On the Sample Complexity of Decentralized Linear Quadratic Regulator with Partially Nested Information Structure

Solving optimal control problems without prior system model knowledge has receive much attention recently. One of the most studied problems is the centralized LQR problem. For this problem, two broad classes of methods have been studied, i.e., model based learning [1, 29, 12], and model-free learning [16, 41, 28, 20]. In the model-based learning approach, a system model is first estimated from observed system trajectories using some system identification method. A control policy can then be obtained based on the estimated system model. In the model-free learning approach, the objective function in the LQR problem is first viewed as a function of the control policies. Based on zeroth-order optimization methods (e.g., [19, 30]), the optimal solution can then be obtained using gradient descent, where the gradient of the objective function is estimated from the data samples from system trajectories. Moreover, the model-based learning approach has also been studied for the centralized linear quadratic Gaussian control problem [43]. In general, compared to model-free learning, model-based learning tends to require less data samples in order to achieve a policy of equivalent performance [38].

Most of the previous works on model-based learning for centralized LQR build on recent advances in non-asymptotic analyses for system identification of linear dynamical systems with full state observations (e.g., [13, 36, 34]). Such non-asymptotic analyses (i.e., sample complexity results) relate the estimation error of the system matrices to the number of samples used for system identification. In particular, it was shown in [36] that when using a single system trajectory, the least squares approach for system identification achieves the optimal sample complexity up to logarithmic factors. In this paper, we utilize a similar least squares approach for estimating the system matrices from a single system trajectory. Although the system matrices in our problem are structured, as dictated by the interconnections among the subsystems, we leverage the results in [2, 10] to provide a non-asymptotic analysis of the resulting estimation error.

There are few results on solving decentralized linear quadratic control problems with information constraints, when the system model is unknown. In [18], the authors studied a decentralized output-feedback linear quadratic control problem, under the assumption that the quadratic invariance condition is satisfied. The authors proposed a model-free approach and provided a sample complexity analysis. They focused on a finite-horizon setting, since gradient-based optimization methods may not converge to the optimal controller for infinite-horizon decentralized linear quadratic control problems with information constraints, even when the system model is known [17, 7]. In [27], the authors proposed a consensus-based model-free learning algorithm for multi-agent decentralized LQR over an infinite horizon, where each agent (i.e., controller) has access to a subset of the global state without delay. They showed that their algorithm converges to a control policy that is a stationary point of the objective function in the LQR problem. In [15], the authors studied model-based learning for LQR with subspace constraints on the closed-loop responses. However, those constraints may not lead to controllers that satisfy the information constraints that we consider in this paper (e.g., [42]).

There is also a line of research on online adaptive control for centralized LQR with unknown system models, using either model-based learning [1, 11, 10], or model-free learning [3, 8]. The goal there is to adaptively design a control policy in an online manner when new data samples from the system trajectory become available, and bound the corresponding regret.

We propose a two-step model-based approach to solving the problem of learning decentralized LQR with a partially nested information structure. Here, we summarize our contributions and technical challenges in the paper.

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  In Section 3, we provide a sample complexity result for estimating the system model from a single system trajectory using a least squares approach. Despite the existence of a sparsity pattern in the system model considered in our problem, we adapt the analyses in [10, 9] for least squares estimation of general linear system models (without any sparsity pattern) to our setting, and show that such a system identification method for general system models suffices for our ensuing analyses.

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  In Section 4, based on the estimated system model, we design a novel decentralized control policy that satisfies the given information structure. Our control policy is inspired by [26], which developed the optimal controller for the decentralized LQR problem with a partially nested information structure and known system model. The optimal controller therein depends on some internal states, each of which evolves according to an auxiliary linear system (characterized by the actual model of the original system with a disturbance term from the original system) and correlates with other internal states. Accordingly, this complicated form of the internal states makes it challenging to extend the design in [26] to the case when the system model is unknown. To tackle this, we capitalize on the observation that the optimal controller proposed in [26] can be viewed as a disturbance-feedback control policy that maps the history of past disturbances (affecting the original system) to the current control input. Thanks to this viewpoint, we put forth a control policy that uses the aforementioned estimated system model and maps the estimates of past disturbances to the current control input via some estimated internal states. Particularly, the estimates of disturbances are obtained using the estimated system model and the state information of original system, and each of the estimated internal states evolves according to a linear system characterized by the estimated system model and the estimated disturbances. More importantly, we show that the proposed control policy can be implemented in a decentralized manner that satisfies the prescribed information structure, which requires a careful investigation of the structure of our problem.

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  In Section 5.2, we characterize the performance guarantee (i.e., suboptimality) of the control policy proposed in Section 4. As we discussed above, our control policy requires obtaining estimates of the past disturbances and maintaining the estimated internal states. When we compare the performance of our control policy to that of the optimal decentralized control policy in [26], both the estimates of the past disturbances and the estimated internal states contribute to the suboptimality of our control policy, which creates the major technical challenge in our analyses. We overcome this challenge by carefully investigating the structure of the proposed control policy, and we show that the suboptimality gap between our control policy and the optimal decentralized control policy (designed based on accurate knowledge of the system model) provided in [26] can be decomposed into two terms, both of which scale linearly with the estimation error of the system model.

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  In Section 5.3, we combine the above results together and provide an end-to-end sample complexity result for learning decentralized LQR with a partially nested information structure. Surprisingly, despite the existence of the information constraints and the fact that the optimal controller is a linear dynamic controller, our sample complexity result matches with that of learning centralized LQR without any information constraints [12].

The sets of integers and real numbers are denoted as $\mathbb{Z}$ and $\mathbb{R}$, respectively. The set of integers (resp., real numbers) that are greater than or equal to $a\in\mathbb{R}$ is denoted as $\mathbb{Z}_{\geq a}$ (resp., $\mathbb{R}_{\geq a}$). For a real number $a$, let $\lceil a\rceil$ be the smallest integer that is greater than or equal to $a$. The space of $m$-dimensional real vectors is denoted by $\mathbb{R}^{m}$, and the space of $m\times n$ real matrices is denoted by $\mathbb{R}^{m\times n}$. For a matrix $P\in\mathbb{R}^{n\times n}$, let $P^{\top}$, $\text{Tr}(P)$, and $\{\sigma_{i}(P):i\in\{1,\dots,n\}\}$ be its transpose, trace, and set of singular values, respectively. Without loss of generality, let the singular values of $P$ be ordered as $\sigma_{1}(P)\geq\cdots\geq\sigma_{n}(P)$. Let $\norm{\cdot}$ denote the $\ell_{2}$ norm, i.e., $\norm{P}=\sigma_{1}(P)$ for a matrix $P\in\mathbb{R}^{n\times n}$, and $\norm{x}=\sqrt{x^{\top}x}$ for a vector $x\in\mathbb{R}^{n}$. Let $\norm{P}_{F}=\sqrt{\text{Tr}(PP^{\top})}$ denote the Frobenius norm of $P\in\mathbb{R}^{n\times m}$. A positive semidefinite matrix $P$ is denoted by $P\succeq 0$, and $P\succeq Q$ if and only if $P-Q\succeq 0$. Let $\mathbb{S}_{+}^{n}$ (resp., $\mathbb{S}_{++}^{n}$) denote the set of $n\times n$ positive semidefinite (resp., positive definite) matrices. Let $I$ denote an identity matrix whose dimension can be inferred from the context. Given any integer $n\geq 1$, we define $[n]=\{1,\dots,n\}$. The cardinality of a finite set $\mathcal{A}$ is denoted by $|\mathcal{A}|$. Let $\mathcal{N}(\mu,\Sigma)$ denote a Gaussian distribution with mean $\mu\in\mathbb{R}^{m}$ and covariance $\Sigma\in\mathbb{S}^{m}_{+}$.

In this section, we sketch the method developed in [26, 35], which presents the optimal solution to a decentralized LQR problem with a partially nested information structure [21], when the system model is known a priori. First, let us consider a networked system that consists of $p\in\mathbb{Z}_{\geq 1}$ interconnected linear-time-invariant (LTI) subsystems. Letting the state, input and disturbance of the subsystem corresponding to node $i\in[p]$ be $x_{i}(t)\in\mathbb{R}^{n_{i}}$, $u_{i}(t)\in\mathbb{R}^{m_{i}}$, and $w_{i}(t)$, respectively, the subsystem corresponding to node $i$ is given by

where $\mathcal{N}_{i}\subseteq[p]$ is the set of subsystems whose states and inputs directly affect the state of subsystem $j$, $A_{ij}\in\mathbb{R}^{n_{i}\times n_{i}}$, $B_{ij}\in\mathbb{R}^{n_{i}\times m_{i}}$, and $w_{i}(t)\in\mathbb{R}^{n_{i}}$ is a white Gaussian noise process with $w_{i}(t)\sim\mathcal{N}(0,\sigma_{w}^{2}I)$ for all $t\in\mathbb{Z}_{\geq 0}$, where $\sigma_{w}\in\mathbb{R}_{>0}$.¹¹1The analysis can be extended to the case when $w_{i}(t)$ is assumed to be a zero-mean white Gaussian noise process with covariance $W\in\mathbb{S}_{++}^{n_{i}}$. In that case, our analysis will depend on $\max_{i\in\mathcal{V}}\sigma_{1}(W_{i})$ and $\min_{i\in\mathcal{V}}\sigma_{n}(W_{i})$. For simplicity, we assume throughout this paper that $n_{i}\geq m_{i}$ for all $i\in\mathcal{V}$. We can also write Eq. (1) as

where $A_{i}\triangleq\begin{bmatrix}A_{ij_{1}}&\cdots A_{ij_{|\mathcal{N}_{i}|}}\end{bmatrix}$, $B_{i}\triangleq\begin{bmatrix}B_{ij_{1}}&\cdots B_{ij_{|\mathcal{N}_{i}|}}\end{bmatrix}$, $x_{\mathcal{N}_{i}}(t)\triangleq\begin{bmatrix}x_{j_{1}}(t)&\cdots x_{j_{|\mathcal{N}_{i}|}}(t)\end{bmatrix}^{\top}$, and $u_{\mathcal{N}_{i}}(t)\triangleq\begin{bmatrix}u_{j_{1}}(t)&\cdots u_{j_{|\mathcal{N}_{i}|}}(t)\end{bmatrix}^{\top}$, with $\mathcal{N}_{i}=\{j_{1},\dots,j_{|\mathcal{N}_{i}|}\}$. Further letting $n=\sum_{i\in\mathcal{V}}n_{i}$ and $m=\sum_{i\in\mathcal{V}}m_{i}$, and defining $x(t)=\begin{bmatrix}x_{1}(t)^{\top}&\cdots&x_{p}(t)^{\top}\end{bmatrix}^{\top}$, $u(t)=\begin{bmatrix}u_{1}(t)^{\top}&\cdots&u_{p}(t)^{\top}\end{bmatrix}^{\top}$ and $w(t)=\begin{bmatrix}w_{1}(t)^{\top}&\cdots&w_{p}(t)^{\top}\end{bmatrix}^{\top}$, we can compactly write Eq. (1) into the following matrix form:

where the $(i,j)$th block of $A\in\mathbb{R}^{n\times n}$ (resp., $B\in\mathbb{R}^{n\times m}$), i.e., $A_{ij}$ (resp., $B_{ij}$) satisfies $A_{ij}=0$ (resp., $B_{ij}=0$) if $j\notin\mathcal{N}_{i}$. We assume that $w_{i}(t_{1})$ and $w_{j}(t_{2})$ are independent for all $i,j\in\mathcal{V}$ with $i\neq j$ and for all $t_{1},t_{2}\in\mathbb{Z}_{\geq 0}$. In other words, $w(t)$ is a white Gaussian noise process with $w(t)\sim\mathcal{N}(0,\sigma_{w}^{2}I)$ for all $t\in\mathbb{Z}_{\geq 0}$. For simplicity, we assume that $x(0)=0$ throughout this paper.²²2The analysis can be extended to the case when $x(0)$ is given by a zero-mean Gaussian distribution, as one may view $x(0)$ as $w(-1)$.

Next, we use a directed graph $\mathcal{G}(\mathcal{V},\mathcal{A})$ with $\mathcal{V}=[p]$ to characterize the information flow among the subsystems in $[p]$ due to communication constraints on the subsystems. Each node in $\mathcal{G}(\mathcal{V},\mathcal{A})$ represents a subsystem in $[p]$, and we assume that $\mathcal{G}(\mathcal{V},\mathcal{A})$ does not have self loops. We associate any edge $(i,j)\in\mathcal{A}$ with a delay of either $0$ or $1$, further denoted as $i\xrightarrow[]{0}j$ or $i\xrightarrow[]{1}j$, respectively.³³3The framework described in this paper can also be used to handle $\mathcal{G}(\mathcal{V},\mathcal{A})$ with larger delays; see [26] for a detailed discussion. Then, we define the delay matrix corresponding to $\mathcal{G}(\mathcal{V},\mathcal{A})$ as $D\in\mathbb{R}^{p\times p}$ such that: (i) If $i\neq j$ and there is a directed path from $j$ to $i$ in $\mathcal{G}(\mathcal{V},\mathcal{A})$, then $D_{ij}$ is equal to the sum of delays along the directed path from node $j$ to node $i$ with the smallest accumulative delay; (ii) If $i\neq j$ and there is no directed path from $j$ to $i$ in $\mathcal{G}(\mathcal{V},\mathcal{A})$, then $D_{ij}=+\infty$; (iii) $D_{ii}=0$ for all $i\in\mathcal{V}$. Here, we consider the scenario where the information (e.g., state information) corresponding to subsystem $j\in\mathcal{V}$ can propagate to subsystem $i\in\mathcal{V}$ with a delay of $D_{ij}$ (in time), if and only if there exists a directed path from $j$ to $i$ with an accumulative delay of $D_{ij}$. Note that as argued in [26], we assume that there is no directed cycle with zero accumulative delay; otherwise, one can first collapse all the nodes in such a directed cycle into a single node, and equivalently consider the resulting directed graph in the framework described above.

To proceed, we consider designing the control input $u(t)$ for the LTI system in Eq. (3). We focus on state-feedback control, i.e., we can view $u(t)$ as a policy that maps the states of the LTI system to a control input. Moreover, we require that $u(t)$ satisfy the information structure according to the directed graph $\mathcal{G}(\mathcal{V},\mathcal{A})$ and the delay matrix $D\in\mathbb{R}^{p\times p}$, described above. Specifically, considering any $i\in\mathcal{V}$ and any $t\in\mathbb{Z}_{\geq 0}$, and noting that the controller corresponding to subsystem $i\in\mathcal{V}$ provides the control input $u_{i}(t)\in\mathbb{R}^{m_{i}}$, the state information that is available to the controller corresponding to $i\in\mathcal{V}$ is given by

where $\mathcal{V}_{i}\triangleq\{j\in\mathcal{V}:D_{ij}\neq+\infty\}$. In other words, the control policy $u_{i}(t)$ maps the states contained in $\mathcal{I}_{i}(t)$ to a control input. In the sequel, we also call $\mathcal{I}_{i}(t)$ the information set of controller $i\in\mathcal{V}$ at time $t\in\mathbb{Z}_{\geq 0}$. Note that $\mathcal{I}_{i}(t)$ contains the states corresponding to the subsystems in $\mathcal{V}$ that have enough time to reach subsystem $i\in\mathcal{V}$ at time $t\in\mathbb{Z}_{\geq 0}$, due to the sparsity and delay constraints described above. Now, based on the information set $\mathcal{I}_{i}(t)$, we further define $\mathcal{S}(\mathcal{I}_{i}(t))$ to be the set that consists of all the policies that map the states in $\mathcal{I}_{i}(t)$ to a control input at node $i$. The goal is then to solve the following constrained optimization problem:

where $Q\in\mathbb{S}_{+}^{n}$ and $R\in\mathbb{S}_{++}^{m}$ are the cost matrices, and the expectation is taken with respect to $w(t)$ for all $t\in\mathbb{Z}_{\geq 0}$. Throughout the paper, we always assume that the following assumption on the information propagation pattern among the subsystems in $\mathcal{V}$ holds (e.g., [26, 40]).

Assumption 1 says that the state of subsystem $i\in\mathcal{V}$ is affected by the state and input of subsystem $j\in\mathcal{V}$, if and only if there is a communication link with a delay of at most $1$ from subsystem $j$ to $i$ in $\mathcal{G}(\mathcal{V},\mathcal{A})$. As shown in [26], Assumption 1 ensures that the information structure associated with the system given in Eq. (1) is partially nested [21]. Assumption 1 is frequently used in decentralized control problems (e.g., [26, 35] and the references therein), and one can see that the assumption is satisfied in networked systems where information propagates at least as fast as dynamics. To illustrate our arguments above, we introduce Example 1.

Now, in order to present the solution to (5) given in, e.g., [26], we need to construct an information graph $\mathcal{P}(\mathcal{U},\mathcal{H})$ (see [26] for more details). Considering any directed graph $\mathcal{G}(\mathcal{V},\mathcal{A})$ with $\mathcal{V}=[p]$, and the delay matrix $D\in\mathbb{R}^{p\times p}$ as we described above, let us first define $s_{j}(k)$ to be the set of nodes in $\mathcal{G}(\mathcal{V},\mathcal{A})$ that are reachable from node $j$ within $k$ time steps, i.e., $s_{j}(k)=\{i\in\mathcal{V}:D_{ij}\leq k\}$. The information graph $\mathcal{P}(\mathcal{U},\mathcal{H})$ is then constructed as

Thus, we see from (7) that each node $s\in\mathcal{U}$ corresponds to a set of nodes from $\mathcal{V}=[p]$ in the original directed graph $\mathcal{G}(\mathcal{V},\mathcal{A})$. Using a similar notation to that for the graph $\mathcal{G}(\mathcal{V},\mathcal{A})$, if there is an edge from $s$ to $r$ in $\mathcal{P}(\mathcal{U},\mathcal{H})$, we denote the edge as $s\to r$. Additionally, considering any $s_{i}(0)\in\mathcal{U}$, we write $w_{i}\xrightarrow[]{}s_{i}(0)$ to indicate the fact that the noise $w_{i}(t)$ is injected to node $i\in\mathcal{V}$ at time $t\in\mathbb{Z}_{\geq 0}$.⁴⁴4Note that we have assumed that there is no directed cycle with zero accumulative delay in $\mathcal{P}(\mathcal{U},\mathcal{H})$. Hence, one can show that for any $s_{i}(0)\in\mathcal{U}$, $w_{i}$ is the only noise term such that $w_{i}\rightarrow s_{i}(0)$. From the above construction of the information graph $\mathcal{P}(\mathcal{U},\mathcal{H})$, one can show that the following properties hold.

To illustrate the construction steps and the properties of the information graph discussed above, we again use Example 1; the resulting information graph $\mathcal{P}(\mathcal{U},\mathcal{H})$ is then given in Fig. 2. Note that the information graph $\mathcal{P}(\mathcal{U},\mathcal{H})$ in Fig. 2 contains two disconnected directed trees, one of which is an isolated node $\{1\}\in\mathcal{U}$ with a self loop. Also notice that $s_{1}(0)=\{1\}$, $s_{2}(0)=\{1,2\}$ and $s_{3}(0)=\{3\}$. In fact, we can check that the results in Lemma 1 hold for $\mathcal{P}(\mathcal{U},\mathcal{H})$ in Fig. 2.

Throughout this paper, we assume that the elements in $\mathcal{V}=[p]$ are ordered in an increasing manner, and that the elements in $s$ are also ordered in an increasing manner for all $s\in\mathcal{U}$. Now, for any $s,r\in\mathcal{U}$, we use $A_{sr}$ (or $A_{s,r}$) to denote the submatrix of $A$ that corresponds to the nodes of the directed graph $\mathcal{G}(\mathcal{V},\mathcal{A})$ contained in $s$ and $r$. For example, $A_{\{1\},\{1,2\}}=\begin{bmatrix}A_{11}&A_{12}\end{bmatrix}$. In the sequel, we will also use similar notations to denote submatrices of $B$, $Q$, $R$ and the identity matrix $I$. We will make the following standard assumptions (see, e.g., [26]).

Leveraging the partial nestedness of (5), the authors in [26] obtained the optimal solution to (5), which we summarize in the following lemma.

Let us use Example 1 to illustrate the results in Lemma 2. First, considering node $\{1\}\in\mathcal{U}$ in the information graph $\mathcal{P}(\mathcal{U},\mathcal{H})$ given in Fig. 2, we have from Eq. (10) that

Next, considering node $2\in\mathcal{V}$ in the directed graph $\mathcal{G}(\mathcal{V},\mathcal{A})$ given in Fig. 1, we see from Eq. (11) and Fig. 2 that

where $K_{r}$ is given by Eq. (8).

We now formally introduce the problem that we will study in this paper. We consider the scenario where the system matrices $A$ and $B$ are unknown. However, we assume that the directed graph $\mathcal{G}(\mathcal{V},\mathcal{A})$ and the associated delay matrix $D$ are known. Similarly to, e.g., [12, 43], we consider the scenario where we can first conduct experiments in order to estimate the unknown system matrices $A$ and $B$. Specifically, starting from the initial state $x(0)=0$, we evolve the system given in Eq. (3) for $N\in\mathbb{Z}_{\geq 1}$ time steps using a given control input sequence $\{u(0),u(1),\dots,u(N-1)\}$, and collect the resulting state sequence $\{x(1),x(2),\dots,x(N)\}$. Based on $\{u(0),\dots,u(N-1)\}$ and $\{x(0),\dots,x(N)\}$, we use a least squares approach to obtain estimates of the system matrices $A$ and $B$, denoted as $\hat{A}$ and $\hat{B}$, respectively. Using the obtained $\hat{A}$ and $\hat{B}$, the goal is still to solve (5). Since the true system matrices $A$ and $B$ are unknown, it may no longer be possible to solve (5) optimally, using the methods introduced in Section 2.2. Thus, we aim to provide a solution to (5) using $\hat{A}$ and $\hat{B}$, and characterize its performance (i.e., suboptimality) guarantees.

In the rest of this paper, we first analyze the estimation error of $\hat{A}$ and $\hat{B}$ obtained from the procedure described above. In particular, we show in Section 3 that the estimation errors $\norm{\hat{A}-A}$ and $\norm{\hat{B}-B}$ scale as $\tilde{\mathcal{O}}(1/\sqrt{N})$ with high probability.⁵⁵5Throughout this paper, we let $\tilde{\mathcal{O}}(\cdot)$ hide logarithmic factors in $N$. Next, in Section 4, we design a control policy $\hat{u}(\cdot)$, based on $\hat{A}$ and $\hat{B}$, which satisfies the information constraints given in (5). Supposing $\norm{\hat{A}-A}\leq\varepsilon$ and $\norm{\hat{B}-B}\leq\varepsilon$, where $\varepsilon\in\mathbb{R}_{>0}$, and denoting the cost of (5) corresponding to $\hat{u}(\cdot)$ as $\hat{J}$, we show in Section 5.2 that

as long as $\varepsilon\leq C_{0}$, where $J_{\star}$ is the optimal cost of (5) given by (12), and $C$ and $C_{0}$ are constants that explicitly depend on the problem parameters of (5). Finally, combining the above results together, we show in Section 5.3 that with high probability and for large enough $N$, the following end-to-end sample complexity of learning decentralized LQR with the partially nested information structure holds:

Let us denote

where $\Theta\in\mathbb{R}^{n\times(n+m)}$ and $z(t)\in\mathbb{R}^{n+m}$. Given the sequences $\{z(0),\dots,z(N-1)\}$ and $\{x(1),\dots,x(N)\}$, we use the regularized least squares to obtain an estimate of $\Theta$, denoted as $\hat{\Theta}(N)$, i.e.,

where $\lambda\in\mathbb{R}_{>0}$ is the regularization parameter. We summarize the above least squares approach in Algorithm 1.

In order to characterize the estimation error of $\hat{\Theta}(N)$ given by (14), we will use the following result from [10], which is a consequence of [2, Theorem 1].

For any $\delta>0$, we now introduce the following probabilistic events that will be useful in our analysis later:

where $\underline{\sigma}\triangleq\min\{\sigma_{w},\sigma_{u}\}$. Denoting

we have the following result; the proof is included in Appendix A.

For the analysis in the sequel, we will make the following assumption, which is also made in related literature (see e.g., [24, 37, 43]).

Note that for any stable matrix $A$, we have from the Gelfand formula (e.g., [22]) that there always exist $\kappa_{0}\in\mathbb{R}_{\geq 1}$ and $\gamma_{0}\in\mathbb{R}$ with $\rho(A)<\gamma_{0}<1$ such that $\norm{A^{k}}\leq\kappa_{0}\gamma_{0}^{k}$ for all $k\in\mathbb{Z}_{\geq 0}$. We then have the following results; the proofs are included in Appendix A.

Several remarks pertaining to Algorithm 1 and the result in Proposition 1 are now in order. First, note that while considering the problem of learning centralized LQR without any information constraints, the authors in [12] proposed to obtain $\hat{A}$ and $\hat{B}$ from multiple system trajectories using least squares, where each trajectory starts from $x(0)=0$. They showed that $\norm{\hat{A}-A}=\mathcal{O}(1/\sqrt{N_{r}})$ and $\norm{\hat{B}-B}=\mathcal{O}(1/\sqrt{N_{r}})$, where $N_{r}\in\mathbb{Z}_{\geq 1}$ is the number of system trajectories. In contrast, we estimate $A$ and $B$ from a single system trajectory, and achieve $\norm{\hat{A}-A}=\tilde{\mathcal{O}}(1/\sqrt{N})$ and $\norm{\hat{B}-B}=\tilde{\mathcal{O}}(1/\sqrt{N})$.

Second, note that we use the regularized least squares in Algorithm 1 to obtain estimates $\hat{A}$ and $\hat{B}$. Although least squares without regularization can also be used to obtain estimates $\hat{A}$ and $\hat{B}$ from a single system trajectory with the same $\tilde{\mathcal{O}}(1/\sqrt{N})$ finite sample guarantee (e.g., [36]), we choose to use regularized least squares considered in, e.g., [2, 10, 9]. The reason is that introducing the regularization into least squares makes the finite sample analysis more tractable (e.g., [10, 9]), which facilitates the adaption of the analysis in [10, 9] to our setting described in this section. Moreover, note that the lower bound on $\lambda$ required in Proposition 1 is merely used to guarantee that the denominator of the right-hand side of Eq. (18) contains the factor $1/\sqrt{N}$; choosing an arbitrary $\lambda\in\mathbb{R}_{>0}$ leads to a factor $1/\sqrt{N-1}$. In general, one can show that choosing any $\lambda\in\mathbb{R}_{>0}$ leads to the same $\tilde{\mathcal{O}}(1/\sqrt{N})$ finite sample guarantee.