×𝑑2 ,78 ‖𝑀‖F : = » ∑︀ 𝑡 ‖𝑀 ( 𝑡 ) ‖ 2 F. Let 𝜎𝑟 ( 𝐴 ) denote the 𝑟th singular value of 𝐴.79 Fourier transform . Given a matrix-valued function 𝐹 : Z → C𝑚×𝑛 , define the ( discrete-time ) 80 Fourier transform as the function “ 𝐹 : R/Z→ C𝑚×𝑛 given by “ 𝐹 ( 𝜔 ) = ∑︀∞𝑡=−∞ 𝐹 ( 𝑡 ) 𝑒−2𝜋𝑖𝜔𝑡.81 Matrices . Given a sequence ( 𝐹 ( 𝑡 ) ) 𝑎+𝑏−1𝑡=1 where each 𝐹 ( 𝑡 ) ∈ C𝑚×𝑛 , define Hankel𝑎×𝑏 ( 𝐹 ) as the82 𝑎𝑚 × 𝑏𝑛 block matrix such that the ( 𝑖 , 𝑗 ) th block is [ Hankel𝑎×𝑏 ( 𝐹 ) ] 𝑖𝑗 = 𝐹 ( 𝑖 + 𝑗 − 1 ) . Given a83 sequence ( 𝐹 ( 𝑡 ) ) 𝑎−1𝑡=0 where each 𝐹 ( 𝑡 ) ∈ C𝑚×𝑛 , define the Toeplitz matrix as the block matrix such84 that the ( 𝑖 , 𝑗 ) th block is [ Toep𝑎×𝑏 ( 𝐹 ) ] 𝑖𝑗 = 𝐹 ( 𝑖 − 𝑗 ) 1𝑖≥𝑗 . For a matrix 𝐴 , let 𝐴⊤ , 𝐴𝐻 , 𝐴† denote85 its transpose , Hermitian ( conjugate transpose ) , and pseudoinverse , respectively . For a vector-valued86 function 𝑣 : { 𝑎 , . . . , 𝑏 } → R𝑛 , let 𝑣𝑎 : 𝑏 ∈ R ( |𝑎−𝑏|+1 ) 𝑛 denote the the vertical concatenation of87 𝑣 ( 𝑎 ) , . . . , 𝑣 ( 𝑏 ) .88 Control theory . For a matrix 𝐴 ∈ C𝑑×𝑑 , define its resolvent as Φ𝐴 ( 𝑧 ) = ( 𝑧𝐼 − 𝐴 ) −1 . For a89 linear dynamical system 𝒟 given by ( 1 ) – ( 2 ) , let Φ𝒟 = Φ𝑢→𝑦 denote the transfer function from90 𝑢 to 𝑦 ( response to input ) . Then Φ𝒟 = Φ𝑢→𝑦 = 𝐶Φ𝐴𝐵 + 𝐷 = 𝐶 ( 𝑧𝐼 − 𝐴 ) −1𝐵 + 𝐷. Let91 T : = { 𝑧 ∈ C : |𝑧| = 1 } be the unit circle in the complex plane . For a matrix-valued function 𝐹 :92 T→ C𝑑1×𝑑2 , define theℋ2 andℋ∞ norms by93 ‖Φ‖ℋ2 = 1 2𝜋 ∫︁ T ‖Φ ( 𝑧 ) ‖2𝐹 𝑑𝑧 ‖Φ‖ℋ∞ = sup 𝑧∈T ‖Φ ( 𝑧 ) ‖ . For a function 𝐹 : N0 → C𝑑1×𝑑2 , define its Z-transform to be 𝒵 [ 𝐹 ] ( 𝑧 ) = ∑︀∞ 𝑛=0 𝐹 ( 𝑛 ) 𝑧 −𝑛 . Con-94 sidered as a function T→ C , we can take its ℋ2 and ℋ∞ norms . Overloading notation , we will let95 ‖𝐹‖ℋ𝑝 : = ‖𝒵𝐹‖ℋ𝑝 for 𝑝 = 2 , ∞ . Theℋ2 andℋ∞ norms can be interpreted as the Frobenius and96 operator norms of the linear operator from input to output , i.e. , they measure the average power of97 the output signal under random or worst-case input , respectively . For background on control theory,98 see e.g. , [ ZDG+96 ] .99 Constants . In proofs , 𝐶 may represent different constants from line to line.100 2 Main results101 We consider the problem of prediction and identification for an unknown linear dynamical sys-102 tem ( 1 ) – ( 2 ) . Our main goal is to obtain error guarantees in ℋ2 norm , which determines prediction103 error under random input [ OO19 , Lemma 3.3 ] .104 Problem 2.1 . Consider the partially-observed LTI system 𝒟 ( 1 ) – ( 2 ) with gaussian inputs 𝑢 ( 𝑡 ) ∼105 𝑁 ( 0 , 𝐼𝑑𝑢 ) for 0 ≤ 𝑡 < 𝑇 . Suppose that the system is stable , that is , 𝜌 ( 𝐴 ) < 1 , and that we observe106 a single trajectory of length 𝑇 started with 𝑥 ( 0 ) = 0 , that is , we observe 𝑢 ( 𝑡 ) ∼ 𝑁 ( 0 , 𝐼𝑑𝑢 ) and 𝑦 ( 𝑡 ) 107 for 0 ≤ 𝑡 < 𝑇 .108 The goal is to learn a LTI system ‹𝒟 with the aim of minimizing ⃦⃦Φ‹𝒟 − Φ𝒟⃦⃦ℋ2 . Equivalently , letting109 𝐹 * ( 𝑡 ) = ® 𝐷 , 𝑡 = 0 𝐶𝐴𝑡−1𝐵 , 𝑡 ≥ 1 denote the impulse response function ( also called the Markov parameters ) of the system , the goal is110 to learn an impulse response ‹𝐹 minimizing ⃦⃦⃦𝐹 * − ‹𝐹 ⃦⃦⃦ ℋ2 = ⃦⃦⃦ 𝐹 * − ‹𝐹 ⃦⃦⃦ F .111 Note that learning 𝐹 * is sufficient to fully understand the input-output behavior of the system , but112 we may also ask to recover the system parameters 𝐴 , 𝐵 , 𝐶 , 𝐷 up to similarity transformation ( see113 Theorem 2.3 ) .114 Previous results [ OO19 , SRD19 ] roughly depend polynomially on the “ memory ” 11−𝜌 ( 𝐴 ) , which115 blows up as the spectral norm of 𝐴 approaches 1 . In the setting of zero process noise , our goal is116 to obtain rates that are ‹𝑂 Äpoly ( 𝑑 , 𝑑𝑢 , 𝑑𝑦 ) √ 𝑇 ä , with only poly-logarithmic dependence on 11−𝜌 ( 𝐴 ) . See117 Figure 1 for a comparison.118 We assume that 𝜌 ( 𝐴 ) < 1 because if 𝒟 is not stable , it is in general impossible to learn ‹𝒟 with119 finite ℋ2 error , as a system with infinite response can have arbitrarily small response on any finite120 time interval . However , it may still be possible to learn the response up to time 𝐿 ≪ 𝑇 in this121 case [ SBR19 ] . The marginally stable case ( 𝜌 ( 𝐴 ) = 1 ) is an important case we leave to future work.122 Method Rollout type Min # samples IR error Least squares ( IR ) [ TBPR17 ] Multi 𝐿 𝜎 » 𝐿 𝑇 Least squares ( IR ) [ OO19 ] Single 𝐿 𝜎 » 𝐿 𝑇 Nuclear norm minimization Multi min { 𝑑2 , 𝐿 } 𝜎 » 𝐿 𝑇 [ SOF20 ] Multi 𝑑 𝜎 » 𝑑𝐿 𝑇 rank-𝑑 SVD ( Theorem 2.2 ) Single 𝐿 𝜎 » 𝑑 𝑇 Figure 1 : Here , 𝐿 is the memory length for the system , which is ‹𝑂 Ä 11−𝜌 ( 𝐴 ) ä for well-conditioned systems . Rollout type refers to whether we have access to a single trajectory or multiple trajectories . Min # samples refers to the minimum number of samples ( up to log factors ) before the bounds are operational . IR error refers to the error in the impulse response in Frobenius/ℋ2 norm . Logarithmic factors are omitted . In our Algorithm 1 , we first use linear regression to obtain a noisy estimate 𝐹 of the impulse re-123 sponse . Next , following standard system identification procedures , we form the Hankel matrix124 Hankel𝐿×𝐿 ( 𝐹 ) with the entries of 𝐹 on its diagonals . Because the true Hankel matrix125 Hankel𝐿×𝐿 ( 𝐹 * ) = à 𝐶𝐵 𝐶𝐴𝐵 · · · 𝐶𝐴𝐿−1𝐵 𝐶𝐴𝐵 𝐶𝐴2𝐵 ... ... . . . ... 𝐶𝐴𝐿−1𝐵 · · · · · · 𝐶𝐴2𝐿−1𝐵 í has rank 𝑑 , we take a low-rank SVD 𝑅𝐿 of the Hankel matrix to “ de-noise ” the impulse response.126 We can then read off the estimated impulse response by averaging over the corresponding diagonal127 of 𝑅𝐿 . For technical reasons , we need to repeat this process for a geometric sequence of sizes128 of the Hankel matrix : 𝐿 × 𝐿 , 𝐿/2 × 𝐿/2 , 𝐿/4 × 𝐿/4 , and so forth . This is because the low-129 rank approximation objective for a ℓ × ℓ Hankel matrix encourages the diagonals that are Θ ( ℓ ) to130 be close—as those are the diagonals with the most entries—and hence estimates 𝐹 * ( 𝑡 ) well when131 𝑡 = Θ ( ℓ ) . In other words , low-rank estimation for Hankelℓ×ℓ ( 𝐹 ) is only sensitive to the portion of132 the signal that is at timescale ℓ . Repeating this process ensures that we cover all scales.133 Our main theorem is the following.134 Theorem 2.2 . There is a constant 𝐶1 such that following holds . In the setting of Problem 2.1,135 suppose that 𝐹 * is the impulse response function , 𝑇 is such that 𝑇 ≥ 𝐶1𝐿𝑑𝑢 log ( ︀ 𝐿𝑑𝑢 𝛿 ) ︀ , 𝜀trunc : =136 ⃦⃦ 𝐹 * 1 [ 2𝐿 , ∞ ) ⃦⃦ ℋ∞ √ 𝑑𝑢 + ⃦⃦ 𝐺 * 1 [ 2𝐿 , ∞ ) ⃦⃦ ℋ∞ ⃦⃦⃦ Σ 1/2 𝑥 ⃦⃦⃦ F , and 𝑀𝑥→𝑦 = ( 𝑂 , 𝐶 , 𝐶𝐴 , . . . , 𝐶𝐴𝐿−1 ) ⊤ ∈137 R ( 𝐿+1 ) 𝑑×𝑑𝑦 . Let 0 < 𝛿 ≤ 12 and 𝜎 = » ‖Σ𝑦‖+ ‖Σ𝑥‖𝐿 log ( ︀ 𝐿𝑑𝑢 𝛿 ) ︀ ‖𝑀𝑥→𝑦‖2 . Then with probabil-138 ity at least 1− 𝛿 , Algorithm 1 learns an impulse response function ‹𝐹 such that139 ⃦⃦⃦‹𝐹 − 𝐹 * ⃦⃦⃦ F = 𝑂 ( ︃ 𝜎 𝑑 ( ︀ 𝑑𝑦 + 𝑑𝑢 + log ( ︀ 𝐿 𝛿 ) ︀ ) ︀ log𝐿 𝑇 + 𝜀trunc √ 𝑑 + ⃦⃦ 𝐹 * 1 ( 𝐿 , ∞ ) ⃦⃦ F ) ︃ . In the absence of process noise ( when Σ𝑥 = 𝑂 ) , when 𝐿 and 𝑇 are chosen large enough , the first140 term dominates , and ignoring log factors , the dependence is 𝑂 ( ︁ » 𝑑 ( 𝑑𝑦+𝑑𝑢 ) 𝑇 ) ︁ . We expect this to be141 Algorithm 1 Learning impulse response through multi-scale low-rank Hankel SVD 1 : Input : Length 𝐿 ( power of 2 ) , time 𝑇 . 2 : Part 1 : Linear regression to recover noisy impulse response 3 : Let 𝑢 ( 𝑡 ) ∼ 𝑁 ( 0 , 𝐼𝑑𝑢 ) for 0 ≤ 𝑡 < 𝑇 , and observe the outputs 𝑦 ( 𝑡 ) ∈ R𝑑𝑦 , 0 ≤ 𝑡 < 𝑇 . 4 : Solve the least squares problem min 𝐹 : Supp ( 𝐹 ) ⊆ [ 0,2𝐿−1 ] 𝑇−1∑︁ 𝑡=0 ‖𝑦 ( 𝑡 ) − 𝐹 * 𝑢 ( 𝑡 ) ‖2 . to obtain the noisy impulse response 𝐹 : [ 0 , 2𝐿− 1 ] ∩ Z→ R𝑑𝑦×𝑑𝑢 . 5 : Part 2 : Low-rank Hankel SVD to de-noise impulse response 6 : Let ‹𝐹 ( 0 ) = 𝐹 ( 0 ) . 7 : for 𝑘 = 0 to log2 𝐿 do 8 : Let ℓ = 2𝑘 . 9 : Let 𝑅ℓ be the rank-𝑑 SVD of Hankelℓ×ℓ ( 𝐹 ) ( i.e. , argminrank ( 𝑅 ) ≤𝑑 ‖𝑅−Hankelℓ×ℓ ( 𝐹 ) ‖ ) . 10 : For ℓ2 < 𝑡 ≤ ℓ , let ‹𝐹 ( 𝑡 ) be the 𝑑𝑦 × 𝑑𝑢 matrix given by ‹𝐹 ( 𝑡 ) = 1𝑡 ∑︀𝑖+𝑗=𝑡 ( 𝑅ℓ ) 𝑖𝑗 , where ( · ) 𝑖𝑗 denotes the ( 𝑖 , 𝑗 ) th block of the matrix . 11 : end for 12 : Output : Estimate of impulse response ‹𝐹 . the optimal sample complexity up to logarithmic factors . However , in the presence of process noise,142 there is an undesirable factor of √ 𝐿 ‖𝑀𝑥→𝑦‖ , which ( for well-conditioned matrices ) is expected to143 be 𝑂 Ä 1 1−𝜌 ( 𝐴 ) ä or 𝑂 ( 𝐿 ) . We leave it an open problem to improve the guarantees in this setting.144 Remark 1 . The 𝐿-factor dependence on the process noise is unavoidable with the current algorithm:145 when the process noise has covariance Σ𝑦 = 𝐼 and decays after 𝐿 steps , it can cause perturbations of146 size 𝑂 ( √ 𝐿 ) compared to the noiseless system . Even in the case 𝑑 = 1 , when the impulse response147 function is 𝑎𝑒−𝑘𝑡/𝐿 for a known 𝑘 , the noise will cause the estimate of 𝑎 to be off by 𝑂 ( √ 𝐿 ) , and148 hence the ℋ2 norm of the impulse response to be off by 𝑂 ( 𝐿 ) . Our algorithm only regresses on149 previous inputs , but in the presence of process noise , a better approach is to regress on both the150 previous inputs 𝑢 ( 𝑡 ) and outputs 𝑦 ( 𝑡 ) and then take a ( weighted ) SVD , as in N4SID [ Qin06 ] .151 Remark 2 . A burn-in time of Ω ( 𝐿 ) is information-theoretically required to get poly ( 𝑑 ) rates . At-152 tempting to extrapolate an impulse response function from time 𝑜 ( 𝐿 ) to time 𝐿 can magnify errors153 by exp ( 𝑑 ) , because the finite impulse response of a system of order 𝑑 can approximate a polynomial154 of degree 𝑑− 1 on [ 0 , 𝐿 ] .155 We also show the following improved rates for learning the system matrices , by combining ℋ∞156 bounds for the learned impulse response with stability results for the Ho-Kalman algorithm [ OO19 ] .157 Because the input-output behavior is unchanged under a similarity transformation ( 𝐴 , 𝐵 , 𝐶 ) ← [ 158 ( 𝑊−1𝐴𝑊 , 𝑊−1𝐵 , 𝐶𝑊 ) , we can only learn the parameters up to similarity transformation.159 Theorem 2.3 . Keep the assumptions and notation of Theorem 2.2 , suppose 𝒟 is observable and160 controllable , and let161 𝜀′ = 𝜎 𝐿 ( ︀ 𝑑𝑦 + 𝑑𝑢 + log ( ︀ 𝐿 𝛿 ) ︀ ) ︀ 𝑇 + 𝜀trunc . Let 𝐻− = Hankel𝐿× ( 𝐿−1 ) ( 𝐹 * ) . Suppose that 𝜀′ = 𝑂 ( 𝜎min ( 𝐻− ) ) . Then with probability at least162 1−𝛿 , the Ho-Kalman algorithm ( Algorithm 1 in [ OO19 ] with 𝑇1 = 𝐿 , 𝑇2 = 𝐿−1 ) returns ̂︀𝐴 , “ 𝐵 , “ 𝐶163 such that there exists a unitary matrix 𝑊 satisfying164 max { ︁⃦⃦⃦ 𝐶 − “ 𝐶𝑊 ⃦⃦⃦ F , ⃦⃦⃦ 𝐵 −𝑊−1 “ 𝐵⃦⃦⃦ F } ︁ = 𝑂 ( √ 𝑑 · 𝜀′ ) ⃦⃦⃦ 𝐴−𝑊−1 ̂︀𝐴𝑊 ⃦⃦⃦ F = 𝑂 Ç 1 𝜎min ( 𝐻− ) · √ 𝑑 · 𝜀′ · Ç ‖Φ𝒟‖ℋ∞ 𝜎min ( 𝐻− ) + 1 åå . As 𝐿 can be chosen to make 𝜀trunc negligible , this gives ‹𝑂 ( ︁ » 𝐿𝑑 ( 𝑑𝑢+𝑑𝑦 ) 𝑇 ) ︁ rates , however,165 with factors depending on the minimum eigenvalue of 𝐻 . This is an improvement over the166 ‹𝑂 ( ︁√𝑑 4 » 𝐿 ( 𝑑𝑢+𝑑𝑦 ) 𝑇 ) ︁ rates in [ OO19 ] .167 We prove Theorem 2.2 in Section 4 and Theorem 2.3 in Appendix B.168 3 Related work169 We survey two classes of methods for learning partially observable LDS ’ s , subspace identification170 and improper learning . With the exception of [ RJR20 ] , all guarantees have sample complexity171 depending on the memory length 𝐿 , which we wish to avoid.172 3.1 Subspace identification173 The basic idea of subspace identification [ Lju98 , Qin06 , VODM12 ] is to learn a certain structured174 matrix ( such as a Hankel matrix ) , take a best rank-𝑘 approximation ( using SVD or another linear di-175 mensionality reduction method ) , and learn the system matrices 𝐴 , 𝐵 , 𝐶 , 𝐷 up to similarity transfor-176 mation . Usage of spectral methods circumvents the fact that the most natural optimization problem177 for 𝐴 , 𝐵 , 𝐶 , 𝐷 is non-convex . However , classical guarantees for these methods are asymptotic.178 Recently , various authors have given non-asymptotic guarantees for system identification algo-179 rithms . [ OO19 ] analyzed the Ho-Kalman algorithm [ HK66 ] in this setting . [ SRD19 ] consider the180 setting where system order is unknown and give an end-to-end result for prediction , while [ TMP20 ] 181 consider the problem of online filtering , that is , recovering 𝑥 ( 𝑡 ) ’ s up to some linear transformation.182 An alternate , empirically successful approach is that of nuclear norm minimization or regulariza-183 tion [ FPST13 ] . [ SOF20 ] ( building on [ CQXY16 ] ) give explicit rates of convergence , and show that184 the algorithm has a lower minimum sample complexity and is easier to tune.185 Our algorithm is based on the classical approach of taking a low-rank approximation of the Hankel186 matrix , but we repeat this process with Hankel matrices of sizes 𝐿×𝐿 , 𝐿/2×𝐿/2 , 𝐿/4×𝐿/4 , and so187 forth ; this is key modification that allows us to obtain better statistical rates . Our analysis builds on188 the analyses given in [ OO19 , SOF20 ] . As essential part of the analysis is analyzing linear regression189 for correlated inputs , where we extend the work of [ DMR19 ] to MIMO systems , as explained below.190 3.1.1 Linear regression with correlated inputs191 An important step in obtaining non-asymptotic rates for system identification is analyzing linear192 regression for correlated inputs . The most challenging step is to lower-bound the sample covariance193 matrix of inputs to the linear regression . A lower bound , rather than a matrix concentration result , is194 sufficient [ Men14 , SMT+18 , MT19 ] ; however , a concentration result is obtainable in our setting.195 [ TBPR17 ] give non-asymptotic bounds for learning the finite impulse response for a SISO system196 in ℓ∞ Fourier norm ; however , they require 𝐿 rollouts of size 𝑂 ( 𝐿 ) and hence Ω ( 𝐿2 ) timesteps.197 Addressing the more challenging single-rollout setting , [ OO19 ] obtain bounds for a single rollout198 of ̃︀Ω ( 𝐿 ) timesteps , by using concentration bounds for random circulant matrices [ KMR14 ] to de-199 rive concentration inequalities for the covariance matrix . These concentration inequalities for the200 covariance matrix were improved ( by logarithmic factors ) by [ DMR19 ] . Although [ DMR19 ] give201 an analysis in the SISO case , as we show in Theorem A.2 , the results can be extended to the MIMO202 case with an 𝜀-net argument.203 3.2 Improper learning using autoregressive methods204 Instead of solving the statistical problem of identifying parameters , another line of work develops205 algorithms for regret minimization in online learning . The goal is simply to do well in predicting206 future observations , with small loss ( regret ) compared to the best predictor in hindsight ; the learned207 predictor is allowed to be improper , that is , take a different functional form . In the stochastic case,208 this allows prediction almost as well as if the actual system parameters were known ; however , the209 framework also allows for adversarial noise.210 One popular strategy for improperly learning the system is to learn a linear autoregressive filter over211 previous inputs and observations , or ARMA model . Naturally , because we are optimizing over a212 larger hypothesis class , the statistical rates depend on 𝐿 rather than the system order 𝑑.213 [ GLS+20 , Theorem 4.7 ] consider the problem of online prediction for a fully or partially observed214 LDS , and give a regret bound that depends polynomially on the memory length 𝐿 . Their approach215 works even for marginally stable systems , that is , systems with 𝜌 ( 𝐴 ) ≤ 1 . See also [ AHMS13,216 HSZ17 , HLS+18 , KMTM19 , TP20 , RJR20 ] for previous work using autoregressive methods.217 Of particular interest to us is [ RJR20 ] , which gives rates independent of spectral radius . Building218 on [ HSZ17 ] , they observe that it suffices to regress on previous inputs and outputs projected to a219 lower-dimensional space . Their algorithm works in the setting of process noise and competes with220 the Kalman filter , but only when 𝐴−𝐾𝐶 has real eigenvalues , where 𝐾 is the Kalman gain.221 4 Proof of main theorem222 In this section , we prove Theorem 2.2 . The proof hinges on the following lemma , which shows223 that if we observe a low-rank matrix plus noise , then taking a low-rank SVD can have a de-noising224 effect , producing a matrix that is closer to the true matrix.225 Lemma 4.1 ( De-noising effect of SVD ) . There exists a constant 𝐶 such that the following holds.226 Suppose that 𝐴 ∈ C𝑚×𝑛 is a rank-𝑟 matrix , ̂︀𝐴 = 𝐴 + 𝐸 , and ̂︀𝐴𝑟 is the rank-𝑟 SVD of ̂︀𝐴 . Then227 ⃦⃦⃦ ̂︀𝐴𝑟 −𝐴⃦⃦⃦ F ≤ 𝐶 √ 𝑟 ‖𝐸‖ . ( 3 ) Compare this with the original error ⃦⃦⃦ ̂︀𝐴−𝐴⃦⃦⃦ F = ‖𝐸‖F , which can only be bounded by228 √︀ min { 𝑚 , 𝑛 } ‖𝐸‖ . When applied to the 𝑑-SVD of the Hankel matrix , this gives a factor of √ 𝑑229 rather than √ 𝐿 for the error.230 Proof . We have231 ⃦⃦⃦ 𝐴𝑟 −𝐴 ⃦⃦⃦ F ≤ √ 2𝑟 ⃦⃦⃦ 𝐴𝑟 −𝐴 ⃦⃦⃦ 2 ( 4 ) ≤ √ 2𝑟 ( ︁⃦⃦⃦ 𝐴𝑟 −𝐴 ⃦⃦⃦ 2 + ⃦⃦⃦ 𝐴−𝐴 ⃦⃦⃦ 2 ) ︁ ( 5 ) ≤ 2 √ 2𝑟 ‖𝐸‖ ( 6 ) where ( 4 ) follows from 𝐴𝑟 − 𝐴 having rank at most 2𝑟 , ( 5 ) follows from the triangle inequality,232 and ( 6 ) follows from Weyl ’ s Theorem : ⃦⃦⃦ 𝐴𝑟 −𝐴 ⃦⃦⃦ 2 ≤ 𝜎𝑟+1 ( 𝐴 ) ≤ 𝜎𝑟+1 ( 𝐴 ) + ‖𝐸‖ = ‖𝐸‖.233 To prove Theorem 2.2 , we will need to obtain bounds for 𝐹 : { 0 , 1 , . . . , 2𝐿−1 } → R𝑑𝑦×𝑑𝑢 learned234 from linear regression inℋ∞ norm . The following is our main technical result.235 Lemma 4.2 . There are 𝐶1 , 𝐶2 such that the following hold . Suppose 𝑦 = 𝐹 * * 𝑢 + 𝐺 * * 236 𝜉 + 𝜂 where 𝑢 ( 𝑡 ) ∼ 𝑁 ( 0 , 𝐼𝑑𝑢 ) , 𝜉 ( 𝑡 ) ∼ 𝑁 ( 0 , Σ𝑥 ) , 𝜂 ( 𝑡 ) ∼ 𝑁 ( 0 , Σ𝑦 ) for 0 ≤ 𝑡 < 𝑇 ,237 and Supp ( 𝐹 * ) , Supp ( 𝐺 * ) ⊆ [ 0 , ∞ ) . Let 𝐹 = argmin𝐹∈ { 0 , ... , 𝐿 } →R𝑑𝑦×𝑑𝑢 ∑︀𝑇−1 𝑡=0 |𝑦 ( 𝑡 ) − ( 𝐹 * 238 𝑢 ) ( 𝑡 ) |2 , 𝑀𝐺 * = ( 𝐺 * ( 0 ) , . . . , 𝐺 * ( 𝐿 ) ) ⊤ ∈ R ( 𝐿+1 ) 𝑑×𝑑𝑦 , and 𝜀trunc = ⃦⃦ 𝐹 * 1 [ 𝐿+1 , ∞ ) ⃦⃦ ℋ∞ √ 𝑑𝑢 +239 ⃦⃦ 𝐺 * 1 [ 𝐿+1 , ∞ ) ⃦⃦ ℋ∞ ⃦⃦⃦ Σ 1/2 𝑥 ⃦⃦⃦ F . For 0 < 𝛿 ≤ 12 , 𝑇 ≥ 𝐶1𝐿𝑑𝑢 log ( ︀ 𝐿𝑑𝑢 𝛿 ) ︀ , −1 ≤ 𝑎 < 𝐿− 𝐿′,240 ⃦⃦ ( 𝐹 − 𝐹 * ) 1 [ 𝑎+1 , 𝑎+𝐿′ ] ⃦⃦ ℋ∞ ≤ 𝐶2 ñ… 1 𝑇 Ç ‖Σ𝑦‖𝐿′ Å 𝑑𝑢 + 𝑑𝑦 + log Å 𝐿′ 𝛿 ãã + ‖Σ𝑥‖𝐿′𝐿𝑑𝑢 log Å 𝐿𝑑𝑢 𝛿 ã ‖𝑀𝐺 * ‖ å + 𝜀trunc ô with probability at least 1− 𝛿.241 In the case Σ𝑥 = 𝑂 , this roughly says that the error in the learned impulse response , 𝐹 − 𝐹 * , over242 any interval of length 𝐿′ , has all Fourier coefficients bounded in spectral norm by ‹𝑂 ( ︁ » 𝐿′ ( 𝑑𝑢+𝑑𝑦 ) 𝑇 ) ︁243 – what we expect if the error from linear regression is uniformly distributed over all frequencies.244 A complete proof is in Appendix A ; we give a brief sketch . First , because the errors are Gaussian,245 the error from linear regression , 𝐹 − 𝐹 * , follows a Gaussian distribution . To bound its covariance,246 we lower-bound the smallest singular value of the sample covariance of the inputs ( Lemma A.1,247 Appendix A.1 ) . Here , the difficulty is that the inputs are correlated – the input at time 𝑡 is 𝑢𝑡 : 𝑡−𝐿.248 Fortunately , the translation structure means it is close to a submatrix of an infinite block Toeplitz249 matrix , which becomes block diagonal in the Fourier domain . This “ decoupling ” allows us to show250 concentration . Compared to the SISO setting in [ DMR19 ] , we require an extra 𝜀-net argument . Once251 we have a bound on the covariance , we can bound any ⃦⃦⃦ ( ◊ 𝐹 − 𝐹 * ) ( 𝜔 ) ⃦⃦⃦ by matrix concentration252 ( Appendix A.2 ) ; to bound theℋ∞ norm it suffices to bound this over a grid of 𝜔 ’ s ( Lemma A.4 ) .253 Bounding the error in ℋ∞ norm of the impulse response allows us to bound the error in operator254 norm of the Hankel matrix , as the following lemma shows.255 Lemma 4.3 . For any 𝐹 : Z→ C𝑚×𝑛 , we have ‖Hankel𝑎×𝑏 ( 𝐹 ) ‖ ≤ ‖𝐹‖ℋ∞ .256 Proof . Note that when 𝑣 : Z → C𝑛 , Supp ( 𝑣 ) ⊆ [ 0 , 𝑏 − 1 ] , we have ( 𝐹 * 𝑣 ) 𝑏 : 𝑏+𝑎−1 =257 Hankel𝑎×𝑏 ( 𝐹 ) 𝑣𝑏−1:0 . Hence for any 𝑣 set to 0 outside of [ 0 , 𝑏 − 1 ] , using Parseval ’ s Theorem258 and the fact that the Fourier transform of a convolution is the product of the Fourier transforms , we259 have260 ‖Hankel𝑎×𝑏 ( 𝐹 ) 𝑣𝑏−1:0‖2 ≤ ‖𝐹 * 𝑣‖2 = ⃦⃦⃦ “ 𝐹̂︀𝑣⃦⃦⃦ 2 ≤ sup 𝜔∈ [ 0,1 ] ⃦⃦⃦ “ 𝐹 ( 𝜔 ) ⃦⃦⃦ 2 ‖̂︀𝑣‖2 = ‖𝐹‖ℋ∞ ‖𝑣‖2 This shows that ‖Hankel𝑎×𝑏 ( 𝐹 ) ‖ ≤ ‖𝐹‖ℋ∞ .261 Theorem 2.2 will follow from the following bound after an application of the triangle inequality.262 Lemma 4.4 . There are 𝐶1 , 𝐶2 such that the following holds for the setting of Problem 2.1.263 Suppose 𝐿 is a power of 2 , and 𝑇 ≥ 𝐶1𝐿𝑑𝑢 log ( ︀ 𝐿𝑑𝑢 𝛿 ) ︀ . Let ⃦⃦ 𝐹 * 1 [ 𝐿+1 , ∞ ) ⃦⃦ ℋ∞ √ 𝑑𝑢 +264 ⃦⃦ 𝐺 * 1 [ 𝐿+1 , ∞ ) ⃦⃦ ℋ∞ ⃦⃦⃦ Σ 1/2 𝑥 ⃦⃦⃦ F and 𝑀𝑥→𝑦 = ( 𝑂 , 𝐶 , 𝐶𝐴 , . . . , 𝐶𝐴𝐿−1 ) ⊤ ∈ R ( 𝐿+1 ) 𝑑×𝑑𝑦 . Then with265 probability at least 1− 𝛿 , the output ‹𝐹 given by Algorithm 1 satisfies266 ⃦⃦⃦ ( ‹𝐹 − 𝐹 * ) 1 [ 1 , 𝐿 ] ⃦⃦⃦ F ≤ 𝐶2 ( ︃ ‖Σ𝑦‖ 𝑑 ( ︀ 𝑑𝑦 + 𝑑𝑢 + log ( ︀ 𝐿 𝛿 ) ︀ ) ︀ log𝐿 𝑇 + ‖Σ𝑥‖𝐿𝑑𝑑𝑢 log ( ︀ 𝐿𝑑𝑢 𝛿 ) ︀ 𝑇 ‖𝑀𝐺 * ‖+ 𝜀trunc √ 𝑑 ) ︃ Proof . We are in the situation of Lemma 4.2 with 𝐺 * ( 𝑡 ) = 𝐶𝐴𝑡−11𝑡≥1 . Let ℋℓ = Hankelℓ×ℓ ( 𝐹 ) 267 andℋ * ℓ = Hankelℓ×ℓ ( 𝐹 * ) . Suppose ℓ ≤ 𝐿 is even . Note that268 ℋℓ = Hankelℓ×ℓ ( 𝐹 * ) ⏟ ⏞ ℋ * ℓ + Hankelℓ×ℓ ( 𝐹 − 𝐹 * ) whereℋ * ℓ = Hankelℓ×ℓ ( 𝐹 * ) is a rank-𝑑 matrix , with error term is bounded by269 ‖Hankelℓ×ℓ ( 𝐹 − 𝐹 * ) ‖ ≤ ⃦⃦ ( 𝐹 − 𝐹 * trunc ) 1 [ 1,2ℓ−1 ] ⃦⃦ ℋ∞ by Lemma 4.3 < 𝐶 [ ︃… 1 𝑇 ( ︃ ‖Σ𝑦‖ ℓ Å 𝑑𝑢 + 𝑑𝑦 + log Å 𝐿′ 𝛿 ãã + ‖Σ𝑥‖ ℓ𝐿𝑑𝑢 log Å 𝐿𝑑𝑢 𝛿 ã ‖𝑀𝐺 * ‖+ 𝜀trunc ] ︃ by Lemma 4.2 ( 7 ) with probability at least 1− 𝛿 . Let 𝑅ℓ be the rank-𝑑 SVD ofℋℓ . Then by Lemma 4.1,270 ‖𝑅ℓ −ℋ * ℓ‖F = 𝑂 Ä√ 𝑑 ‖Hankelℓ×ℓ ( 𝐹 − 𝐹 * ) ‖ ä . ( 8 ) Now letting ‹𝐹 ( 𝑡 ) = 1𝑡 ∑︀𝑖+𝑗=𝑡 ( 𝑅ℓ ) 𝑖𝑗 when ℓ2 < 𝑡 ≤ ℓ we have ( using the fact that the mean271 minimizes the sum of squared errors ) 272 ‖𝑅ℓ −ℋ * ℓ‖ 2 F ≥ ℓ∑︁ 𝑡= ℓ2+1 ∑︁ 𝑖+𝑗=𝑡 ‖ ( 𝑅ℓ ) 𝑖𝑗 − 𝐹 * ( 𝑡 ) ‖2F ≥ ℓ∑︁ 𝑡= ℓ2+1 Å 𝑡 · ⃦⃦⃦‹𝐹 ( 𝑡 ) − 𝐹 * ( 𝑡 ) ⃦⃦⃦2 F ã ≥ Å ℓ 2 + 1 ã ℓ∑︁ 𝑡= ℓ2+1 Å⃦⃦⃦‹𝐹 ( 𝑡 ) − 𝐹 * ( 𝑡 ) ⃦⃦⃦2 F ã . Note that we only get a lower bound with a factor of ℓ if we restrict to 𝑡 that is Θ ( ℓ ) , i.e. , restrict to273 diagonals that have many entries . This is the reason we will have to repeat this process for multiple274 sizes . Hence275 ⃦⃦⃦ ( ‹𝐹 − 𝐹 * ) 1 [ ℓ2+1 , ℓ ] ⃦⃦⃦2F ≤ 1ℓ/2 ‖𝑅ℓ −ℋ * ℓ‖2F . Together with ( 8 ) and ( 7 ) this gives with probability ≥ 1− 𝛿 that276 ⃦⃦⃦ ( ‹𝐹 − 𝐹 * ) 1 [ ℓ 2 +1 , ℓ ] ⃦⃦⃦ F ≤ 𝐶 ( ︃ ‖Σ𝑦‖ 𝑑 ( ︀ 𝑑𝑦 + 𝑑𝑢 + log ( ︀ 𝐿 𝛿 ) ︀ ) ︀ 𝑇 + ‖Σ𝑥‖𝐿𝑑𝑢 log ( ︀ 𝐿𝑑𝑑𝑢 𝛿 ) ︀ 𝑇 ‖𝑀𝐺 * ‖+ 𝜀trunc √ 𝑑√ ℓ ) ︃ Replacing 𝛿 by 𝛿log2 𝐿 , using a union bound over powers of 2 , and summing gives277 ⃦⃦⃦ ( ‹𝐹 − 𝐹 * ) 1 [ 1 , 𝐿 ] ⃦⃦⃦ F = 𝑂 ( ︃ ‖Σ𝑦‖ 𝑑 ( ︀ 𝑑𝑦 + 𝑑𝑢 + log ( ︀ 𝐿 𝛿 ) ︀ ) ︀ log𝐿 𝑇 + ‖Σ𝑥‖𝐿𝑑𝑑𝑢 log ( ︀ 𝐿𝑑𝑢 𝛿 ) ︀ log𝐿 𝑇 ‖𝑀𝐺 * ‖+ 𝜀trunc √ 𝑑 ) ︃ . 278 Proof of Theorem 2.2 . We have the bound in Lemma 4.4 , and also the same bound279 for ⃦⃦⃦ ( ‹𝐹 − 𝐹 * ) ( 0 ) ⃦⃦⃦ F after applying Lemma 4.2 to ( 𝐹 − 𝐹 * ) 𝛿0 . Finally , note that280 ⃦⃦⃦ ( ‹𝐹 − 𝐹 * ) 1 ( 𝐿 , ∞ ) ⃦⃦⃦ F = ⃦⃦ 𝐹 * 1 ( 𝐿 , ∞ ) ⃦⃦ F and use the triangle inequality.281 5 Experiments282 We compared three algorithms for learning the impulse response function : least-squares , and low-283 rank Hankel SVD with and without the multi-scale repetition . We include details of the experimental284 setup in Appendix D. 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Summary: This work studies the estimation of linear dynamical systems with unobserved state. The authors propose a novel algorithm which relies on obtaining low-rank approximations to the Hankel matrix. They provide several results on the performance of their algorithm, including finite time bounds on the error it obtains in estimating the system’s impulse response. As compared to previous work, the main novelty of this paper is eliminating a factor of the horizon in the case where there is no process noise.