Logarithmic Regret for Episodic Continuous-Time Linear-Quadratic Reinforcement Learning over a Finite-Time Horizon

In the area of adaptive control with unknown dynamics parameters, the goal is to find optimal stationary policy that stabilizes the unknown dynamics and minimizes the long term average cost (Ioannou and Fidan (2006); Landau et al. (2011)). For an infinite-time horizon LQ system, it has been shown that persistent excitation conditions Green and Moore (1986) are critical to the parameter identification. Meanwhile, algorithms with asymptotic convergence in both the parameter estimation and the optimal control have been developed in Goodwin et al. (1981), Kumar (1983) and Campi and Kumar (1998): the first one assumes that costs only depend on state variables and the other two consider both state and control costs and use a cost-biased least-squared estimation method. See Faradonbeh et al. (2018a, 2019) and references therein for recent developments of (randomised) adaptive control algorithms for LQ systems.

Following the seminal works of Auer and Ortner (2007); Auer et al. (2009) and Osband et al. (2013), non-asymptotic regret bound analysis for RL algorithms has been one of the main topics, and has been developed for tabular Markov decision problems.

The non-asymptotic analysis of adaptive LQ problem by Abbasi-Yadkori and Szepesvári (2011) utilizes the Optimism in the Face of Uncertainty principle to construct a sequence of improving confidence regions for the unknown model parameters, and solves a non-convex constrained optimization problem for each confidence region; their algorithm achieves an $\mathcal{O}(\sqrt{T})$ regret bound, with $T$ being the number of time steps. To reduce the computational complexity and to avoid the non-convexity issue, Abeille and Lazaric (2018) and Ouyang et al. (2017) propose Thompson-sampling-based algorithms and derive $\mathcal{O}(\sqrt{T})$ regret bounds in the Bayesian setting; Dean et al. (2018) proposes a robust adaptive control algorithm to solve a convex sub-problem in each step and achieves an $\mathcal{O}(T^{2/3})$ regret bound. The gap between these regret bounds is removed by Mania et al. (2019) and Cohen et al. (2019) via two different approaches for the same $\mathcal{O}(\sqrt{T})$ frequentist regret bound. Later, Simchowitz and Foster (2020) establishes a lower bound on the regret of order $\mathcal{O}(\sqrt{d_{u}^{2}d_{x}T})$, where $d_{u}$ and $d_{x}$ are the dimensions of the actions and the states, and shows that a simple variant of certainty equivalent control matches the lower bound in both $T$ and the dimensions. Similar regret bounds have also been established under different settings and assumptions, such as Chen and Hazan (2021) in the adversarial setting and Lale et al. (2020a) without a stabilizing controller at the early stages of agent-environment interaction.

All the analyses are in discrete-time with an infinite time horizon. In all these problems, adaptive control algorithms are shown to achieve logarithmic regret bounds when additional information regarding the parameters of the system (often referred to as identifiability conditions) is available. Indeed, Faradonbeh et al. (2018b, 2020) prove that certainty equivalent adaptive regulator achieves logarithmic regret bounds if the system parameter satisfies certain sparsity or low-rankness conditions. Cassel et al. (2020) establishes logarithmic regret bounds when either the state transition matrix is unknown, or when the state-action transition matrix is unknown and the optimal policy is non-degenerate. In partially observable linear dynamical systems, which takes linear-quadratic Gaussian problem as a special case, Lale et al. (2020b) proposes an algorithm with a logarithmic regret bound, under the assumption that one has access to a set in which all controllers persistently excite the system to approximate the optimal control. Logarithmic regret bounds in the adversarial setting with known dynamics parameters have been established in Agarwal et al. (2019); Foster and Simchowitz (2020).

Most real-world control systems, such as those in aerospace, automotive industry and robotics, are naturally continuous-time dynamical systems. So are their related physical tasks, such as inverted pendulum problems, cart-pole balancing problems, and legged robot problems. Continuous-time finite-time horizon LQ control problems can be found in portfolio optimization Wang and Zhou (2020), algorithmic trading Cartea et al. (2018), production management of exhaustible resources Graber (2016), and biological movement systems Bailo et al. (2018).

Analysis for continuous-time LQ-RL and general RL problems, however, is fairly limited. The primary approach is to develop learning algorithms after discretizing both the time and the space spaces, and establish the convergence as discretization parameters tend to zero. For instance, Munos (2006) proposes a policy gradient algorithm and shows the convergence of the policy gradient estimate to the true gradient. Munos and Bourgine (1998); Munos (2000) design learning algorithms by discretizing Bellman equations of the underlying control problems and prove the asymptotic convergence of their algorithms. For the LQ system, attentions have been mostly on algorithms designs, including the integral reinforcement learning algorithm in Modares and Lewis (2014), and the policy iteration algorithm in Rizvi and Lin (2018). Yet, very little is known regarding the convergence rate or the regret bound of all these algorithms. Indeed, despite the natural analytical connection between LQ control and RL, the best known theoretical work for continuous-time LQ-RL is still due to Duncan et al. (1999), where an asymptotically sublinear regret for an ergodic model has been derived via a weighted least-squares-based estimation approach. Nevertheless, the exact order of the regret bound has not been studied.

In this section, we consider the linear-quadratic reinforcement learning (LQ-RL) problem, where the drift coefficient of the state dynamics is unknown to the controller.

More precisely, let $T\in(0,\infty)$ be a given terminal time, $W$ be an $n$-dimensional standard Brownian motion defined on a complete probability space $(\Omega,\mathcal{F},\mathbb{P})$, and ${\mathbb{F}}=(\mathcal{F}_{t})_{t\in[0,T]}$ be the filtration generated by $W$ augmented by the $\mathbb{P}$-null sets. Let $x_{0}\in{\mathbb{R}}^{n}$ be a given initial state and $(A^{\star},B^{\star})\in{\mathbb{R}}^{n\times n}\times{\mathbb{R}}^{n\times d}$ be fixed but unknown matrices, consider the following problem:

where for each $U\in\mathcal{H}^{2}({\mathbb{R}}^{d})$, the process $X^{\theta^{\star},U}\in\mathcal{S}^{2}({\mathbb{R}}^{n})$ satisfies the following controlled dynamics associated with the parameter $\theta^{\star}=(A^{\star},B^{\star})^{\top}$:

with given matrices $Q\in{\mathbb{S}}^{n}_{0}$ and $R\in{\mathbb{S}}^{d}_{+}$. Note that we assume the loss functional (2.1) only involves a time homogeneous running cost to allow a direct comparison with infinite-time horizon RL problems (see e.g., Duncan et al. (1992)), but similar analysis can be performed if the cost functions are time inhomogeneous, a terminal cost is included, or the Brownian motion $W$ in (2.2) is scaled by an known nonsingular diffusion matrix.

If the parameter $\theta^{\star}=(A^{\star},B^{\star})^{\top}$ are known to the controller, then (2.1)-(2.2) reduces to the classical LQ control problems. In this case, it is well known that (see e.g., Yong and Zhou (1999) and the references therein), the optimal control $U^{\theta^{\star}}$ of (2.1)-(2.2) is given in a feedback form by

where $K^{\theta^{\star}}_{t}=-R^{-1}(B^{\star})^{\top}P^{\theta^{\star}}_{t}$ for all $t\in[0,T]$, $(P_{t}^{\theta^{\star}})_{t\in[0,T]}$ solves the Riccati equation

and $X^{\theta^{\star}}$ is the state process governed by the following dynamics:

To solve the LQ-RL problem (2.1)-(2.2) with unknown $\theta^{\star}$, the controller searches for the optimal control while simultaneously learning the system, i.e., the matrices $A^{\star},B^{\star}$. In an episodic (also known as reset or restart) learning framework, the controller improves her knowledge of the underlying dynamics $X_{t}$ through successive learning episodes, in order to find a control that is close to the optimal one.

Mathematically, it goes as follows. Let $M\in\mathbb{N}$ be the total number of learning episodes. In the $i$-th learning episode, $i=1,\ldots,M$, a feedback control $\psi^{i}$ is exercised, and the state process $X^{\psi^{i}}$ evolves according to the dynamics (2.2) controlled by the policy $\psi^{i}$:

Here $W^{i},i=1,2,\ldots,M$ are independent $n$-dimensional Brownian motions defined on the same probability space $(\Omega,\mathcal{F},\mathbb{P})$. The (expected) cost of learning in the $i$-th episode is then given by

and the (expected) regret of learning up to $M\in{\mathbb{N}}$ episodes (with the sequence of controls $(U^{\psi^{i}})_{i=1}^{M}$) is defined as follows:

where $J^{\theta^{\star}}(U^{\theta^{\star}})$ is the optimal cost of (2.1)-(2.2) when $A^{\star},B^{\star}$ are known. Intuitively, the regret characterizes the cumulative loss from taking sub-optimal policies in all episodes.

In the following, we shall propose several least-squares-based learning algorithms to solve (2.1)-(2.2), and prove that they achieve logarithmic regrets if $\theta^{\star}$ is identifiable (see Remark 2.1 for details).

In this section, we consider a continuous-time least-squares algorithm, which chooses the optimal feedback control based on the current estimation of the parameter, and updates the parameter estimation based on the whole trajectories of the state dynamics.

More precisely, let $\theta=(A,B)^{\top}\in{\mathbb{R}}^{(n+d)\times n}$ be the current estimate of the unknown parameter $\theta^{\star}$, then the controller would exercise the optimal feedback control $\psi^{\theta}$ for (2.1)-(2.2) with $\theta^{\star}$ replaced by $\theta$, i.e.,

where $P^{\theta}$ satisfies the Riccati equation (2.4) with $\theta^{\star}$ replaced by $\theta$:

This leads to the state process $X^{\psi^{\theta}}$ satisfying (cf. (2.6)):

We proceed to derive an $\ell_{2}$-regularized least-squares estimation for $\theta^{\star}$ based on sampled trajectories of $X^{\psi^{\theta}}$. Observing from (2.11) that

Hence the martingale property of the Itô integral implies that

provided that $\mathbb{E}\big{[}\int_{0}^{T}Z^{\psi^{\theta}}_{t}(Z^{\psi^{\theta}}_{t})^{\top}\,{\mathrm{d}}t\big{]}$ is invertible. This suggests a practical rule to improve one’s estimate $\theta$ for the true parameter $\theta^{\star}$, by replacing the expectations in (2.12) with empirical averages over independent realizations. More precisely, let $m\in{\mathbb{N}}$ and $(X^{\psi^{\theta},i}_{t},\psi^{\theta}(t,X^{\psi^{\theta},i}_{t}))_{t\in[0,T]}$, $i=1,\ldots,m$, be trajectories of $m$ independent realizations of the state and control processes, we shall update the estimate $\theta$ by the following rule, inspired by (2.12):

where $Z^{\psi^{\theta},i}_{t}\coloneqq\begin{pmatrix}X^{\psi^{\theta},i}_{t}\\ \psi^{\theta}(t,X^{\psi^{\theta},i}_{t})\end{pmatrix}$ for all $t\in[0,T]$ and $i=1,\ldots,m$, and $I$ is the $(n+d)\times(n+d)$ identity matrix.

The regularization term $\frac{1}{m}I$ in (2.13) guarantees the required matrix inverse and vanishes as $m\rightarrow\infty$. The estimator (2.13) can be equivalently expressed as an $\ell_{2}$-regularized least-squares estimator, as pointed out in Duncan et al. (1992) for the ergodic LQ-RL problem.

We summarize the continuous-time least-squares algorithm as follows.

Note that Algorithm 1 operates in cycles, with $m_{\ell}$ the number of episodes in the $\ell$-th cycle. Hence, the regret of learning up to $M$ episodes (cf. (2.8)) can be upper bounded by the accumulated regret at the end of the $L$-th cycle, where $L$ is the smallest integer such that $\sum_{\ell=0}^{L}m_{\ell}\geq M$.

In this section, we analyze the regret of Algorithm 1 based on the following assumptions of the learning problem (2.1)-(2.2).

Before discussing the regret of Algorithm 1, we make the following remark of (H.1).

Moreover, the following proposition provides sufficient conditions of (H.12), which are special cases of Proposition 3.11.

Proposition 2.1 provides two sets of conditions for (H.12) under two different scenarios: Item 1 applies to an arbitrary finite $T>0$, and Item 2 only applies to sufficiently large $T$. Item 2 assumes the asymptotic behavior of solutions to Riccati differential equations, which can be ensured by the stabilizability of the pair $(A^{\star},B^{\star})$ and detectability of the pair $(A^{\star},Q^{1/2})$ (see (Bitmead and Gevers, 1991, Theorems 10.9 and 10.10)). Note that our subsequent analysis is based on (H.1), and does not require stabilizability assumptions.

We are now ready to state the main result of this section, which shows that the regret of Algorithm 1 grows logarithmically with respect to the number of episodes.

To simplify the presentation, we analyze the performance of Algorithm 1 by assuming the number of learning episodes $\{m_{\ell}\}_{\ell}$ is doubled between two successive updates of the estimation of $\theta^{\star}$. Similar regret results can be established for Algorithm 1 with different choices of $\{m_{\ell}\}_{\ell}$. Under this specific choice of $\{m_{\ell}\}_{\ell}$, for any $M\in{\mathbb{N}}$, Algorithm 1 splits $M$ episodes into $L=\lceil\log_{2}(\frac{M}{m_{0}}+1)\rceil-1$ cycles, where the $\ell$-th cycle, $\ell=0,1,\ldots,L-1$, contains $m_{\ell}$ episodes, and the remaining $M-\sum_{\ell=0}^{L-1}m_{\ell}$ episodes are in the last cycle.

Note that Algorithm 1 in Section 2.2 requires executing feedback controls and observing corresponding state trajectories continuously. A common practice to solve continuous-time RL problems is by assuming that at each learning episode the dynamics only evolves in discrete time, and then estimate parameters according to discrete-time RL algorithms (see e.g., Munos and Bourgine (1998); Munos (2000, 2006); Tallec et al. (2019)). As the true dynamics evolves continuously, it is necessary to quantify the impact of reaction stepsize on the algorithm performance.

In this section, we analyze the performance of the above procedure for solving (2.1)-(2.2). We adapt regularized least-squares algorithms for discrete-time LQ problems to the present setting, and establish their regret bounds in terms of the discretization stepsize. Our analysis shows that a proper scaling of the regularization term in the least-squares estimation in terms of stepsize is critical for a robust performance with respect to different timescales.

More precisely, for a given cycle (i.e., the index $\ell$ in Algorithm 1), let $\theta=(A,B)^{\top}\in{\mathbb{R}}^{(n+d)\times n}$ be the current estimate of $\theta^{\star}$ in (2.2), and let $\{t_{i}\}_{i=0}^{N}$, $N\in{\mathbb{N}}$, be a uniform partition of $[0,T]$ with stepsize $\tau=T/N$. We then assume that (2.1)-(2.2) is piecewise constant between any two grid points $\{t_{i}\}_{i=0}^{N}$, choose actions and make observations every $\tau$, and update the estimated parameter based on these observations. To this end, we consider the following discrete-time LQ control problem with parameter $\theta$:

where $\mathcal{H}^{2}_{N}({\mathbb{R}}^{d})=\{U\in\mathcal{H}^{2}({\mathbb{R}}^{d})\mid U_{t}=U_{t_{i}},t\in[t_{i},t_{i+1}),i=0,\ldots,N-1\}$, and $(X^{U,\tau}_{t_{i}})_{i=0}^{N-1}$ are defined by

Note that for simplicity, our strategy is constructed by assuming a discrete-time dynamics arising from an Euler discretization of (2.2) (with the estimated parameter $\theta$); similar analysis can be performed with a high-order approximation of (2.1)-(2.2).

It is well-known that (see e.g., Bitmead and Gevers (1991)), the optimal control of (2.17)-(2.18) is given by the following feedback form:

where $K^{\theta,\tau}:[0,T)\rightarrow{\mathbb{R}}^{d\times n}$ is the piecewise constant function (with stepsize $\tau=T/N$) defined by

We then implement the piecewise constant strategy $\psi^{\theta,\tau}$ defined in (2.19) on the original system (2.2) for $m$ episodes, and update the estimated parameter $\theta$ by observing (2.2) with stepsize $\tau=T/N$. More precisely, let $X^{\psi^{\theta,\tau}}\in\mathcal{S}^{2}({\mathbb{R}}^{n})$ be the state process associated with $\psi^{\theta,\tau}$:

and $(X^{\psi^{\theta,\tau},j}_{t})_{t\in[0,T]}$, $j=1,\ldots,m$, $m\in{\mathbb{N}}$, be $m$ independent trajectories of $X^{\psi^{\theta,\tau}}\in\mathcal{S}^{2}({\mathbb{R}}^{n})$, we update the parameter $\theta$ according to the following discrete-time least-squares estimator:

with $Z^{\psi^{\theta,\tau},j}_{t_{i}}\coloneqq\begin{pmatrix}X^{\psi^{\theta,\tau},j}_{t_{i}}\\ K^{\theta,\tau}_{t_{i}}X^{\psi^{\theta,\tau},j}_{t_{i}}\end{pmatrix}$ for all $i,j$. The update (2.22) is consistent with the agent’s assumption that the state evolves according to (2.18) between two grid points. Setting the derivative (with respect to $\theta$) of the right-hand side of (2.22) to zero leads to

Dividing both sides by $\tau/m$ and rearranging the terms give the following equivalent expression of the discrete-time least squares estimator (2.22):

We now summarize the discrete-time least-squares algorithm as follows.

Again, as the $\ell$-th cycle of Algorithm 2 contains $m_{\ell}$ episodes, for each $M\in{\mathbb{N}}$, the regret of learning up to $M$ episodes (cf. (2.8)) can be upper bounded by the accumulated regret at the end of the $L$-th cycle, where $L$ is the smallest integer such that $\sum_{\ell=0}^{L}m_{\ell}\geq M$. The following theorem is an analogue of Theorem 2.2 for Algorithm 2.