Regret Bounds for Episodic Risk-Sensitive Linear Quadratic Regulator

## Abstract

Abstract Risk-sensitive linear quadratic regulator is one of the most fundamental problems in risk-sensitive optimal control.
In this paper, we study online adaptive control of risk-sensitive linear quadratic regulator in the finite horizon episodic setting.
We propose a simple least-squares greedy algorithm and show that it achieves 𝒪 ~ ​ ( log ⁡ N ) ~ 𝒪 𝑁 \widetilde{\mathcal{O}}(\log N) regret under a specific identifiability assumption, where N 𝑁 N is the total number of episodes. If the identifiability assumption is not satisfied, we propose incorporating exploration noise into the least-squares-based algorithm, resulting in an algorithm with 𝒪 ~ ​ ( N ) ~ 𝒪 𝑁 \widetilde{\mathcal{O}}(\sqrt{N}) regret.
To our best knowledge, this is the first set of regret bounds for episodic risk-sensitive linear quadratic regulator.
Our proof relies on perturbation analysis of less-standard Riccati equations for risk-sensitive linear quadratic control, and a delicate analysis of the loss in the risk-sensitive performance criterion due to applying the suboptimal controller in the online learning process.

## 1 Introduction

In classical reinforcement learning (RL), one optimizes the expected cumulative rewards in an unknown environment modeled by a Markov decision
process (MDP, Sutton and Barto, ()). However, this risk-neutral performance criterion
may not be the most suitable one in applications such as finance, robotics and healthcare. Hence, a large body of literature has studied risk-sensitive RL, incorporating the notion of risk into the decision criteria, see, e.g., Mihatsch and Neuneier, (), Shen et al., (), Chow et al., (), Prashanth L and Fu, ().

In this paper, we study online learning and adaptive control for a risk-sensitive linear quadratic control problem, referred to as
the Linear Exponential-of-Quadratic Regulator (LEQR) problem. The LEQR problem is one of the most fundamental problems in risk-sensitive optimal control (Jacobson, , Whittle, ). In this control problem, the system dynamics is linear in the state and control variables, and it is disturbed with additive Gaussian noise. The cost in each period is convex quadratic in both the state and the control/action variables,
and the performance criteria is the logarithm of the expectation of the exponential functions of the cumulative costs. When the system parameters are known, the optimal control at each stage is linear in state with the coefficient determined by certain Riccati equation. Different from the risk-neutral setting, the solution to the Riccati equation for LEQR explicitly depends on the risk parameter and the covariance matrix of the additve Gaussian noise in the system dynamics (Jacobson, ).
For general risk-sensitive nonlinear control, one does not have such closed-form solutions. However, one can use LEQR as a local approximation model and solve risk-sensitive control problems by iteratively solving LEQR problems, see e.g. Roulet et al., ().

We consider the standard finite-horizon episodic RL setting, where the system matrices of LEQR are unknown to the agent.
The learning agent repeatedly interacts with the unknown system over $N$ episodes, the time horizon of each episode is fixed, and the system resets to a fixed initial state distribution at the beginning of each
episode. We focus on the finite horizon LEQR model because it is widely used as a model of locally linear dynamics.
The performance of the agent or the online algorithm is often quantified by the total regret, which measures the cumulative suboptimality of the algorithm accrued over time as compared to the optimal policy. We seek algorithms with (finite-time) regret that is sublinear in $N$, which means the per episode regret converges to zero and the agent can act near optimally as $N$ grows.

Regret bounds for the risk-neutral linear quadratic regulator (LQR) in the infinite-horizon average reward setting have been extensively studied in the literature,
see e.g. Abbasi-Yadkori and Szepesvári, (), Mania et al., (), Cohen et al., (), Simchowitz and Foster, ().
It has been shown that in this average reward setting, the certainty-equivalent controller where the agent selects control inputs according to the optimal controller for her estimate of the system, together with a simple random-search type exploration strategy, is (rate-)optimal for the online adaptive control of risk-neutral LQR (Simchowitz and Foster, ). However, non-asymptotic regret analysis of the finite-horizon episodic LQR has received much less attention. Basei et al., () is among the first to establish regret bounds for the risk-neutral continuous time finite-horizon LQR. They proposed a greedy least-squares algorithm and established a regret bound that is logarithmic in the number of episodes $N$ under a specific identifiability condition.

On the other hand, there is a surge of interest recently on studying finite-time regret bounds for risk-sensitive RL. The first regret bound for risk-sensitive tabular MDP is due to Fei et al., (), who study episodic RL with the goal of optimizing the exponential utility of the cumulative
rewards. There is now a rapidly growing body of literature on this topic, see, e.g. (Fei et al., , , Du et al., , Bastani et al., , Liang and Luo, , Xu et al., , Wang et al., , Wu and Xu, , Chen et al., ). Most of the studies consider learning in risk-sensitive MDPs with finite state and action spaces.

Inspired by these studies, in this paper we study regret bounds for online adaptive control of the (discrete-time) risk-sensitive LEQR in the finite-horizon episodic setting, where both the state and the action spaces are continuous. In particular, we obtain two main results:

- •
First, we propose a simple least-squares greedy algorithm without exploration noise (Algorithm [1](#alg1)), and show that it achieves a regret of order $\log N$ under a certain identifiability condition (Assumption [1](#Thmassumption1)) on the LEQR model.
- •
Second, without Assumption [1](#Thmassumption1), we propose another algorithm with actively injected exploration noise (Algorithm [2](#alg2)), and show that it achieves a regret of order $\sqrt{N}$.

To the best of our knowledge, this is the first set of regret bounds for finite-horizon episodic LEQR. When the risk parameter in the LEQR model approaches zero, the LEQR model reduces to the risk-neutral LQR model, and our results still hold. In the learning theory community, there has been a significant interest in the questions of whether logarithmic regret is possible for what type of linear systems and under what assumptions. See e.g. Agarwal et al., (), Cassel et al., (), Faradonbeh et al., (), Foster and Simchowitz, (), Lale et al., (). Our first result provides an answer to these questions in the setting of risk-sensitive LEQR models. In addition, there appears to be an absence of published results on $\sqrt{N}$-regret bounds for episodic LQR even in the risk-neutral setting. Our second result indicates that such $\sqrt{N}$-regret bound can be established and it holds in greater generality in the sense that it extends to risk-sensitive LQRs.

We briefly discuss the technical challenges and highlight the novelty of our regret analysis. Even though our proposed algorithms are fairly simple, the analysis is nontrivial and it builds on two new components: (a) perturbation analysis of Riccati equation for LEQR; and (b) analysis of risk-sensitive performance loss due to the suboptimal controller applied in the online control process. For the perturbation analysis in (a), we cannot use the existing techniques from the literature on online learning in risk-neutral LQR (Mania et al., , Simchowitz and Foster, , Basei et al., ). This is because the Riccati equation for LEQR is less standard and more complicated: there are some extra parameters (see $\widetilde{P}_{t},t=0,\ \cdots,T-1$ in ([2.1](#S2.Ex2))) involved in the equation, and the risk-sensitive parameter impacts the solution to the Riccati equation. To overcome this challenge, we first analyze one-step perturbation bound for the solution to Riccati equation, and then leverage the recursive structure of Riccati equation from our finite-horizon LEQR problem to establish a bound on the controller mismatch in terms of the error in the estimated system matrices. For the performance loss in (b), we can not employ the existing approach in online control of risk-neutral LQR as well. This is because the performance objective in LEQR is nonlinear in terms of the random cumulative costs (unlike the expectation which is a linear operator). Indeed, this type of non-linearity has been one of the key challenges in regret analysis for risk-sensitive tabular MDPs (Fei et al., ). To address this challenge, we leverage results from Jacobson, () for LEQR to express the performance loss in terms of the controller mismatch (i.e. the gap between the executed controller and the optimal controller).

We conclude this introduction by mentioning several recent studies on RL for LEQR.
proposes model-free policy gradient methods for
solving the finite-horizon LEQR problem and provides a sample complexity result. Sample complexity is another popular performance metric for RL algorithms in addition to the regret. Note that the controller in
is assumed to have simulation access to the model, i.e., the controller can execute multiple policies within each episode. By contrast, our work considers online
control of LEQG with regret guarantees, where we do not assume access to a simulator and the agent can only execute
one policy within each episode. Other related works include , which proposes a nested natural actor-critic algorithm for LEQR with the average reward criteria, and
Cui et al., (), which proposes a robust policy optimization algorithm for solving the LEQR problem to handle model disturbances and mismatches. These studies do not consider regret bounds for LEQR, and hence are different from our work.

## 2 Problem Formulation

### 2.1 The LEQR problem

We first provide a brief review of the LEQR problem (Jacobson, ).
We consider the following linear discrete-time dynamic system:

$$ $\displaystyle x_{t+1}=Ax_{t}+Bu_{t}+w_{t},\quad t=0,1,\cdots,T-1,$ (1) $$

where the state vector $x_{t}\in\mathbb{R}^{n}$, the control vector $u_{t}\in\mathbb{R}^{m}$, the matrices $A\in\mathbb{R}^{n\times n}$, $B\in\mathbb{R}^{n\times m}$, and the process noise $w_{t}\in\mathbb{R}^{n}$ form a sequence of i.i.d. Gaussian random vectors. For the simplicity of presentation, we assume the noise
$w_{t}\sim\mathcal{N}(0,I)$ where $I$ is the identity matrix. The goal in the finite-horizon LEQR problem is to choose a control policy $\pi=\{u_{0},u_{1},\cdots,u_{T-1}\}$ so as to minimize the exponential risk-sensitive cost given by

$$ $\displaystyle J^{\pi}(x_{0})=\frac{1}{\gamma}\log\mathbb{E}\exp\left(\frac{\gamma}{2}\left(\sum_{t=0}^{T-1}c_{t}(x_{t},u_{t})+c_{T}(x_{T})\right)\right),$ (2) $$

where $c_{t}(x_{t},u_{t})=x_{t}^{\top}Qx_{t}+u_{t}^{\top}Ru_{t}$, $c_{T}(x_{T})=x_{T}^{\top}Q_{T}x_{T}$, $Q\succeq 0,Q_{T}\succeq 0$ (i.e. positive semidefinite), $R\succ 0$ (i.e. positive definite), and $\gamma$ is the risk-sensitivity parameter.

Note that when $\gamma$ is small, we have by Taylor expansion:

$$ $\displaystyle\frac{1}{\gamma}\log\mathbb{E}\exp(\gamma Z)=\mathbb{E}[Z]+\frac{\gamma}{2}Var(Z)+O(\gamma^{2}),$ $$

for a random variable $Z$ with a finite moment generating function.
Hence, when $\gamma\rightarrow 0,$ the LEQR problem reduces to the conventional risk-neutral linear quadratic control where one minimizes the expected total quadratic cost. For concreteness, we focus on the case where $\gamma>0$ and the optimizer is risk-averse (our analysis extends to $\gamma\leq 0$).
The corresponding optimal performance is denoted by

$$ $\displaystyle J^{\star}(x_{0})=\inf_{\pi}J^{\pi}(x_{0}).$ (3) $$

When the system parameters are all known, Jacobson, () shows that under the assumption that $I-\gamma P_{t+1}\succ 0$ for all $t=0,1,\cdots,T-1$ (Note that if $\gamma$ is too large, we have $J^{\pi}(x_{0})=\infty$ for all policies),
the optimal feedback control for ([3](#S2.E3)) is a linear function of the system state

$$ $\displaystyle u_{t}^{\star}=K_{t}x_{t},\quad t=0,1,\cdots,T-1,$ (4) $$

where $(K_{t})$ can be solved from the following discrete-time (modified) Riccati equation:

$$ $\displaystyle P_{T}$ $\displaystyle=Q_{T},$ $\displaystyle\widetilde{P}_{t+1}$ $\displaystyle=P_{t+1}+\gamma P_{t+1}\left(I_{n}-\gamma P_{t+1}\right)^{-1}P_{t+1},$ $\displaystyle K_{t}$ $\displaystyle=-(B^{\top}\widetilde{P}_{t+1}B+R)^{-1}B^{\top}\widetilde{P}_{t+1}A,$ $\displaystyle P_{t}$ $\displaystyle=Q+K_{t}^{\top}RK_{t}+(A+BK_{t})^{\top}\widetilde{P}_{t+1}(A+BK_{t}),$ $\displaystyle t$ $\displaystyle=0,1,\cdots,T-1.$ (5) $$

One can see that scaling all the cost matrices $Q,Q_{T},$ and $R$ does not change the optimal controller, and hence we assume $R\succeq I_{m}$ without loss of generality. Note that in the risk-neutral setting where $\gamma=0$, we have $\widetilde{P}_{t}={P}_{t}$ in the Riccati equation ([2.1](#S2.Ex2)). However, in the risk-sensitive setting, we have extra matrices $(\widetilde{P}_{t})$ in the Riccati equation. This is one of the difficulties we need to overcome when we study perturbation analysis of Riccati equations for the LEQR problem.

### 2.2 Finite-horizon Episodic RL in LEQR

In this paper, we consider the online learning/control setting for LEQR, where the system matrices $(A,B)$ are unknown to the agent.
The learning agent repeatedly interacts with the linear system ([1](#S2.E1)) over $N$ episodes, where the time horizon of each episode is $T$.
In each episode $i=1,2,\cdots,N$, an arbitrary fixed initial state $x_{0}^{k}=x_{0}\in\mathbb{R}^{n}$ is picked.(^4^44The results of the paper can also be extended to the case where the initial states are drawn from a fixed distribution over $\mathbb{R}^{n}$.)
An online learning algorithm executes policy $\pi^{i}$ throughout episode $i$ based on the observed past data (states, actions and costs) up to the end of episode $i-1$. The performance of an online algorithm over $N$ episodes of interaction with the linear system ([1](#S2.E1)) is the (total) regret:

$$ $\operatorname{Regret}(N)=\sum_{i=1}^{N}\left(J^{\pi^{i}}(x_{0}^{i})-J^{\star}(x_{0}^{i})\right),$ $$

where the term $J^{\pi^{i}}(x_{0}^{i})-J^{\star}(x_{0}^{i})$ (see ([2](#S2.E2)) and ([3](#S2.E3))) measures the performance loss when the agent executes the suboptimal policy $\pi^{i}$ in episode $i$. In the next section, we propose a greedy algorithm for Episodic RL in LEQR.

## 3 A logarithmic regret bound

In this section, we propose a simple least-squares greedy algorithm and show that it achieves a regret that is logarithmic in $N$, under a specific identifiability assumption.

### 3.1 A Least-Squares Greedy Algorithm

We now present the details of the least-squares greedy algorithm, which combines least-squares estimation for the unknown system matrices $(A,B)$ with a greedy strategy.

We divide the $N$ episodes into $L$ epochs. The $l$-th epoch has $m_{l}$ episodes, thus $\sum_{l=1}^{L}m_{l}=N$. At the beginning of the $l$-th epoch, we estimate the system matrices $(A,B)$ by using the data from the $(l-1)$-th epoch, and the obtained estimator is denoted by $(A^{l},B^{l})$. Then we select the control inputs according
to the optimal controller for the estimate $(A^{l},B^{l})$ of the system, and execute such a policy throughout epoch $l.$ The feedback control $K_{t}^{l}$ is obtained by replacing $(A,B)$ in ([2.1](#S2.Ex2)) with the estimate $(A^{l},B^{l})$. Then, in the $k$-th episode of epoch $l$, We play the greedy policy $u_{t}^{l,k}$ by taking $K_{t}^{l}$ into ([4](#S2.E4)).

It remains to discuss the estimation procedure for $(A^{l},B^{l})$ which is conducted at the beginning of epoch $l$. Within the $l$-th epoch, we note that the same policy is executed in each episode. Because we consider the episodic setting where the system state reset to the same state at $t=0$, we obtain that the state-action trajectories across different episodes are i.i.d within the same epoch.
Note that the random linear dynamical system in epoch $l$ is given by

$$ $\displaystyle x_{t+1}^{l}=Ax_{t}^{l}+Bu_{t}^{l}+w_{t}^{l},\quad t=0,1,\cdots,T-1,$ (6) $$

where $u_{t}^{l}=K_{t}^{l}x_{t}^{l}$.
For simplicity of notation, we denote by $z^{l}_{t}=\left[x_{t}^{l\top}\ u_{t}^{l\top}\right]^{\top}$, which is the state-action random vector at step $t$ in epoch $l$. We also denote by $\theta=[A\ B]^{\top}$ for the system matrices. Taking the transpose of ([6](#S3.E6)) and multiplying $z_{t}^{l}$ on both sides of ([6](#S3.E6)), we can get $z_{t}^{l}x_{t+1}^{l\top}=z_{t}^{l}z_{t}^{l\top}\theta+z_{t}^{l}w_{t}^{l\top}.$
Summing over $T$ steps and taking the expectation, we obtain $\mathbb{E}\left[Y^{l}\right]=\mathbb{E}\left[V^{l}\right]\theta,$
where $V^{l}=\sum_{t=0}^{T-1}z_{t}^{l}z_{t}^{l\top}$ and $Y^{l}=\sum_{t=0}^{T-1}z_{t}^{l}x_{t+1}^{l\top}.$
It follows that

$$ $\displaystyle\theta=[A\ B]^{\top}=\left(\mathbb{E}\left[V^{l}\right]\right)^{-1}\left(\mathbb{E}\left[Y^{l}\right]\right),$ (7) $$

provided that the matrix $\mathbb{E}[V^{l}]$ is invertible. The formula ([7](#S3.E7)) and the fact that state-action trajectories across different episodes are i.i.d. within the same epoch provide the basis for our estimation procedure. Given the data in epoch $l$, we now discuss the construction of the estimator $\theta^{l+1}:=\left[A^{l+1},B^{l+1}\right]^{\top}.$

Consider the sample state process in the $k$-th episode of epoch $l$:

$$ $\displaystyle x_{t+1}^{l,k}=Ax_{t}^{l,k}+Bu_{t}^{l,k}+w_{t}^{l,k},t=0,1,\cdots,T-1.$ (8) $$

Denote the sample state-action vector by $z_{t}^{l,k}=\left[x_{t}^{l,k\top}\ u_{t}^{l,k\top}\right]^{\top}$.
Then, we can design the $l_{2}$-regularized least-squares estimation for $\theta$ by replacing the expectation in ([7](#S3.E7)) with the sample average over the $m_{l}$ episodes in epoch $l$ and adding the regularized term $\frac{1}{m_{l}}I_{n+m}$:

$$ $\displaystyle\theta^{l+1}=\left(\bar{V}^{l}+\frac{1}{m_{l}}I_{n+m}\right)^{-1}\bar{Y}^{l},$ (9) $$

where $\bar{V}^{l}=\frac{1}{m_{l}}\sum_{k=1}^{m_{l}}\sum_{t=0}^{T-1}z_{t}^{l,k}z_{t}^{l,k\top}$ and $\bar{Y}^{l}=\frac{1}{m_{l}}\sum_{k=1}^{m_{l}}\sum_{t=0}^{T-1}z_{t}^{l,k}x_{t+1}^{l,k\top}.$

We now summarize the details of the least-squares greedy algorithm in Algorithm [1](#alg1). Note that the input parameter $\theta^{1}$ denotes the initial guess of the true system matrices $(A,B)$.

Figure: Algorithm 1 The Least-Squares Greedy Algorithm

### 3.2 Logarithmic Regret

In this section, we state our first main result. We first introduce the following assumption.

###### Assumption 1 .

For the sequence of the controller $(K_{t})$ defined in ([2.1](#S2.Ex2)), we assume that

$$ $\displaystyle\left\{v\in\mathbb{R}^{n+m}\Big{|}\left[I_{n}\ K_{t}^{\top}\right]v=0,\forall t=0,\cdots,T-1\right\}=\{0\}.$ (10) $$

Assumption [1](#Thmassumption1) is essentially Assumption H.1(2) in Basei et al., () for learning finite-horizon continuous-time risk-netural LQR, and it is referred to as the self-exploration property therein (i.e., exploration is ‘automatic’ due to the system noise and the time-dependent optimal feedback matrix $(K_{t})_{t=0,\ldots,T-1}$ ). One can show that Assumption [1](#Thmassumption1) is equivalent to the condition (see Lemma [7](#Thmlemma7))

$$ $\displaystyle\mathbb{E}\left[\sum_{t=0}^{T-1}z_{t}z_{t}^{\top}\right]$ $\displaystyle=\sum_{t=0}^{T-1}\left[\begin{array}[]{cc}I_{n}\\ K_{t}\end{array}\right]\mathbb{E}\left[x_{t}x_{t}^{\top}\right]\left[I_{n}\ K_{t}^{\top}\right]\succ 0.$ (13) $$

In view of ([7](#S3.E7)) and ([13](#S3.E13)), Assumption [1](#Thmassumption1) essentially guarantees the identifiability of the true system matrices when the time-dependent optimal control in ([2.1](#S2.Ex2)) is executed. This is important for the proposed greedy least-squares algorithm to achieve a logarithmic regret bound. It is in sharp contrast with RL for infinite-horizon average reward LQR, where
in the certainty equivalent approach one often needs to add exploration noises to achieve sublinear regret (Simchowitz and Foster, ), precisely because the true parameters are not identifiable under the optimal closed-loop policy (which is characterized by a time-independent feedback matrix) in that average reward setting (Tsiamis et al., ).
Assumption [1](#Thmassumption1) can be satisfied under various sufficient conditions. We provide one set of sufficient conditions in Proposition [4](#Thmproposition4).

We now present our first main result, which provides a logarithmic regret bound of Algorithm [1](#alg1). We denote $\|\cdot\|$ as the spectral norm for matrices.

###### Theorem 1 .

Suppose Assumption [1](#Thmassumption1) holds and assume the optimal controller for the initial estimate $\theta^{1}$ also satisfy ([10](#S3.E10)).
Fix $\delta\in(0,\frac{3}{\pi^{2}})$.
Then we can choose $m_{1}=\mathcal{C}_{0}(-\log\delta)$ for some positive constant $\mathcal{C}_{0}$ such that with probability at least $1-\frac{\pi^{2}\delta}{3}$, the regret of Algorithm [1](#alg1) satisfies

$$ $\displaystyle\operatorname{Regret}(N)\leq\mathcal{C}\left(\sum_{t=0}^{T-1}\psi_{t}\right)\left[\log\left(\frac{m+n}{\sqrt{\delta}}\right)L+L\log L\right],$ (14) $$

where $\mathcal{C}$ is a constant independent of $N$
and $(\psi_{t})$ is a sequence recursively defined by

$$ $\displaystyle\psi_{T-1}=2\widetilde{\Gamma}^{3},\quad\psi_{t}=2\widetilde{\Gamma}^{3}(10\mathcal{V}^{2}\mathcal{L}\widetilde{\Gamma}^{4})^{2(T-t-1)}+12\widetilde{\Gamma}^{4}\psi_{t+1},\ t\in[T-2],$ $$

with

$$ $\displaystyle\Gamma_{t}=\max\left\{\left\|A\right\|,\|B\|,\|Q\|,\|Q_{T}\|,\|R\|,\|P_{t}\|,\|\widetilde{P}_{t}\|,\|K_{t-1}\|\right\},\quad\widetilde{\Gamma}=1+\max_{t}\Gamma_{t},$ (15) $\displaystyle\mathcal{V}=2(\mathcal{L}+1)\widetilde{\Gamma}^{3},\quad\mathcal{L}=\frac{1}{(1-\gamma\sigma^{2}\widetilde{\Gamma})^{2}}.$ $$

Because $\sum_{l=1}^{L}m_{l}=N$ and $m_{l}=2^{l-1}m_{1}$, we infer that
$L=\left\lceil\log_{2}\left(\frac{N}{m_{1}}+1\right)\right\rceil\lesssim\log N$, where $\lesssim$ means the inequality holds with a multiplicative constant. Hence, Theorem [1](#Thmtheorem1) implies that the regret of Algorithm [1](#alg1) satisfies $\operatorname{Regret}(N)={O}\left(\log N\cdot\log\log(N)\right),$ where ${O}$ hides dependency on other constants. A few remarks are in order.

###### Remark 1 .

Algorithm [1](#alg1) requires $L$, or equivalently $N$ (the total number of episodes) as input. For unknown $N$, one can use the well-known doubling trick in online learning (Besson and Kaufmann, ) and sill obtain a logarithmic regret bound.

###### Remark 2 .

As $\gamma\rightarrow 0,$ our LEQR problem reduces to the discrete-time risk-neutral LQR problem, and Theorem [1](#Thmtheorem1) implies a logarithmic regret bound for this risk neutral episodic setting. This is consistent with the logarithmic regret bound proved in Basei et al., () for continuous-time episodic risk-neutral LQR. While our analysis of estimation error of system matrices builds on Basei et al., (), our proof of Theorem [1](#Thmtheorem1) is substantially different from the proof in Basei et al., () in terms of the perturbation analysis of a less-standard Reccati equation ([2.1](#S2.Ex2)) and the analysis of risk-sensitive performance loss in the online control process.

###### Remark 3 .

Similar as in Basei et al., (), the regret bound in Theorem 3.3 has an exponential dependence on the horizon length $T$ in general. Such exponential dependency on horizon length is common in regret bounds for risk-sensitive RL, see e.g. Fei et al., (). Indeed, the lower bound in Fei et al., () shows that such exponential dependency is unavoidable for any algorithm with $\tilde{O}(\sqrt{N})$ regret in tabular MDPs with exponential utility.
See Appendix [A.4.1](#A1.SS4.SSS1) for further discussions.

###### Assumption 1 .

###### Theorem 1 .

###### Remark 1 .

###### Remark 2 .

###### Remark 3 .

### 3.3 Proof Sketch of Theorem 1

In this section, we provide the proof sketch of Theorem [1](#Thmtheorem1). The full proof is given in Appendix [A](#A1).

Step 1: We adapt the analysis in Basei et al., () and use Bernstein inequality for the sub-exponential random variables to derive the following bound on estimation errors of system matrices.

###### Proposition 1 (Informal) .

Fix $\delta\in(0,\frac{3}{\pi^{2}})$. Let $\delta_{l}=\delta/l^{2}$. For $m_{l}\gtrsim\log\left(\frac{(m+n)^{2}}{\delta_{l}}\right)$, we have with probability at least $1-2\delta_{l}$,

$$ $\displaystyle\left\|\theta^{l+1}-\theta\right\|\lesssim\sqrt{\frac{\log\left(\frac{(m+n)^{2}}{\delta_{l}}\right)}{m_{l}}}+\frac{\log\left(\frac{(m+n)^{2}}{\delta_{l}}\right)}{m_{l}}+\frac{\log^{2}\left(\frac{(m+n)^{2}}{\delta_{l}}\right)}{m_{l}^{2}}.$ $$

For a complete rigorous statement, see Proposition [3](#Thmproposition3) in Appendix.

Step 2: We recursively carry out the perturbation analysis of less-standard Riccati equation ([2.1](#S2.Ex2)) and prove that the perturbation of the controller $\Delta K_{t}^{l}:=K_{t}^{l}-K_{t}$ is on the order of $O(\epsilon_{l})$, where $\epsilon_{l}=\max\left\{\|A^{l}-A\|,\|B^{l}-B\|\right\}$. The formal statement is presented in Lemma [8](#Thmlemma8).

Step 3: We use a result of Jacobson, () (see Lemma [10](#Thmlemma10)) and the proof technique in Fazel et al., () to prove that

$$ $J^{\pi^{l,k}}(x_{0}^{l,k})-J^{\star}(x_{0}^{l,k})=-\frac{1}{2\gamma}\sum_{t=1}^{T-1}\log\left(\det\left(I_{n}-\gamma D_{t}^{l}\right)\right)+\frac{1}{2}x_{0}^{l,k\top}D_{0}^{l}x_{0}^{l,k},$ $$

where $D_{t}^{l}$ is a function of $\Delta K_{t}^{l},\cdots,\Delta K_{T-1}^{l}$, with $\|D_{t}^{l}\|\leq\psi_{t}\mathcal{V}^{2}\epsilon_{l}^{2}+o(\epsilon_{l}^{2})$. See Proposition [5](#Thmproposition5). Here, $\pi^{l,k}$ is the sub-optimal controller $\pi^{l,k}$ executed in the $k$-th episode of the $l$-th epoch.

Step 4:
We can then bound the regret: $\operatorname{Regret}(N)=\sum_{l=1}^{L}\sum_{k=1}^{m_{l}}\left(J^{\pi^{l,k}}(x_{0}^{l,k})-J^{\star}(x_{0}^{l,k})\right)\lesssim\sum_{l=1}^{L}m_{l}\epsilon_{l}^{2}\lesssim\sum_{l=1}^{L}\log(l)\lesssim O(\log N\cdot\log\log(N)).$

###### Proposition 1 (Informal) .

## 4 A square-root regret bound

Theorem [1](#Thmtheorem1) shows that the logarithmic regret bound is achievable for episodic LEQR under Assumption [1](#Thmassumption1). One may wonder how does the regret bound changes after removing Assumption [1](#Thmassumption1). In particular, is
$\sqrt{N}$ regret achievable without Assumption [1](#Thmassumption1)? This section provides an affirmative answer to this question, by proposing and analyzing a least-squares-based algorithm with actively injected exploration noise.

### 4.1 A Least-Squares-Based Algorithm with Exploration Noise

We now introduce the least-squares-based algorithm with exploration noise, see Algorithm [2](#alg2). Algorithm [2](#alg2) is different from Algorithm [1](#alg1). We no longer divide the $N$ episodes into epochs of increasing lengths to estimate the system matrices. Instead, in the $k$-th episode, the algorithm updates the estimation of the system matrices $(A,B)$ by using the data from the previous $k-1$ episodes, which is denoted by $(A^{k},B^{k})$. Similar to $K_{t}^{l}$ in Section [3.1](#S3.SS1), we can obtain the feedback control $K_{t}^{k}$ by replacing the true system matrices in ([2.1](#S2.Ex2)) with $(A^{k},B^{k})$. Then, we execute the control with exploration noise $(g_{t}^{k})$ that follows a Gaussian distribution in the $k$-th episode.

The estimation of system matrices $(A,B)$ in Algorithm [2](#alg2) is different from that in Algorithm [1](#alg1). In Algorithm [2](#alg2), the estimator $(A^{k+1},B^{k+1})$ is obtained by solving the following $l_{2}$-regularized least-squares problem (based on the linear dynamics ([1](#S2.E1))):

$$ $\displaystyle\theta^{k+1}\in\arg\min_{y}\left\{\lambda\|y\|^{2}+\sum_{i=1}^{k}\sum_{t=0}^{T-1}\|x_{t+1}^{i}-y^{\top}z_{t}^{i}\|^{2}\right\},$ (16) $$

where $\theta^{k+1}=\left[A^{k+1},B^{k+1}\right]^{\top},z_{t}^{i}=[x_{t}^{i\top},\ u_{t}^{i\top}]^{\top}$ and $\lambda>0$ is the regularization parameter. By solving ([16](#S4.E16)), we can get

$$ $\theta^{k+1}=\left(\bar{V}^{k}\right)^{-1}\left(\sum_{i=1}^{k}\sum_{t=0}^{T-1}z_{t}^{i}x_{t+1}^{i\top}\right),$ (17) $$

where $\bar{V}^{k}=\lambda I+\sum_{i=1}^{k}\sum_{t=0}^{T-1}z_{t}^{i}z_{t}^{i\top}.$

Figure: Algorithm 2 The Least-Squares-Based Algorithm with Exploration Noise

### 4.2 Square-root Regret

In this section, we present the second main result of our paper, which demonstrates that Algorithm [2](#alg2) can attain $\sqrt{N}$-regret.

###### Theorem 2 .

Fix $\delta\in(0,1)$. When $N\geq 200\left(3(n+m)+\log\left(\frac{4N}{\delta}\right)\right)$, with probability at least $1-\delta$, the regret of Algorithm [2](#alg2) satisfies

$$ $\operatorname{Regret}(N)\leq\widetilde{\mathcal{C}}\sum_{t=0}^{T-1}\left(\alpha_{t}C_{N}+\beta_{t}\right)\sqrt{N},$ $$

where $\widetilde{\mathcal{C}}$ is a constant independent of $N$, $C_{N}$ exhibits a logarithmic dependence on $N$ and depends on $\lambda$, and $(\alpha_{t}),(\beta_{t})$ are two sequences recursively defined by

$$ $\displaystyle\alpha_{T-1}=2\widetilde{\Gamma}^{3},\quad\alpha_{t}=2\widetilde{\Gamma}\left(10\mathcal{V}^{2}\mathcal{L}\widetilde{\Gamma}^{4}\right)^{2(T-t-1)}+12\widetilde{\Gamma}^{4}\alpha_{t+1},$ $\displaystyle\beta_{T-1}=0,\quad\beta_{t}=12\widetilde{\Gamma}^{4}+12\widetilde{\Gamma}^{4}\beta_{t+1},$ $$

with $\widetilde{\Gamma},\mathcal{V},\mathcal{L}$ defined in ([15](#S3.E15)).

###### Remark 4 .

The design of Algorithm [2](#alg2) is inspired by (Mania et al., , Simchowitz and Foster, ), which establish $\sqrt{T}$ regret bounds for risk-neutral LQR in the infinite-horizon average reward setting, where $T$ is the number of time steps. The proof of Theorem [2](#Thmtheorem2), however, is significantly different from these studies because we consider the episodic setting with a risk-sensitive objective.

###### Remark 5 .

There appears to be an absence of published results on $\sqrt{N}-$regret bounds for episodic LQR even in the risk-neutral setting. While Basei et al., () established a logarithmic regret bound for continuous-time episodic risk-neutral LQR, they did not provide square-root regret bounds. Theorem [2](#Thmtheorem2)
shows that such $\sqrt{N}-$regret bound can be established both in the risk neutral ($\gamma=0$) and risk sensitive cases ($\gamma\neq 0.$)

###### Remark 6 .

The proof of Theorem [2](#Thmtheorem2) shares some similarities to the proof of Theorem [1](#Thmtheorem1). The main differences lie in (a) the removal of Assumption [1](#Thmassumption1), which leads to different estimation procedures and error analysis, and (b) the introduction of the exploration noise $g_{t}^{k}$ in the executed control, which leads to a more complicated analysis of performance loss.

###### Theorem 2 .

###### Remark 4 .

###### Remark 5 .

###### Remark 6 .

### 4.3 Proof Sketch of Theorem 2

We provide a proof outline for Theorem [2](#Thmtheorem2). The complete proof is given in Appendix [B](#A2).

Step 1: We adapt the self-normalized martingale analysis framework (Abbasi-Yadkori et al., , Cohen et al., , Simchowitz and Foster, ) to derive the following high probability bound for the estimation error. See Proposition [6](#Thmproposition6) for the complete statement.

###### Proposition 2 (informal) .

When $k$ is large enough, with probability at least $1-\delta$,

$$ $\left\|\theta^{k+1}-\theta\right\|\lesssim k^{-\frac{1}{4}}\sqrt{\log\left(1+k\log\left(\frac{N}{\delta}\right)\right)}.$ (18) $$

Step 2: We conduct perturbation analysis of the Riccati equation ([2.1](#S2.Ex2)) and show that $\Delta K_{t}^{k}:=K_{t}^{k}-K_{t}$ is on the order of $O(\epsilon_{k})$, where $\epsilon_{k}$ denotes the estimation error of system matrices, i.e. right-hand-side of ([18](#S4.E18)). This step is essentially the same as Step 2 in Section [3.3](#S3.SS3).

Step 3: Because of the additional exploration noise added to the online control, we show that the loss in the risk-sensitive performance becomes

$$ $J^{\pi^{k}}(x_{0}^{k})-J^{\star}(x_{0}^{k})=-\frac{1}{2\gamma}\sum_{t=0}^{T-1}\log\det\left(I_{n}-\gamma F_{t}^{k}\right)-\frac{1}{2\gamma}\sum_{t=1}^{T-1}\log\det\left(I_{m}-\gamma U_{t}^{k}\right)+\frac{1}{2}x_{0}^{k\top}U_{0}^{k}x_{0}^{k},$ $$

where $F_{t}^{k}$ and $U_{t}^{k}$ are functions of $\Delta K_{i}^{k}=K_{i}^{k}-K_{i},i=t,\cdots,T-1$, with $F_{t}^{k}\leq\frac{2\widetilde{\Gamma}^{3}}{\sqrt{k}}+o(\epsilon_{k}^{2})$ and $U_{t}^{k}\leq\alpha_{t}\mathcal{V}^{2}\epsilon_{k}^{2}+\frac{5\widetilde{\Gamma}^{5}(1+\beta_{t})}{\sqrt{k}}+o(\epsilon_{k}^{2})$.
See Proposition [7](#Thmproposition7).

Step 4: Finally we can bound the regret:
$\operatorname{Regret}(N)=\sum_{k=1}^{N}\left(J^{\pi^{k}}(x_{0}^{k})-J^{\star}(x_{0}^{k})\right)\lesssim\sum_{k=1}^{N}\epsilon_{k}^{2}\lesssim\sum_{k=1}^{N}\frac{1}{\sqrt{k}}\log\left(1+N\log\left(\frac{N}{\delta}\right)\right)\lesssim\widetilde{\mathcal{O}}(\sqrt{N})$.

###### Proposition 2 (informal) .

## 5 Simulation studies

We carry out simulation studies to illustrate the regret performances of
Algorithm [1](#alg1) and Algorithm [2](#alg2). The experiments are conducted on a PC with 2.10 GHz Intel Processor and 16 GB of RAM.

We consider an LEQR model with horizon $T=10$ steps with an initial state $x_{0}=0$ for simplicity. The dimensions of the state vector and control vector are both set to be $n=m=5$. The system and cost matrices $A,B,Q,R$ are all positive definite matrices randomly generated, and we also set $Q_{T}=0$ so that Assumption [1](#Thmassumption1) is satisfied according to Proposition [4](#Thmproposition4). The risk coefficient is set to be $\gamma=0.02$.
We consider the total number of episodes $N=5\times 10^{5}$ in the online control of the LEQR model. To implement Algorithm [1](#alg1), we also choose the input parameter $m_{1}=800$, set $L=\left\lceil\log_{2}\left(\frac{N}{m_{1}}+1\right)\right\rceil$, and randomly generate positive definite matrix $A^{1},B^{1}$ for initial estimates $\theta^{1}=(A^{1},B^{1})$ of true system matrices. To implement Algorithm 2, we choose the regularized parameter $\lambda=0.1$.

The numerical results are presented in Figure [5](#S5) and Figure [5](#S5). We compute the expected regret of each algorithm by averaging over 30 independent runs, but we do not plot the confidence intervals since the confidence intervals estimated from the 30 samples are very narrow compared with the magnitude of the regret and are almost invisible in the figures. Figure [5](#S5) demonstrates that the regret of Algorithm [1](#alg1) on the LEQR instance increases (approximately) logarithmically with the number of episodes, aligning with the findings of Theorem [1](#Thmtheorem1). Figure [5](#S5) is plotted on a log-log scale and shows that the regret of Algorithm [2](#alg2) on the LEQR instance grows (approximately) as a square-root function as the number of episodes increase, which is consistent with Theorem [2](#Thmtheorem2).

Figure: (a) Regret Performance of Algorithm [1](#alg1)
Refer to caption: /html/2406.05366/assets/lognoisefree.png

## 6 Conclusion and Future Work

This paper proposes two simple least-squares-based algorithm for online adaptive control of LEQR in the finite-horizon episodic setting. We prove that the least-squares greedy algorithm can achieve a regret bound that is logarithmic in the number of episodes under a identifiability condition of the system. We also prove that the least-squares-based algorithm with exploration noise can achieve $\sqrt{N}$-regret when the identifiability condition is not satisfied. To the best of our knowledge, this is the first set of regret bounds for LEQR.

The study of regret analysis for risk-sensitive control with continuous state and action spaces is still uncommon, and there are many open questions. For instance, it would be interesting to study regret bounds for LEQR in the infinite-horizon average-reward setting, for LEQR with partially observable states, and for more general risk-sensitive nonlinear control problems. We leave them for future work.

## Appendix A Regret Analysis for the Least-Squares Greedy Algorithm

In this section, we carry out the regret analysis for the least-squares greedy algorithm in Section [3](#S3). We derive the high-probability bounds for the estimation error of system matrices in Appendix [A.1](#A1.SS1). We do the perturbation analysis of Riccati equations in Appendix [A.2](#A1.SS2). We simplify the suboptimality gap due to the controller mismatch in Appendix [A.3](#A1.SS3). Finally, we combine the results derived/proved above and prove Theorem [1](#Thmtheorem1).

### A.1 Bounds for the Estimation Error of System Matrices

In this section, we discuss the high probability bound for the estimation error of system matrices in Algorithm [1](#alg1). We adapt the analysis framework in Basei et al., () and use the Bernstein inequality for the sub-exponential random variables to derive the desired error bound.

To facilitate the presentation, we first introduce some notations.
We fix the $l$-th epoch and define the following set

$$ $\displaystyle\Theta=\left\{\hat{\theta}\in\mathbb{R}^{(n+m)\times n}\Bigg{|}\left\|\hat{\theta}-\theta\right\|\leq\rho\right\}\cup\left\{\theta^{1}\right\},$ $$

where $\rho>0$ is a constant such that for any $\theta^{l}\in\Theta$, $\left\|\left(\mathbb{E}[V^{l}]\right)^{-1}\right\|\leq\mathcal{C}_{2}$ and $\left\|\mathbb{E}\left[Y^{l}\right]\right\|\leq\mathcal{C}_{2}$ for some constant $\mathcal{C}_{2}\geq 1$. We choose the initial number of episodes $m_{1}$ such that

$$ $\displaystyle\rho\geq 3\mathcal{C}_{1}\sqrt{\frac{\log\left(\frac{(n+m)^{2}}{\delta_{j-1}}\right)}{m_{j-1}}},\forall j\in\mathbb{N}^{+}\backslash\{1\},$ $$

where $\delta_{j-1}=\frac{\delta}{(j-1)^{2}}$, $m_{j-1}=2^{j-2}m_{1}$ and $\mathcal{C}_{1}$ is a constant independent of $m_{j},\forall j\in\mathbb{N}^{+}\backslash\{1\}$, but may depend on other constants including $m,n,T$. We will show how to choose $m_{1}$ in Section [A.4](#A1.SS4). We also define the event

$$ $\displaystyle\mathcal{G}^{l}=\left\{\theta^{j}\in\Theta,\forall j=1,\cdots,l\right\}.$ $$

We will prove that $\mathbb{P}(\mathcal{G}^{l})\geq 1-\sum_{j=1}^{l-1}\delta_{j}$ in Section [A.4](#A1.SS4). The following proposition is the main result of this section. Recall that $\theta^{l+1}=[A^{l+1}\ B^{l+1}]^{\top}$ are the estimated system matrices and $\theta=[A\ B]^{\top}$ are the true system matrices.

###### Proposition 3 .

Conditional on event $\mathcal{G}^{l}$, there exists a constant $\mathcal{C}_{3}\geq 1$ such that for $m_{l}\geq\mathcal{C}_{3}\log\left(\frac{(n+m)^{2}}{\delta_{l}}\right)$, with probability at least $1-2\delta_{l}$,

$$ $\displaystyle\left\|\theta^{l+1}-\theta\right\|\leq\mathcal{C}_{1}\left(\sqrt{\frac{\log\left(\frac{(n+m)^{2}}{\delta_{l}}\right)}{m_{l}}}+\frac{\log\left(\frac{(n+m)^{2}}{\delta_{l}}\right)}{m_{l}}+\frac{\log^{2}\left(\frac{(n+m)^{2}}{\delta_{l}}\right)}{m_{l}^{2}}\right).$ $$

The proof of Proposition [3](#Thmproposition3) is long, and we discuss it in the new few sections.

###### Proposition 3 .

#### A.1.1 Preliminaries

In this section, we recall the definition of sub-exponential random variables and state several well-known results about such random variables that will be used in our analysis later.

###### Definition 1 (Definition 2.7 of Wainwright, ( 2019 ) ) .

A random variable $X$ with mean $\mu=\mathbb{E}X$ is sub-exponential if there are non-negative parameters $(\nu,\alpha)$ such that $\mathbb{E}[e^{\lambda(X-\mu)}]\leq e^{\frac{\nu^{2}\lambda^{2}}{2}}$ for all $|\lambda|<\frac{1}{\alpha}$. Denote the set of such random variables as $SE(\nu^{2},\alpha)$.

###### Lemma 1 (Bernstein Inequality, Proposition 2.9 of Wainwright, ( 2019 ) ) .

Suppose that $X\in SE(\nu^{2},\alpha)$, and let $\mu=\mathbb{E}X$. Then
for any $\zeta>0$, we have

$$ $\displaystyle\mathbb{P}\left(|X-\mu|\geq\zeta\right)\leq 2\exp\left(-\min\left\{\frac{\zeta^{2}}{2\nu^{2}},\frac{\zeta}{2\alpha}\right\}\right).$ $$

###### Lemma 2 (Lemma 5.1 of Alessandro, ( 2018 ) ) .

If $X_{i}\in SE(\nu_{i}^{2},\alpha_{i}),i\in[n]$, then

$$ $\displaystyle\sum_{i=1}^{n}X_{i}\in\begin{cases}SE\left(\sum_{i=1}^{n}\nu_{i}^{2},\max_{i\in[n]}\alpha_{i}\right)&\text{if $X_{i}$ are independent},\\ SE\left(\left(\sum_{i=1}^{n}\nu_{i}\right)^{2},\max_{i\in[n]}\alpha_{i}\right)&\text{if $X_{i}$ are not independent}.\end{cases}$ $$

###### Lemma 3 (Lemma 2.7.7 of Vershynin, ( 2018 ) ) .

Let $X$ and $Y$ be sub-Gaussian random variables. Then, $XY$ is sub-exponential.

###### Definition 1 (Definition 2.7 of Wainwright, ( 2019 ) ) .

###### Lemma 1 (Bernstein Inequality, Proposition 2.9 of Wainwright, ( 2019 ) ) .

###### Lemma 2 (Lemma 5.1 of Alessandro, ( 2018 ) ) .

###### Lemma 3 (Lemma 2.7.7 of Vershynin, ( 2018 ) ) .

#### A.1.2 Properties of Estimated System Matrix

In this section, we use the properties of sub-exponential random variables to derive some statistical properties for the estimated system matrix $\theta^{l}$.

The following lemma shows that every element of the state-action random sample vector $z_{t}^{l,k}=\left[x_{t}^{l,k\top}\ u_{t}^{l,k\top}\right]^{\top}$ is sub-Gaussian.

###### Lemma 4 .

Consider the sample state ([8](#S3.E8)) in section [3.1](#S3.SS1), conditional on event $\mathcal{G}^{l}$, we can prove that every element of the sample state $x_{t}^{l,k}$ and action vector $u_{t}^{l,k}$ is sub-Gaussian for any step $t$, episode $k$, epoch $l$.

###### Proof.

By the definition ([8](#S3.E8)), we have

$$ $\displaystyle x_{t}^{l,k}$ $\displaystyle=Ax_{t-1}^{l,k}+Bu_{t-1}^{l,k}+w_{t-1}^{l,k}$ $\displaystyle=(A+BK_{t-1}^{l})x_{t-1}^{l,k}+w_{t-1}^{l,k}$ $\displaystyle=(A+BK_{t-1}^{l})(A+BK_{t-2}^{l})x_{t-2}^{l,k}+(A+BK_{t-1}^{l})w_{t-2}^{l,k}+w_{t-1}^{l,k}.$ $$

Repeating this procedure, we can get

$$ $\displaystyle x^{l,k}_{t}$ $\displaystyle=\left(\prod_{i=t-1}^{0}(A+BK^{l}_{i})\right)x^{l,k}_{0}+\sum_{j=0}^{t-1}\left(\prod_{i=t-1}^{j+1}(A+BK^{l}_{i})\right)w^{l,k}_{j},$ (19) $$

where $\prod_{i=t-1}^{t}(A+BK^{l}_{i}):=I_{n}$. Recall the definition of $K_{t}^{l}$ in ([2.1](#S2.Ex2)) by using $\theta^{l}$. It’s continuous in $\theta^{l}\in\Theta$, so $K_{t}^{l}$ is uniformly bounded for any $\theta^{l}\in\Theta$ by the boundedness of $\Theta$, i.e. there exists some constant $M>0$ such that $\sup_{t}\|K_{t}^{l}\|\leq M$. Because $x_{0}^{l,k}=x_{0}$ and $\{w_{i}^{l,k}\}_{i=0}^{t-1}$ are independent zero-mean normal random variables, we can then readily obtain from ([19](#A1.E19)) that every element of $x_{t}^{l,k}$ is sub-Gaussian by the uniform boundedness of $K_{t}^{l}$. Similarly, we can prove that every element of $u_{t}^{l,k}=K_{t}^{l}x_{t}^{l,k}$ is sub-Gaussian, which completes the proof.
∎

Recall the following matrices in ([7](#S3.E7)) and ([9](#S3.E9)):

$$ $\displaystyle V^{l}=\sum_{t=0}^{T-1}z^{l}_{t}z_{t}^{l\top}\qquad\qquad\qquad\qquad Y^{l}=\sum_{t=0}^{T-1}z^{l}_{t}x_{t+1}^{l\top}$ (20) $\displaystyle\bar{V}^{l}=\frac{1}{m_{l}}\sum_{k=1}^{m_{l}}\sum_{t=0}^{T-1}z_{t}^{l,k}z_{t}^{l,k\top}\qquad\qquad\bar{Y}^{l}=\frac{1}{m_{l}}\sum_{k=1}^{m_{l}}\sum_{t=0}^{T-1}z_{t}^{l,k}x_{t+1}^{l,k\top},$ $$

where $V^{l},\bar{V}^{l}\in\mathbb{R}^{(n+m)\times(n+m)}$ and $Y^{l},\bar{Y}^{l}\in\mathbb{R}^{(n+m)\times n}$. We denote the elements of $V^{l},Y^{l},\bar{V}^{l},\bar{Y}^{l}$ as

$$ $\displaystyle V^{l}_{i,j}=\sum_{t=0}^{T-1}z^{l}_{t,i}z^{l}_{t,j},i,j\in[n+m]$ $\displaystyle Y^{l}_{i,j}=\sum_{t=0}^{T-1}z^{l}_{t,i}x^{l}_{t+1,j},i\in[n+m],j\in[n]$ $\displaystyle\bar{V}^{l}_{i,j}=\frac{1}{m_{l}}\sum_{k=1}^{m_{l}}\sum_{t=0}^{T-1}z_{t,i}^{l,k}z_{t,j}^{l,k},i,j\in[n+m]$ $\displaystyle\bar{Y}^{l}_{i,j}=\frac{1}{m_{l}}\sum_{k=1}^{m_{l}}\sum_{t=0}^{T-1}z_{t,i}^{l,k}x_{t+1,j}^{l,k},i\in[n+m],j\in[n].$ $$

Then we have the following result from Lemma [4](#Thmlemma4).

###### Lemma 5 .

There exist non-negative parameters $\iota$ and $\eta$
such that $\bar{V}_{i,j}^{l},\bar{Y}_{i,j}^{l}\in SE\left(\frac{\iota^{2}}{m_{l}},\frac{\eta}{m_{l}}\right)$ for all $i,j$ and $l\in[L]$.

###### Proof.

By Lemma [3](#Thmlemma3) and Lemma [4](#Thmlemma4), we know that every element of $z_{t}^{l,k}z_{t}^{l,k\top}$ and $z_{t}^{l,k}x_{t+1}^{l,k\top}$ are sub-exponential random variables. That is,
$z_{t,i}^{l,k}z_{t,j}^{l,k}\in SE\left(\left(\nu_{t,i,j}\right)^{2},\alpha_{t,i,j}\right),i,j\in[n+m]$ and $z_{t,i}^{l,k}x_{t+1,j}^{l,k}\in SE\left(\left(\omega_{t,i,j}\right)^{2},\beta_{t,i,j}\right),i\in[n+m],j\in[n]$. The subexponential parameters can be chosen independent of $l$ and $k$ by the proof of Lemma [4](#Thmlemma4).
If we denote by

$$ $\displaystyle\nu_{t}=\max_{i,j}\nu_{t,i,j},\qquad\qquad\alpha_{t}=\max_{i,j}\alpha_{t,i,j},$ $\displaystyle\omega_{t}=\max_{i,j}\omega_{t,i,j},\qquad\qquad\beta_{t}=\max_{i,j}\beta_{t,i,j},$ $$

then we have $z_{t,i}^{l,k}z_{t,j}^{l,k}\in SE\left(\left(\nu_{t}\right)^{2},\alpha_{t}\right)$ for any $k\in[m_{l}]$, $i,j\in[n+m]$ and $z_{t,i}^{l,k}x_{t+1,j}^{l,k}\in SE\left(\left(\omega_{t}\right)^{2},\beta_{t}\right)$ for any $k\in[m_{l}]$, $i\in[n+m],j\in[n]$.

By Lemma [2](#Thmlemma2), for non-independent sub-exponential random variables, we obtain

$$ $\displaystyle\sum_{t=0}^{T-1}z_{t,i}^{l,k}z_{t,j}^{l,k}\in SE\left(\left(\sum_{t=0}^{T-1}\nu_{t}\right)^{2},\max_{t}\alpha_{t}\right),$ $\displaystyle\sum_{t=0}^{T-1}z_{t,i}^{l,k}x_{t+1,j}^{l,k}\in SE\left(\left(\sum_{t=0}^{T-1}\omega_{t}\right)^{2},\max_{t}\beta_{t}\right).$ $$

Applying Lemma [2](#Thmlemma2) again, but for independent sub-exponential random variables, we infer that

$$ $\displaystyle\bar{V}_{i,j}^{l}=\frac{1}{m_{l}}\sum_{k=1}^{m_{l}}\sum_{t=0}^{T-1}z_{t,i}^{l,k}z_{t,j}^{l,k}$ $\displaystyle\in SE\left(\frac{\sum_{k=1}^{m_{l}}\left(\sum_{t=0}^{T-1}\nu_{t}\right)^{2}}{m_{l}^{2}},\frac{\max_{t}\alpha_{t}}{m_{l}}\right)$ $\displaystyle=SE\left(\frac{\left(\sum_{t=0}^{T-1}\nu_{t}\right)^{2}}{m_{l}},\frac{\max_{t}\alpha_{t}}{m_{l}}\right),$ $\displaystyle\bar{Y}_{i,j}^{l}=\frac{1}{m_{l}}\sum_{k=1}^{m_{l}}\sum_{t=0}^{T-1}z_{t,i}^{l,k}x_{t+1,j}^{l,k}$ $\displaystyle\in SE\left(\frac{\sum_{k=1}^{m_{l}}\left(\sum_{t=0}^{T-1}\omega_{t}\right)^{2}}{m_{l}^{2}},\frac{\max_{t}\beta_{t}^{l}}{m_{l}}\right)$ $\displaystyle=SE\left(\frac{\left(\sum_{t=0}^{T-1}\omega_{t}\right)^{2}}{m_{l}},\frac{\max_{t}\beta_{t}}{m_{l}}\right).$ $$

The proof is complete by letting

$$ $\displaystyle\iota=\max\left\{\sqrt{\left(\sum_{t=0}^{T-1}\nu_{t}\right)^{2}},\sqrt{\left(\sum_{t=0}^{T-1}\omega_{t}\right)^{2}}\right\},$ $\displaystyle\eta=\max\left\{\max_{t}\alpha_{t},\max_{t}\beta_{t}\right\}.$ $$

∎

We can now derive the concentration inequalities for $\bar{V}^{l}$ and $\bar{Y}^{l}$.

###### Lemma 6 .

Conditional on event $\mathcal{G}^{l}$, we can derive that for any $\zeta>0$,

$$ $\displaystyle\max\left\{\mathbb{P}\left(\left|\bar{V}^{l}-\mathbb{E}V^{l}\right|\geq\zeta\right),\mathbb{P}\left(\left|\bar{Y}^{l}-\mathbb{E}Y^{l}\right|\geq\zeta\right)\right\}$ (21) $\displaystyle\leq 2(n+m)^{2}\exp\left(-\min\left\{\frac{m_{l}\zeta^{2}}{2\iota^{2}(n+m)^{4}},\frac{m_{l}\zeta}{2\eta(n+m)^{2}}\right\}\right),$ $$

where $|\cdot|$ is a matrix norm that represents the summation of the absolute value of all the elements of the matrix, e.g. $|A|=\sum_{i,j}|a_{i,j}|$.

###### Proof.

We first consider one element of the matrix $\bar{V}^{l}$. By Lemma [1](#Thmlemma1) and Lemma [5](#Thmlemma5), we have

$$ $\displaystyle\mathbb{P}\left(\left|\bar{V}^{l}_{i,j}-\mathbb{E}V_{i,j}^{l}\right|\geq\zeta\right)\leq 2\exp\left(-\min\left\{\frac{m_{l}\zeta^{2}}{2\iota^{2}},\frac{m_{l}\zeta}{2\eta}\right\}\right).$ $$

Then, by the fact that $\mathbb{P}\left(\sum_{i=1}^{M}\left|X_{i}\right|\geq\zeta\right)\leq\sum_{i=1}^{M}\mathbb{P}\left(|X_{i}|\geq\frac{\zeta}{M}\right)$ for all $M\in\mathbb{N}$ and random variables $(X_{i})_{i=1}^{M}$, we can derive the concentration inequality for $\left|\bar{V}^{l}-\mathbb{E}V^{l}\right|$:

$$ $\displaystyle\mathbb{P}\left(\left|\bar{V}^{l}-\mathbb{E}V^{l}\right|\geq\zeta\right)$ $\displaystyle=\mathbb{P}\left(\sum_{i=1}^{n+m}\sum_{j=1}^{n+m}\left|\bar{V}^{l}_{i,j}-\mathbb{E}V_{i,j}^{l}\right|\geq\zeta\right)$ $\displaystyle\leq\sum_{i=1}^{n+m}\sum_{j=1}^{n+m}\mathbb{P}\left(\left|\bar{V}^{l}_{i,j}-\mathbb{E}V_{i,j}^{l}\right|\geq\frac{\zeta}{(n+m)^{2}}\right)$ $\displaystyle\leq 2(n+m)^{2}\exp\left(-\min\left\{\frac{m_{l}\zeta^{2}}{2\iota^{2}(n+m)^{4}},\frac{m_{l}\zeta}{2\eta(n+m)^{2}}\right\}\right).$ $$

Similarly, we can derive the concentration probability for $\bar{Y}^{l}$:

$$ $\displaystyle\mathbb{P}\left(\left|\bar{Y}^{l}-\mathbb{E}Y^{l}\right|\geq\zeta\right)$ $\displaystyle=\mathbb{P}\left(\sum_{i=1}^{n+m}\sum_{j=1}^{n}\left|\bar{Y}^{l}_{i,j}-\mathbb{E}Y_{i,j}^{l}\right|\geq\zeta\right)$ $\displaystyle\leq\sum_{i=1}^{n+m}\sum_{j=1}^{n}\mathbb{P}\left(\left|\bar{Y}^{l}_{i,j}-\mathbb{E}Y_{i,j}^{l}\right|\geq\frac{\zeta}{(n+m)n}\right)$ $\displaystyle\leq 2(n+m)n\exp\left(-\min\left\{\frac{m_{l}\zeta^{2}}{2\iota^{2}(n+m)^{2}n^{2}},\frac{m_{l}\zeta}{2\eta(n+m)n}\right\}\right)$ $\displaystyle\overset{(1)}{\leq}2(n+m)^{2}\exp\left(-\min\left\{\frac{m_{l}\zeta^{2}}{2\iota^{2}(n+m)^{4}},\frac{m_{l}\zeta}{2\eta(n+m)^{2}}\right\}\right),$ $$

where inequality (1) follows from the fact that $n+m\geq n$.

Finally, combining the two probability inequalities above, we can obtain ([21](#A1.E21)).
∎

In order to derive the probability bounds in Proposition [3](#Thmproposition3), we prove that $\left\|\left(\mathbb{E}[V^{l}]\right)^{-1}\right\|$ and $\left\|\mathbb{E}[Y^{l}]\right\|$ are bounded by a positive constant for any $\theta^{l}$ lies in $\Theta$. The boundedness of $\left\|\mathbb{E}[Y^{l}]\right\|$ can be proved directly, because $\mathbb{E}[Y^{l}]$ is continuous in terms of $\theta^{l}$ according to the definition of $Y^{l}$ in ([20](#A1.E20)). In terms of $\left\|\left(\mathbb{E}[V^{l}]\right)^{-1}\right\|$, we will use the following lemma to show that it’s bounded when $\theta^{l}\in\Theta$. Similar results can be found in Proposition 3.10 of Basei et al., ().

###### Lemma 7 .

The following properties are equivalent:

- 1.
For the sequence of the controller $K_{t},t=0,\cdots,T-1$ defined in ([2.1](#S2.Ex2)),
$\left\{v\in\mathbb{R}^{n+m}\Big{|}\left[I\ K_{t}^{\top}\right]v=0,\forall t=0,\cdots,T-1\right\}=\{0\};$
- 2.
$\mathbb{E}[V]\succ 0$, where $V=\sum_{t=0}^{T-1}z_{t}z_{t}^{\top}$ is generated by the optimal policy in ([2.1](#S2.Ex2));
- 3.
There exists $\lambda_{0}>0$ such that $\lambda_{\min}\left(\mathbb{E}\left[V^{l}\right]\right)\geq\lambda_{0}$ for any estimated $\theta^{l}\in\Theta.$

###### Proof.

We first prove property 1 $\iff$ property 2.

For simplicity of notation, let $h_{t}=[I\ K_{t}^{\top}]^{\top}$ and $H=[h_{0},h_{1},\cdots,h_{T-1}]$. Property 1 is equivalent to that there exists no nonzero $v$ such that $H^{\top}v=0$, which is also equivalent to that for any $v\neq 0$, $v^{\top}HH^{\top}v>0$, i.e.

$$ $\displaystyle HH^{\top}=\sum_{t=0}^{T-1}h_{t}h_{t}^{\top}=\sum_{t=0}^{T-1}\left[\begin{array}[]{c}I_{n}\\ K_{t}\end{array}\right]\left[I_{n}\ K_{t}^{\top}\right]\succ 0.$ $$

One can readily compute that

$$ $\displaystyle\mathbb{E}[V]$ $\displaystyle=\mathbb{E}\left[\sum_{t=0}^{T-1}z_{t}z_{t}^{\top}\right]$ $\displaystyle=\sum_{t=0}^{T-1}\left[\begin{array}[]{c}I_{n}\\ K_{t}\end{array}\right]\mathbb{E}\left[x_{t}x_{t}^{\top}\right]\left[I_{n}\ K_{t}^{\top}\right]=H\text{diag}\left(\mathbb{E}\left[x_{0}x_{0}^{\top}\right],\cdots,\mathbb{E}\left[x_{T-1}x_{T-1}^{\top}\right]\right)H^{\top},$ (24) $$

where $\text{diag}(\cdot)$ is the notation of a diagonal block matrix. Next we show that $\mathbb{E}\left[x_{t}x_{t}^{\top}\right]$ is positive definite for each $t$. Similar to ([19](#A1.E19)), we can expand the system dynamics under the true system matrix $\theta=(A,B)$ as

$$ $\displaystyle x_{t}$ $\displaystyle=\left(\prod_{i=t-1}^{0}(A+BK_{i})\right)x_{0}+\sum_{j=0}^{t-1}\left(\prod_{i=t-1}^{j+1}(A+BK_{i})\right)w_{j},$ (25) $$

where $\prod_{i=t-1}^{j+1}(A+BK_{i})$ means $(A+BK_{t-1})(A+BK_{t-2})\cdots(A+BK_{j+1})$, and $\prod_{i=t-1}^{t}(A+BK_{i})=I_{n}$. For simplicity of notation, let

$$ $\displaystyle\Phi_{t_{1},t_{0}}=(A+BK_{t_{1}})(A+BK_{t_{1}-1})\cdots(A+BK_{t_{0}}),\quad\text{for any $t_{1}\geq t_{0}$.}$ (26) $$

When $t_{1}<t_{0}$, we set $\Phi_{t_{1},t_{0}}=I_{n}$. Then we have
$x_{t}=\Phi_{t-1,0}x_{0}+\sum_{j=0}^{t-1}\Phi_{t-1,j+1}w_{j}.$ It follows that

$$ $\displaystyle\mathbb{E}\left[x_{t}x_{t}^{\top}\right]$ $\displaystyle=\Phi_{t-1,0}\mathbb{E}\left[x_{0}x_{0}^{\top}\right]\Phi_{t-1,0}^{\top}+\sum_{j=0}^{t-1}\Phi_{t-1,j+1}\mathbb{E}\left[w_{j}w_{j}^{\top}\right]\Phi_{t-1,j+1}^{\top}$ $\displaystyle\overset{(1)}{=}\Phi_{t-1,0}x_{0}x_{0}^{\top}\Phi_{t-1,0}^{\top}+\sum_{j=1}^{t}\Phi_{t-1,j}\Phi_{t-1,j}^{\top}$ $\displaystyle\overset{(2)}{=}\Phi_{t-1,0}x_{0}x_{0}^{\top}\Phi_{t-1,0}^{\top}+I_{n}+\sum_{j=1}^{t-1}\Phi_{t-1,j}\Phi_{t-1,j}^{\top}$ $\displaystyle\succeq I_{n},$ $$

where the equality (1) follows from the fact that $w_{j}\sim\mathcal{N}\left(0,I_{n}\right),j=0,\cdots,t-1$, and equality (2) holds by the fact that $\Phi_{t-1,t}=I_{n}$. Then, we can prove that property 1 is equivalent to that for any $v\neq 0$,

$$ $\displaystyle v^{\top}\mathbb{E}[V]v=v^{\top}H\text{diag}\left(\mathbb{E}\left[x_{0}x_{0}^{\top}\right],\cdots,\mathbb{E}\left[x_{T-1}x_{T-1}^{\top}\right]\right)H^{\top}v>0,$ $$

which is equivalent to property 2.

We next prove property 2 $\iff$ property 3.

In terms of property 3 $\Longrightarrow$ property 2, it’s obvious that when $\theta^{l}=\theta$, we have $\lambda_{\min}\left(\mathbb{E}[V]\right)\geq\lambda_{0}>0$, i.e. $\mathbb{E}[V]\succ 0$.

In order to prove property 2 $\Longrightarrow$ property 3, we prove the continuity of $\mathbb{E}[V]$ in terms of the system matrices $\theta$. By the recursive formula of the discrete-time Riccati equations and the optimal controller in ([2.1](#S2.Ex2)), we can find that $P_{t},\widetilde{P}_{t}$ and $K_{t}$ are continuous in terms of $\theta\in\Theta$. Recall that

$$ $\displaystyle\mathbb{E}[x_{t}x_{t}^{\top}]=\Phi_{t-1,0}x_{0}x_{0}^{\top}\Phi_{t-1,0}^{\top}+I_{n}+\sum_{j=1}^{t-1}\Phi_{t-1,j}\Phi_{t-1,j}^{\top},$ (27) $$

where $\Phi_{t-1,j}$ is defined in ([26](#A1.E26)). Plugging ([27](#A1.E27)) into ([A.1.2](#A1.Ex47)), we can see that $\mathbb{E}[V]$ is continuous in terms of $\theta$. So for any $\theta^{l}\in\Theta$, there exists $\lambda_{0}>0$ such that $\lambda_{\min}\left(\mathbb{E}[V^{l}]\right)\geq\lambda_{0}$.
∎

Now we are ready for the proof of Proposition [3](#Thmproposition3).

###### Proof of Proposition 3 .

Recall the definition of $\bar{V}^{l},\bar{Y}^{l},V^{l}$ and $Y^{l}$ in ([20](#A1.E20)), we have

$$ $\displaystyle\left\|\theta^{l+1}-\theta\right\|$ (28) $\displaystyle=\left\|\left(\bar{V}^{l}+\frac{1}{m_{l}}I_{n+m}\right)^{-1}\bar{Y}^{l}-\left(\mathbb{E}\left[V^{l}\right]\right)^{-1}\mathbb{E}\left[Y^{l}\right]\right\|$ $\displaystyle\leq\left\|\left(\bar{V}^{l}+\frac{1}{m_{l}}I_{n+m}\right)^{-1}-\left(\mathbb{E}\left[V^{l}\right]\right)^{-1}\right\|\cdot\left\|\bar{Y}^{l}\right\|+\left\|\left(\mathbb{E}\left[V^{l}\right]\right)^{-1}\right\|\cdot\left\|\bar{Y}^{l}-\mathbb{E}[Y^{l}]\right\|$ $\displaystyle\overset{(1)}{\leq}\left\|\left(\bar{V}^{l}+\frac{1}{m_{l}}I_{n+m}\right)^{-1}\right\|\cdot\left\|\left(\mathbb{E}\left[V^{l}\right]\right)^{-1}\right\|\cdot\left\|\bar{Y}^{l}\right\|\cdot\left\|\bar{V}^{l}+\frac{1}{m_{l}}I_{n+m}-\mathbb{E}\left[V^{l}\right]\right\|$ $\displaystyle\quad+\left\|\left(\mathbb{E}\left[V^{l}\right]\right)^{-1}\right\|\cdot\left\|\bar{Y}^{l}-\mathbb{E}\left[Y^{l}\right]\right\|$ $\displaystyle\overset{(2)}{\leq}\mathcal{C}_{2}\left(\left\|\left(\bar{V}^{l}+\frac{1}{m_{l}}I_{n+m}\right)^{-1}\right\|\cdot\left\|\bar{Y}^{l}\right\|\cdot\left\|\bar{V}^{l}+\frac{1}{m_{l}}I_{n+m}-\mathbb{E}\left[V^{l}\right]\right\|+\left\|\bar{Y}^{l}-\mathbb{E}\left[Y^{l}\right]\right\|\right),$ $$

where inequality (1) holds by the fact that $E^{-1}-F^{-1}=E^{-1}(F-E)F^{-1}$, inequality (2) follows from the results in Lemma [7](#Thmlemma7) that $\left\|\left(\mathbb{E}[V^{l}]\right)^{-1}\right\|\leq\mathcal{C}_{2}$. By Lemma [6](#Thmlemma6) and the equivalence of matrix norms, with probability at least $1-2\delta_{l}$, we have $\left\|\bar{V}^{l}-\mathbb{E}\left[V^{l}\right]\right\|\leq\Delta_{l}$ and $\left\|\bar{Y}^{l}-\mathbb{E}\left[Y^{l}\right]\right\|\leq\Delta_{l}$, where

$$ $\displaystyle\Delta_{l}:=\max\left\{\sqrt{\frac{2\iota^{2}(n+m)^{5}\log\left(\frac{(n+m)^{2}}{\delta_{l}}\right)}{m_{l}}},\frac{2\eta(n+m)^{2.5}\log\left(\frac{(n+m)^{2}}{\delta_{l}}\right)}{m_{l}}\right\}.$ $$

For notational simplicity, we denote by

$$ $\displaystyle\mathcal{C}_{4}:=\max\left\{\sqrt{2\iota^{2}(n+m)^{5}},2\eta(n+m)^{2.5}\right\}.$ $$

$\mathcal{C}_{4}$ is a constant depending on $m,n$ polynomially and depending on $\gamma$ exponentially. For simplicity, we ignore the $T$-dependence of $\mathcal{C}_{4}$. Then, we have

$$ $\displaystyle\Delta_{l}\leq\mathcal{C}_{4}\max\left\{\sqrt{\frac{\log\left(\frac{(n+m)^{2}}{\delta_{l}}\right)}{m_{l}}},\frac{\log\left(\frac{(n+m)^{2}}{\delta_{l}}\right)}{m_{l}}\right\}.$ $$

Now we can use $\Delta_{l}$ to further bound the terms in ([28](#A1.E28)). Let $m_{l}$ be large enough so that $\Delta_{l}+\frac{1}{m_{l}}\leq\frac{1}{2\mathcal{C}_{2}}$, i.e. $m_{l}\geq\mathcal{C}_{3}\log\left(\frac{(n+m)^{2}}{\delta_{l}}\right)$ for some constant $\mathcal{C}_{3}\geq 1$. Then, with probability at least $1-2\delta_{l}$, we have $\left\|\bar{V}^{l}-\mathbb{E}\left[V^{l}\right]+\frac{1}{m_{l}}I_{n+m}\right\|\leq\Delta_{l}+\frac{1}{m_{l}}\leq\frac{1}{2\mathcal{C}_{2}}$, and thus

$$ $\displaystyle\lambda_{\min}\left(\bar{V}^{l}+\frac{1}{m_{l}}I_{n+m}\right)\geq\lambda_{\min}\left(\mathbb{E}\left[V^{l}\right]\right)-\left\|\bar{V}^{l}-\mathbb{E}\left[V^{l}\right]+\frac{1}{m_{l}}I_{n+m}\right\|\geq\frac{1}{2\mathcal{C}_{2}}.$ $$

Then, we get $\left\|\left(\bar{V}^{l}+\frac{1}{m_{l}}I_{n+m}\right)^{-1}\right\|\leq 2\mathcal{C}_{2}$. In terms of $\left\|\bar{Y}^{l}\right\|$, we have

$$ $\displaystyle\left\|\bar{Y}^{l}\right\|=\left\|\bar{Y}^{l}-\mathbb{E}\left[Y^{l}\right]+\mathbb{E}\left[Y^{l}\right]\right\|\overset{(3)}{\leq}\mathcal{C}_{2}+\left\|\bar{Y}^{l}-\mathbb{E}\left[Y^{l}\right]\right\|\leq\mathcal{C}_{2}+\Delta_{l},$ $$

where inequality (3) follows from the fact that $\left\|\mathbb{E}\left[Y^{l}\right]\right\|\leq\mathcal{C}_{2}$.
Finally, substituting all the elements into ([28](#A1.E28)), we can get

$$ $\displaystyle\left\|\theta^{l+1}-\theta\right\|$ $\displaystyle\leq\mathcal{C}_{2}\left(2\mathcal{C}_{2}\cdot\left(\mathcal{C}_{2}+\left\|\bar{Y}^{l}-\mathbb{E}\left[Y^{l}\right]\right\|\right)\cdot\left(\Delta_{l}+\frac{1}{m_{l}}\right)+\Delta_{l}\right)$ $\displaystyle\leq\mathcal{C}_{2}\left(2\mathcal{C}_{2}\cdot\left(\mathcal{C}_{2}+\Delta_{l}\right)\cdot\left(\Delta_{l}+\frac{1}{m_{l}}\right)+\Delta_{l}\right)$ $\displaystyle\overset{(4)}{\leq}2\mathcal{C}_{2}^{3}\left(\left(1+\Delta_{l}\right)\left(\Delta_{l}+\frac{1}{m_{l}}\right)+\Delta_{l}\right)$ $\displaystyle\overset{(5)}{\leq}8\mathcal{C}_{2}^{3}\left(\Delta_{l}+\Delta_{l}^{2}+\frac{1}{m_{l}}\right)$ $\displaystyle\overset{(6)}{\leq}16\mathcal{C}_{2}^{3}\mathcal{C}_{4}^{2}\left(\sqrt{\frac{\log\left(\frac{(n+m)^{2}}{\delta_{l}}\right)}{m_{l}}}+\frac{\log\left(\frac{(n+m)^{2}}{\delta_{l}}\right)}{m_{l}}+\frac{\log^{2}\left(\frac{(n+m)^{2}}{\delta_{l}}\right)}{m_{l}^{2}}\right),$ $$

where inequality (4) follows from $\mathcal{C}_{2}\geq 1$, inequality (5) holds by the fact that $m_{l}\geq 1$ and inequality (6) holds because $\log\left(\frac{(n+m)^{2}}{\delta_{l}}\right)\geq 1$. The proof is hence complete. ∎

Lemma [7](#Thmlemma7) shows that Assumption [1](#Thmassumption1) can be extended to the neighbourhood of the true system matrices $\theta$, and thus guarantee the well-posedness of the sample variance of the estimated system matrices within the neighbourhood. The following proposition provides a sufficient condition for Assumption [1](#Thmassumption1).

###### Proposition 4 .

If the parameters defined in Section [2.1](#S2.SS1) satisfies

- 1.
$A\in\mathbb{R}^{n\times n}$ has full rank;
- 2.
$Q\succ 0$ and $Q_{T}=0$;
- 3.
$B\in\mathbb{R}^{n\times m}$ has full column rank,

then for the sequence of the controller $K_{t},t=0,\cdots,T-1$ defined in ([2.1](#S2.Ex2)),
we have

$$ $\left\{v\in\mathbb{R}^{n+m}\bigg{|}[I\ K_{t}^{\top}]v=0,\forall t=0,\cdots,T-1\right\}=\{0\}.$ $$

###### Proof.

Let $v=[v_{1}^{\top}\ v_{2}^{\top}]^{\top}$ satisfying $[I\ K_{t}^{\top}]v=0,\forall t=0,\cdots,T-1$, where $v_{1}\in\mathbb{R}^{n}$, $v_{2}\in\mathbb{R}^{m}$. Recall the optimal control defined in ([2.1](#S2.Ex2)), by the condition $Q_{T}=0$, we have $K_{T-1}=0$, and thus $v_{1}=0$. Then, $[I\ K_{t}^{\top}]v=0,\forall t=0,\cdots,T-1$ is equivalent to
$K_{t}^{\top}v_{2}=0,\forall t=0,\cdots,T-1$. Substitute ([2.1](#S2.Ex2)) into it, we can obtain

$$ $\displaystyle K_{t}^{\top}v_{2}$ $\displaystyle=-A^{\top}\widetilde{P}_{t+1}B\left(B^{\top}\widetilde{P}_{t+1}B+R\right)^{-1}v_{2}=0.$ $$

Recall that

$$ $\displaystyle\widetilde{P}_{t+1}=P_{t+1}+\gamma P_{t+1}\left(I_{n}-\gamma P_{t+1}\right)^{-1}P_{t+1},$ $\displaystyle P_{t}=Q+K_{t}^{\top}RK_{t}+(A+BK_{t})^{\top}\widetilde{P}_{t+1}(A+BK_{t}),t=0,\cdots,T-1.$ $$

We can prove that $P_{t}\succ 0$ for any $t=0,\cdots,T-1$ by the mathematical induction. When $t=T$, $\widetilde{P}_{T}=0$, and thus $P_{T-1}\succ 0$ by $Q\succ 0$ and $K_{T-1}^{\top}RK_{T-1}\succeq 0$. For any $t=1,\cdots,T-1$, assume that $P_{t+1}\succ 0$, we can prove that $\widetilde{P}_{t+1}\succ 0$, and thus $P_{t}\succ 0$ by $Q\succ 0$, $K_{t}^{\top}RK_{t}\succeq 0$ and $(A+BK_{t})^{\top}\widetilde{P}_{t+1}(A+BK_{t})\succeq 0$, which finish the mathematical induction. And by the condition $A\in\mathbb{R}^{n\times n}$ has full rank, we have

$$ $\displaystyle B\left(B^{\top}\widetilde{P}_{t+1}B+R\right)^{-1}v_{2}=0.$ $$

According to the setting in Section [2.1](#S2.SS1), $R\succ 0$, so $B^{\top}\widetilde{P}_{t+1}B+R\succ 0$. So $B\left(B^{\top}\widetilde{P}_{t+1}B+R\right)^{-1}$ has full column rank by the condition that $B\in\mathbb{R}^{n\times m}$ has full column rank, and thus $v_{2}=0$, which completes the proof.
∎

###### Lemma 4 .

###### Proof.

###### Lemma 5 .

###### Proof.

###### Lemma 6 .

###### Proof.

###### Lemma 7 .

###### Proof.

###### Proof of Proposition 3 .

###### Proposition 4 .

###### Proof.

### A.2 Perturbation Analysis of Riccati Equation

In this section, we discuss perturbation analysis of Riccati Equation, i.e., how the solutions to Riccati Equation ([2.1](#S2.Ex2)) change when we perturb the system matrices.

The main result of this section is the following lemma. We fix epoch $l$ in the analysis below and recall that $(A^{l},B^{l})$ are the estimators for the true system matrices $(A,B)$.

###### Lemma 8 .

Assume $1-\gamma\widetilde{\Gamma}>0$ and fix any $\epsilon_{l}>0$. Suppose $\|A^{l}-A\|\leq\epsilon_{l},\|B^{l}-B\|\leq\epsilon_{l}$, then for any $t=0,1,\cdots,T-1$, we have

$$ $\displaystyle\|K^{l}_{t}-K_{t}\|\leq(10\mathcal{V}^{2}\mathcal{L}\widetilde{\Gamma}^{4})^{T-t-1}\mathcal{V}\epsilon_{l},$ $\displaystyle\|P^{l}_{t}-P_{t}\|\leq(10\mathcal{V}^{2}\mathcal{L}\widetilde{\Gamma}^{4})^{T-t}\epsilon_{l},$ $$

where $\widetilde{\Gamma},\mathcal{V}$ and $\mathcal{L}$ are defined in ([15](#S3.E15)).

To prove Lemma [8](#Thmlemma8), we need the following result, which provides ‘one-step’ perturbation bounds for the solutions to Riccati equations.

###### Lemma 9 .

Assume $1-\gamma\widetilde{\Gamma}>0.$
For any $\epsilon_{l}>0,W\geq 1$, assume $\|A^{l}-A\|\leq\epsilon_{l},\|B^{l}-B\|\leq\epsilon_{l}$ and $\|P^{l}_{t+1}-P_{t+1}\|\leq W\epsilon_{l}\leq 1$ for a given $t\in\{0,\cdots,T-1\}$. Then we have

$$ $\displaystyle\|K^{l}_{t}-K_{t}\|\leq\mathcal{V}W\epsilon_{l},$ $\displaystyle\|P^{l}_{t}-P_{t}\|\leq 10\mathcal{V}^{2}\mathcal{L}\widetilde{\Gamma}^{4}W\epsilon_{l},$ $$

where $\widetilde{\Gamma},\mathcal{V}$ and $\mathcal{L}$ are given in ([15](#S3.E15)).

###### Proof.

We first bound the perturbation of the optimal controller, i.e., $\Delta K_{t}^{l}=K_{t}^{l}-K_{t}$.
Recall that

$$ $\displaystyle K_{t}=-(B^{\top}\widetilde{P}_{t+1}B+R)^{-1}B^{\top}\widetilde{P}_{t+1}A,\quad\text{and}\quad K_{t}^{l}=-\left(B^{l\top}\widetilde{P}^{l}_{t+1}B^{l}+R\right)^{-1}B^{l\top}\widetilde{P}^{l}_{t+1}A^{l}.$ (29) $$

To bound $\Delta K_{t}^{l}$, we first bound $\left\|\widetilde{P}^{l}_{t+1}-\widetilde{P}_{t+1}\right\|$ as follows:

$$ $\displaystyle\left\|\widetilde{P}^{l}_{t+1}-\widetilde{P}_{t+1}\right\|$ $\displaystyle\overset{(1)}{=}\left\|(I_{n}-\gamma P_{t+1}^{l})^{-1}P_{t+1}^{l}-(I_{n}-\gamma P_{t+1})^{-1}P_{t+1}\right\|$ $\displaystyle=\big{\|}(I_{n}-\gamma P_{t+1}^{l})^{-1}P_{t+1}^{l}-(I_{n}-\gamma P_{t+1})^{-1}P_{t+1}^{l}+(I_{n}-\gamma P_{t+1})^{-1}P_{t+1}^{l}-(I_{n}-\gamma P_{t+1})^{-1}P_{t+1}\big{\|}$ $\displaystyle\leq\left\|(I_{n}-\gamma P_{t+1}^{l})^{-1}P_{t+1}^{l}-(I_{n}-\gamma P_{t+1})^{-1}P_{t+1}^{l}\right\|+\left\|I_{n}-\gamma P_{t+1})^{-1}P_{t+1}^{l}-(I_{n}-\gamma P_{t+1})^{-1}P_{t+1}\right\|$ $\displaystyle\leq\left\|(I_{n}-\gamma P_{t+1}^{l})^{-1}-(I_{n}-\gamma P_{t+1})^{-1}\right\|\cdot\|P_{t+1}^{l}\|+\|(I_{n}-\gamma P_{t+1})^{-1}\|\cdot\|P_{t+1}^{l}-P_{t+1}\|.$ $$

Here, the equality (1) follows by the definition of $\widetilde{P}_{t+1}^{l}$ and $\widetilde{P}_{t+1}$, and the fact that

$$ $\displaystyle\widetilde{P}_{t}$ $\displaystyle=P_{t}+\gamma P_{t}(I_{n}-\gamma P_{t})^{-1}P_{t}$ $\displaystyle=\left[I_{n}+\gamma P_{t}(I-\gamma P_{t})^{-1}\right]P_{t}$ $\displaystyle=\left[(I_{n}-\gamma P_{t})(I_{n}-\gamma P_{t})^{-1}+\gamma P_{t}(I_{n}-\gamma P_{t})^{-1}\right]P_{t}$ $\displaystyle=(I_{n}-\gamma P_{t})^{-1}P_{t}.$ $$

It follows from the fact that $E^{-1}-F^{-1}=E^{-1}(F-E)F^{-1}$ for any invertible matrix $E$ and $F$,

$$ $\displaystyle\left\|\widetilde{P}^{l}_{t+1}-\widetilde{P}_{t+1}\right\|$ $\displaystyle\leq\left\|(I_{n}-\gamma P_{t+1}^{l})^{-1}\gamma(P_{t+1}^{l}-P_{t+1})(I_{n}-\gamma P_{t+1})^{-1}\right\|\cdot\|P_{t+1}^{l}\|+\|(I_{n}-\gamma P_{t+1})^{-1}\|\cdot\|P_{t+1}^{l}-P_{t+1}\|$ $\displaystyle\overset{(2)}{\leq}\frac{1}{1-\gamma\|P_{t+1}^{l}\|}\cdot\frac{1}{1-\gamma\|P_{t+1}\|}\cdot\gamma\|P_{t+1}^{l}-P_{t+1}\|\cdot\|P_{t+1}^{l}\|+\frac{1}{1-\gamma\|P_{t+1}\|}\cdot\|P_{t+1}^{l}-P_{t+1}\|$ $\displaystyle\leq\frac{1}{1-\gamma(W\epsilon_{l}+\Gamma)}\cdot\frac{1}{1-\gamma\Gamma}\cdot\gamma W\epsilon_{l}(W\epsilon_{l}+\Gamma)+\frac{1}{1-\gamma\Gamma}\cdot W\epsilon_{l}$ $\displaystyle\overset{(3)}{\leq}\frac{1}{(1-\gamma\widetilde{\Gamma})^{2}}\gamma\widetilde{\Gamma}W\epsilon_{l}+\frac{1}{1-\gamma\widetilde{\Gamma}}W\epsilon_{l}$ $\displaystyle=\left[\frac{\gamma\widetilde{\Gamma}}{(1-\gamma\widetilde{\Gamma})^{2}}+\frac{1}{1-\gamma\widetilde{\Gamma}}\right]W\epsilon_{l}$ $\displaystyle=\frac{W\epsilon_{l}}{(1-\gamma\widetilde{\Gamma})^{2}},$ $$

where the inequality (2) holds by the fact that for any matrix $E\in R^{n\times n}$, if $\|E\|<1$, then $\|(I_{n}-E)^{-1}\|\leq\frac{1}{1-\|E\|}$, and the inequality (3) holds because we assume that $\|P_{t+1}^{l}-P_{t+1}\|\leq W\epsilon_{l}\leq 1$.

To bound $\Delta K_{t}^{l}$, we next bound $\left\|B^{\top}\widetilde{P}_{t+1}B-B^{l\top}\widetilde{P}^{l}_{t+1}B^{l}\right\|$ in view of the expressions in ([29](#A1.E29)):

$$ $\displaystyle\left\|B^{\top}\widetilde{P}_{t+1}B-B^{l\top}\widetilde{P}^{l}_{t+1}B^{l}\right\|$ $\displaystyle\leq\left\|B^{\top}\widetilde{P}_{t+1}B-B^{\top}\widetilde{P}_{t+1}B^{l}\right\|+\left\|B^{\top}\widetilde{P}_{t+1}B^{l}-B^{\top}\widetilde{P}^{l}_{t+1}B^{l}\right\|+\left\|B^{\top}\widetilde{P}^{l}_{t+1}B^{l}-B^{l\top}\widetilde{P}^{l}_{t+1}B^{l}\right\|$ $\displaystyle\leq\|B^{\top}\widetilde{P}_{t+1}\|\cdot\|B-B^{l}\|+\|B\|\cdot\|\widetilde{P}_{t+1}-\widetilde{P}_{t+1}^{l}\|\cdot\|B^{l}\|+\|B-B^{l}\|\cdot\|\widetilde{P}_{t+1}^{l}B^{l}\|$ $\displaystyle\leq\epsilon_{l}\Gamma^{2}+\Gamma\mathcal{L}W\epsilon_{l}(\Gamma+\epsilon_{l})+\epsilon_{l}(\mathcal{L}W\epsilon_{l}+\Gamma)(\Gamma+\epsilon_{l})$ $\displaystyle\overset{(4)}{\leq}W\epsilon_{l}(\widetilde{\Gamma}^{2}+\widetilde{\Gamma}^{2}\mathcal{L}+(\epsilon_{l}\mathcal{L}+\Gamma)\widetilde{\Gamma})$ $\displaystyle\leq 2(\mathcal{L}+1)\widetilde{\Gamma}^{2}W\epsilon_{l},$ $$

where inequality (4) holds by the fact that $W\epsilon_{l}\leq 1$. Similarly, we can derive that

$$ $\displaystyle\left\|B^{\top}\widetilde{P}_{t+1}A-B^{l\top}\widetilde{P}^{l}_{t+1}A^{l}\right\|$ $\displaystyle\leq\left\|B^{\top}\widetilde{P}_{t+1}A-B^{\top}\widetilde{P}_{t+1}A^{l}\right\|+\left\|B^{\top}\widetilde{P}_{t+1}A^{l}-B^{\top}\widetilde{P}^{l}_{t+1}A^{l}\right\|+\left\|B^{\top}\widetilde{P}^{l}_{t+1}A^{l}-B^{l\top}\widetilde{P}^{l}_{t+1}A^{l}\right\|$ $\displaystyle\leq 2(\mathcal{L}+1)\widetilde{\Gamma}^{2}W\epsilon_{l}.$ $$

Then, following a similar argument as in Lemma 2 of Mania et al., (), we can obtain

$$ $\displaystyle\|\Delta K_{t}^{l}\|=\|K_{t}-K_{t}^{l}\|\leq 2(L+1)\widetilde{\Gamma}^{3}W\epsilon_{l}.$ $$

Next we proceed to bound $\|P^{l}_{t}-P_{t}\|$. Recall that

$$ $\displaystyle P_{t}=Q+K_{t}^{\top}RK_{t}+(A+BK_{t})^{\top}\widetilde{P}_{t+1}(A+BK_{t}),$ $\displaystyle P_{t}^{l}=Q+K_{t}^{l\top}RK_{t}^{l}+(A^{l}+B^{l}K_{t}^{l})^{\top}\widetilde{P}^{l}_{t+1}(A^{l}+B^{l}K_{t}^{l}).$ $$

We can directly compute that

$$ $\displaystyle\left\|A+BK_{t}-A^{l}-B^{l}K^{l}_{t}\right\|$ $\displaystyle\leq\left\|A-A^{l}\right\|+\left\|BK_{t}-BK^{l}_{t}\right\|+\left\|BK^{l}_{t}-B^{l}K^{l}_{t}\right\|$ $\displaystyle\leq\epsilon_{l}+\Gamma\mathcal{V}W\epsilon_{l}+\epsilon_{l}(\mathcal{V}W\epsilon_{l}+\Gamma)$ $\displaystyle\overset{(5)}{\leq}\epsilon_{l}+\Gamma\mathcal{V}W\epsilon_{l}+\widetilde{\Gamma}\mathcal{V}W\epsilon_{l}$ $\displaystyle\leq 2\mathcal{V}\widetilde{\Gamma}W\epsilon_{l},$ $$

where inequality (5) holds by the fact that $\epsilon_{l}(\mathcal{V}W\epsilon_{l}+\Gamma)\leq\mathcal{V}W\epsilon_{l}+\Gamma\epsilon_{l}W\mathcal{V}$ when both $W$ and $\mathcal{V}$ are larger than $1$. Similarly, we can derive that

$$ $\displaystyle\left\|K_{t}^{\top}RK_{t}-K_{t}^{l\top}RK^{l}_{t}\right\|$ $\displaystyle\leq\left\|K_{t}^{\top}RK_{t}-K_{t}^{\top}RK^{l}_{t}\right\|+\left\|K_{t}^{\top}RK^{l}_{t}-K_{t}^{l\top}RK^{l}_{t}\right\|$ $\displaystyle\leq\Gamma^{2}\mathcal{V}W\epsilon_{l}+\mathcal{V}W\epsilon_{l}\Gamma(\mathcal{V}W\epsilon_{l}+\Gamma)$ $\displaystyle\leq 2\mathcal{V}^{2}\widetilde{\Gamma}^{2}W\epsilon_{l}.$ $$

In addition, we can derive that

$$ $\displaystyle\left\|(A+BK_{t})^{\top}\widetilde{P}_{t+1}(A+BK_{t})-(A+BK^{l}_{t})^{\top}\widetilde{P}^{l}_{t+1}(A+BK^{l}_{t})\right\|$ $\displaystyle\leq\left\|(A+BK_{t})^{\top}\widetilde{P}_{t+1}(A+BK_{t})-(A+BK^{l}_{t})^{\top}\widetilde{P}_{t+1}(A+BK_{t})\right\|$ $\displaystyle\quad+\left\|(A+BK_{t}^{l})^{\top}\widetilde{P}_{t+1}(A+BK_{t})-(A+BK^{l}_{t})^{\top}\widetilde{P}^{l}_{t+1}(A+BK_{t})\right\|$ $\displaystyle\quad+\left\|(A+BK^{l}_{t})^{\top}\widetilde{P}^{l}_{t+1}(A+BK_{t})-(A+BK^{l}_{t})^{\top}\widetilde{P}^{l}_{t+1}(A+BK^{l}_{t})\right\|$ $\displaystyle\leq 2\mathcal{V}\widetilde{\Gamma}^{4}W\epsilon_{l}+2\mathcal{L}\mathcal{V}\widetilde{\Gamma}^{4}W\epsilon_{l}+4\mathcal{V}^{2}\mathcal{L}\widetilde{\Gamma}^{4}W\epsilon_{l}$ $\displaystyle\leq 8\mathcal{V}^{2}\mathcal{L}\widetilde{\Gamma}^{4}W\epsilon_{l}.$ $$

It then follows that

$$ $\displaystyle\|P^{l}_{t}-P_{t}\|\leq 10\mathcal{V}^{2}\mathcal{L}\widetilde{\Gamma}^{4}W\epsilon_{l}.$ $$

The proof is therefore complete.
∎

With Lemma [9](#Thmlemma9), we are now ready to prove Lemma [8](#Thmlemma8).

###### Proof of Lemma 8 .

By definition we know that $P_{T}^{l}=P_{T}=Q_{T}$, and thus we have $\|P_{T}^{l}-P_{T}\|\leq\epsilon_{l}$. By Lemma [9](#Thmlemma9), we can derive that at time $T-1$,

$$ $\displaystyle\|K_{T-1}^{l}-K_{T-1}\|\leq\mathcal{V}\epsilon_{l},$ $\displaystyle\|P_{T-1}^{l}-P_{T-1}\|\leq(10\mathcal{V}^{2}\mathcal{L}\widetilde{\Gamma}^{4})\epsilon_{l},$ $$

which implies that

$$ $\displaystyle\|K_{T-2}^{l}-K_{T-2}\|\leq(10\mathcal{V}^{2}\mathcal{L}\widetilde{\Gamma}^{4})\mathcal{V}\epsilon_{l},$ $\displaystyle\|P_{T-2}^{l}-P_{T-2}\|\leq(10\mathcal{V}^{2}\mathcal{L}\widetilde{\Gamma}^{4})^{2}\epsilon_{l}.$ $$

Applying Lemma [9](#Thmlemma9) recursively, we obtain for any $t=0,\cdots,T-1$.

$$ $\displaystyle\|K_{t}^{l}-K_{t}\|\leq(10\mathcal{V}^{2}\mathcal{L}\widetilde{\Gamma}^{4})^{T-t-1}\mathcal{V}\epsilon_{l},$ $\displaystyle\|P_{t}^{l}-P_{t}\|\leq(10\mathcal{V}^{2}\mathcal{L}\widetilde{\Gamma}^{4})^{T-t}\epsilon_{l},$ $$

which completes the proof.
∎

###### Lemma 8 .

###### Lemma 9 .

###### Proof.

###### Proof of Lemma 8 .

### A.3 Suboptimality Gap Due to the Controller Mismatch

In this section, we will simplify the performance gap between the total cost under policy $\pi^{l,k}$ and the total cost under the optimal policy. We recall the corresponding total cost under entropic risk,

$$ $\displaystyle J_{0}^{\pi^{l,k}}\left(x^{l,k}_{0}\right)=\frac{1}{\gamma}\log\mathbb{E}\exp\left(\frac{\gamma}{2}\left(\sum_{t=0}^{T-1}\left(x_{t}^{l,k\top}Qx^{l,k}_{t}+u_{t}^{l,k\top}Ru^{l,k}_{t}\right)+x_{T}^{l,k\top}Q_{T}x_{T}^{l,k}\right)\right),$ $$

where $u_{t}^{l,k}=K_{t}^{l}x_{t}^{l,k}$, and $K_{t}^{l}$ is obtained by substituting $(A^{l},B^{l})$ into ([2.1](#S2.Ex2)).

Let $\mathbb{H}_{t}^{l,k}$ be the set of possible histories up to the $t$-th step in the $k$-th episode of epoch $l$. Then, one sample of the history up to the $t$-th step in the $k$-th episode of epoch $l$ is

$$ $\displaystyle H_{t}^{l,k}=\left(x_{0}^{1,1},u_{0}^{1,1},\cdots,x_{T}^{1,1},x_{0}^{1,2},\cdots,x_{0}^{2,1},\cdots,x_{T}^{2,1},\cdots,x_{0}^{l,k},\cdots,x_{t}^{l,k},u_{t}^{l,k}\right).$ $$

We also introduce some new notations, which will be heavily used in the regret analysis. For any $t=1,\cdots,T-2$, we define the following recursive equations:

$$ $\displaystyle D_{T-1}^{l}=\Delta K_{T-1}^{l\top}(R+B^{\top}\widetilde{P}_{T}B)\Delta K_{T-1}^{l},$ (30) $\displaystyle\widetilde{D}_{T-1}^{l}=(I_{n}-\gamma D_{T-1}^{l})^{-1}D_{T-1}^{l},$ $\displaystyle D_{t}^{l}=\Delta K_{t}^{l\top}\left(R+B^{\top}\widetilde{P}_{t+1}B\right)\Delta K_{t}^{l}+(A+BK_{t}^{l})^{\top}\widetilde{D}_{t+1}^{l}(A+BK_{t}^{l}),$ $\displaystyle\widetilde{D}_{t}^{l}=(I_{n}-\gamma D_{t}^{l})^{-1}D_{t}^{l},$ $\displaystyle D_{0}^{l}=\Delta K_{0}^{l\top}\left(R+B^{\top}\widetilde{P}_{1}B\right)\Delta K_{0}^{l}+(A+BK_{0}^{l})^{\top}\widetilde{D}_{1}^{l}(A+BK_{0}^{l}),$ $$

where $\Delta K_{t}^{l}=K_{t}^{l}-K_{t}$ and $\widetilde{P}_{T}$ is defined in ([2.1](#S2.Ex2)). In the following parts, we still consider the risk-averse setting, where $\gamma>0$. The following proposition is the key result of this section.

###### Proposition 5 .

We can simplify the performance gap in the $k$-th episode of epoch $l$ to

$$ $\displaystyle J_{0}^{\pi^{l,k}}(x_{0}^{l,k})-J^{\star}_{0}(x_{0}^{l,k})=-\frac{1}{2\gamma}\sum_{t=1}^{T-1}\log\left(\det\left(I_{n}-\gamma D_{t}^{l}\right)\right)+\frac{1}{2}x_{0}^{l,k\top}D_{0}^{l}x_{0}^{l,k}.$ (31) $$

where $D_{t}^{l}$ is defined in ([30](#A1.E30)).

In order to prove Proposition [5](#Thmproposition5), we introduce Lemma [10](#Thmlemma10), see p.8 of Jacobson, ().

###### Lemma 10 ( Jacobson, ( 1973 ) ) .

Consider the linear dynamic system $x_{t+1}=Ax_{t}+Bu_{t}+w_{t},w_{t}\sim\mathcal{N}(0,I_{n}),t=0,\cdots,T-1$. For any sequence of positive semidefinite matrix $E_{t+1}$ satisfying $I_{n}-\gamma E_{t+1}\succ 0$, we have

$$ $\displaystyle\mathbb{E}\left[\exp\left(\frac{\gamma}{2}x_{t+1}^{\top}E_{t+1}x_{t+1}\right)\Big{|}x_{t},u_{t}\right]$ $\displaystyle=\left(\det(I_{n}-\gamma E_{t+1})\right)^{-\frac{1}{2}}\exp\left(\frac{\gamma}{2}(Ax_{t}+Bu_{t})^{\top}\widetilde{E}_{t+1}(Ax_{t}+Bu_{t})\right),$ $$

where $\widetilde{E}_{t+1}=E_{t+1}+\gamma E_{t+1}(I_{n}-\gamma E_{t+1})^{-1}E_{t+1}$.

We apply Lemma [10](#Thmlemma10) to simplify the performance gap $J_{0}^{\pi^{l,k}}(x^{l,k}_{0})-J_{0}^{\star}(x^{l,k}_{0})$ in the $k$-th episode of epoch $l$ in the following lemma.

###### Lemma 11 .

We can simplify the performance gap as

$$ $\displaystyle J_{0}^{\pi^{l,k}}(x^{l,k}_{0})-J_{0}^{\star}(x^{l,k}_{0})=\frac{1}{\gamma}\log\mathbb{E}\Bigg{[}\exp\bigg{(}\frac{\gamma}{2}\sum_{t=0}^{T-1}x_{t}^{l,k\top}\Delta K_{t}^{l\top}(R+B^{\top}\widetilde{P}_{t+1}B)\Delta K_{t}^{l}x_{t}^{l,k}\bigg{)}\Bigg{|}x_{0}^{l,k},H_{T}^{l,k-1}\Bigg{]},$ (32) $$

where $\Delta K_{t}^{l}=K^{l}_{t}-K_{t}$.

###### Proof.

Denote $J_{t}(x_{t}^{l,k})=\frac{1}{2}\left(x_{t}^{l,k\top}P_{t}x^{l,k}_{t}-\sum_{i=t}^{T-1}\frac{1}{\gamma}\log\det\left(I-\gamma P_{t+1}\right)\right),t=0,\cdots,T-1$, which is the dynamic programming equations of LEQR problem. When $t=T$, $J_{T}(x_{T}^{l,k})=x_{T}^{l,k\top}Q_{T}x_{T}^{l,k}$.

By the definition of $J_{0}^{\pi^{l,k}}(x^{l,k}_{0})$ and $J_{0}^{\star}(x^{l,k}_{0})$, we have

$$ $\displaystyle J_{0}^{\pi^{l,k}}(x^{l,k}_{0})-J_{0}^{\star}(x^{l,k}_{0})$ (33) $\displaystyle=\frac{1}{\gamma}\log\mathbb{E}\left[\exp\left(\frac{\gamma}{2}\left(\sum_{t=0}^{T-1}\left(x_{t}^{l,k\top}Qx^{l,k}_{t}+u_{t}^{l,k\top}Ru^{l,k}_{t}\right)+x_{T}^{l,k\top}Q_{T}x^{l,k}_{T}\right)\right)\Bigg{|}x_{0}^{l,k},H_{T}^{l,k-1}\right]-J_{0}(x^{l,k}_{0})$ $\displaystyle=\frac{1}{\gamma}\log\mathbb{E}\Bigg{[}\exp\Bigg{(}\frac{\gamma}{2}\Bigg{(}\sum_{t=0}^{T-1}\left(\left(x_{t}^{l,k\top}Qx^{l,k}_{t}+u_{t}^{l,k\top}Ru^{l,k}_{t}\right)+J_{t}(x^{l,k}_{t})-J_{t}(x^{l,k}_{t})\right)$ $\displaystyle\qquad\qquad\qquad\quad+x_{T}^{l,k\top}Q_{T}x^{l,k}_{T}\Bigg{)}\Bigg{)}\Bigg{|}x_{0}^{l,k},H_{T}^{l,k-1}\Bigg{]}-J_{0}(x^{l,k}_{0}).$ $$

Recall that $J_{T}(x_{T}^{l,k})=x_{T}^{l,k\top}Q_{T}x_{T}^{l,k}$, we have

$$ $\displaystyle J_{0}^{\pi^{l,k}}(x^{l,k}_{0})-J_{0}^{\star}(x^{l,k}_{0})$ $\displaystyle=\frac{1}{\gamma}\log\mathbb{E}\Bigg{[}\exp\Bigg{(}\frac{\gamma}{2}\Bigg{(}\sum_{t=0}^{T-1}\left(\left(x_{t}^{l,k\top}Qx^{l,k}_{t}+u_{t}^{l,k\top}Ru^{l,k}_{t}\right)+J_{t}(x^{l,k}_{t})-J_{t}(x^{l,k}_{t})\right)$ $\displaystyle\qquad\qquad\qquad\quad+J_{T}(x_{T}^{l,k})\Bigg{)}\Bigg{)}\Bigg{|}x_{0}^{l,k},H_{T}^{l,k-1}\Bigg{]}-\frac{1}{\gamma}\log\left(\exp(J_{0}(x_{0}^{l,k}))\right)$ $\displaystyle\overset{(1)}{=}\frac{1}{\gamma}\log\mathbb{E}\left[\exp\left(\frac{\gamma}{2}\sum_{t=0}^{T-1}\left(\left(x_{t}^{l,k\top}Qx^{l,k}_{t}+u_{t}^{l,k\top}Ru^{l,k}_{t}\right)+J_{t+1}(x^{l,k}_{t+1})-J_{t}(x^{l,k}_{t})\right)\right)\Bigg{|}x_{0}^{l,k},H_{T}^{l,k-1}\right]$ $\displaystyle\overset{(2)}{=}\frac{1}{\gamma}\log\mathbb{E}\Bigg{[}\exp\Bigg{(}\gamma\sum_{t=0}^{T-1}\Bigg{(}\frac{1}{2}x_{t}^{l,k\top}(Q+K_{t}^{l\top}RK^{l}_{t})x^{l,k}_{t}+\frac{1}{2}x^{l,k\top}_{t+1}P_{t+1}x^{l,k}_{t+1}-\frac{1}{2}x^{l,k\top}_{t}P_{t}x^{l,k}_{t}$ $\displaystyle\qquad\qquad\qquad\quad+\frac{1}{2\gamma}\log\det(I_{n}-\gamma P_{t+1})\Bigg{)}\Bigg{)}\Bigg{|}x_{0}^{l,k},H_{T}^{l,k-1}\Bigg{]},$ $$

where equality (1) holds by canceling out the $J_{0}(x_{0}^{l,k})$ inside and outside the entropic risk, and equality (2) follows from the definition of the total cost under entropic risk and $u_{t}^{l,k}=K_{t}^{l}x_{t}^{l,k}$. By the law of total expectation, i.e. $\mathbb{E}[X|Z]=\mathbb{E}[\mathbb{E}[X|Y,Z]|Z]$ for any random variables $X,Y,Z$, we consider the conditional expectation

$$ $\displaystyle\mathbb{E}\Bigg{[}\exp\Bigg{(}\gamma\Big{(}\frac{1}{2}x_{t}^{l,k\top}(Q+K_{t}^{l\top}RK^{l}_{t})x^{l,k}_{t}+\frac{1}{2}x^{l,k\top}_{t+1}P_{t+1}x^{l,k}_{t+1}-\frac{1}{2}x^{l,k\top}_{t}P_{t}x^{l,k}_{t}$ (34) $\displaystyle\qquad\quad+\frac{1}{2\gamma}\log\det(I_{n}-\gamma P_{t+1})\Big{)}\Bigg{)}\Bigg{|}H_{t}^{l,k}\Bigg{]}$ $\displaystyle=\exp\Bigg{(}\gamma\Bigg{(}\frac{1}{2}x_{t}^{l,k\top}(Q+K_{t}^{l\top}RK^{l}_{t})x^{l,k}_{t}-\frac{1}{2}x^{l,k\top}_{t}P_{t}x^{l,k}_{t}+\frac{1}{2\gamma}\log\det(I_{n}-\gamma P_{t+1})\Bigg{)}\Bigg{)}$ $\displaystyle\quad\times\mathbb{E}\left[\exp\left(\frac{\gamma}{2}x_{t+1}^{l,k\top}P_{t+1}x^{l,k}_{t+1}\right)\Bigg{|}H_{t}^{l,k}\right]$ $\displaystyle\overset{(3)}{=}\exp\Bigg{(}\gamma\Bigg{(}\frac{1}{2}x_{t}^{l,k\top}(Q+K_{t}^{l\top}RK^{l}_{t})x^{l,k}_{t}-\frac{1}{2}x^{l,k\top}_{t}P_{t}x^{l,k}_{t}+\frac{1}{2\gamma}\log\det(I_{n}-\gamma P_{t+1})\Bigg{)}\Bigg{)}$ $\displaystyle\quad\times(\det(I_{n}-\gamma P_{t+1}))^{-1/2}\exp\Bigg{[}\frac{\gamma}{2}\Bigg{(}x_{t}^{l,k\top}(A+BK^{l}_{t})^{\top}\widetilde{P}_{t+1}(A+BK^{l}_{t})x^{l,k}_{t}\Bigg{)}\Bigg{]}$ $\displaystyle=\exp\left[\frac{\gamma}{2}\Big{(}x_{t}^{l,k\top}(Q+K_{t}^{l\top}RK^{l}_{t}+(A+BK^{l}_{t})^{\top}\widetilde{P}_{t+1}(A+BK^{l}_{t}))x^{l,k}_{t}-x_{t}^{l,k\top}P_{t}x^{l,k}_{t}\Big{)}\right],$ $$

where the equality (3) follows from Lemma [10](#Thmlemma10).

Recall that $\Delta K_{t}^{l}=K_{t}^{l}-K_{t}$ and $P_{t}=Q+K_{t}^{\top}RK_{t}+(A+BK_{t})^{\top}\widetilde{P}_{t+1}(A+BK_{t})$. Then the RHS of Equation ([34](#A1.E34)) becomes

$$ $\displaystyle\exp\Big{[}\frac{\gamma}{2}x_{t}^{l,k\top}\Big{(}Q+(\Delta K_{t}^{l}+K_{t})^{\top}R(\Delta K_{t}^{l}+K_{t})$ (35) $\displaystyle\qquad+(A+B(\Delta K_{t}^{l}+K_{t}))^{\top}\widetilde{P}_{t+1}(A+B(\Delta K_{t}^{l}+K_{t}))\Big{)}x_{t}^{l,k}-\frac{\gamma}{2}x_{t}^{l,k\top}P_{t}x_{t}^{l,k}\Big{]}$ $\displaystyle=\exp\Big{[}\frac{\gamma}{2}x_{t}^{l,k\top}\Delta K_{t}^{l\top}(R+B^{\top}\widetilde{P}_{t+1}B)\Delta K_{t}^{l}x_{t}^{l,k}+\gamma x_{t}^{l,k\top}\Delta K_{t}^{l\top}\big{(}(R+B^{\top}\widetilde{P}_{t+1}B)K_{t}+B^{\top}\widetilde{P}_{t+1}A\big{)}x_{t}^{l,k}\Big{]}$ $\displaystyle\overset{(4)}{=}\exp\Big{[}\frac{\gamma}{2}x_{t}^{l,k\top}\Delta K_{t}^{l\top}(R+B^{\top}\widetilde{P}_{t+1}B)\Delta K_{t}^{l}x_{t}^{l,k}\Big{]},$ $$

where the equality (4) holds by the fact that $K_{t}=-(R+B^{\top}\widetilde{P}_{t+1}B)^{-1}B^{\top}\widetilde{P}_{t+1}A$. Finally, substituting ([35](#A1.E35)) into ([34](#A1.E34)) and then substituting ([34](#A1.E34)) into ([33](#A1.E33)), we can get ([32](#A1.E32))
∎

With Lemma [11](#Thmlemma11), we are now ready to prove Proposition [5](#Thmproposition5).

###### Proof of Proposition 5 .

We prove the result recursively. When $t=T-1$, we have

$$ $\displaystyle\mathbb{E}\left[\exp\left(\frac{\gamma}{2}x_{T-1}^{l,k\top}\Delta K_{T-1}^{l\top}(R+B^{\top}\widetilde{P}_{T}B)\Delta K_{T-1}^{l}x_{T-1}^{l,k}\right)\bigg{|}H_{T-2}^{l,k}\right]$ $\displaystyle=\mathbb{E}\left[\exp\left(\frac{\gamma}{2}x_{T-1}^{l,k\top}D_{T-1}^{l}x_{T-1}^{l,k}\right)\bigg{|}H_{T-2}^{l,k}\right]$ $\displaystyle\overset{(1)}{=}\left(\det(I_{n}-\gamma D_{T-1}^{l})\right)^{-\frac{1}{2}}\exp\left\{\frac{\gamma}{2}\Big{[}x_{T-2}^{l,k\top}(A+BK_{T-2}^{l})^{\top}\widetilde{D}_{T-1}^{l}(A+BK_{T-2}^{l})x_{T-2}^{l,k}\Big{]}\right\},$ $$

where equality (1) follows from Lemma [10](#Thmlemma10) and $u_{T-1}^{l,k}=K_{T-1}^{l}x_{T-1}^{l,k}$. When $t=T-2$, we have

$$ $\displaystyle\mathbb{E}\left[\exp\bigg{(}\frac{\gamma}{2}\sum_{t=T-2}^{T-1}\Big{(}x_{t}^{l,k\top}\Delta K_{t}^{l\top}(R+B^{\top}\widetilde{P}_{t+1}B)\Delta K_{t}^{l}x_{t}^{l,k}\Big{)}\bigg{)}\bigg{|}H_{T-3}^{l,k}\right]$ $\displaystyle=\left(\det(I_{n}-\gamma D_{T-1}^{l})\right)^{-\frac{1}{2}}\mathbb{E}\bigg{[}\exp\bigg{(}\frac{\gamma}{2}\Big{[}x_{T-2}^{l,k\top}\big{(}\Delta K_{T-2}^{l\top}(R+B^{\top}\widetilde{P}_{T-1}B)\Delta K_{T-2}^{l}$ $\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad+(A+BK_{T-2}^{l})^{\top}\widetilde{D}_{T-1}^{l}(A+BK_{T-2}^{l})\big{)}x_{T-2}^{l,k}\Big{]}\bigg{)}\bigg{|}H_{T-3}^{l,k}\bigg{]}$ $\displaystyle=\left(\det(I_{n}-\gamma D_{T-1}^{l})\right)^{-\frac{1}{2}}\mathbb{E}\left[\exp\left(\frac{\gamma}{2}x_{T-2}^{l,k\top}D_{T-2}^{l}x_{T-2}^{l,k}\right)\Big{|}H_{T-3}^{l,k}\right]$ $\displaystyle=\prod_{t=T-2}^{T-1}\left(\det(I_{n}-\gamma D_{t}^{l})\right)^{-\frac{1}{2}}\exp\left(\frac{\gamma}{2}\left[x_{T-3}^{l,k\top}(A+BK_{T-3}^{l})^{\top}\widetilde{D}_{T-2}^{l}(A+BK_{T-3}^{l})x_{T-3}^{l,k}\right]\right).$ $$

When $t=i,i=1,\cdots,T-1$, similarly, we have

$$ $\displaystyle\mathbb{E}\Bigg{[}\exp\bigg{(}\frac{\gamma}{2}\sum_{t=i}^{T-1}\Big{(}x_{t}^{l,k\top}\Delta K_{t}^{l\top}(R+B^{\top}\widetilde{P}_{t+1}B)\Delta K_{t}^{l}x_{t}^{l,k}\Big{)}\bigg{)}\Bigg{|}H_{i-1}^{l,k}\Bigg{]}$ $\displaystyle=\prod_{t=i}^{T-1}\left(\det(I_{n}-\gamma D_{t}^{l})\right)^{-\frac{1}{2}}\exp\Bigg{(}\frac{\gamma}{2}\Big{[}x_{i-1}^{l,k\top}(A+BK_{i-1}^{l})^{\top}\widetilde{D}_{i}^{l}(A+BK_{i-1}^{l})x_{i-1}^{l,k}\Big{]}\Bigg{)}.$ $$

Repeating this procedure, we get

$$ $\displaystyle J_{0}^{\pi^{l,k}}(x_{0}^{l,k})-J^{\star}_{0}(x_{0}^{l,k})$ $\displaystyle=\frac{1}{\gamma}\log\mathbb{E}\Bigg{[}\prod_{t=1}^{T-1}\left(\det(I_{n}-\gamma D_{t}^{l})\right)^{-\frac{1}{2}}$ $\displaystyle\quad\times\exp\Bigg{(}\frac{\gamma}{2}\Big{[}x_{0}^{l,k\top}\left((A+BK_{0}^{l})^{\top}\widetilde{D}_{1}^{l}(A+BK_{0}^{l})+\Delta K_{0}^{l\top}(R+B^{\top}\widetilde{P}_{1}B)\Delta K_{0}^{l}\right)x_{0}^{l,k}\Big{]}\Bigg{)}\Bigg{|}x_{0}^{l,k},H_{T}^{l,k-1}\Bigg{]}$ $\displaystyle=\frac{1}{\gamma}\log\mathbb{E}\left[\prod_{t=1}^{T-1}\left(\det(I_{n}-\gamma D_{t}^{l})\right)^{-\frac{1}{2}}\exp\left(\frac{\gamma}{2}x_{0}^{l,k\top}D_{0}^{l}x_{0}^{l,k}\right)\bigg{|}x_{0}^{l,k},H_{T}^{l,k-1}\right]$ $\displaystyle\overset{(2)}{=}\frac{1}{\gamma}\log\left(\prod_{t=1}^{T-1}\left(\det(I_{n}-\gamma D_{t}^{l})\right)^{-\frac{1}{2}}\exp\left(\frac{\gamma}{2}x_{0}^{l,k\top}D_{0}^{l}x_{0}^{l,k}\right)\right)$ $\displaystyle=-\frac{1}{2\gamma}\sum_{t=1}^{T-1}\log\left(\det\left(I_{n}-\gamma D_{t}^{l}\right)\right)+\frac{1}{2}x_{0}^{l,k\top}D_{0}^{l}x_{0}^{l,k},$ $$

where inequality (2) holds because $D_{t}^{l},t=0,\cdots,T-1$ is based on the data from epoch $1$ to epoch $l-1$.
∎

###### Proposition 5 .

###### Lemma 10 ( Jacobson, ( 1973 ) ) .

###### Lemma 11 .

###### Proof.

###### Proof of Proposition 5 .

### A.4 Proof of Theorem 1

Now, we can derive the regret upper bound for Algorithm [1](#alg1). Before we derive the high probability bounds for ([31](#A1.E31)), we introduce some new notations and provide the bounds for $D_{t}^{l}$ in ([30](#A1.E30)). Recall that

$$ $\displaystyle\psi_{T-1}=2\widetilde{\Gamma}^{3},$ (36) $\displaystyle\psi_{t}=2\widetilde{\Gamma}^{3}(10\mathcal{V}^{2}\mathcal{L}\widetilde{\Gamma}^{4})^{2(T-t-1)}+12\widetilde{\Gamma}^{4}\psi_{t+1},\ t=0,\cdots,T-2,$ $$

where the definitions of $\mathcal{V}$ and $\mathcal{L}$ are given in ([15](#S3.E15)).
Assume that for any $t=1,\cdots,T-1,\ l\in[L],$ we have

$$ $\displaystyle\gamma\leq\frac{1}{2\psi_{t}\mathcal{V}^{2}\epsilon_{l}^{2}}.$ (37) $$

We can choose a proper constant $\mathcal{C}_{0}$ for the initial epoch size $m_{1}$ in Theorem [1](#Thmtheorem1) so that $\gamma$ can satisfy assumptions in ([37](#A1.E37)) when it satisfies the assumption of $I_{n}-\gamma P_{t+1}\succ 0$ and $I_{n}-\gamma P^{l}_{t+1}\succ 0$ in ([2.1](#S2.Ex2)). Because $D_{t}^{l}$ are defined recursively, we obtain the bounds recursively from step $T-1$ to step $1$. At step $T-1$,

$$ $\displaystyle\left\|D_{T-1}^{l}\right\|$ $\displaystyle=\left\|\Delta K_{T-1}^{l\top}\left(R+B^{\top}\widetilde{P}_{T}B\right)\Delta K_{T-1}^{l}\right\|$ $\displaystyle\leq\left\|R+B^{\top}\widetilde{P}_{T}B\right\|\cdot\left\|\Delta K_{T-1}^{l}\right\|^{2}$ $\displaystyle\overset{(1)}{\leq}2\widetilde{\Gamma}^{3}\mathcal{V}^{2}\epsilon_{l}^{2}$ $\displaystyle=\psi_{T-1}\mathcal{V}^{2}\epsilon_{l}^{2},$ $$

where inequality (1) follows from the definition of $\widetilde{\Gamma}$ in ([15](#S3.E15)) and Lemma [8](#Thmlemma8). In terms of the bound for $\widetilde{D}_{T-1}^{l}$, we have

$$ $\displaystyle\left\|\widetilde{D}_{T-1}^{l}\right\|$ $\displaystyle=\left\|\left(I_{n}-\gamma D_{T-1}^{l}\right)^{-1}D_{T-1}\right\|$ $\displaystyle\leq\left\|\left(I_{n}-\gamma D_{T-1}^{l}\right)^{-1}\right\|\cdot\left\|D_{T-1}^{l}\right\|$ $\displaystyle\overset{(2)}{\leq}\frac{\|D_{T-1}^{l}\|}{1-\gamma\|D_{T-1}^{l}\|}$ $\displaystyle\overset{(3)}{\leq}2\|D_{T-1}^{l}\|$ $\displaystyle=2\psi_{T-1}\mathcal{V}^{2}\epsilon_{l}^{2},$ $$

where inequality (2) holds by the fact that for any matrix $M\in R^{n\times n}$, if $\|M\|<1$, then $\|(I_{n}-M)^{-1}\|\leq\frac{1}{1-\|M\|}$ and inequality (3) follows from the assumption in ([37](#A1.E37)). At step $T-2$, we have

$$ $\displaystyle\left\|D_{T-2}^{l}\right\|$ $\displaystyle=\left\|\Delta K_{T-2}^{l\top}\left(R+B^{\top}\widetilde{P}_{T-1}B\right)\Delta K_{T-2}^{l}+\left(A+BK_{T-2}^{l}\right)^{\top}\widetilde{D}_{T-1}^{l}\left(A+BK_{T-2}^{l}\right)\right\|$ $\displaystyle\leq 2\widetilde{\Gamma}^{3}\left(10\mathcal{V}^{2}\mathcal{L}\widetilde{\Gamma}^{4}\right)^{2}\mathcal{V}^{2}\epsilon_{l}^{2}+\left\|A+B(\Delta K_{T-2}^{l}+K_{T-2})\right\|^{2}\cdot\left\|\widetilde{D}_{T-1}^{l}\right\|$ $\displaystyle\overset{(4)}{\leq}2\widetilde{\Gamma}^{3}\left(10\mathcal{V}^{2}\mathcal{L}\widetilde{\Gamma}^{4}\right)^{2}\mathcal{V}^{2}\epsilon_{l}^{2}+\left(6\widetilde{\Gamma}^{4}+3\widetilde{\Gamma}^{2}\left(10\mathcal{V}^{2}\mathcal{L}\widetilde{\Gamma}^{4}\right)^{2}\mathcal{V}^{2}\epsilon_{l}^{2}\right)\cdot 2\psi_{T-1}\mathcal{V}^{2}\epsilon_{l}^{2}$ $\displaystyle=\left(2\widetilde{\Gamma}^{3}\left(10\mathcal{V}^{2}\mathcal{L}\widetilde{\Gamma}^{4}\right)^{2}+12\widetilde{\Gamma}^{4}\psi_{T-1}\right)\mathcal{V}^{2}\epsilon_{l}^{2}+o(\epsilon_{l}^{2})$ $\displaystyle=\psi_{T-2}\mathcal{V}^{2}\epsilon_{l}^{2}+o(\epsilon_{l}^{2}),$ $$

where inequality (4) follows from the fact that $\left\|\sum_{t=1}^{K}x_{t}\right\|^{2}\leq K\sum_{t=1}^{K}\left\|x_{t}\right\|^{2}$. Similarly, we have

$$ $\displaystyle\left\|\widetilde{D}_{T-2}^{l}\right\|\leq 2\psi_{T-2}\mathcal{V}^{2}\epsilon_{l}^{2}+o(\epsilon_{l}^{2}).$ $$

For $t=T-2,\cdots,1$, we can recursively derive that

$$ $\displaystyle\|D_{t}^{l}\|\leq\psi_{t}\mathcal{V}^{2}\epsilon_{l}^{2}+o(\epsilon_{l}^{2}),$ $\displaystyle\left\|\widetilde{D}_{t}^{l}\right\|\leq 2\psi_{t}\mathcal{V}^{2}\epsilon_{l}^{2}+o(\epsilon_{l}^{2}),$ $\displaystyle\|D_{0}^{l}\|\leq\psi_{0}\mathcal{V}^{2}\epsilon_{l}^{2}+o(\epsilon_{l}^{2}).$ (38) $$

According to Lemma [5](#Thmproposition5), the performance loss in the $k$-th episode of epoch $l$ is

$$ $\displaystyle J_{0}^{\pi^{l,k}}(x_{0}^{l,k})-J^{\star}_{0}(x_{0}^{l,k})$ (39) $\displaystyle=-\frac{1}{2\gamma}\sum_{t=1}^{T-1}\log\left(\det\left(I_{n}-\gamma D_{t}^{l}\right)\right)+\frac{1}{2}x_{0}^{l,k\top}D_{0}^{l}x_{0}^{l,k}$ $\displaystyle\overset{(5)}{\leq}-\frac{1}{2\gamma}\sum_{t=1}^{T-1}\log\left(1-\gamma\|D_{t}^{l}\|\right)^{n}+\frac{1}{2}\|x_{0}^{l,k}\|^{2}\|D_{0}^{l}\|.$ $$

Here, inequality (5) holds because $I_{n}-\gamma D_{t}^{l}\succeq\left(1-\gamma\|D_{t}^{l}\|\right)I_{n}\succeq(1-\gamma(\psi_{t}\mathcal{V}^{2}\epsilon_{l}^{2}+o(\epsilon_{l}^{2})))I_{n}\succ 0$ and $\det\left(\left(1-\gamma\|D_{t}^{l}\|\right)I_{n}\right)=\left(1-\gamma\|D_{t}^{l}\|\right)^{n}$.
Substituting the inequalities in ([A.4](#A1.Ex146)) into ([39](#A1.E39)), we obtain

$$ $\displaystyle J_{0}^{\pi^{l,k}}(x_{0}^{l,k})-J^{\star}_{0}(x_{0}^{l,k})$ $\displaystyle\overset{(6)}{\leq}-\frac{n}{2\gamma}\sum_{t=1}^{T-1}\log\left(1-\gamma\left(\psi_{t}\mathcal{V}^{2}\epsilon_{l}^{2}+o\left(\epsilon_{l}^{2}\right)\right)\right)+\frac{1}{2}\|x_{0}\|^{2}\left(\psi_{0}\mathcal{V}^{2}\epsilon_{l}^{2}+o\left(\epsilon_{l}^{2}\right)\right)$ $\displaystyle\overset{(7)}{\leq}\frac{n}{2}\left(\sum_{t=1}^{T-1}\psi_{t}\mathcal{V}^{2}\epsilon_{l}^{2}+o\left(\epsilon_{l}^{2}\right)\right)+\frac{1}{2}\|x_{0}\|^{2}\left(\psi_{0}\mathcal{V}^{2}\epsilon_{l}^{2}+o\left(\epsilon_{l}^{2}\right)\right)$ $\displaystyle=\frac{n}{2}\sum_{t=1}^{T-1}\psi_{t}\mathcal{V}^{2}\epsilon_{l}^{2}+\frac{1}{2}\|x_{0}\|^{2}\psi_{0}\mathcal{V}^{2}\epsilon_{l}^{2}+o(\epsilon_{l}^{2}),$ $$

inequality (6) holds by the inequalities in ([A.4](#A1.Ex146)), and inequality (7) follows from the fact that $\log(1+y)\leq y$ for any $y>-1$.

Now, we can substitute the high probability bounds derived in Section [A.1](#A1.SS1) into ([39](#A1.E39)). Recall that conditional on event $\mathcal{G}^{l-1}$ in Lemma [3](#Thmproposition3), with probability at least $1-2\delta_{l-1}$, we have

$$ $\displaystyle\left\|\theta^{l}-\theta\right\|\leq\epsilon_{l}:=\mathcal{C}_{1}\left(\sqrt{\frac{\log\left(\frac{(m+n)^{2}}{\delta_{l-1}}\right)}{m_{l-1}}}+\frac{\log\left(\frac{(m+n)^{2}}{\delta_{l-1}}\right)}{m_{l-1}}+\frac{\log^{2}\left(\frac{(m+n)^{2}}{\delta_{l-1}}\right)}{m_{l-1}^{2}}\right).$ $$

Similar to the procedure in page 26 in Basei et al., (), we set $\delta_{l-1}=\frac{\delta}{(l-1)^{2}},\ m_{l-1}=2^{l-2}m_{1},\ m_{1}=\mathcal{C}_{0}(-\log\delta)$, where $\delta\in(0,\frac{3}{\pi^{2}})$ and $\mathcal{C}_{0}$ is a finite positive constant that satisfies

$$ $\displaystyle\mathcal{C}_{0}\geq\mathcal{C}_{3}\sup_{l\in\mathbb{N}^{+}\backslash\{1\},\delta\in\left(0,\frac{2}{\pi^{2}}\right)}\left\{\left\{\frac{\log\left(\frac{(m+n)^{2}}{\delta_{l-1}}\right)}{2^{l-2}(-\log\delta)}\right\}\Bigg{/}\min\left\{\left(\frac{\rho}{3\mathcal{C}_{1}}\right)^{2},1\right\}\right\},$ $$

where $\rho$ is defined at the beginning of Appendix [A.1](#A1.SS1). Then, we have $m_{l-1}\geq\mathcal{C}_{3}\log\left(\frac{(m+n)^{2}}{\delta_{l-1}}\right)$ and thus

$$ $\displaystyle\mathcal{C}_{1}\left(\sqrt{\frac{\log\left(\frac{(m+n)^{2}}{\delta_{l-1}}\right)}{m_{l-1}}}+\frac{\log\left(\frac{(m+n)^{2}}{\delta_{l-1}}\right)}{m_{l-1}}+\frac{\log^{2}\left(\frac{(m+n)^{2}}{\delta_{l-1}}\right)}{m_{l-1}^{2}}\right)\overset{(8)}{\leq}3\mathcal{C}_{1}\sqrt{\frac{\log\left(\frac{(m+n)^{2}}{\delta_{l-1}}\right)}{m_{l-1}}}\leq\rho,\forall l\in\mathbb{N^{+}}\backslash\{1\},$ $$

where inequality (8) holds because $\mathcal{C}_{3}\geq 1$ in Proposition [3](#Thmproposition3). By a similar mathematical induction on page 27 in Basei et al., (), we can prove the following event

$$ $\displaystyle\mathcal{G}=\left\{\left\|\theta^{l}-\theta\right\|\leq 3\mathcal{C}_{1}\sqrt{\frac{\log\left(\frac{(m+n)^{2}}{\delta_{l-1}}\right)}{m_{l-1}}},\forall l\in\mathbb{N^{+}}\backslash\{1\}\right\}\cup\left\{\theta^{1}\in\Theta\right\}$ (40) $$

holds with probability at least $1-2\sum_{l=2}^{\infty}\delta_{l-1}=1-\frac{\pi^{2}\delta}{3}$, i.e. $\mathbb{P}(\mathcal{G})\geq 1-\frac{\pi^{2}\delta}{3}$.

Under the event $\mathcal{G}$, which satisfies $\mathbb{P}(\mathcal{G})\geq 1-\frac{\pi^{2}\delta}{3}$, we can derive that

$$ $\displaystyle\operatorname{Regret}(N)$ $\displaystyle=\sum_{i=1}^{N}\left(J^{\pi^{i}}(x^{i}_{0})-J^{\star}(x^{i}_{0})\right)$ $\displaystyle=\sum_{l=1}^{L}\sum_{k=1}^{m_{l}}\left(J^{\pi^{l,k}}(x_{0}^{l,k})-J^{\star}(x_{0}^{l,k})\right)$ $\displaystyle\overset{(9)}{\leq}m_{1}\left[\frac{n}{2}\sum_{t=1}^{T-1}\psi_{t}\mathcal{V}^{2}\epsilon_{1}^{2}+\frac{1}{2}\|x_{0}\|^{2}\psi_{0}\mathcal{V}^{2}\epsilon_{1}^{2}+o(\epsilon_{1}^{2})\right]+\sum_{l=2}^{L}m_{l}\left[\frac{n}{2}\sum_{t=1}^{T-1}\psi_{t}\mathcal{V}^{2}\epsilon_{l}^{2}+\frac{1}{2}\|x_{0}\|^{2}\psi_{0}\mathcal{V}^{2}\epsilon_{l}^{2}+o(\epsilon_{l}^{2})\right]$ $\displaystyle\overset{(10)}{\leq}m_{1}\left[\frac{n}{2}\sum_{t=1}^{T-1}\psi_{t}\mathcal{V}^{2}\epsilon_{1}^{2}+\frac{1}{2}\|x_{0}\|^{2}\psi_{0}\mathcal{V}^{2}\epsilon_{1}^{2}+o(\epsilon_{1}^{2})\right]+\sum_{l=2}^{L}m_{l}\Bigg{[}\frac{n}{2}\sum_{t=1}^{T-1}\psi_{t}\mathcal{V}^{2}\cdot 9\mathcal{C}_{1}^{2}\cdot\frac{\log\left(\frac{(m+n)^{2}}{\delta_{l-1}}\right)}{m_{l-1}}$ $\displaystyle+\frac{1}{2}\|x_{0}\|^{2}\psi_{0}\mathcal{V}^{2}\cdot 9\mathcal{C}_{1}^{2}\cdot\frac{\log\left(\frac{(m+n)^{2}}{\delta_{l-1}}\right)}{m_{l-1}}+o\left(\frac{\log\left(\frac{(m+n)^{2}}{\delta_{l-1}}\right)}{m_{l-1}}\right)\Bigg{]}$ $\displaystyle\overset{(11)}{\leq}\mathcal{C}_{\operatorname{high}}+m_{1}\left[\frac{n}{2}\sum_{t=1}^{T-1}\psi_{t}\mathcal{V}^{2}\epsilon_{1}^{2}+\frac{1}{2}\|x_{0}\|^{2}\psi_{0}\mathcal{V}^{2}\epsilon_{1}^{2}\right]$ $\displaystyle+\left[9\mathcal{C}_{1}^{2}\mathcal{V}^{2}\left(n\sum_{t=1}^{T-1}\psi_{t}+\|x_{0}\|^{2}\psi_{0}\right)\right]\cdot\sum_{l=2}^{L}\left(\log\left(\frac{m+n}{\sqrt{\delta}}\right)+\log(l-1)\right)$ $\displaystyle\overset{(12)}{\leq}\mathcal{C}\left(\sum_{t=0}^{T-1}\psi_{t}\right)\left[\log\left(\frac{m+n}{\sqrt{\delta}}\right)L+L\log L\right],$ $$

where inequality (9) follows from ([39](#A1.E39)), inequality (10) follows from the definition of the event $\mathcal{G}$ in ([40](#A1.E40)), $\psi_{t}$ is defined in ([36](#A1.E36)), $\mathcal{C}_{\operatorname{high}}$ in inequality (11) is a constant depends on $T,\gamma,m,n,\mathcal{V},\widetilde{\Gamma}$ polynomially and it can bound the higher order term in inequality (10), and inequality (12) holds by Stirling’s formula: $\sum_{l=2}^{L}\log(l-1)=\log((L-1)!)\leq\mathcal{C}^{\prime}(L-1)\log(L-1)$, where $\mathcal{C}^{\prime}$ is a positive constant. The expression of $\mathcal{C}$ is given by

$$ $\displaystyle\mathcal{C}:=\operatorname{Polynomial}\left(\mathcal{C}_{1},\mathcal{C}^{\prime},\mathcal{V},n,\epsilon_{1},n,m_{1},\|x_{0}\|\right),$ $$

where $\epsilon_{1}$ is the estimation error in the first epoch, $m_{1}$ is the number of episodes in the first epoch, $\psi_{t}$ is defined in ([36](#A1.E36)), $\mathcal{C}_{1}=16\mathcal{C}_{2}^{3}\mathcal{C}_{4}^{2}$ is from the proof of Proposition [3](#Thmproposition3), and $\mathcal{V}$ is defined in ([15](#S3.E15)).

#### A.4.1 Dependency of the regret bound ( 14 ) on other parameters

In this section, we provide some further discussions on the dependency of the regret bound on other problem parameters, including the horizon length $T$, and the risk parameter $\gamma$ of the LEQR model. Spelling out the explicit dependency is generally difficult, due to the implicit dependency of $\tilde{\Gamma}$ and constant $\mathcal{C}$ on the model parameters. Hence, in the following we focus our discussion on the term $\sum_{t=0}^{T-1}\psi_{t}$ in view of the bound ([14](#S3.E14)).

Since $\{\psi_{t}\}_{t=0}^{T-1}$ is defined in a recursive manner, one can directly verify that

$$ $\displaystyle 2\widetilde{\Gamma}^{3}\left(10\mathcal{V}^{2}\mathcal{L}\widetilde{\Gamma}^{4}\right)^{2(T-1)}\leq$ $\displaystyle\sum_{t=0}^{T-1}\psi_{t}\leq 2\widetilde{\Gamma}^{3}T^{2}\left(10\mathcal{V}^{2}\mathcal{L}\widetilde{\Gamma}^{4}\right)^{2(T-1)}.$ (41) $$

The formula ([41](#A1.E41)) implies that the term $\sum_{t=0}^{T-1}\psi_{t}$ has exponential dependence on the horizon length $T$. When $\gamma\widetilde{\Gamma}>0$ is small, according to Taylor’s Theorem, we have

$$ $\displaystyle\frac{1}{1-\gamma\widetilde{\Gamma}}=1+\gamma\widetilde{\Gamma}+o\left(\gamma\widetilde{\Gamma}\right)\approx\exp\left(\gamma\widetilde{\Gamma}\right).$ (42) $$

Using the formula of $\mathcal{L}$ in ([15](#S3.E15)) and plugging ([42](#A1.E42)) into ([41](#A1.E41)), we find that the dependence of the term $\sum_{t=0}^{T-1}\psi_{t}$ on $\gamma$ is on the order of $\exp\left(12\gamma\widetilde{\Gamma}(T-1)\right).$ This also suggests that the regret bound in Theorem [1](#Thmtheorem1) has exponential dependence on $\gamma$ (ignoring the possible dependency of the constants $\mathcal{C}$ and $\tilde{\Gamma}$ on these parameters).

Note that Basei et al., () proved a regret bound that is logarithmic in the number of episodes $N$ for continuous-time risk-neutral LQR problem, also in the finite-horizon episodic setting. They also mentioned (see Remark 2.2 in their paper) that the regret bound of their algorithm in general depends exponentially on the time horizon $T$. So our previous discussion is consistent with their findings.
Note that they did not make explicit of the dependency of their regret bound on the horizon length $T$.

We also compare our results with Fei and Xu, (), which proved gap-dependent logarithmic regret bounds for tabular MDPs under the entropic risk criteria. In particular, they showed their algorithms can achieve the regret of $\frac{\left(\exp(|\beta|H)-1\right)^{2}}{|\beta|^{2}\Delta_{\min}}\cdot\text{poly}(H,S,A)\cdot\log\left(\frac{HSAK}{\delta}\right)$
with probability at least $1-\delta$,
where $\text{poly}(\cdot)$ represents the polynomial function, $H$ is the length of the episode, $S$ is the size of the state space, $A$ is the size of the action space, $\beta$ is the risk coefficient and $\Delta_{\min}$ is the minimum value of the sub-optimality gap of the value functions.
Their regret bound also has exponential dependency on the risk coefficient $\beta$ and the length of the episode $H$, which is similar as our regret bound. While there are some similarities, it is also important to emphasize we consider LEQR which has continuous state and action spaces, which are different from tabular MDPs with finite state and action spaces.

## Appendix B Regret Analysis of the Least-Squares-Based Algorithm with Exploration Noise

In this section, we prove Theorem [2](#Thmtheorem2) discussed in Section [4](#S4). The proof structure of Theorem [2](#Thmtheorem2) is similar to the proof structure of Theorem [1](#Thmtheorem1). We present the high probability bounds for the estimation error of system matrices in Section [B.1](#A2.SS1), the perturbation analysis of Riccati equations in Section [B.2](#A2.SS2), and the simplification of the suboptimality gap resulting from controller mismatch in Section [B.3](#A2.SS3).

### B.1 Bounds for the Estimation Error of System Matrices

In this section, we derive the high probability bound for the estimation error of system matrices in Algorithm [2](#alg2). Different from Section [A.1](#A1.SS1), we adapt the classical self-normalized martingale analysis framework to derive the desired error bound.

Similar as in Section [A.1](#A1.SS1), we fix the $k$-th episode and define the following compact set

$$ $\Xi=\left\{\hat{\theta}\in\mathbb{R}^{(n+m)\times n}\Big{|}\left\|\hat{\theta}-\theta\right\|\leq\varpi\right\}\cup\{\theta^{1}\},$ $$

where $\varpi>0$ is a constant that satisfies

$$ $\displaystyle\varpi$ $\displaystyle\geq\max\Bigg{\{}\frac{2n}{\lambda}\left(\log\left(\frac{3n^{2}N}{\delta^{2}}\right)+(n+m)\log\left(1+\frac{\tilde{c}N\log\left(\frac{3TN^{2}}{\delta}\right)}{\lambda}\right)\right)+2(n+m)^{2}\widetilde{\Gamma}^{2},$ (43) $\displaystyle\qquad\qquad\frac{80n}{cT}\left(\log\left(\frac{4n^{2}N}{\delta^{2}}\right)+(n+m)\log\left(1+\frac{\tilde{c}N\log\left(\frac{4TN^{2}}{\delta}\right)}{\lambda}\right)\right)+\frac{80\lambda(n+m)^{2}\widetilde{\Gamma}^{2}}{cT}\Bigg{\}}.$ $$

Here, $\widetilde{\Gamma}$ is defined in ([15](#S3.E15)), $\lambda$ is the regularization parameter and $c,\tilde{c}>0$ are two constants independent of $k$ and $N$ but may depend on other constants including $n,m,\gamma$. The explicit expression of $c$ and $\tilde{c}$ can be found in ([50](#A2.E50)) and ([59](#A2.E59)). For any estimated $\tilde{\theta}\in\Xi$, there exists a universal constant $C_{K}>0$ such that

$$ $\left\|\widetilde{K}_{t}\right\|\leq C_{K},\quad\forall t,$ (44) $$

where $\widetilde{K}_{t}$ is the control corresponding to $\tilde{\theta}$ and it’s continuous in terms of $\tilde{\theta}$ according to ([2.1](#S2.Ex2)). We also define the following event

$$ $\widetilde{\mathcal{G}}^{k}=\{\theta^{i}\in\Xi,\forall i=1,\cdots,k\}.$ (45) $$

We will prove $\mathbb{P}(\widetilde{\mathcal{G}}^{k})\geq 1-\sum_{i=1}^{k-1}\frac{\delta}{N-1}=1-\frac{(k-1)\delta}{N-1}$ in Section [B.4](#A2.SS4).

The main result of this section is the following proposition,
which provides the high probability bound for the estimation error of system matrices estimated in Algorithm [2](#alg2).

###### Proposition 6 .

Let $\delta\in\left(0,\frac{1}{4}\right)$. Conditional on event $\widetilde{\mathcal{G}}^{k}$, when $kT\geq 200\left(3(n+m)+\log\left(\frac{1}{\delta}\right)\right)$, with probability at least $1-4\delta$,

$$ $\displaystyle\left\|\theta^{k+1}-\theta\right\|^{2}\leq\frac{80n}{cT\sqrt{k}}\left(\log\left(\frac{n^{2}}{\delta^{2}}\right)+(n+m)\log\left(1+\frac{\tilde{c}k\log\left(\frac{TN}{\delta}\right)}{\lambda}\right)\right)+\frac{80\lambda(n+m)^{2}\widetilde{\Gamma}^{2}}{cT\sqrt{k}},$ $$

where $\widetilde{\Gamma}$ is defined in ([15](#S3.E15)), the explicit expressions of $c$ and $\tilde{c}$ can be found in ([50](#A2.E50)) and ([59](#A2.E59)), $n$ is the
dimension of the system state vector, $m$ is the dimension
of the control vector and $\lambda$ is the regularization parameter.
When $kT<200\left(3(n+m)+\log\left(\frac{1}{\delta}\right)\right)$, with probability at least $1-3\delta$,

$$ $\displaystyle\left\|\theta^{k+1}-\theta\right\|^{2}\leq\frac{2n}{\lambda}\left(\log\left(\frac{n^{2}}{\delta^{2}}\right)+(n+m)\log\left(1+\frac{\tilde{c}k\log\left(\frac{TN}{\delta}\right)}{\lambda}\right)\right)+2(n+m)^{2}\widetilde{\Gamma}^{2}.$ $$

The proof of Proposition [6](#Thmproposition6) is long, and we will discuss it in the following subsections.

###### Proposition 6 .

#### B.1.1 Preliminaries

In this section, we recall an important high probability bound, known as self-normalized bound for vector-valued martingales. It will be
used in the derivation of the bounds for the estimation error of system matrices.

###### Lemma 12 (Theorem 1 in Abbasi-Yadkori et al., ( 2011 ) ) .

Let $\left\{\mathcal{F}_{t}\right\}_{t=0}^{\infty}$ be a filtration. Let $\left\{\eta_{t}\right\}_{t=0}^{\infty}$ be a real-valued stochastic process such that $\eta_{t}$ is $\mathcal{F}_{t+1}$-measurable and $\eta_{t}$ is conditionally $R$-sub-Gaussian for some $R\geq 0$ i.e.

$$ $\displaystyle\mathbb{E}\left[e^{\lambda\eta_{t}}\Big{|}\mathcal{F}_{t}\right]\leq\exp\left(\frac{\lambda^{2}R^{2}}{2}\right),\forall\lambda\in\mathbb{R}.$ $$

Let $\left\{X_{t}\right\}_{t=0}^{\infty}$ be an $\mathbb{R}^{d}$-valued stochastic process such that $X_{t}$ is $\mathcal{F}_{t}$-measurable. Assume that $V$ is a $d\times d$ positive definite matrix. For any $t\geq 0$, define

$$ $\displaystyle\bar{V}_{t}=V+\sum_{s=0}^{t}X_{s}X_{s}^{\top},\quad\quad S_{t}=\sum_{s=0}^{t}\eta_{s}X_{s}.$ $$

Then, for any $\delta>0$, with probability at least $1-\delta$, for all $t\geq 0$,

$$ $\displaystyle\left\|S_{t}\right\|_{\bar{V}_{t}^{-1}}^{2}\leq 2R^{2}\log\left(\frac{\operatorname{det}\left(\bar{V}_{t}\right)^{1/2}\operatorname{det}(V)^{-1/2}}{\delta}\right),$ $$

where $\|S_{t}\|_{\bar{V}_{t}^{-1}}^{2}=S_{t}^{\top}\left(\bar{V}_{t}\right)^{-1}S_{t}$.

###### Lemma 12 (Theorem 1 in Abbasi-Yadkori et al., ( 2011 ) ) .

#### B.1.2 Self-Normalized Bounds for the Estimation Error of System Matrices

In this section, we analyze the estimation error based on bounds for the self-normalized martingale. Similar to Section [A.3](#A1.SS3), let $\mathbb{H}_{t}^{k}$ be the set of possible histories up to step $t$ in the $k$-th episode. Denote the history up to step $t$ in the $k$-th episode by

$$ $\mathcal{H}_{t}^{k}=\left(x_{0}^{1},u_{0}^{1},\cdots,x_{T}^{1},x_{0}^{2},\cdots,x_{0}^{k},\cdots,x_{t-1}^{k},u_{t-1}^{k},x_{t}^{k},u_{t}^{k}\right).$ (46) $$

The following lemma is a modified version of Theorem 2 in Abbasi-Yadkori et al., () and Lemma 6 in Cohen et al., (), which provides a coarse self-normalized bound for the estimation error.

###### Lemma 13 .

For any $\delta\in(0,1)$, with probability at least $1-\delta$, we have

$$ $\operatorname{Tr}\left((\theta^{k+1}-\theta)^{\top}\bar{V}^{k}(\theta^{k+1}-\theta)\right)\leq 2n\log\left(\frac{n^{2}}{\delta^{2}}\frac{\det(\bar{V}^{k})}{\det(\lambda I)}\right)+2\lambda\|\theta\|_{F}^{2},$ (47) $$

where $\theta^{k+1}$ is the estimated system matrix defined in ([17](#S4.E17)), $\theta$ is the true system matrix, $\lambda$ is the regularization parameter and

$$ $\displaystyle\bar{V}^{k}=\lambda I+\sum_{i=1}^{k}\sum_{t=0}^{T-1}z_{t}^{i}z_{t}^{i\top}.$ $$

###### Proof.

We first follow Lemma 6 in Cohen et al., () to simplify $\theta^{k+1}-\theta$. Recall that

$$ $x_{t+1}^{i}=\theta^{\top}z_{t}^{i}+w_{t}^{i},\quad w_{t}^{i}\sim\mathcal{N}\left(0,I_{n}\right)$ $$

where $z_{t}^{i}=\left[x_{t}^{i\top}\ u_{t}^{i\top}\right]^{\top}$. Together with ([17](#S4.E17)), we can obtain

$$ $\displaystyle\theta^{k+1}$ $\displaystyle=\left(\bar{V}^{k}\right)^{-1}\left(\sum_{i=1}^{k}\sum_{t=0}^{T-1}z_{t}^{i}\left(z_{t}^{i\top}\theta+w_{t}^{i\top}\right)\right)$ $\displaystyle=\left(\bar{V}^{k}\right)^{-1}\left(\lambda\theta+\sum_{i=1}^{k}\sum_{t=0}^{T-1}z_{t}^{i}z_{t}^{i\top}\theta+\sum_{i=1}^{k}\sum_{t=0}^{T-1}z_{t}^{i}w_{t}^{i\top}-\lambda\theta\right)$ $\displaystyle=\theta+\left(\bar{V}^{k}\right)^{-1}\left(S_{T-1}^{k}-\lambda\theta\right),$ $$

where we denote $S_{T-1}^{k}=\sum_{i=1}^{k}\sum_{t=0}^{T-1}z_{t}^{i}w_{t}^{i\top}$ for the simplicity of notation. Then, we obtain

$$ $\displaystyle\operatorname{Tr}\left(\left(\theta^{k+1}-\theta\right)^{\top}\bar{V}^{k}\left(\theta^{k+1}-\theta\right)\right)$ (48) $\displaystyle=\operatorname{Tr}\left(\left(S_{T-1}^{k}-\lambda\theta\right)^{\top}\left(\bar{V}^{k}\right)^{-1}\left(S^{k}_{T-1}-\lambda\theta\right)\right)$ $\displaystyle=\operatorname{Tr}\left(S_{T-1}^{k\top}\left(\bar{V}^{k}\right)^{-1}S_{T-1}^{k}+\lambda^{2}\theta^{\top}\left(\bar{V}^{k}\right)^{-1}\theta-\lambda S_{T-1}^{k\top}\left(\bar{V}^{k}\right)^{-1}\theta-\lambda\theta^{\top}\left(\bar{V}^{k}\right)^{-1}S_{T-1}^{k}\right)$ $\displaystyle\overset{(1)}{\leq}\operatorname{Tr}\left(S_{T-1}^{k\top}\left(\bar{V}^{k}\right)^{-1}S_{T-1}^{k}+\lambda^{2}\theta^{\top}\left(\bar{V}^{k}\right)^{-1}\theta\right)+2\left\|\lambda\theta^{\top}\left(\bar{V}^{k}\right)^{-\frac{1}{2}}\right\|_{F}\cdot\left\|\left(\bar{V}^{k}\right)^{-\frac{1}{2}}S_{T-1}^{k}\right\|_{F}$ $\displaystyle\overset{(2)}{\leq}2\operatorname{Tr}\left(S_{T-1}^{k\top}\left(\bar{V}^{k}\right)^{-1}S_{T-1}^{k}\right)+2\lambda^{2}\operatorname{Tr}\left(\theta^{\top}\left(\bar{V}^{k}\right)^{-1}\theta\right)$ $\displaystyle\overset{(3)}{\leq}2\operatorname{Tr}\left(S_{T-1}^{k\top}\left(\bar{V}^{k}\right)^{-1}S_{T-1}^{k}\right)+2\lambda\|\theta\|_{F}^{2}.$ $$

Here, we use Cauchy–Schwarz inequality $\left|\operatorname{Tr}(EF)\right|\leq\|E\|_{F}\|F\|_{F}$ for any matrix $E$ and $F$ to obtain inequality (1), we use the inequality $2ab\leq a^{2}+b^{2}$ for any $a$ and $b$ to obtain inequality (2), and we use the fact that $\bar{V}^{k}\succeq\lambda I$ to obtain inequality (3).

We further bound $\operatorname{Tr}\left(S_{T-1}^{k\top}\left(\bar{V}^{k}\right)^{-1}S_{T-1}^{k}\right)$ in ([48](#A2.E48)) to get the result in ([47](#A2.E47)). Let $S_{t}^{k}(j)=\sum_{i=1}^{k}\sum_{s=0}^{t}z_{s}^{i}w_{s}^{i}(j),j=1,\cdots,n,\ t=0,\cdots,T-1,\ k=1,\cdots,N$, where $w_{s}^{i}(j)$ is the $j$-th element of the random vector $w_{s}^{i}$. Recall the trajectory in ([46](#A2.E46)), $z_{s}^{i}$ is $\mathcal{H}_{s}^{i}$-measurable for any step $s$ in the $i$-th episode and $w_{s}^{i}(j)$ is $\mathcal{H}_{s+1}^{i}$-measurable for any step $s$ in the $i$-th episode. Therefore, we can apply Lemma [12](#Thmlemma12) and obtain that with probability at least $1-\frac{\delta}{n}$,

$$ $\displaystyle S_{T-1}^{k}(j)^{\top}\left(\bar{V}^{k}\right)^{-1}S_{T-1}^{k}(j)\leq 2\log\left(\frac{n}{\delta}\frac{\det(\bar{V}^{k})^{1/2}}{\det(\lambda I)^{1/2}}\right).$ $$

By a union bound, we can obtain that with probability at least $1-\delta$,

$$ $\displaystyle\operatorname{Tr}\left(S_{T-1}^{k\top}\left(\bar{V}^{k}\right)^{-1}S_{T-1}^{k}\right)$ $\displaystyle=\sum_{j=1}^{n}S_{T-1}^{k}(j)^{\top}\left(\bar{V}^{k}\right)^{-1}S_{T-1}^{k}(j)\leq n\log\left(\frac{n^{2}}{\delta^{2}}\frac{\det(\bar{V}^{k})}{\det(\lambda I)}\right).$ (49) $$

On combining ([48](#A2.E48)) with ([49](#A2.E49)), we can obtain ([47](#A2.E47)).
∎

After deriving the coarse self-normalized bounds in ([47](#A2.E47)), we need to find the upper and lower bounds for $\bar{V}^{k}$ to obtain the result in Proposition [6](#Thmproposition6). We follow the proof of Theorem 20 in Cohen et al., () to derive the high probability lower bound for $\bar{V}^{k}$. The main difference is that we consider a decaying exploration noise while they consider a nondecaying exploration noise. The next lemma provides a lower bound for the conditional expectation of $z_{t}^{k}z_{t}^{k\top},\forall k,t$, which is a modification of Lemma 34 in Cohen et al., ().

###### Lemma 14 .

For all episode $k$ and step $t$, conditional on event $\widetilde{\mathcal{G}}^{k}$, we have

$$ $\displaystyle\mathbb{E}\left[z_{t}^{k}z_{t}^{k\top}\big{|}\mathcal{H}_{t-1}^{k}\right]\succeq\frac{c}{\sqrt{k}}I_{m+n},\quad\text{$t\neq 0$},$ $$

where $c>0$ is a constant satisfying

$$ $c\leq\frac{C_{K}^{2}-C_{K}\sqrt{C_{K}^{2}+4}+2}{2}$ (50) $$

with $C_{K}$ defined in ([44](#A2.E44)) and $\frac{C_{K}^{2}-C_{K}\sqrt{C_{K}^{2}+4}+2}{2}\in(0,1)$.

###### Proof.

Recall that $z_{t}^{k}=\left[x_{t}^{k\top},u_{t}^{k\top}\right]^{\top}$, we have

$$ $\displaystyle\mathbb{E}\left[z_{t}^{k}z_{t}^{k\top}\big{|}\mathcal{H}_{t-1}^{k}\right]$ $\displaystyle=\left[\begin{array}[]{l}I_{n}\\ K_{t}^{k}\end{array}\right]\mathbb{E}\left[x_{t}^{k}x_{t}^{k\top}\big{|}\mathcal{H}_{t-1}^{k}\right]\left[\begin{array}[]{ll}I_{n}&K_{t}^{k\top}\end{array}\right]+\left[\begin{array}[]{cc}0&0\\ 0&\frac{1}{\sqrt{k}}I_{m}\end{array}\right]$ $\displaystyle\overset{(1)}{\succeq}\left[\begin{array}[]{l}I_{n}\\ K_{t}^{k}\end{array}\right]\left[\begin{array}[]{ll}I_{n}&K_{t}^{k\top}\end{array}\right]+\left[\begin{array}[]{cc}0&0\\ 0&\frac{1}{\sqrt{k}}I_{m}\end{array}\right]$ $\displaystyle=\left[\begin{array}[]{cc}\left(1-\frac{c}{\sqrt{k}}\right)I_{n}&K_{t}^{k\top}\\ K_{t}^{k}&K_{t}^{k}K_{t}^{k\top}+\frac{1}{\sqrt{k}}(1-c)I_{m}\end{array}\right]+\frac{c}{\sqrt{k}}I_{n+m}$ $\displaystyle\overset{(2)}{\succeq}\left[\begin{array}[]{l}\sqrt{1-\frac{c}{\sqrt{k}}}I_{n}\\ \frac{K_{t}^{k}}{\sqrt{1-\frac{c}{\sqrt{k}}}}\end{array}\right]\left[\begin{array}[]{ll}\sqrt{1-\frac{c}{\sqrt{k}}}I_{n}&\frac{K_{t}^{k\top}}{\sqrt{1-\frac{c}{\sqrt{k}}}}\end{array}\right]+\frac{c}{\sqrt{k}}I_{n+m}$ $\displaystyle\succeq\frac{c}{\sqrt{k}}I_{n+m}.$ $$

Here, inequality (1) follows from the fact that $z_{t-1}^{k}=\left[x_{t-1}^{k\top},u_{t-1}^{k\top}\right]^{\top}$ is $\mathcal{H}_{t-1}^{k}$-measurable and

$$ $\displaystyle\mathbb{E}\left[x_{t}^{k}x_{t}^{k\top}\big{|}\mathcal{H}_{t-1}^{k}\right]$ $\displaystyle=\mathbb{E}\left[\left(Ax_{t-1}^{k}+Bu_{t-1}^{k}+w_{t-1}^{k}\right)\left(Ax_{t-1}^{k}+Bu_{t-1}^{k}+w_{t-1}^{k}\right)^{\top}\Big{|}\mathcal{H}_{t-1}^{k}\right]$ $\displaystyle=\left(Ax_{t-1}^{k}+Bu_{t-1}^{k}\right)\left(Ax_{t-1}^{k}+Bu_{t-1}^{k}\right)^{\top}+\mathbb{E}\left[w_{t-1}^{k}w_{t-1}^{k\top}\big{|}\mathcal{H}_{t-1}^{k}\right]$ $\displaystyle\succeq I_{n}.$ $$

For inequality (2), when $0<c\leq\frac{C_{K}^{2}-C_{K}\sqrt{C_{K}^{2}+4}+2}{2}$, we can obtain

$$ $\frac{1}{1-\frac{c}{\sqrt{k}}}K_{t}^{k}K_{t}^{k\top}\preceq K_{t}^{k}K_{t}^{k\top}+\frac{1}{\sqrt{k}}(1-c)I_{m}.$ (51) $$

We can prove that ([51](#A2.E51)) is equivalent to $\frac{c}{1-\frac{c}{\sqrt{k}}}K_{t}^{k}K_{t}^{k\top}\preceq\frac{c}{1-c}K_{t}^{k}K_{t}^{k\top}\preceq\frac{c}{1-c}C_{K}^{2}I_{m}\preceq(1-c)I_{m}$. Solving the inequality $\frac{c}{1-c}C_{K}^{2}\leq 1-c$, we can obtain $0<c\leq\frac{C_{K}^{2}-C_{K}\sqrt{C_{K}^{2}+4}+2}{2}$.
∎

With the lower bound for the conditional expectation of $z_{t}^{k}z_{t}^{k\top},\forall k,t$, we can derive the high probability lower bound as Lemma 33 in Cohen et al., ().

###### Lemma 15 .

Let $\delta\in(0,1)$. Conditional on event $\widetilde{\mathcal{G}}^{k}$, when $kT\geq 200\left(600(n+m)+\log\left(\frac{1}{\delta}\right)\right)$, with probability at least $1-\delta$, we have

$$ $\bar{V}^{k}\succeq\frac{cT\sqrt{k}}{40}I_{n+m}.$ (52) $$

###### Proof.

Let $e\in\mathbb{S}^{n+m-1},$ where $\mathbb{S}^{n+m-1}=\left\{v\in\mathbb{R}^{n+m}|\|v\|_{2}=1\right\}$. Let $I_{t}^{k}=e^{\top}z_{t}^{k}$ and let $\mathcal{Y}_{t}^{k}$ be an indicator random variable that equals $1$ if $(I_{t}^{k})^{2}>\frac{c}{2\sqrt{k}}$ and 0 0 otherwise. By the similar arguments as in the proof of Lemma 35 in Cohen et al., (), we can prove that

$$ $\displaystyle\mathbb{E}\left[\mathcal{Y}_{t}^{k}\big{|}\mathcal{H}_{t-1}^{k}\right]=\mathbb{P}\left(\mathcal{Y}_{t}^{k}=1\big{|}\mathcal{H}_{t-1}^{k}\right)\geq\frac{1}{5},\quad\text{if $t\neq 0$}.$ (53) $$

Let $U^{k}_{t}=\mathcal{Y}_{t}^{k}-\mathbb{E}\left[\mathcal{Y}_{t}^{k}|\mathcal{H}_{t-1}^{k}\right]$. Then, $(U_{t}^{k})$ is a martingale difference sequence with $|U_{t}^{k}|\leq 1,\forall k,t$. So we can use Azuma-Hoeffding inequality to derive the high probability bound: with probability at least $1-\delta$,

$$ $\sum_{i=1}^{k}\sum_{t=1}^{T-1}U_{t}^{i}\geq-\sqrt{2kT\log\left(\frac{1}{\delta}\right)}\overset{(1)}{\geq}-\frac{kT}{10},$ (54) $$

where inequality (1) holds when $kT\geq 200\log\left(\frac{1}{\delta}\right)$. On combining $U^{i}_{t}=\mathcal{Y}_{t}^{i}-\mathbb{E}\left[\mathcal{Y}_{t}^{i}|\mathcal{H}_{t-1}^{i}\right]$ with ([54](#A2.E54)), we can obtain with probability at least $1-\delta$,

$$ $\displaystyle\sum_{i=1}^{k}\sum_{t=1}^{T-1}\mathcal{Y}_{t}^{i}$ $\displaystyle\geq\sum_{i=1}^{k}\sum_{t=1}^{T-1}\mathbb{E}\left[\mathcal{Y}_{t}^{i}\big{|}\mathcal{H}_{t-1}^{i}\right]-\frac{kT}{10}\overset{(2)}{\geq}\frac{kT}{5}-\frac{kT}{10}=\frac{kT}{10},$ (55) $$

where inequality (2) follows from ([53](#A2.E53)). Denote $V^{k}=\sum_{i=1}^{k}\sum_{t=1}^{T-1}z_{t}^{i}z_{t}^{i\top}$. Then, we can get with probability at least $1-\delta$,

$$ $\displaystyle e^{\top}V^{k}e$ $\displaystyle=\sum_{i=1}^{k}\sum_{t=1}^{T-1}\left(I_{t}^{i}\right)^{2}\overset{(3)}{\geq}\sum_{i=1}^{k}\sum_{t=1}^{T-1}\mathcal{Y}_{t}^{i}\frac{c}{2\sqrt{i}}\geq\sum_{i=1}^{k}\sum_{t=1}^{T-1}\mathcal{Y}_{t}^{i}\frac{c}{2\sqrt{k}}\overset{(4)}{\geq}\frac{kT}{10}\cdot\frac{c}{2\sqrt{k}}=\frac{\sqrt{k}Tc}{20},$ $$

where inequality (3) follows from the definition of $I_{t}^{i}$, inequality (4) holds by ([55](#A2.E55)). Finally, by the similar $\frac{1}{4}$-net argument in the proof of Theorem 20 in Cohen et al., (), we can prove that when $kT\geq 200\left(3(n+m)+\log\left(\frac{1}{\delta}\right)\right)$, with probability at least $1-\delta$,

$$ $\left\|\left(V^{k}\right)^{-1}\right\|\leq\frac{40}{cT\sqrt{k}},$ $$

which is equivalent to

$$ $\bar{V}^{k}\succeq V^{k}\succeq\sum_{i=1}^{k}\sum_{t=1}^{T-1}z_{t}^{i}z_{t}^{i\top}\succeq\frac{cT\sqrt{k}}{40}I_{n+m}.$ $$

∎

In addition to the lower bound of $\bar{V}^{k}$, we also need to find the upper bound of $\bar{V}^{k}$ to get the final high probability bound for the estimation error of system matrices. In the following lemma, we provide the high probability upper bound for $\|x_{t}^{k}\|$, which plays a vital role in deriving the high probability upper bound of $\bar{V}^{k}$.

###### Lemma 16 .

Let $\delta\in\left(0,\frac{1}{2}\right)$. Conditional on the event $\widetilde{\mathcal{G}}^{k}$, with probability at least $1-2\delta$, for all $0\leq t\leq T$, we have

$$ $\|x_{t}^{k}\|\leq 6\left(\widetilde{\Gamma}(1+C_{K})\right)^{t}\left(n^{\frac{3}{4}}+m^{\frac{3}{4}}\right)\max\left\{\|x_{0}\|,1\right\}\widetilde{\Gamma}\log^{\frac{1}{2}}\left(\frac{TN}{\delta}\right),$ (56) $$

where $\widetilde{\Gamma}$ is defined in ([15](#S3.E15)) and $C_{K}$ is defined ([44](#A2.E44)).

###### Proof.

Recall that $x_{t}^{k}=Ax_{t-1}^{k}+Bu_{t-1}^{k}+w_{t-1}^{k}=(A+BK_{t-1}^{k})x_{t-1}^{k}+Bg_{t-1}^{k}+w_{t-1}^{k}$. Similar to ([25](#A1.E25)), we can simplify $x_{t}^{k}$ to

$$ $\displaystyle x_{t}^{k}=\left(\prod_{j=t-1}^{0}\left(A+BK_{j}^{k}\right)\right)x_{0}^{k}+\sum_{r=0}^{t-1}\left(\prod_{j=t-1}^{r+1}\left(A+BK_{j}^{k}\right)\right)\left(Bg_{r}^{k}+w_{r}^{k}\right),$ $$

where $\prod_{j=t-1}^{r+1}(A+BK_{j}^{k})=(A+BK_{t-1}^{k})(A+BK_{t-2}^{k})\cdots(A+BK_{r+1}^{k})$, and $\prod_{i=t-1}^{t}(A+BK_{i}^{k})=I_{n}$.
Similar to Theorem 21 and Lemma 32 of Cohen et al., (), we can use Hanson-Wright inequality in Proposition 1.1 of Hsu et al., () to derive that

$$ $\displaystyle\mathbb{P}\left(\|w_{r}^{k}\|^{2}\leq 5n^{\frac{3}{2}}\log\left(\frac{TN}{\delta}\right),\|g_{r}^{k}\|^{2}\leq 5m^{\frac{3}{2}}\frac{1}{\sqrt{k}}\log\left(\frac{TN}{\delta}\right),\forall k,r\right)\geq 1-2\delta$ (57) $$

Then, we can bound the state vector by

$$ $\displaystyle\|x_{t}^{k}\|$ $\displaystyle\leq\left\|\prod_{j=t-1}^{0}(A+BK_{j}^{k})\right\|\cdot\|x_{0}^{k}\|+\sum_{r=0}^{T-1}\left\|\prod_{j=t-1}^{r+1}(A+BK_{j}^{k})\right\|\cdot\left\|Bg_{r}^{k}+w_{r}^{k}\right\|$ $\displaystyle\leq\prod_{j=t-1}^{0}\left(\|A\|+\|B\|\cdot\|K_{j}^{k}\|\right)\|x_{0}^{k}\|+\sum_{r=0}^{t-1}\prod_{j=t-1}^{r+1}\left(\|A\|+\|B\|\cdot\|K_{j}^{k}\|\right)\left(\|B\|\cdot\|g_{r}^{k}\|+\|w_{r}^{k}\|\right)$ $\displaystyle\overset{(1)}{\leq}\widetilde{\Gamma}^{t}(1+C_{K})^{t}\|x_{0}\|+\sum_{r=0}^{t-1}\sqrt{5}\widetilde{\Gamma}^{t-r-1}(1+C_{K})^{t-r-1}\left(\frac{1}{k^{\frac{1}{4}}}m^{\frac{3}{4}}\widetilde{\Gamma}+n^{\frac{3}{4}}\right)\log^{\frac{1}{2}}\left(\frac{TN}{\delta}\right)$ $\displaystyle=\widetilde{\Gamma}^{t}(1+C_{K})^{t}\|x_{0}\|+\frac{\widetilde{\Gamma}^{t}(1+C_{K})^{t}-1}{\widetilde{\Gamma}(1+C_{K})-1}\cdot\sqrt{5}\left(\frac{1}{k^{\frac{1}{4}}}m^{\frac{3}{4}}\widetilde{\Gamma}+n^{\frac{3}{4}}\right)\log^{\frac{1}{2}}\left(\frac{TN}{\delta}\right)$ $\displaystyle\overset{(2)}{\leq}\left(\widetilde{\Gamma}(1+C_{K})\right)^{t}\|x_{0}\|+5\left(\widetilde{\Gamma}(1+C_{K})\right)^{t-1}\left(m^{\frac{3}{4}}+n^{\frac{3}{4}}\right)\widetilde{\Gamma}\log^{\frac{1}{2}}\left(\frac{TN}{\delta}\right)$ $\displaystyle\leq 6\left(\widetilde{\Gamma}(1+C_{K})\right)^{t}\left(n^{\frac{3}{4}}+m^{\frac{3}{4}}\right)\max\left\{\|x_{0}\|,1\right\}\widetilde{\Gamma}\log^{\frac{1}{2}}\left(\frac{TN}{\delta}\right),$ $$

where inequality (1) holds by the inequalities in ([57](#A2.E57)) and inequality (2) follows from the fact that $\widetilde{\Gamma}(1+C_{K})-1\geq\widetilde{\Gamma}(1+C_{K})-\frac{1}{2}\widetilde{\Gamma}(1+C_{K})=\frac{1}{2}\widetilde{\Gamma}(1+C_{K})$ and $k\geq 1$.
∎

With the result in Lemma [16](#Thmlemma16), we can derive the high probability bound for $\left\|\bar{V}^{k}\right\|$.

###### Lemma 17 .

Let $\delta\in(0,1)$. Conditional on event $\widetilde{\mathcal{G}}^{k}$, with probability at least $1-2\delta$, we have

$$ $\displaystyle\left\|\bar{V}^{k}\right\|\leq\lambda+(1+2C_{K}^{2})\left[\frac{72\left((\widetilde{\Gamma}(1+C_{K}))^{2T}-1\right)}{\left(\widetilde{\Gamma}(1+C_{K})\right)^{2}-1}\cdot(n^{\frac{3}{2}}+m^{\frac{3}{2}})\cdot\max\left\{\|x_{0}\|^{2},1\right\}\widetilde{\Gamma}^{2}k\log\left(\frac{TN}{\delta}\right)\right]$ $\displaystyle\qquad\quad+20m^{\frac{3}{2}}\sqrt{k}T\log\left(\frac{TN}{\delta}\right).$ $$

###### Proof.

Recall that $\bar{V}^{k}=\lambda I_{n+m}+\sum_{i=1}^{k}\sum_{t=0}^{T-1}z_{t}^{i}z_{t}^{i\top}$, $z_{t}^{i}=[x_{t}^{i\top},u_{t}^{i\top}]^{\top}$. We have

$$ $\displaystyle\|\bar{V}^{k}\|$ $\displaystyle\leq\lambda+\sum_{i=1}^{k}\sum_{t=0}^{T-1}\|z_{t}^{i}\|^{2}$ (58) $\displaystyle=\lambda+\sum_{i=1}^{k}\sum_{t=0}^{T-1}\left(\|x_{t}^{i}\|^{2}+\|K_{t}^{i}x_{t}^{i}+g_{t}^{i}\|^{2}\right)$ $\displaystyle\overset{(1)}{\leq}\lambda+\sum_{i=1}^{k}\sum_{t=0}^{T-1}\left(\|x_{t}^{i}\|^{2}+2\|K_{t}^{i}\|^{2}\cdot\|x_{t}^{i}\|^{2}+2\|g_{t}^{i}\|^{2}\right)$ $\displaystyle\overset{(2)}{\leq}\lambda+\sum_{i=1}^{k}\sum_{t=0}^{T-1}\left((1+2C_{K}^{2})\|x_{t}^{i}\|^{2}+2\|g_{t}^{i}\|^{2}\right),$ $$

where inequality (1) follows from the fact that $\|u+v\|^{2}\leq 2\|u\|^{2}+2\|v\|^{2},\forall u,v$, inequality (2) holds by ([44](#A2.E44)). Combine the results in Lemma [16](#Thmlemma16) with ([58](#A2.E58)), with probability at least $1-2\delta$, we can get

$$ $\displaystyle\|\bar{V}^{k}\|\overset{(3)}{\leq}\lambda+\sum_{i=1}^{k}\sum_{t=0}^{T-1}\Bigg{[}(1+2C_{K}^{2})\bigg{(}72\left(\widetilde{\Gamma}\left(1+C_{K}\right)\right)^{2t}\left(n^{\frac{3}{2}}+m^{\frac{3}{2}}\right)\max\left\{\|x_{0}\|^{2},1\right\}\widetilde{\Gamma}^{2}\log\left(\frac{TN}{\delta}\right)\bigg{)}$ $\displaystyle\qquad\qquad\qquad\qquad\quad+\frac{10m^{\frac{3}{2}}}{\sqrt{i}}\log\left(\frac{TN}{\delta}\right)\Bigg{]}$ $\displaystyle=\lambda+(1+2C_{K}^{2})\left[\frac{72\left(\left(\widetilde{\Gamma}(1+C_{K})\right)^{2T}-1\right)}{\left(\widetilde{\Gamma}(1+C_{K})\right)^{2}-1}\cdot(n^{\frac{3}{2}}+m^{\frac{3}{2}})\cdot\max\left\{\|x_{0}\|^{2},1\right\}\widetilde{\Gamma}^{2}k\log\left(\frac{TN}{\delta}\right)\right]$ $\displaystyle\quad+20m^{\frac{3}{2}}\sqrt{k}T\log\left(\frac{TN}{\delta}\right).$ $$

where inequality (3) follows from ([56](#A2.E56)) and ([57](#A2.E57)) in Lemma [16](#Thmlemma16) and the fact that $\|u+v\|^{2}\leq 2\|u\|^{2}+2\|v\|^{2},\forall u,v$.
∎

For the simplicity of notation, we denote

$$ $\displaystyle\tilde{c}$ $\displaystyle=(1+2C_{K}^{2})\left[\frac{72\left(\left(\widetilde{\Gamma}(1+C_{K})\right)^{2T}-1\right)}{\left(\widetilde{\Gamma}(1+C_{K})\right)^{2}-1}\cdot(n^{\frac{3}{2}}+m^{\frac{3}{2}})\cdot\max\left\{\|x_{0}\|^{2},1\right\}\widetilde{\Gamma}^{2}\right]+20m^{\frac{3}{2}}T,$ (59) $$

which is a constant independent of $k$ and $N$.
Then, we can get

$$ $\|\bar{V}^{k}\|\leq\lambda+\tilde{c}k\log\left(\frac{TN}{\delta}\right).$ (60) $$

Now we are ready to prove Proposition [6](#Thmproposition6).

###### Proof of Proposition 6 .

We can simplify ([47](#A2.E47)) as follows:

$$ $\displaystyle\lambda_{\min}\left(\bar{V}^{k}\right)\left\|\theta^{k+1}-\theta\right\|_{F}^{2}$ $\displaystyle\leq\operatorname{Tr}\left(\left(\theta^{k+1}-\theta\right)^{\top}\bar{V}^{k}\left(\theta^{k+1}-\theta\right)\right)$ (61) $\displaystyle\leq 2n\log\left(\frac{n^{2}}{\delta^{2}}\frac{\det(\bar{V}^{k})}{\det(\lambda I)}\right)+2\lambda\|\theta\|_{F}^{2}$ $\displaystyle\overset{(1)}{\leq}2n\left(\log\left(\frac{n^{2}}{\delta^{2}}\right)+(n+m)\log\left(\frac{\|\bar{V}^{k}\|}{\lambda}\right)\right)+2\lambda\|\theta\|_{F}^{2},$ $$

where inequality (1) follows from the fact that $\det(M)\leq\det(\lambda_{\max}(M)I_{n})=\lambda_{\max}^{n}(M),\forall M\in\mathbb{R}^{n\times n}$. When $kT\geq 200\left(3(n+m)+\log\left(\frac{1}{\delta}\right)\right)$, substituting ([52](#A2.E52)) and ([60](#A2.E60)) into ([61](#A2.E61)), with probability at least $1-4\delta$, we have

$$ $\displaystyle\left\|\theta^{k+1}-\theta\right\|^{2}\leq\left\|\theta^{k+1}-\theta\right\|_{F}^{2}$ $\displaystyle\leq\frac{80n}{cT\sqrt{k}}\left(\log\left(\frac{n^{2}}{\delta^{2}}\right)+(n+m)\log\left(1+\frac{\lambda+\tilde{c}k\log\left(\frac{TN}{\delta}\right)}{\lambda}\right)\right)+\frac{80\lambda(n+m)^{2}\widetilde{\Gamma}^{2}}{cT\sqrt{k}}.$ $$

When $kT<200\left(3(n+m)+\log\left(\frac{1}{\delta}\right)\right)$, because $\bar{V}^{k}=\lambda I+\sum_{i=1}^{k}\sum_{t=0}^{T-1}z_{t}^{i}z_{t}^{i\top}\succeq\lambda I$, with probability at least $1-3\delta$,

$$ $\displaystyle\left\|\theta^{k+1}-\theta\right\|^{2}\leq\left\|\theta^{k+1}-\theta\right\|_{F}^{2}$ $\displaystyle\leq\frac{2n}{\lambda}\left(\log\left(\frac{n^{2}}{\delta^{2}}\right)+(n+m)\log\left(1+\frac{\lambda+\tilde{c}k\log\left(\frac{TN}{\delta}\right)}{\lambda}\right)\right)+2(n+m)^{2}\widetilde{\Gamma}^{2}.$ $$

The proof is therefore complete.
∎

###### Lemma 13 .

###### Proof.

###### Lemma 14 .

###### Proof.

###### Lemma 15 .

###### Proof.

###### Lemma 16 .

###### Proof.

###### Lemma 17 .

###### Proof.

###### Proof of Proposition 6 .

### B.2 Perturbation Analysis of Riccati Equation

The perturbation analysis of Riccati equation under Algorithm [1](#alg1) and Algorithm [2](#alg2) is the same. So we can get the similar bounds of Riccati perturbation by replacing $\epsilon_{l}$ with $\epsilon_{k}$ in Lemma [8](#Thmlemma8), where $\epsilon_{k}=\max\{\|A^{k}-A\|,\|B^{k}-B\|\}$. The modified version of Lemma [8](#Thmlemma8) is presented in the following lemma.

###### Lemma 18 .

Assume $1-\gamma\widetilde{\Gamma}>0$ and fix any $\epsilon_{k}>0$. Suppose $\|A^{k}-A\|\leq\epsilon_{k},\|B^{k}-B\|\leq\epsilon_{k}$, then for any $t=0,1,\cdots,T-1$, we have

$$ $\displaystyle\|K^{k}_{t}-K_{t}\|\leq(10\mathcal{V}^{2}\mathcal{L}\widetilde{\Gamma}^{4})^{T-t-1}\mathcal{V}\epsilon_{k},$ $\displaystyle\|P^{k}_{t}-P_{t}\|\leq(10\mathcal{V}^{2}\mathcal{L}\widetilde{\Gamma}^{4})^{T-t}\epsilon_{k},$ $$

where the definitions of $\mathcal{V}$, $\mathcal{L}$ and $\widetilde{\Gamma}$ can be found in ([15](#S3.E15)).

###### Lemma 18 .

### B.3 Suboptimality Gap Due to the Controller Mismatch

In this section, we will connect the gap between the total cost under policy $\pi^{k}$ and the total cost under the optimal policy with the estimation error and the perturbation of Riccati equation in Appendix [B.1](#A2.SS1) and [B.2](#A2.SS2). The proof framework is similar to the framework in Appendix [A.3](#A1.SS3) except that we need to analyse the additional exploration noise added to the control. We define the total cost under entropic risk following policy $\pi^{k}$ (with slight abuse of notations) by

$$ $\displaystyle J_{0}^{\pi^{k}}\left(x^{k}_{0}\right)=\frac{1}{\gamma}\log\mathbb{E}\exp\left(\frac{\gamma}{2}\left(\sum_{t=0}^{T-1}\left(x_{t}^{k\top}Qx^{k}_{t}+u_{t}^{k\top}Ru^{k}_{t}\right)+x_{T}^{k\top}Q_{T}x_{T}^{k}\right)\right),$ $$

where $u_{t}^{k}=K_{t}^{k}x_{t}^{k}+g_{t}^{k},g_{t}^{k}\sim\mathcal{N}\left(0,\frac{1}{\sqrt{k}}I_{m}\right)$, $K_{t}^{k}$ is obtained by substituting $(A^{k},B^{k})$ into ([2.1](#S2.Ex2)). Similar to Appendix [A.3](#A1.SS3), we introduce the following new notations used in the regret analysis. For any $t=0,1,\cdots,T-2$, we define the following recursive equations:

$$ $\displaystyle D_{T-1}^{k}=\Delta K_{T-1}^{k\top}(R+B^{\top}\widetilde{P}_{T}B)\Delta K_{T-1}^{k},$ (62) $\displaystyle E_{T-1}^{k}=\sigma_{k}\left(RK_{T-1}^{k}+B^{\top}\widetilde{P}_{T}(A+BK_{T-1}^{k})\right),$ $\displaystyle F_{T-1}^{k}=\sigma_{k}^{2}\left(R+B^{\top}\widetilde{P}_{T}B\right),$ $\displaystyle U_{T-1}^{k}=D_{T-1}^{k}+\gamma E_{T-1}^{k\top}(I_{m}-\gamma F_{T-1}^{k})^{-1}E_{T-1}^{k},$ $\displaystyle\widetilde{U}_{T-1}^{k}=(I_{n}-\gamma U_{T-1}^{k})^{-1}U_{T-1}^{k},$ $\displaystyle D_{t}^{k}=\Delta K_{t}^{k\top}\left(R+B^{\top}\widetilde{P}_{t+1}B\right)\Delta K_{t}^{k}+(A+BK_{t}^{k})^{\top}\widetilde{U}_{t+1}^{k}(A+BK_{t}^{k}),$ $\displaystyle E_{t}^{k}=\sigma_{k}\left(RK_{t}^{k}+B^{\top}(\widetilde{P}_{t+1}+\widetilde{U}_{t+1}^{k})(A+BK_{t}^{k})\right),$ $\displaystyle F_{t}^{k}=\sigma_{k}^{2}\left(R+B^{\top}(\widetilde{P}_{t+1}+\widetilde{U}_{t+1}^{k})B\right),$ $\displaystyle U_{t}^{k}=D_{t}^{k}+\gamma E_{t}^{k\top}(I_{m}-\gamma F_{t}^{k})^{-1}E_{t}^{k},$ $\displaystyle\widetilde{U}_{t}^{k}=(I_{n}-\gamma U_{t}^{k})^{-1}U_{t}^{k},$ $$

where $\Delta K_{t}^{k}:=K_{t}^{k}-K_{t}$, $\sigma_{k}:=k^{-\frac{1}{4}}$ and $\widetilde{P}_{T}$ is defined in ([2.1](#S2.Ex2)).

We then follow the proof framework of Appendix [A.3](#A1.SS3) to derive the bounds for the suboptimality gap due to the controller mismatch. The key result of this section is the following proposition.

###### Proposition 7 .

We have

$$ $\displaystyle J_{0}^{\pi^{k}}(x_{0}^{k})-J^{\star}_{0}(x_{0}^{k})$ $\displaystyle=-\frac{1}{2\gamma}\sum_{t=0}^{T-1}\log\det\left(I_{n}-\gamma F_{t}^{k}\right)-\frac{1}{2\gamma}\sum_{t=1}^{T-1}\log\det\left(I_{m}-\gamma U_{t}^{k}\right)+\frac{1}{2}x_{0}^{k\top}U_{0}^{k}x_{0}^{k},$ (63) $$

where $F_{t}^{k}$ and $U_{t}^{k}$ are defined in ([62](#A2.E62)).

In order to prove Proposition [7](#Thmproposition7), we extend Lemma [10](#Thmlemma10) and prove the following result. Recall that $\sigma_{k}=k^{-\frac{1}{4}}$.

###### Lemma 19 .

For any $t\in[T-1]$, we have

$$ $\displaystyle E\left[\exp\left(\frac{\gamma}{2}\left[\begin{array}[]{l}x_{t}^{k}\\ \sigma_{k}^{-1}g_{t}^{k}\end{array}\right]^{\top}\left[\begin{array}[]{ll}D_{t}^{k}&E_{t}^{k\top}\\ E_{t}^{k}&F_{t}^{k}\end{array}\right]\left[\begin{array}[]{l}x_{t}^{k}\\ \sigma_{k}^{-1}g_{t}^{k}\end{array}\right]\right)\Bigg{|}\mathcal{H}_{t-1}^{k}\right]$ (64) $\displaystyle=\left(\det(I_{n}-\gamma U_{t}^{k})\right)^{-1/2}\cdot\left(\det(I_{m}-\gamma F_{t}^{k})\right)^{-1/2}$ $\displaystyle\quad\times\exp\Bigg{\{}\frac{\gamma}{2}\bigg{[}x_{t-1}^{k\top}(A+BK_{t-1}^{k})^{\top}\widetilde{U}_{t}^{k}(A+BK_{t-1}^{k})x_{t-1}^{k}$ $\displaystyle\qquad\qquad+2g_{t-1}^{k\top}B^{\top}\widetilde{U}_{t}^{k}(A+BK_{t-1}^{k})x_{t-1}^{k}+g_{t-1}^{k\top}B^{\top}\widetilde{U}_{t}^{k}Bg_{t-1}^{k}\bigg{]}\Bigg{\}},$ $$

where given $(x_{t-1}^{k},u_{t-1}^{k})$,

$$ $\left[\begin{array}[]{l}x_{t}^{k}\\ \sigma_{k}^{-1}g_{t}^{k}\end{array}\right]\sim\mathcal{N}\left(\left[\begin{array}[]{c}Ax_{t-1}^{k}+Bu_{t-1}^{k}\\ 0\end{array}\right],I_{n+m}\right).$ $$

###### Proof.

We obtain from Lemma [10](#Thmlemma10) that

$$ $\displaystyle\mathbb{E}\left[\exp\left(\frac{\gamma}{2}\left[\begin{array}[]{l}x_{t}^{k}\\ \sigma_{k}^{-1}g_{t}^{k}\end{array}\right]^{\top}\left[\begin{array}[]{cc}D_{t}^{k}&E_{t}^{k\top}\\ E_{t}^{k}&F_{t}^{k}\end{array}\right]\left[\begin{array}[]{l}x_{t}^{k}\\ \sigma_{k}^{-1}g_{t}^{k}\end{array}\right]\right)\Bigg{|}\mathcal{H}_{t-1}^{k}\right]$ (65) $\displaystyle=\left(\det\left(I_{m+n}-\gamma\left[\begin{array}[]{cc}D_{t}^{k}&E_{t}^{k\top}\\ E_{t}^{k}&F_{t}^{k}\end{array}\right]\right)\right)^{-\frac{1}{2}}$ $\displaystyle\quad\times\exp\Bigg{(}\frac{\gamma}{2}\left[\begin{array}[]{c}Ax_{t-1}^{k}+Bu_{t-1}^{k}\\ 0\end{array}\right]^{\top}\left(I_{m+n}-\gamma\left[\begin{array}[]{cc}D_{t}^{k}&E_{t}^{k\top}\\ E_{t}^{k}&F_{t}^{k}\end{array}\right]\right)^{-1}$ $\displaystyle\qquad\qquad\cdot\left[\begin{array}[]{cc}D_{t}^{k}&E_{t}^{k\top}\\ E_{t}^{k}&F_{t}^{k}\end{array}\right]\left[\begin{array}[]{c}Ax_{t-1}^{k}+Bu^{k}_{t-1}\\ 0\end{array}\right]\Bigg{)}.$ $$

We perform a block Gauss–Jordan elimination, take the inverse of the matrix in the determinant of ([65](#A2.E65)), and obtain

$$ $\displaystyle\left(I_{n+m}-\gamma\left[\begin{array}[]{cc}D_{t}^{k}&E_{t}^{k\top}\\ E_{t}^{k}&F_{t}^{k}\end{array}\right]\right)^{-1}$ $\displaystyle=\left[\begin{array}[]{cc}I_{n}-\gamma D_{t}^{k}&-\gamma E_{t}^{k\top}\\ -\gamma E_{t}^{k}&I_{m}-\gamma F_{t}^{k}\end{array}\right]^{-1}$ $\displaystyle=\left[\begin{array}[]{cc}I_{n}&0\\ (I_{m}-\gamma F_{t}^{k})^{-1}\gamma E_{t}^{k}&I_{m}\end{array}\right]\times\left[\begin{array}[]{cc}\left(I_{n}-\gamma D_{t}^{k}-\gamma^{2}E_{t}^{k\top}(I_{m}-\gamma F_{t}^{k})^{-1}E_{t}^{k}\right)^{-1}&0\\ 0&(I_{m}-\gamma F_{t}^{k})^{-1}\end{array}\right]$ $\displaystyle\quad\times\left[\begin{array}[]{cc}I_{n}&\gamma E_{t}^{k\top}(I_{m}-\gamma F_{t}^{k})^{-1}\\ 0&I_{m}\end{array}\right]$ $\displaystyle=\left[\begin{array}[]{cc}EL_{1}&EL_{2}\\ EL_{3}&EL_{4}\end{array}\right],$ $$

where

$$ $\displaystyle EL_{1}$ $\displaystyle=\left(I_{n}-\gamma D_{t}^{k}-\gamma^{2}E_{t}^{k\top}(I_{m}-\gamma F_{t}^{k})^{-1}E_{t}^{k}\right)^{-1},$ $\displaystyle EL_{2}$ $\displaystyle=(I_{n}-\gamma D_{t}^{k}-\gamma^{2}E_{t}^{k\top}(I_{m}-\gamma F_{t}^{k})^{-1}E_{t}^{k})^{-1}\gamma E_{t}^{k\top}(I_{m}-\gamma F_{t}^{k})^{-1}),$ $\displaystyle EL_{3}$ $\displaystyle=(I_{m}-\gamma F_{t}^{k})^{-1}\gamma E_{t}^{k}(I_{n}-\gamma D_{t}^{k}-\gamma^{2}E_{t}^{k\top}(I_{m}-\gamma F_{t}^{k})^{-1}E_{t}^{k})^{-1},$ $\displaystyle EL_{4}$ $\displaystyle=(I_{m}-\gamma F_{t}^{k})^{-1}\gamma E_{t}^{k}(I_{n}-\gamma D_{t}^{k}-\gamma^{2}E_{t}^{k\top}(I_{m}-\gamma F_{t}^{k})^{-1}E_{t}^{k})^{-1}\gamma E_{t}^{k\top}(I_{m}-\gamma F_{t}^{k})^{-1}$ $\displaystyle+(I_{m}-\gamma F_{t}^{k})^{-1}.$ $$

Then, we can obtain

$$ $\displaystyle\left(\det\left(I_{n+m}-\gamma\left[\begin{array}[]{cc}D_{t}^{k}&E_{t}^{k\top}\\ E_{t}^{k}&F_{t}^{k}\end{array}\right]\right)\right)^{-\frac{1}{2}}$ (66) $\displaystyle=\left(\det\left(I_{n}-\gamma D_{t}^{k}-\gamma^{2}E_{t}^{k\top}(I_{m}-\gamma F_{t}^{k})^{-1}E_{t}^{k}\right)\right)^{-\frac{1}{2}}\cdot\left(\det(I_{m}-\gamma F_{t}^{k})\right)^{-\frac{1}{2}}$ $\displaystyle=\left(\det\left(I_{n}-\gamma U_{t}^{k}\right)\right)^{-\frac{1}{2}}\cdot\left(\det(I_{m}-\gamma F_{t}^{k})\right)^{-\frac{1}{2}},$ $$

where $U_{t}^{k}=D_{t}^{k}+\gamma E_{t}^{k\top}(I_{m}-\gamma F_{t}^{k})^{-1}E_{t}^{k}$, and

$$ $\displaystyle\left[\begin{array}[]{c}Ax_{t-1}^{k}+Bu_{t-1}^{k}\\ 0\end{array}\right]^{\top}\left(I_{n+m}-\gamma\left[\begin{array}[]{cc}D_{t}^{k}&E_{t}^{k\top}\\ E_{t}^{k}&F_{t}^{k}\end{array}\right]\right)^{-1}\left[\begin{array}[]{cc}D_{t}^{k}&E_{t}^{k\top}\\ E_{t}^{k}&F_{t}^{k}\end{array}\right]\left[\begin{array}[]{c}Ax_{t-1}^{k}+Bu^{k}_{t-1}\\ 0\end{array}\right]$ (67) $\displaystyle=(Ax_{t-1}^{k}+Bu_{t-1}^{k})^{\top}\Bigg{[}\left(I_{n}-\gamma D_{t}^{k}-\gamma E_{t}^{k\top}(I_{m}-\gamma F_{t}^{k})^{-1}E_{t}^{k}\right)^{-1}$ $\displaystyle\times\left(D_{t}^{k}+\gamma E_{t}^{k\top}(I_{m}-\gamma F_{t}^{k})^{-1}E_{t}^{k}\right)\Bigg{]}(Ax_{t-1}^{k}+Bu_{t-1}^{k})$ $\displaystyle=\left((A+BK_{t-1}^{k})x_{t-1}^{k}+Bg_{t-1}^{k}\right)^{\top}\widetilde{U}_{t}^{k}\left((A+BK_{t-1}^{k})x_{t-1}^{k}+Bg_{t-1}^{k}\right)$ $\displaystyle=x_{t-1}^{k\top}(A+BK_{t-1}^{k})^{\top}\widetilde{U}_{t}^{k}(A+BK_{t-1}^{k})x_{t-1}^{k}+2g_{t-1}^{k\top}B^{\top}\widetilde{U}_{t}^{k}(A+BK_{t-1}^{k})x_{t-1}^{k}+g_{t-1}^{k\top}B^{\top}\widetilde{U}_{t}^{k}Bg_{t-1}^{k},$ $$

where $\widetilde{U}_{t}^{k}=(I_{n}-\gamma U_{t}^{k})^{-1}U_{t}^{k}$.
On combining ([66](#A2.E66)) and ([67](#A2.E67)) with ([65](#A2.E65)), we can obtain ([64](#A2.E64)).
∎

The following lemma is an extension of Lemma [11](#Thmlemma11), which provides a coarse simplification of the performance gap in one episode.

###### Lemma 20 .

With Lemma [10](#Thmlemma10), we can simplify the performance gap to

$$ $\displaystyle J_{0}^{\pi^{k}}(x^{k}_{0})-J_{0}^{\star}(x^{k}_{0})$ $\displaystyle=\frac{1}{\gamma}\log\mathbb{E}\Bigg{[}\exp\bigg{(}\frac{\gamma}{2}\sum_{t=0}^{T-1}\Big{(}x_{t}^{k\top}\Delta K_{t}^{k\top}(R+B^{\top}\widetilde{P}_{t+1}B)\Delta K_{t}^{k}x_{t}^{k}$ $\displaystyle\qquad\qquad\qquad\quad+2g_{t}^{k\top}(RK_{t}^{k}+B^{\top}\widetilde{P}_{t+1}(A+BK_{t}^{k}))x_{t}^{k}$ $\displaystyle\qquad\qquad\qquad\quad+g_{t}^{k\top}(R+B^{\top}\widetilde{P}_{t+1}B)g_{t}^{k}\Big{)}\bigg{)}\Bigg{|}x_{0}^{k},\mathcal{H}_{T}^{k-1}\Bigg{]},$ $$

where $\Delta K_{t}^{k}=K^{k}_{t}-K_{t}$ and $\mathcal{H}_{T}^{k-1}$ is defined in ([46](#A2.E46)).

###### Proof.

By a similar procedure as in ([33](#A1.E33)), we can derive that

$$ $\displaystyle J_{0}^{\pi^{k}}(x^{k}_{0})-J_{0}^{\star}(x^{k}_{0})$ (68) $\displaystyle=\frac{1}{\gamma}\log\mathbb{E}\left[\exp\left(\frac{\gamma}{2}\sum_{t=0}^{T-1}\left(c_{t}(x_{t}^{k},u_{t}^{k})+J_{t+1}(x^{k}_{t+1})-J_{t}(x^{k}_{t})\right)\right)\Bigg{|}x_{0}^{k},\mathcal{H}_{T}^{k-1}\right]$ $\displaystyle\overset{(1)}{=}\frac{1}{\gamma}\log\mathbb{E}\Bigg{[}\exp\Bigg{(}\gamma\sum_{t=0}^{T-1}\Bigg{(}\frac{1}{2}x_{t}^{k\top}(Q+K_{t}^{k\top}RK^{k}_{t})x^{k}_{t}+g_{t}^{k\top}RK_{t}^{k}x_{t}^{k}+\frac{1}{2}g_{t}^{k\top}Rg_{t}^{k}$ $\displaystyle\qquad\qquad\qquad\quad+\frac{1}{2}x^{k\top}_{t+1}P_{t+1}x^{k}_{t+1}-\frac{1}{2}x^{k\top}_{t}P_{t}x^{k}_{t}+\frac{1}{2\gamma}\log\det(I_{n}-\gamma P_{t+1})\Bigg{)}\Bigg{)}\Bigg{|}x_{0}^{k},\mathcal{H}_{T}^{k-1}\Bigg{]},$ $$

where equality (1) follows from the definition of the total cost under entropic risk and $u_{t}^{k}=K_{t}^{k}x_{t}^{k}+g_{t}^{k}$. Again, we apply the law of total expectation, and compute

$$ $\displaystyle\mathbb{E}\Bigg{[}\exp\Bigg{(}\gamma\Big{(}\frac{1}{2}x_{t}^{k\top}(Q+K_{t}^{k\top}RK^{k}_{t})x^{k}_{t}+g_{t}^{k\top}RK_{t}^{k}x_{t}^{k}+\frac{1}{2}g_{t}^{k\top}Rg_{t}^{k}$ (69) $\displaystyle\qquad\quad+\frac{1}{2}x^{k\top}_{t+1}P_{t+1}x^{k}_{t+1}-\frac{1}{2}x^{k\top}_{t}P_{t}x^{k}_{t}+\frac{1}{2\gamma}\log\det(I_{n}-\gamma P_{t+1})\Big{)}\Bigg{)}\Bigg{|}\mathcal{H}_{t}^{k}\Bigg{]}$ $\displaystyle=\exp\Bigg{(}\gamma\Bigg{(}\frac{1}{2}x_{t}^{k\top}(Q+K_{t}^{k\top}RK^{k}_{t})x^{k}_{t}-\frac{1}{2}x^{k\top}_{t}P_{t}x^{k}_{t}+g_{t}^{k\top}RK_{t}^{k}x_{t}^{k}+\frac{1}{2}g_{t}^{k\top}Rg_{t}^{k}$ $\displaystyle\qquad\quad+\frac{1}{2\gamma}\log\det(I_{n}-\gamma P_{t+1})\Bigg{)}\Bigg{)}\times\mathbb{E}\left[\exp\left(\frac{\gamma}{2}x_{t+1}^{k\top}P_{t+1}x^{k}_{t+1}\right)\Bigg{|}\mathcal{H}_{t}^{k}\right]$ $$

It follows from Lemma [10](#Thmlemma10) that

$$ $\displaystyle\eqref{eq:gap2}$ $\displaystyle=\exp\Bigg{(}\gamma\Bigg{(}\frac{1}{2}x_{t}^{k\top}(Q+K_{t}^{k\top}RK^{k}_{t})x^{k}_{t}-\frac{1}{2}x^{k\top}_{t}P_{t}x^{k}_{t}+g_{t}^{k\top}RK_{t}^{k}x_{t}^{k}+\frac{1}{2}g_{t}^{k\top}Rg_{t}^{k}$ $\displaystyle\qquad\quad+\frac{1}{2\gamma}\log\det(I_{n}-\gamma P_{t+1})\Bigg{)}\Bigg{)}\times(\det(I_{n}-\gamma P_{t+1}))^{-1/2}$ $\displaystyle\quad\times\exp\Bigg{[}\frac{\gamma}{2}x_{t}^{k\top}(A+BK^{k}_{t})^{\top}\widetilde{P}_{t+1}(A+BK^{k}_{t})x^{k}_{t}+\gamma g_{t}^{k\top}B^{\top}\widetilde{P}_{t+1}(A+BK_{t}^{k})x_{t}^{k}$ $\displaystyle\qquad\qquad+\frac{\gamma}{2}g_{t}^{k\top}B^{\top}\widetilde{P}_{t+1}Bg_{t}^{k}\Bigg{]}$ $\displaystyle=\exp\bigg{[}\frac{\gamma}{2}\Big{(}x_{t}^{k\top}(Q+K_{t}^{k\top}RK^{k}_{t}+(A+BK^{k}_{t})^{\top}\widetilde{P}_{t+1}(A+BK^{k}_{t}))x^{k}_{t}-x_{t}^{k\top}P_{t}x^{k}_{t}\Big{)}$ $\displaystyle\qquad\quad+\gamma g_{t}^{k\top}(RK_{t}^{k}+B^{\top}\widetilde{P}_{t+1}(A+BK_{t}^{k}))x_{t}^{k}+\frac{\gamma}{2}g_{t}^{k\top}(R+B^{\top}\widetilde{P}_{t+1}B)g_{t}^{k}\bigg{]},$ $$

Substituting $\Delta K_{t}^{k}=K_{t}^{k}-K_{t}$ and the Riccati equation $P_{t}=Q+K_{t}^{\top}RK_{t}+(A+BK_{t})^{\top}\widetilde{P}_{t+1}(A+BK_{t})$ into ([69](#A2.E69)), we obtain

$$ $\displaystyle\exp\Big{[}\frac{\gamma}{2}x_{t}^{k\top}\Big{(}Q+(\Delta K_{t}^{k}+K_{t})^{\top}R(\Delta K_{t}^{k}+K_{t})+(A+B(\Delta K_{t}^{k}+K_{t}))^{\top}\widetilde{P}_{t+1}(A+B(\Delta K_{t}^{k}+K_{t}))\Big{)}x_{t}^{k}$ (70) $\displaystyle\qquad+\gamma g_{t}^{k\top}(RK_{t}^{k}+B^{\top}\widetilde{P}_{t+1}(A+BK_{t}^{k}))x_{t}^{k}+\frac{\gamma}{2}g_{t}^{k\top}(R+B^{\top}\widetilde{P}_{t+1}B)g_{t}^{k}-\frac{\gamma}{2}x_{t}^{k\top}P_{t}x_{t}^{k}\Big{]}$ $\displaystyle=\exp\Big{[}\frac{\gamma}{2}x_{t}^{k\top}\Delta K_{t}^{k\top}(R+B^{\top}\widetilde{P}_{t+1}B)\Delta K_{t}^{k}x_{t}^{k}+\gamma x_{t}^{k\top}\Delta K_{t}^{k\top}\big{(}(R+B^{\top}\widetilde{P}_{t+1}B)K_{t}+B^{\top}\widetilde{P}_{t+1}A\big{)}x_{t}^{k}$ $\displaystyle\qquad\quad+\gamma g_{t}^{k\top}(RK_{t}^{k}+B^{\top}\widetilde{P}_{t+1}(A+BK_{t}^{k}))x_{t}^{k}+\frac{\gamma}{2}\ g_{t}^{k\top}(R+B^{\top}\widetilde{P}_{t+1}B)g_{t}^{k}\Big{]}$ $\displaystyle\overset{(2)}{=}\exp\Big{[}\frac{\gamma}{2}x_{t}^{k\top}\Delta K_{t}^{k\top}(R+B^{\top}\widetilde{P}_{t+1}B)\Delta K_{t}^{k}x_{t}^{k}+\gamma g_{t}^{k\top}(RK_{t}^{k}+B^{\top}\widetilde{P}_{t+1}(A+BK_{t}^{k}))x_{t}^{k}$ $\displaystyle\qquad\quad+\frac{\gamma}{2}g_{t}^{k\top}(R+B^{\top}\widetilde{P}_{t+1}B)g_{t}^{k}\Big{]},$ $$

where the equality (2) holds by the fact that $K_{t}=-(R+B^{\top}\widetilde{P}_{t+1}B)^{-1}B^{\top}\widetilde{P}_{t+1}A$. Substituting ([70](#A2.E70)) into ([69](#A2.E69)) and then substituting ([69](#A2.E69)) into ([68](#A2.E68)), we can get

$$ $\displaystyle J_{0}^{\pi^{k}}(x^{k}_{0})-J_{0}^{\star}(x^{k}_{0})$ $\displaystyle=\frac{1}{\gamma}\log\mathbb{E}\Bigg{[}\exp\bigg{(}\frac{\gamma}{2}\sum_{t=0}^{T-1}\Big{(}x_{t}^{k\top}\Delta K_{t}^{k\top}(R+B^{\top}\widetilde{P}_{t+1}B)\Delta K_{t}^{k}x_{t}^{k}$ $\displaystyle\quad+2g_{t}^{k\top}(RK_{t}^{k}+B^{\top}\widetilde{P}_{t+1}(A+BK_{t}^{k}))x_{t}^{k}+g_{t}^{k\top}(R+B^{\top}\widetilde{P}_{t+1}B)g_{t}^{k}\Big{)}\bigg{)}\Bigg{|}x_{0}^{k},\mathcal{H}_{T}^{k-1}\Bigg{]}.$ $$

The proof is complete.
∎

Combining Lemma [19](#Thmlemma19) with Lemma [20](#Thmlemma20), we can prove Proposition [7](#Thmproposition7).

###### Proof of Proposition 7 .

We prove the result recursively. Recall that $\sigma_{k}=k^{-\frac{1}{4}}$. When $t=T-1$,

$$ $\displaystyle\mathbb{E}\Bigg{[}\exp\bigg{(}\frac{\gamma}{2}\Big{(}x_{T-1}^{k\top}\Delta K_{T-1}^{k\top}(R+B^{\top}\widetilde{P}_{T}B)\Delta K_{T-1}^{k}x_{T-1}^{k}$ (71) $\displaystyle\quad+2\sigma_{k}\sigma_{k}^{-1}g_{T-1}^{k\top}(RK_{T-1}^{k}+B^{\top}\widetilde{P}_{T}(A+BK_{T-1}^{k}))x_{T-1}^{k}+\sigma_{k}^{2}\sigma_{k}^{-2}g_{T-1}^{k\top}(R+B^{\top}\widetilde{P}_{T}B)g_{T-1}^{k}\Big{)}\bigg{)}\Bigg{|}\mathcal{H}_{T-2}^{k}\Bigg{]}$ $\displaystyle\overset{(1)}{=}\mathbb{E}\left[\exp\left(\frac{\gamma}{2}\left(x_{T-1}^{k\top}D_{T-1}^{k}x_{T-1}^{k}+2\sigma_{k}^{-1}g_{T-1}^{k\top}E_{T-1}^{k}x_{T-1}^{k}+\sigma_{k}^{-2}g_{T-1}^{k\top}F_{T-1}^{k}g_{T-1}^{k}\right)\right)\Big{|}\mathcal{H}_{T-2}^{k}\right]$ $\displaystyle\overset{(2)}{=}\left(\det(I_{n}-\gamma U_{T-1}^{k})\right)^{-\frac{1}{2}}\left(\det(I_{m}-\gamma F_{T-1}^{k})\right)^{-\frac{1}{2}}$ $\displaystyle\quad\times\exp\bigg{\{}\frac{\gamma}{2}\Big{[}x_{T-2}^{k\top}(A+BK_{T-2}^{k})^{\top}\widetilde{U}_{T-1}^{k}(A+BK_{T-2}^{k})x_{T-2}^{k}$ $\displaystyle\qquad\qquad+2g_{T-2}^{k\top}B^{\top}\widetilde{U}_{T-1}^{k}(A+BK_{T-2}^{k})x_{T-2}^{k}+g_{T-2}^{k\top}B^{\top}\widetilde{U}_{T-1}^{k}Bg_{T-2}^{k}\Big{]},$ $$

where equality (1) holds by ([62](#A2.E62)), and equality (2) follows from Lemma [19](#Thmlemma19). When $t=T-2$, we have

$$ $\displaystyle\mathbb{E}\Bigg{[}\exp\bigg{(}\frac{\gamma}{2}\sum_{t=T-2}^{T-1}\Big{(}x_{t}^{k\top}\Delta K_{t}^{k\top}(R+B^{\top}\widetilde{P}_{t+1}B)\Delta K_{t}^{k}x_{t}^{k}+2\sigma_{k}\sigma_{k}^{-1}g_{t}^{k\top}(RK_{t}^{k}+B^{\top}\widetilde{P}_{t+1}(A+BK_{t}^{k}))x_{t}^{k}$ $\displaystyle\quad+\sigma_{k}^{2}\sigma_{k}^{-2}g_{t}^{k\top}(R+B^{\top}\widetilde{P}_{t+1}B)g_{t}^{k}\Big{)}\bigg{)}\Bigg{|}\mathcal{H}_{T-3}^{k}\Bigg{]}$ $\displaystyle\overset{(3)}{=}\left(\det(I_{n}-\gamma U_{T-1}^{k})\right)^{-\frac{1}{2}}\left(\det(I_{m}-\gamma F_{T-1}^{k})\right)^{-\frac{1}{2}}$ $\displaystyle\quad\times\mathbb{E}\Bigg{[}\exp\bigg{(}\frac{\gamma}{2}\bigg{[}x_{T-2}^{k\top}\Big{(}\Delta K_{T-2}^{k\top}(R+B^{\top}\widetilde{P}_{T-1}B)\Delta K_{T-2}^{k}+(A+BK_{T-2}^{k})^{\top}\widetilde{U}_{T-1}^{k}(A+BK_{T-2}^{k})\Big{)}x_{T-2}^{k}$ $\displaystyle\qquad\qquad\quad+2\sigma_{k}\sigma_{k}^{-1}g_{T-2}^{k\top}(RK_{T-2}^{k}+B^{\top}(\widetilde{P}_{T-1}+\widetilde{U}_{T-1}^{k})(A+BK_{T-2}^{l}))x_{T-2}^{k}$ $\displaystyle\qquad\qquad\quad+\sigma_{k}^{2}\sigma_{k}^{-2}g_{T-2}^{k\top}(R+B^{\top}(\widetilde{P}_{T-1}+\widetilde{U}_{T-1}^{k})B)g_{T-2}^{k}\bigg{]}\bigg{)}\Bigg{|}\mathcal{H}_{T-3}^{k}\Bigg{]}$ $\displaystyle\overset{(4)}{=}\left(\det(I_{n}-\gamma U_{T-1}^{k})\right)^{-\frac{1}{2}}\left(\det(I_{m}-\gamma F_{T-1}^{k})\right)^{-\frac{1}{2}}$ $\displaystyle\quad\times\mathbb{E}\left[\exp\left(\frac{\gamma}{2}\left(x_{T-2}^{k\top}D_{T-2}^{k}x_{T-2}^{k}+2\sigma_{k}^{-1}g_{T-2}^{k\top}E_{T-2}^{k}x_{T-2}^{k}+\sigma_{k}^{-2}g_{T-2}^{k\top}F_{T-2}^{k}g_{T-2}^{k}\right)\right)\Big{|}\mathcal{H}_{T-3}^{k}\right]$ $\displaystyle\overset{(5)}{=}\prod_{t=T-2}^{T-1}\left(\det(I_{n}-\gamma U_{t}^{k})\right)^{-\frac{1}{2}}\left(\det(I_{m}-\gamma F_{t}^{k})\right)^{-\frac{1}{2}}$ $\displaystyle\quad\times\exp\Bigg{(}\frac{\gamma}{2}\Big{[}x_{T-3}^{k\top}(A+BK_{T-3}^{k})^{\top}\widetilde{U}_{T-2}^{k}(A+BK_{T-3}^{k})x_{T-3}^{k}$ $\displaystyle\qquad\qquad+2g_{T-3}^{k\top}B^{\top}\widetilde{U}_{T-2}^{k}(A+BK_{T-3}^{k})x_{T-3}^{k}+g_{T-3}^{k\top}B^{\top}\widetilde{U}_{T-2}^{k}Bg_{T-3}^{k}\Big{]}\Bigg{)},$ $$

where equality (3) holds by applying the law of total expectation and applying ([71](#A2.E71)), equality (4) follows from ([62](#A2.E62)) and equality (5) still holds by applying Lemma [19](#Thmlemma19). When $t=i$, we can similarly derive

$$ $\displaystyle\mathbb{E}\Bigg{[}\exp\bigg{(}\frac{\gamma}{2}\sum_{t=i}^{T-1}\Big{(}x_{t}^{k\top}\Delta K_{t}^{k\top}(R+B^{\top}\widetilde{P}_{t+1}B)\Delta K_{t}^{k}x_{t}^{k}$ $\displaystyle\quad+2\sigma_{k}\sigma_{k}^{-1}g_{t}^{k\top}(RK_{t}^{k}+B^{\top}\widetilde{P}_{t+1}(A+BK_{t}^{k}))x_{t}^{k}+\sigma_{k}^{2}\sigma_{k}^{-2}g_{t}^{k\top}(R+B^{\top}\widetilde{P}_{t+1}B)g_{t}^{k}\Big{)}\bigg{)}\Bigg{|}\mathcal{H}_{i-1}^{k}\Bigg{]}$ $\displaystyle=\prod_{t=i}^{T-1}\left(\det(I_{n}-\gamma U_{t}^{k})\right)^{-\frac{1}{2}}\left(\det(I_{m}-\gamma F_{t}^{k})\right)^{-\frac{1}{2}}$ $\displaystyle\quad\times\exp\Bigg{(}\frac{\gamma}{2}\Big{[}x_{i-1}^{k\top}(A+BK_{i-1}^{k})^{\top}\widetilde{U}_{i}^{k}(A+BK_{i-1}^{k})x_{i-1}^{k}$ $\displaystyle\qquad\qquad+2g_{i-1}^{k\top}B^{\top}\widetilde{U}_{i}^{k}(A+BK_{i-1}^{k})x_{i-1}^{k}+g_{i-1}^{k\top}B^{\top}\widetilde{U}_{i}^{k}Bg_{i-1}^{k}\Big{]}\Bigg{)}.$ $$

Repeating this procedure, we can obtain

$$ $\displaystyle J_{0}^{\pi^{k}}(x_{0}^{k})-J^{\star}_{0}(x_{0}^{k})$ $\displaystyle=\frac{1}{\gamma}\log\left(\prod_{t=1}^{T-1}\left(\det(I_{n}-\gamma U_{t}^{k})\right)^{-\frac{1}{2}}\left(\det(I_{m}-\gamma F_{t}^{k})\right)^{-\frac{1}{2}}\right)$ $\displaystyle\quad+\frac{1}{\gamma}\log\mathbb{E}\Bigg{[}\exp\Bigg{(}\frac{\gamma}{2}\bigg{[}x_{0}^{k\top}\left(\Delta K_{0}^{k\top}(R+B^{\top}\widetilde{P}_{1}B)\Delta K_{0}^{k}+(A+BK_{0}^{k})^{\top}\widetilde{U}_{1}^{k}(A+BK_{0}^{k})\right)x_{0}^{k}$ $\displaystyle\qquad+2g_{0}^{k\top}(RK_{0}^{k}+B^{\top}(\widetilde{P}_{1}+\widetilde{U}_{1}^{k})(A+BK_{0}^{l}))x_{0}^{k}+\sigma_{k}^{2}\sigma_{k}^{-2}g_{0}^{k\top}(R+B^{\top}(\widetilde{P}_{1}+\widetilde{U}_{1}^{k})B)g_{0}^{k}\bigg{]}\Bigg{)}\Bigg{|}x_{0}^{k},H_{T}^{k-1}\Bigg{]}$ $\displaystyle=\frac{1}{\gamma}\log\left(\prod_{t=1}^{T-1}\left(\det(I_{n}-\gamma U_{t}^{k})\right)^{-\frac{1}{2}}\left(\det(I_{m}-\gamma F_{t}^{k})\right)^{-\frac{1}{2}}\right)$ $\displaystyle\quad+\frac{1}{\gamma}\log\mathbb{E}\left[\exp\left(\frac{\gamma}{2}\left(x_{0}^{k\top}D_{0}^{k}x_{0}^{k}+2\sigma_{k}^{-1}g_{0}^{k\top}E_{0}^{k}x_{0}^{k}+\sigma_{k}^{-2}g_{0}^{k\top}F_{0}^{k}g_{0}^{k}\right)\right)\Big{|}x_{0}^{k},H_{T}^{k-1}\right]$ $\displaystyle\overset{(6)}{=}-\frac{1}{2\gamma}\sum_{t=1}^{T-1}\left[\log\det\left(I_{n}-\gamma U_{t}^{k}\right)+\log\det\left(I_{m}-\gamma F_{t}^{k}\right)\right]$ $\displaystyle\quad+\frac{1}{2}x_{0}^{k\top}D_{0}^{k}x_{0}^{k}-\frac{1}{2\gamma}\log\det\left(I_{m}-\gamma F_{0}^{k}\right)+\frac{1}{2}\gamma x_{0}^{k\top}E_{0}^{k\top}\left(I_{m}-\gamma F_{0}^{k}\right)^{-1}E_{0}^{k}x_{0}^{k}$ $\displaystyle\overset{(7)}{=}-\frac{1}{2\gamma}\sum_{t=0}^{T-1}\log\det\left(I_{n}-\gamma F_{t}^{k}\right)-\frac{1}{2\gamma}\sum_{t=1}^{T-1}\log\det\left(I_{m}-\gamma U_{t}^{k}\right)+\frac{1}{2}x_{0}^{k\top}U_{0}^{k}x_{0}^{k},$ $$

where equality (6) holds by directly calculating the conditional expectation of quadratic function of $g_{0}^{k}$ in the second term of the previous equality and equality (7) follows from the definition of $U_{0}^{k}$ in ([62](#A2.E62)).
∎

###### Proposition 7 .

###### Lemma 19 .

###### Proof.

###### Lemma 20 .

###### Proof.

###### Proof of Proposition 7 .

### B.4 Proof of Theorem 2

Now, we can derive the regret upper bound for Algorithm 2. Similar to Appendix [A.4](#A1.SS4), we derive the bounds for the equations in ([62](#A2.E62)). We recursively define the following constants similarly as ([36](#A1.E36)) in Appendix [A.4](#A1.SS4). For any $t=0,\cdots,T-2$,

$$ $\displaystyle\alpha_{T-1}=2\widetilde{\Gamma}^{3},$ $\displaystyle\beta_{T-1}=0,$ $\displaystyle\alpha_{t}=2\widetilde{\Gamma}\left(10\mathcal{V}^{2}\mathcal{L}\widetilde{\Gamma}^{4}\right)^{2(T-t-1)}+12\widetilde{\Gamma}^{4}\alpha_{t+1},$ $\displaystyle\beta_{t}=12\widetilde{\Gamma}^{4}+12\widetilde{\Gamma}^{4}\beta_{t+1},$ $$

where $\mathcal{V}$ and $\mathcal{L}$ are defined in ([15](#S3.E15)). To derive the regret bounds, we assume that for any $t=0,\cdots,T-1$,

$$ $\displaystyle\gamma\leq\frac{1}{2\alpha_{t}\mathcal{V}^{2}\epsilon_{k}^{2}+2\beta_{t}\cdot 5\widetilde{\Gamma}^{5}\sigma_{k}^{2}+10\widetilde{\Gamma}^{5}\sigma_{k}^{2}}.$ (72) $$

We are now ready to derive the bounds for $D_{t}^{k},E_{t}^{k},F_{t}^{k},U_{t}^{k},\widetilde{U}_{t}^{k}$ recursively from step $T-1$ to step 0 0 conditional on event $\widetilde{\mathcal{G}}^{k}$ in ([45](#A2.E45)). At step $T-1$,

$$ $\displaystyle\|D_{T-1}^{k}\|$ $\displaystyle=\left\|\Delta K_{T-1}^{k\top}(R+B^{\top}\widetilde{P}_{T}B)\Delta K_{T-1}^{k}\right\|$ (73) $\displaystyle\leq\|R+B^{\top}\widetilde{P}_{T}B\|\cdot\|\Delta K_{T-1}^{k}\|^{2}$ $\displaystyle\overset{(1)}{\leq}2\widetilde{\Gamma}^{3}\mathcal{V}^{2}\epsilon_{k}^{2}$ $\displaystyle=\alpha_{T-1}\mathcal{V}^{2}\epsilon_{k}^{2}+5\widetilde{\Gamma}^{5}\beta_{T-1}\sigma_{k}^{2},$ $$

where inequality (1) holds by the definition of $\widetilde{\Gamma}$ in ([15](#S3.E15)) and Lemma [18](#Thmlemma18). Similarly,

$$ $\displaystyle\left\|E_{T-1}^{k}\right\|$ $\displaystyle=\left\|\sigma_{k}(RK_{T-1}^{k}+B^{\top}\widetilde{P}_{T}(A+BK_{T-1}^{k})\right\|$ (74) $\displaystyle\leq\sigma_{k}\|R(\Delta K_{T-1}^{k}+K_{T-1})\|$ $\displaystyle\quad+\sigma_{k}\|B\|\cdot\|\widetilde{P}_{T}\|\cdot\left(\|A\|+\|B\|\cdot\|K_{T-1}\|+\|B\|\cdot\|\Delta K_{T-1}^{k}\|\right)$ $\displaystyle\overset{(2)}{\leq}\sigma_{k}\widetilde{\Gamma}(\mathcal{V}\epsilon_{k}+\widetilde{\Gamma})+\sigma_{k}\widetilde{\Gamma}^{2}(2\widetilde{\Gamma}^{2}+\widetilde{\Gamma}\mathcal{V}\epsilon_{k})$ $\displaystyle=2\sigma_{k}\widetilde{\Gamma}^{4}+\sigma_{k}\widetilde{\Gamma}^{2}+\sigma_{k}(\widetilde{\Gamma}+\widetilde{\Gamma}^{3})\mathcal{V}\epsilon_{k}$ $\displaystyle\leq 3\sigma_{k}\widetilde{\Gamma}^{4}+2\sigma_{k}\widetilde{\Gamma}^{3}\mathcal{V}\epsilon_{k}$ $$

where inequality (2) holds by the definition of $\widetilde{\Gamma}$ in ([15](#S3.E15)) and Lemma [18](#Thmlemma18). We also have

$$ $\displaystyle\left\|F_{T-1}^{k}\right\|$ $\displaystyle=\left\|\sigma_{k}^{2}(R+B^{\top}\widetilde{P}_{T}B)\right\|\leq 2\widetilde{\Gamma}^{3}\sigma_{k}^{2}.$ (75) $$

Substitute ([73](#A2.E73)), ([74](#A2.E74)) and ([75](#A2.E75)) into $U_{T-1}^{k}$, we can derive that

$$ $\displaystyle\left\|U_{T-1}^{k}\right\|$ $\displaystyle=\left\|D_{T-1}^{k}+\gamma E_{T-1}^{k\top}(I_{m}-\gamma F_{T-1}^{k})^{-1}E_{T-1}^{k}\right\|$ (76) $\displaystyle\leq\|D_{T-1}^{k}\|+\gamma\left\|E_{T-1}^{k\top}(I_{m}-\gamma F_{T-1}^{k})^{-1}E_{T-1}^{k}\right\|$ $\displaystyle\leq\|D_{T-1}^{k}\|+\gamma\|(I_{m}-\gamma F_{T-1}^{k})^{-1}\|\cdot\|E_{T-1}^{k}\|^{2}$ $\displaystyle\overset{(3)}{\leq}2\widetilde{\Gamma}^{3}\mathcal{V}^{2}\epsilon_{k}^{2}+\gamma\left(3\sigma_{k}\widetilde{\Gamma}^{4}+2\sigma_{k}\widetilde{\Gamma}^{3}\mathcal{V}\epsilon_{k}\right)^{2}\cdot\frac{1}{1-2\widetilde{\Gamma}^{3}\gamma\sigma_{k}^{2}}$ $\displaystyle=2\widetilde{\Gamma}^{3}\mathcal{V}^{2}\epsilon_{k}^{2}+\gamma\left(9\sigma_{k}^{2}\widetilde{\Gamma}^{8}+4\sigma_{k}^{2}\widetilde{\Gamma}^{6}\mathcal{V}^{2}\epsilon_{k}^{2}+12\sigma_{k}^{2}\widetilde{\Gamma}^{7}\mathcal{V}\epsilon_{k}\right)\cdot\frac{1}{1-2\widetilde{\Gamma}^{3}\gamma\sigma_{k}^{2}}$ $\displaystyle\overset{(4)}{\leq}2\widetilde{\Gamma}^{3}\mathcal{V}^{2}\epsilon_{k}^{2}+18\gamma\sigma_{k}^{2}\widetilde{\Gamma}^{8}+o(\epsilon_{k}^{2})$ $\displaystyle\overset{(5)}{\leq}2\widetilde{\Gamma}^{3}\mathcal{V}^{2}\epsilon_{k}^{2}+5\sigma_{k}^{2}\widetilde{\Gamma}^{5}+o(\epsilon_{k}^{2})$ $\displaystyle=\alpha_{T-1}\mathcal{V}^{2}\epsilon_{k}^{2}+5\widetilde{\Gamma}^{5}\beta_{T-1}\sigma_{k}^{2}+5\widetilde{\Gamma}^{5}\sigma_{k}^{2}+o(\epsilon_{k}^{2}),$ $$

where inequality (3) follows by substituting ([73](#A2.E73)), ([74](#A2.E74)) and ([75](#A2.E75)) into ([76](#A2.E76)) and the fact that for any matrix $M\in R^{n\times n}$, if $\|M\|<1$, then $\|(I_{n}-M)^{-1}\|\leq\frac{1}{1-\|M\|}$, inequality (4) holds by the assumption in ([72](#A2.E72)), i.e. $1-2\widetilde{\Gamma}^{3}\gamma\sigma_{k}^{2}\geq 1-\frac{1}{2}\sigma_{k}^{2}\geq\frac{1}{2}$, and inequality (5) still holds by assumption ([72](#A2.E72)), i.e. $18\gamma\sigma_{k}^{2}\widetilde{\Gamma}^{8}\leq 18\sigma_{k}^{2}\widetilde{\Gamma}^{8}\cdot\frac{1}{4\widetilde{\Gamma}^{3}}\leq 5\widetilde{\Gamma}^{5}\sigma_{k}^{2}$. Note that conditional on event $\widetilde{\mathcal{G}}^{k}$, $\epsilon_{k}^{2}$ is of order $\frac{1}{\sqrt{k}}$, so $\epsilon_{k}^{2}$ and $\sigma_{k}^{2}$ share the same order conditional on event $\widetilde{\mathcal{G}}^{k}$. Then, we can obtain

$$ $\displaystyle\left\|\widetilde{U}_{T-1}^{k}\right\|$ $\displaystyle=\left\|(I_{n}-\gamma U_{t}^{k})^{-1}U_{t}^{k}\right\|\overset{(6)}{\leq}\frac{\|U_{T-1}^{k}\|}{1-\gamma\|U_{T-1}^{k}\|}\overset{(7)}{\leq}2\|U_{T-1}^{k}\|,$ (77) $$

where inequality (6) still holds by the fact that for any matrix $M\in R^{n\times n}$, if $\|M\|<1$, then $\|(I_{n}-M)^{-1}\|\leq\frac{1}{1-\|M\|}$, and inequality (7) holds by the assumption in ([72](#A2.E72)), i.e. $1-\gamma\|U_{T-1}^{k}\|\geq 1-\frac{1}{2}=\frac{1}{2}$. It follows from ([76](#A2.E76)) that

$$ $\left\|\widetilde{U}_{T-1}^{k}\right\|\leq 2\alpha_{T-1}\mathcal{V}^{2}\epsilon_{k}^{2}+10\widetilde{\Gamma}^{5}\beta_{T-1}\sigma_{k}^{2}+10\widetilde{\Gamma}^{5}\sigma_{k}^{2}+o(\epsilon_{k}^{2}).$ $$

With the bounds in the $(T-1)$-th step, we can recursively derive the bounds in the $(T-2)$-th step. At step $T-2$, by the similar arguments in step $T-1$, we can obtain

$$ $\displaystyle\left\|D_{T-2}^{k}\right\|$ $\displaystyle=\left\|\Delta K_{T-2}^{k\top}\left(R+B^{\top}\widetilde{P}_{T-1}B\right)\Delta K_{T-2}^{k}+(A+BK_{T-2}^{k})^{\top}\widetilde{U}_{T-1}^{k}(A+BK_{T-2}^{k})\right\|$ $\displaystyle\leq\left\|R+B^{\top}\widetilde{P}_{T-1}B\right\|\cdot\left\|\Delta K_{T-2}^{k}\right\|^{2}+\left\|A+B\Delta K_{T-2}^{k}+BK_{T-2}\right\|^{2}\cdot\left\|\widetilde{U}_{T-1}^{k}\right\|$ $\displaystyle\overset{(8)}{\leq}2\widetilde{\Gamma}^{3}\left(10\mathcal{V}^{2}\mathcal{L}\widetilde{\Gamma}^{4}\right)^{2}\mathcal{V}^{2}\epsilon_{k}^{2}$ $\displaystyle\quad+3\left(\widetilde{\Gamma}^{2}+\widetilde{\Gamma}^{4}+\widetilde{\Gamma}^{2}\left(10\mathcal{V}^{2}\mathcal{L}\widetilde{\Gamma}^{4}\right)^{2}\mathcal{V}^{2}\epsilon_{k}^{2}\right)\left(4\widetilde{\Gamma}^{3}\mathcal{V}^{2}\epsilon_{k}^{2}+10\sigma_{k}^{2}\widetilde{\Gamma}^{5}+o(\epsilon_{k}^{2})\right)$ $\displaystyle\leq 2\widetilde{\Gamma}^{3}\left(10\mathcal{V}^{2}\mathcal{L}\widetilde{\Gamma}^{4}\right)^{2}\mathcal{V}^{2}\epsilon_{k}^{2}+12\widetilde{\Gamma}^{4}\left(2\widetilde{\Gamma}^{3}\mathcal{V}^{2}\epsilon_{k}^{2}+5\widetilde{\Gamma}^{5}\sigma_{k}^{2}\right)+o(\epsilon_{k}^{2})$ $\displaystyle=\alpha_{T-2}\mathcal{V}^{2}\epsilon_{k}^{2}+5\widetilde{\Gamma}^{5}\beta_{T-2}\sigma_{k}^{2}+o(\epsilon_{k}^{2}),$ $$

where inequality (8) holds by Lemma [18](#Thmlemma18), ([77](#A2.E77)) and the fact that $\left\|\sum_{t=1}^{K}x_{t}\right\|^{2}\leq K\sum_{t=1}^{K}\left\|x_{t}\right\|^{2}$. Similar to ([74](#A2.E74)), we have

$$ $\displaystyle\left\|E_{T-2}^{k}\right\|$ $\displaystyle=\left\|\sigma_{k}\left(RK_{T-2}^{k}+B^{\top}\left(\widetilde{P}_{T-1}+\widetilde{U}_{T-1}^{k}\right)(A+BK_{T-2}^{k})\right)\right\|$ $\displaystyle\leq\sigma_{k}\left\|RK_{T-2}^{k}+B^{\top}\widetilde{P}_{T-1}(A+BK_{T-2}^{k})\right\|+\sigma_{k}\left\|B^{\top}\widetilde{U}_{T-1}^{k}(A+BK_{T-2}^{k})\right\|$ $\displaystyle\leq\sigma_{k}\left\|R\Delta K_{T-2}^{k}+RK_{T-2}\right\|+\sigma_{k}\|B\|\cdot\left\|\widetilde{P}_{T-1}\right\|\cdot\left\|A+B\Delta K_{T-2}^{k}+BK_{T-2}\right\|$ $\displaystyle\quad+\sigma_{k}\|B\|\cdot\left\|\widetilde{U}_{T-1}^{k}\right\|\cdot\left\|A+B\Delta K_{T-2}^{k}+BK_{T-2}\right\|$ $\displaystyle\overset{(9)}{\leq}\sigma_{k}\widetilde{\Gamma}\left(\left(10\mathcal{V}^{2}\mathcal{L}\widetilde{\Gamma}^{4}\right)\mathcal{V}\epsilon_{k}+\widetilde{\Gamma}\right)+\sigma_{k}\widetilde{\Gamma}^{2}\left(2\widetilde{\Gamma}^{2}+\widetilde{\Gamma}\left(10\mathcal{V}^{2}\mathcal{L}\widetilde{\Gamma}^{4}\right)\mathcal{V}\epsilon_{k}\right)$ $\displaystyle\quad+\sigma_{k}\widetilde{\Gamma}\left(4\widetilde{\Gamma}^{3}\mathcal{V}^{2}\epsilon_{k}^{2}+10\sigma_{k}^{2}\widetilde{\Gamma}^{5}\right)\cdot\left(2\widetilde{\Gamma}^{2}+\widetilde{\Gamma}\left(10\mathcal{V}^{2}\mathcal{L}\widetilde{\Gamma}^{4}\right)\mathcal{V}\epsilon_{k}\right)$ $\displaystyle=\sigma_{k}\left(\widetilde{\Gamma}\left(10\mathcal{V}^{2}\mathcal{L}\widetilde{\Gamma}^{4}\right)+\widetilde{\Gamma}^{3}\left(10\mathcal{V}^{2}\mathcal{L}\widetilde{\Gamma}^{4}\right)\right)\mathcal{V}\epsilon_{k}+\sigma_{k}\left(\widetilde{\Gamma}^{2}+2\widetilde{\Gamma}^{4}\right)+o(\epsilon_{k}^{2})$ $\displaystyle\leq 2\sigma_{k}\widetilde{\Gamma}^{3}\left(10\mathcal{V}^{2}\mathcal{L}\widetilde{\Gamma}^{4}\right)\mathcal{V}\epsilon_{k}+3\widetilde{\Gamma}^{4}\sigma_{k}+o(\epsilon_{k}^{2}),$ $$

where inequality (9) follows from Lemma [18](#Thmlemma18) and ([76](#A2.E76)). Similar to ([75](#A2.E75)), we have

$$ $\displaystyle\left\|F_{T-2}^{k}\right\|$ $\displaystyle=\left\|\sigma_{k}^{2}\left(R+B^{\top}\left(\widetilde{P}_{T-1}+\widetilde{U}_{T-1}^{k}\right)B\right)\right\|\leq 2\widetilde{\Gamma}^{3}\sigma_{k}^{2}+o(\epsilon_{k}^{2}).$ $$

Similar to ([76](#A2.E76)), we have

$$ $\displaystyle\left\|U_{T-2}^{k}\right\|$ $\displaystyle\leq\|D_{T-2}^{k}\|+\gamma\|(I_{m}-\gamma F_{T-2}^{k})^{-1}\|\cdot\|E_{T-2}^{k}\|^{2}$ (78) $\displaystyle\leq 2\widetilde{\Gamma}^{3}\left(10\mathcal{V}^{2}\mathcal{L}\widetilde{\Gamma}^{4}\right)^{2}\mathcal{V}^{2}\epsilon_{k}^{2}+6\widetilde{\Gamma}^{4}\left(4\widetilde{\Gamma}^{3}\mathcal{V}^{2}\epsilon_{k}^{2}+10\widetilde{\Gamma}^{5}\sigma_{k}^{2}\right)+18\gamma\sigma^{2}\widetilde{\Gamma}^{8}\sigma_{k}^{2}+o(\epsilon_{k}^{2})$ $\displaystyle\overset{(10)}{\leq}2\widetilde{\Gamma}^{3}\left(10\mathcal{V}^{2}\mathcal{L}\widetilde{\Gamma}^{4}\right)^{2}\mathcal{V}^{2}\epsilon_{k}^{2}+6\widetilde{\Gamma}^{4}\left(4\widetilde{\Gamma}^{3}\mathcal{V}^{2}\epsilon_{k}^{2}+10\widetilde{\Gamma}^{5}\sigma_{k}^{2}\right)+5\widetilde{\Gamma}^{5}\sigma_{k}^{2}+o(\epsilon_{k}^{2})$ $\displaystyle=\alpha_{T-2}\mathcal{V}^{2}\epsilon_{k}^{2}+5\widetilde{\Gamma}^{5}\beta_{T-2}\sigma_{k}^{2}+5\widetilde{\Gamma}^{5}\sigma_{k}^{2}+o(\epsilon_{k}^{2}),$ $$

where inequality (10) follows from the assumption ([72](#A2.E72)) and the similar arguments in ([76](#A2.E76)). Then, by ([78](#A2.E78)) and assumption ([72](#A2.E72)), we can get

$$ $\displaystyle\left\|\widetilde{U}^{k}_{T-2}\right\|\leq 2\alpha_{T-2}\mathcal{V}^{2}\epsilon_{k}^{2}+10\widetilde{\Gamma}^{5}\beta_{T-2}\sigma_{k}^{2}+10\widetilde{\Gamma}^{5}\sigma_{k}^{2}+o(\epsilon_{k}^{2}).$ $$

Repeat this procedure from step $T-1$ to step 0 0, we can get the following recursive inequalities. For any $t=0,\cdots,T-2$,

$$ $\displaystyle\left\|D_{T-1}^{k}\right\|$ $\displaystyle\leq\alpha_{T-1}\mathcal{V}^{2}\epsilon_{k}^{2}+5\widetilde{\Gamma}^{5}\beta_{T-1}\sigma_{k}^{2},$ (79) $\displaystyle\left\|F_{T-1}^{k}\right\|$ $\displaystyle\leq 2\widetilde{\Gamma}^{3}\sigma_{k}^{2},$ $\displaystyle\left\|U_{T-1}^{k}\right\|$ $\displaystyle\leq\alpha_{T-1}\mathcal{V}^{2}\epsilon_{k}^{2}+5\widetilde{\Gamma}^{5}\beta_{T-1}\sigma_{k}^{2}+5\widetilde{\Gamma}^{5}\sigma_{k}^{2}+o(\epsilon_{k}^{2}),$ $\displaystyle\left\|D_{t}^{k}\right\|$ $\displaystyle\leq\alpha_{t}\mathcal{V}^{2}\epsilon_{k}^{2}+5\widetilde{\Gamma}^{5}\beta_{t}\sigma_{k}^{2}+o(\epsilon_{k}^{2}),$ $\displaystyle\left\|F_{t}^{k}\right\|$ $\displaystyle\leq 2\widetilde{\Gamma}^{3}\sigma_{k}^{2}+o(\epsilon_{k}^{2}),$ $\displaystyle\left\|U_{t}^{k}\right\|$ $\displaystyle\leq\alpha_{t}\mathcal{V}^{2}\epsilon_{k}^{2}+5\widetilde{\Gamma}^{5}\beta_{t}\sigma_{k}^{2}+5\widetilde{\Gamma}^{5}\sigma_{k}^{2}+o(\epsilon_{k}^{2}).$ $$

Substituting ([79](#A2.E79)) into ([7](#A2.Ex181)) in Lemma [7](#Thmproposition7), we have

$$ $\displaystyle J_{0}^{\pi^{k}}(x_{0}^{k})-J^{\star}_{0}(x_{0}^{k})$ (80) $\displaystyle=-\frac{1}{2\gamma}\sum_{t=1}^{T-1}\log\det\left(I_{n}-\gamma U_{t}^{k}\right)-\frac{1}{2\gamma}\sum_{t=0}^{T-1}\log\det\left(I_{m}-\gamma F_{t}^{k}\right)+\frac{1}{2}x_{0}^{k\top}U_{0}^{k}x_{0}^{k}$ $\displaystyle\overset{(11)}{\leq}-\frac{1}{2\gamma}\sum_{t=1}^{T-1}\log\left(1-\gamma\left\|U_{t}^{k}\right\|\right)^{n}-\frac{1}{2\gamma}\sum_{t=0}^{T-1}\log\left(1-\gamma\left\|F_{t}^{k}\right\|\right)^{m}+\frac{1}{2}\left\|U_{0}^{k}\right\|\cdot\left\|x_{0}^{k}\right\|^{2}$ $\displaystyle\overset{(12)}{\leq}-\frac{1}{2\gamma}\sum_{t=1}^{T-1}n\log\left(1-\gamma\left(\alpha_{t}\mathcal{V}^{2}\epsilon_{k}^{2}+5\widetilde{\Gamma}^{5}\beta_{t}\sigma_{k}^{2}+5\widetilde{\Gamma}^{5}\sigma_{k}^{2}+o(\epsilon_{k}^{2})\right)\right)$ $\displaystyle\quad-\frac{1}{2\gamma}\sum_{t=0}^{T-1}m\log\left(1-\gamma\left(2\widetilde{\Gamma}^{3}\sigma_{k}^{2}+o(\epsilon_{k}^{2})\right)\right)+\frac{1}{2}\left(\alpha_{0}\mathcal{V}^{2}\epsilon_{k}^{2}+5\widetilde{\Gamma}^{5}\beta_{0}\sigma_{k}^{2}+5\widetilde{\Gamma}^{5}\sigma_{k}^{2}+o(\epsilon_{k}^{2})\right)\left\|x_{0}\right\|^{2}$ $\displaystyle\overset{(13)}{\leq}\frac{n}{2}\sum_{t=1}^{T-1}\left(\alpha_{t}\mathcal{V}^{2}\epsilon_{k}^{2}+5\widetilde{\Gamma}^{5}\beta_{t}\sigma_{k}^{2}+5\widetilde{\Gamma}^{5}\sigma_{k}^{2}+o(\epsilon_{k}^{2})\right)+\frac{m}{2}\sum_{t=0}^{T-1}\left(2\widetilde{\Gamma}^{3}\sigma_{k}^{2}+o(\epsilon_{k}^{2})\right)$ $\displaystyle\quad+\frac{1}{2}\left(\alpha_{0}\mathcal{V}^{2}\epsilon_{k}^{2}+5\widetilde{\Gamma}^{5}\beta_{0}\sigma_{k}^{2}+5\widetilde{\Gamma}^{5}\sigma_{k}^{2}+o(\epsilon_{k}^{2})\right)\left\|x_{0}\right\|^{2},$ $$

where inequality (11) holds because

$$ $I_{n}-\gamma U_{t}^{k}\succeq\left(1-\gamma\left\|U_{t}^{k}\right\|\right)I_{n}\succeq\left(1-\gamma\left(\alpha_{t}\mathcal{V}^{2}\epsilon_{k}^{2}+5\widetilde{\Gamma}^{5}\beta_{t}\sigma_{k}^{2}+5\widetilde{\Gamma}^{5}\sigma_{k}^{2}+o(\epsilon_{k}^{2})\right)\right)I_{n}\succ 0,$ $$

and

$$ $I_{m}-\gamma F_{t}^{k}\succeq\left(1-\gamma\left\|F_{t}^{k}\right\|\right)I_{m}\succeq\left(1-\gamma\left(2\widetilde{\Gamma}^{3}\sigma_{k}^{2}+o(\epsilon_{k}^{2})\right)\right)I_{m}\succ 0,$ $$

inequality (12) follows from ([79](#A2.E79)), inequality (13) holds because $\log(1+x)\leq x$ for any $x>-1$.

Then, substituting the high probability bounds derived in Appendix [B.1](#A2.SS1) into ([80](#A2.E80)), we can further bound $J_{0}^{\pi^{k}}(x_{0}^{k})-J^{\star}_{0}(x_{0}^{k})$. According to Proposition [6](#Thmproposition6), conditional on event $\widetilde{\mathcal{G}}^{k}$ defined in ([45](#A2.E45)), when $kT\geq 200\left(3(n+m)+\log\left(\frac{4N}{\delta}\right)\right)$, with probability at least $1-\frac{\delta}{N-1}$, we have

$$ $\left\|\theta^{k+1}-\theta\right\|^{2}\leq\epsilon_{k}:=\frac{C_{N}}{\sqrt{k}}.$ (81) $$

where

$$ $C_{N}=\frac{160n}{cT}\left(\log\left(\frac{4nN}{\delta}\right)+(n+m)\log\left(1+\frac{\tilde{c}N\log\left(\frac{4TN^{2}}{\delta}\right)}{\lambda}\right)\right)+\frac{80\lambda(n+m)^{2}\widetilde{\Gamma}^{2}}{cT},$ (82) $$

and $c,\tilde{c}$ are defined in ([50](#A2.E50)) and ([59](#A2.E59)). Denote $\tilde{k}=\left\lceil\frac{200\left(3(n+m)+\log\left(\frac{4N}{\delta}\right)\right)}{T}\right\rceil$. When $k>\tilde{k}$, the estimation error bounds are given by ([81](#A2.E81)).

By a similar mathematical induction as discussed in Section [A.4](#A1.SS4) and page 27 in Basei et al., (), we can prove that the event $\widetilde{\mathcal{G}}=\left\{\|\theta^{k}-\theta\|\leq\varpi,\forall k=2,\cdots,N\right\}\cup\left\{\theta^{1}\in\Xi\right\}$ holds with probability at least $1-\sum_{i=2}^{N}\frac{\delta}{N-1}=1-\delta$, i.e. $\mathbb{P}\left(\widetilde{\mathcal{G}}\right)\geq 1-\delta$, where $\varpi$ is defined in ([43](#A2.E43)).

Finally, conditional on the event $\widetilde{\mathcal{G}}$, we can derive an upper bound for $\operatorname{Regret}(N)$. Note that

$$ $\displaystyle\operatorname{Regret}(N)=\sum_{k=1}^{\tilde{k}}\left(J^{\pi^{k}}(x_{0}^{k})-J^{\star}(x_{0}^{k})\right)+\sum_{k=\tilde{k}+1}^{N}\left(J^{\pi^{k}}(x_{0}^{k})-J^{\star}(x_{0}^{k})\right),$ (83) $$

where $\tilde{k}=\left\lceil\frac{200\left(3(n+m)+\log\left(\frac{4N}{\delta}\right)\right)}{T}\right\rceil$. We bound the two terms in ([83](#A2.E83)) separately. We first bound the regret incurred up to the $\tilde{k}$-th episode. We have

$$ $\displaystyle\sum_{k=1}^{\tilde{k}}\left(J^{\pi^{k}}(x_{0}^{k})-J^{\star}(x_{0}^{k})\right)\leq\sum_{k=1}^{\tilde{k}}J^{\pi^{k}}(x_{0}^{k})\leq\sum_{k=1}^{\tilde{k}}\frac{1}{\gamma}\log\mathbb{E}\exp\left(\frac{\gamma\widetilde{\Gamma}}{2}\left(\sum_{t=0}^{T-1}\left(\|x_{t}^{k}\|^{2}+\|u_{t}^{k}\|^{2}\right)+\|x_{T}^{k}\|^{2}\right)\right).$ (84) $$

It follows from ([57](#A2.E57)) in Lemma [16](#Thmlemma16) that

$$ $\displaystyle\sum_{k=1}^{\tilde{k}}\left(J^{\pi^{k}}(x_{0}^{k})-J^{\star}(x_{0}^{k})\right)$ $\displaystyle\leq\frac{\widetilde{\Gamma}\tilde{k}}{2}\Bigg{(}\tilde{c}\log\left(\frac{TN}{\delta}\right)+72T\left(\widetilde{\Gamma}(1+C_{K})\right)^{2T}\left(n^{\frac{3}{2}}+m^{\frac{3}{2}}\right)\max\{\|x_{0}\|^{2},1\}\widetilde{\Gamma}^{2}\log\left(\frac{TN}{\delta}\right)\Bigg{)}.$ $$

We next bound the regret in the remaining episodes as follows:

$$ $\displaystyle\sum_{k=\tilde{k}+1}^{N}\left(J^{\pi^{k}}(x_{0}^{k})-J^{\star}(x_{0}^{k})\right)$ (85) $\displaystyle\leq\sum_{k={\tilde{k}+1}}^{N}\Bigg{[}\frac{n}{2}\sum_{t=1}^{T-1}\left(\alpha_{t}\mathcal{V}^{2}\epsilon_{k}^{2}+5\widetilde{\Gamma}^{5}\beta_{t}\sigma_{k}^{2}+5\widetilde{\Gamma}^{5}\sigma_{k}^{2}+o(\epsilon_{k}^{2})\right)+\frac{m}{2}\sum_{t=0}^{T-1}\left(2\widetilde{\Gamma}^{3}\sigma_{k}^{2}+o(\epsilon_{k}^{2})\right)$ $\displaystyle\qquad\qquad+\frac{1}{2}\left(\alpha_{0}\mathcal{V}^{2}\epsilon_{k}^{2}+5\widetilde{\Gamma}^{5}\beta_{0}\sigma_{k}^{2}+5\widetilde{\Gamma}^{5}\sigma_{k}^{2}+o(\epsilon_{k}^{2})\right)\left\|x_{0}\right\|^{2}\Bigg{]}$ $\displaystyle\leq\sum_{k=\tilde{k}+1}^{N}\Bigg{[}\frac{n}{2}\sum_{t=1}^{T-1}\left(\frac{\alpha_{t}\mathcal{V}^{2}C_{N}}{\sqrt{k}}+\frac{5\widetilde{\Gamma}^{5}\beta_{t}}{\sqrt{k}}+\frac{5\widetilde{\Gamma}^{5}}{\sqrt{k}}+o(\epsilon_{k}^{2})\right)+\frac{m}{2}\sum_{t=0}^{T-1}\left(\frac{2\widetilde{\Gamma}^{3}}{\sqrt{k}}+o(\epsilon_{k}^{2})\right)$ $\displaystyle\qquad\qquad+\frac{1}{2}\left(\frac{\alpha_{0}\mathcal{V}^{2}C_{N}}{\sqrt{k}}+\frac{5\widetilde{\Gamma}^{5}\beta_{0}}{\sqrt{k}}+\frac{5\widetilde{\Gamma}^{5}}{\sqrt{k}}+o(\epsilon_{k}^{2})\right)\left\|x_{0}\right\|^{2}\Bigg{]}$ $\displaystyle\leq\Bigg{[}n\sum_{t=1}^{T-1}\left(\alpha_{t}\mathcal{V}^{2}C_{N}+5\widetilde{\Gamma}^{5}\beta_{t}+5\widetilde{\Gamma}^{5}\right)+2mT\widetilde{\Gamma}^{3}+\left(\alpha_{0}\mathcal{V}^{2}C_{N}+5\widetilde{\Gamma}^{5}\beta_{0}+5\widetilde{\Gamma}^{5}\right)\|x_{0}\|^{2}\Bigg{]}\sqrt{N}+o\left(\sqrt{N}\right),$ $$

where the first inequality follows from ([80](#A2.E80)), the second inequality follows from ([81](#A2.E81)) and $C_{N}$ is given in ([82](#A2.E82)). On combining ([84](#A2.E84)) with ([85](#A2.E85)), we can obtain

$$ $\displaystyle\operatorname{Regret}(N)\leq\widetilde{\mathcal{C}}\sum_{t=0}^{T-1}\left(\alpha_{t}C_{N}+\beta_{t}\right)\sqrt{N},$ $$

where $\widetilde{\mathcal{C}}:=\operatorname{Polynomial}\left(n,m,\tilde{c},\mathcal{V},\widetilde{\Gamma},T,\tilde{k},\|x_{0}\|,\epsilon_{1}\right)\cdot\left(\widetilde{\Gamma}(1+C_{K})\right)^{2T}.$

## Peer Reviews

confidence: 4
contribution: 2
initial_rating: 5
presentation: 2
questions: Please refer to Weakness.
soundness: 3
strengths: 1.	This paper initiate the consideration of risk-sensitive formulation of LQR in an online episodic setting.
2.	The paper rigorously characterize the regret bounds of the proposed algorithms.
summary: This paper considers the risk-sensitive LQR problem with unknown system matrices. The authors consider the standard LEQR formulation to capture the risk-aware scenario in the control problem and study the episodic problem setting where the system needs to be reset to the initial condition at the end of each episode. The authors propose online learning algorithms to solve the problem and show that the regret of the algorithm is bounded as \sqrt{N}, where N is the number of episodes in the problem. Under some additional assumption, the authors show that the regret bound becomes \logN.
weaknesses: 1. Although the authors mentioned a previous work Basei et al. 2022 that considers an episodic control problem, it would be good to further explain why it is important/of practical interest to consider the episodic setting in the control problem. In addition, it would be good to discuss what the major challenge is if we move to the non-episodic setting which is the most standard setting in control problems (i.e., the system involves continuously and does not reset).
2. It would be good to explain more how the LEQR formulation captures the risk-aware scenario in Section 2.1. The authors also mention that their analysis extends to \gamma<0. What does \gamma<0 stand for in the risk-aware scenario? In general, it would be good to provide an intuitive explanation of how different values of γ (positive, negative, or zero) correspond to different risk attitudes. This would help readers better understand the practical implications of the LEQR formulation.
3. The authors should justify their choice to use only data from epoch l-1 for estimation in Algorithm 1, and discuss whether using data from multiple previous epochs could improve accuracy. Additionally, they should clarify whether the i.i.d. assumption on initial states is necessary for their results to hold.
4. The optimal controller based on the initial estimate \theta^1 needs to satisfy Assumption~1 as well in Theorem 1. Could you justify this assumption on \theta^1, i.e., how do you choose \theta^1 so that the assumption is satisfied? 
5. Regarding the regret bounds provided in Theorems~1 and 2, they depend exponentially on the horizon length T for each episode. However, in episodic MDP problems, the regret bound depends on T only polynomially (e.g., https://proceedings.mlr.press/v70/azar17a/azar17a.pdf). Could the authors comment on this issue:  whether this is fundamental to the risk-sensitive setting or if it's an artifact of the regret analysis? Moreover, would it be possible to improve this dependence to polynomial in T?
6. Could you explain why the least squares approach in Algoirthm~2 needs a regularization term?
7. Please explicitly specify how the initial estimate \theta^1 is chosen in Algorithm~2.

---

confidence: 3
contribution: 3
initial_rating: 8
presentation: 3
questions: Please address the weaknesses of the paper. I have a few more questions/comments:
1. Instead of using exponential noise, can a UCB (upper confidence bound)-based exploration strategy in Algorithm 2?

2. How to extend these results for other risk measures, e.g., coherent risk measures (VaR, CVaR, EVaR, etc.) as studies in the following paper:
	1. Lam et al., [Risk-Aware Reinforcement Learning with Coherent Risk Measures and Non-linear Function Approximation](https://openreview.net/pdf?id=-RwZOVybbj)

3. What is the overall dependence of regret bounds given in Theorem 1 and 2 on horizon length ($T$) and risk-sensitivity parameter ($\gamma$)?


**Minor comment:**

Adding the layman's explanation and consequences of Assumption 1 will help readers identify when this assumption can hold in practice. Also, details (in the appendix and refer to the main paper) can added on how the standard Riccati equations are modified to incorporate exponential risk-sensitive costs.

I am open to changing my score based on the authors' responses.
soundness: 3
strengths: **The following are the strengths of the paper:**
1. This paper is the first to provide regret bounds for the episodic finite-horizon LEQR problem, which has applications in risk-sensitive control problems in areas like finance and healthcare.

2. The authors proposed two algorithms with sub-linear regret bounds guarantees. The regret bounds are derived using perturbation analysis of modified Riccati equations, which incorporate exponential risk-sensitive cost (defined in Eq. 2).

3. Finally, the authors have demonstrated the different performance aspects (sub-linear regret and effect of risk parameter and horizon on regret) of the proposed algorithms on two synthetic LEQR systems.
summary: This paper studies the online adaptive control of risk-sensitive Linear Quadratic Regulator (LQR) problem in a finite-horizon episodic setting, which is referred to as the Linear Exponential-of-Quadratic Regulator (LEQR) problem. The goal is to design the online algorithm that selects control that minimizes the total regret, i.e., the difference between the sum of minimum exponential risk-sensitive cost (defined in Eq. 2) and the algorithm's cost in each episode.

The authors introduce two algorithms: a least-squares greedy algorithm (Algorithm 1) that achieves $\tilde{O}(\log N)$ regret under identifiability condition (Assumption 1) and a second algorithm (Algorithm 2: Least-Squares-Based Algorithm with Exploration Noise) that incorporates exploration noise to achieve $\tilde{O}(\sqrt{N​})$ regret without the identifiability assumption. Finally, the authors have shown the different performance aspects of the proposed algorithms on two synthetic LEQR systems.
weaknesses: **The following are the weaknesses of the paper:**
1. Since verifying the identifiability assumption (Assumption 1) for a given problem may not be possible, the first algorithm may not be useful in practice. 

2. Both proposed algorithms are restricted to fixed finite-horizon settings and linear dynamics, which limits their real-world application, where horizon length can vary across episodes and problems with non-linear dynamics.

3. The following parts of the paper are not clear enough:
    - Why are the $N$ episodes in Algorithm 1 divided into epochs of increasing lengths to estimate the system matrices?
    - Why does the paper only consider the exponential risk-sensitive cost? Are there any real-world motivating examples for this choice?

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confidence: 3
contribution: 2
initial_rating: 6
presentation: 3
questions: *1.* Do the two theorems hold when gamma goes to 0, i.e., the limit is right-continuous in gamma? 

*2.* Typo: line 218-219, "We" play.
soundness: 3
strengths: The paper is well-written and easy to follow. All the assumptions are well-motivated and standard in the literature. The proposed algorithms are simple and can be easily implemented by practitioners. 

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summary: The paper studies an online linear quadratic regulator (LQR) problem with risk-sensitive cost functions. In particular, when the environment can provide enough exploration noise, a least-square model fitting algorithm is provided and it achieves log N regret. When the environment is not "noisy" enough, the authors present a gaussian exploration method that achieves square root N regret. Numerical experiments are provided to validate the proposed algorithms.
weaknesses: *1.* I think the authors should provide an explicit dependence of gamma in the two theorems, i.e., the big O should contain gamma orders, as gamma is the key in the studied risk-sensitive LQR problems. With gamma dependence in the results, the authors could compare the regret order with risk-neural MDP/LQR problems. Discussions/guidance on gamma selections could be included.

*2.* Literature/discussions on risk-averse RL/LQR problems are limited, in particular, the choice of risk-averse metrics and the reason for studying the LEQR problem.

*3.* The authors could provide a more interesting real-world control problem in the simulation. 

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confidence: 3
contribution: 3
initial_rating: 8
presentation: 3
questions: - As the authors noted, studying regret bounds for LEQR in the infinite-horizon average-reward (non-episodic) setting would be interesting. What challenges arise in extending the current proof methods to this setting?
soundness: 3
strengths: - The authors introduce two algorithms based on least-squares methods tailored for LEQR. The first least-squares greedy algorithm without exploration noise achieves a logarithmic regret bound $O(\log n)$ under an identifiability condition, and the second algorithm, designed for scenarios where the identifiability assumption does not hold, achieves a $O(\sqrt{n})$ regret bound.
- This work provides the first regret bounds for finite-horizon episodic LEQR, showing the existence of logarithmic regret under identifiability assumption.
- The theoretical analysis relies on perturbation analysis of special Riccati equations for the risk-sensitive LEQR setting, where additional matrices ($\widetilde P$) are present in the Riccati equations due to the risk sensitivity. The authors show nontrivial analysis techniques addressing the challenges in the analysis.
- The authors also introduce novel analysis on risk-sensitive loss.
summary: This paper proposes a least-squares greedy algorithm for the online adaptive control of the risk-sensitive linear quadratic regulator (LQR) in a finite-horizon, episodic setting, which achieves a regret bound of $O(\log n)$ under identifiability assumption. It also proposes adding exploration noise to the least-squares algorithm when the identifiability assumption does not hold, resulting in a modified approach with a regret bound of $O(\sqrt{n})$.
weaknesses: - It would be beneficial to establish matching lower bounds for both algorithms to provide a more comprehensive understanding of their optimality.
- Typo/Inconsistency of the plot descriptions: Regret performance of Algorithm 1 (System 1) vs Effect of the risk parameter (Algorithm 1 System 1)

Meta-review: This paper introduces a problem setting for online learning in risk-sensitive episodic LQR optimal control, proposes an algorithm, and establishes its regret upper bound. While the paper has weaknesses, such as the absence of a regret lower bound and insufficient explanation of its motivation and applications, the authors have adequately addressed these issues raised by the reviewers. The reviewers have reached a consensus with positive opinions, and therefore, I support the acceptance of this paper.

Decision: Accept (Poster)