{"title": "Robust exploration in linear quadratic reinforcement learning", "book": "Advances in Neural Information Processing Systems", "page_first": 15336, "page_last": 15346, "abstract": "Learning to make decisions in an uncertain and dynamic environment is a task of fundamental performance in a number of domains.\nThis paper concerns the problem of learning control policies for an unknown linear dynamical system so as to minimize a quadratic cost function.\nWe present a method, based on convex optimization, that accomplishes this task \u2018robustly\u2019, i.e., the worst-case cost, accounting for system uncertainty given the observed data, is minimized.\nThe method balances exploitation and exploration, exciting the system in such a way so as to reduce uncertainty in the model parameters to which the worst-case cost is most sensitive.\nNumerical simulations and application to a hardware-in-the-loop servo-mechanism are used to demonstrate the approach, with appreciable performance and robustness gains over alternative methods observed in both.", "full_text": "Robust exploration in linear quadratic\n\nreinforcement learning\n\nJack Umenberger\n\nDepartment of Information Technology\n\nUppsala University, Sweden\n\njack.umenberger@it.uu.se\n\nThomas B. Sch\u00f6n\n\nDepartment of Information Technology\n\nUppsala University, Sweden\nthomas.schon@it.uu.se\n\nMina Ferizbegovic\n\nSchool of Electrical Engineering\n\nand Computer Science\n\nKTH, Sweden\nminafe@kth.se\n\nH\u00e5kan Hjalmarsson\n\nSchool of Electrical Engineering\n\nand Computer Science\n\nKTH, Sweden\n\nhjalmars@kth.se\n\nAbstract\n\nThis paper concerns the problem of learning control policies for an unknown\nlinear dynamical system to minimize a quadratic cost function. We present a\nmethod, based on convex optimization, that accomplishes this task robustly: i.e.,\nwe minimize the worst-case cost, accounting for system uncertainty given the\nobserved data. The method balances exploitation and exploration, exciting the\nsystem in such a way so as to reduce uncertainty in the model parameters to which\nthe worst-case cost is most sensitive. Numerical simulations and application to a\nhardware-in-the-loop servo-mechanism demonstrate the approach, with appreciable\nperformance and robustness gains over alternative methods observed in both.\n\n1\n\nIntroduction\n\nLearning to make decisions in an uncertain and dynamic environment is a task of fundamental\nimportance in a number of domains. Though it has been the subject of intense research activity since\nthe formulation of the \u2018dual control problem\u2019 in the 1960s[17], the recent success of reinforcement\nlearning (RL), particularly in games [33, 37], has inspired a resurgence in interest in the topic.\nProblems of this nature require decisions to be made with respect to two objectives. First, there is\na goal to be achieved, typically quanti\ufb01ed as a reward function to be maximized. Second, due to\nthe inherent uncertainty there is a need to gather information about the environment, often referred\nto as \u2018learning\u2019 via \u2018exploration\u2019. These two objectives are often competing, a fact known as the\nexploration/exploitation trade-off in RL, and the \u2018dual effect\u2019 (of decision) in control.\nIt is important to recognize that the second objective (exploration) is important only in so far as it\nfacilitates the \ufb01rst (maximizing reward); there is no intrinsic value in reducing uncertainty. As a\nconsequence, exploration should be targeted or application speci\ufb01c; it should not excite the system\narbitrarily, but rather in such a way that the information gathered is useful for achieving the goal.\nFurthermore, in many real-world applications, it is desirable that exploration does not compromise\nthe safe and reliable operation of the system.\nThis paper is concerned with control of uncertain linear dynamical systems, with the goal of maxi-\nmizing (minimizing) rewards (costs) that are a quadratic function of states and actions; cf. \u00a72 for\na detailed problem formulation. We derive methods to synthesize control policies that balance the\n\n33rd Conference on Neural Information Processing Systems (NeurIPS 2019), Vancouver, Canada.\n\n\fexploration/exploitation tradeoff by performing robust, targeted exploration: robust in the sense\nthat we optimize for worst-case performance given uncertainty in our knowledge of the system, and\ntargeted in the sense that the policy excites the system so as to reduce uncertainty in such a way\nthat speci\ufb01cally minimizes the worst-case cost. To this end, this paper makes the following speci\ufb01c\ncontributions. We derive a high-probability bound on the spectral norm of the system parameter\nestimation error, in a form that is applicable to both robust control synthesis and design of targeted\nexploration; cf. \u00a73. We also derive a convex approximation of the worst-case (w.r.t. parameter\nuncertainty) in\ufb01nite-horizon linear quadratic regulator (LQR) problem; cf. \u00a74.2. We then combine\nthese two developments to present an approximate solution, via convex semide\ufb01nite programing\n(SDP), to the problem of minimizing the worst-case quadratic costs for an uncertain linear dynamical\nsystem; cf. \u00a74. For brevity, we will refer to this as a \u2018robust reinforcement learning\u2019 (RRL) problem.\n\n1.1 Related work\nInspired, perhaps in part, by the success of RL in games [33, 37], there has been a \ufb02urry of recent\nresearch activity in the analysis and design of RL methods for linear dynamical systems with quadratic\nrewards. Works such as [1, 23, 15] employ the so-called \u2018optimism in the face of uncertainty\u2019 (OFU)\nprinciple, which selects control actions assuming that the true system behaves as the \u2018best-case\u2019\nmodel in the uncertain set. This leads to optimal regret but requires the solution of intractable non-\nconvex optimization problems. Alternatively, the works of [35, 3, 4] employ Thompson sampling,\nwhich optimizes the control action for a system drawn randomly from the posterior distribution over\nthe set of uncertain models, given data. The work of [30] eschews uncertainty quanti\ufb01cation, and\ndemonstrates that \u2018so-called\u2019 certainty equivalent control attains optimal regret. There has also been\nconsiderable interest in \u2018model-free\u2019 methods[39] for direct policy optimization [16, 29], as well\npartially model-free methods based on spectral \ufb01ltering [22, 21]. Unlike the present paper, none\nof the works above consider robustness which is essential for implementation on physical systems.\nRobustness is studied in the so-called \u2018coarse-ID\u2019 family of methods, c.f. [12, 13, 11]. In [11], sample\nconvexity bounds are derived for LQR with unknown linear dynamics. This approach is extended to\nadaptive LQR in [12], however, unlike the present paper, the policies are not optimized for exploration\nand exploitation jointly; exploration is effectively random. Also of relevance is the \ufb01eld of so-called\n\u2018safe RL\u2019 [19] in which one seeks to respect certain safety constraints during exploration and/or policy\noptimization [20, 2], as well as \u2018risk-sensitive RL\u2019, in which the search for a policy also considers the\nvariance of the reward [32, 14]. Other works seek to incorporate notions of robustness commonly\nencountered in control theory, e.g. stability [34, 7, 10]. In closing, we mention that related problems\nof simultaneous learning and control have a long history in control theory, beginning with the study\nof \u2018dual control\u2019 [17, 18] in the 1960s. Many of these formulations relied on a dynamic programing\n(DP) solution and, as such, were applicable only in special cases [6, 8]. Nevertheless, these early\nefforts [9] established the importance of balancing \u2018probing\u2019 (exploration) with \u2018caution\u2019 (robustness).\nFor subsequent developments from the \ufb01eld of control theory, cf. e.g. [24, 25, 5].\n\n2 Problem statement\n\nIn this section we describe in detail the problem addressed in this paper. Notation is as follows: A>\nt=1. max(A) denotes the\ndenotes the transpose of a matrix A. x1:n is shorthand for the sequence {xt}n\nmaximum eigenvalue of a matrix A. \u2326 denotes the Kronecker product. vec (A) stacks the columns of\nA to form a vector. Sn\n++) denotes the cone(s) of n \u21e5 n symmetric positive semide\ufb01nite (de\ufb01nite)\nmatrices. w.p. means \u2018with probability.\u2019 2\nn(p) denotes the value of the Chi-squared distribution with\nn degrees of freedom and probability p. blkdiag is the block diagonal operator.\n\n+ (Sn\n\nDynamics and cost function We are concerned with control of linear time-invariant systems\n\nxt+1 = Axt + But + wt, wt \u21e0N 0, 2\n\n(1)\nwhere xt 2 Rnx, ut 2 Rnu and wt 2 Rn denote the state (which is assumed to be observed directly,\nwithout noise), input and process noise, respectively, at time t. The objective is to design a feedback\ncontrol policy ut = ({x1:t, u1:t1}) so as to minimize the cost functionPT\nt=i c(xt, ut), where\nc(xt, ut) = x>t Qxt + u>t Rut for user-speci\ufb01ed positive semide\ufb01nite matrices Q and R. When the\nparameters of the true system, denoted {Atr, Btr}, are known this is exactly the \ufb01nite-horizon LQR\nproblem, the optimal solution of which is well-known. We assume that {Atr, Btr} are unknown.\n\nwInx ,\n\nx0 = 0,\n\n2\n\n\fapply 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\n\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\n\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\n\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\n\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\n\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\n\n. 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\n\ntN1\n\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\n\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\n\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\n\nFigure 1: Cartoon depiction of the\nproblem addressed in this paper.\nThe goal is to design N policies,\ni=1, so as to minimize the\n{Ki}N\nworst-case cost (blue area) over the\ntime horizon [0, T ]; cf. \u00a72.\n\nModeling and data As {Atr, Btr} are unknown, all knowledge about the true system dynamics\nmust be inferred from observed data, Dn := {xt, ut}n\nt=1. We assume that w is known, or has been\nestimated, and that we have access to initial data, denoted (with slight notational abuse) D0, obtained,\ne.g. during a preliminary experiment. For the model (1), parameter uncertainty can be quanti\ufb01ed as:\nProposition 2.1. Given observed data Dn from (1), and a uniform prior over the parameters\n\u2713 = vec ([A B]), i.e., p(\u2713) / 1, the posterior distribution p(\u2713|Dn) is given by N (\u00b5\u2713, \u2303\u2713), where\n\u00b5\u2713 = vec\u21e3[ \u02c6A \u02c6B]\u2318 = arg min\u27132Rn2\nt=1 |xt+1 [x>t u>t ] \u2326 Inx\u2713|2, i.e., the ordinary least\nut \uf8ff xt\nut >\n\nsquares estimator, and \u2303\u27131 = 1\n2\n\nx+nxnuPn1\nt=1 \uf8ff xt\nwPn1\n\nProof : cf. \u00a7A.1.1. The uniform prior, p(\u2713) / 1, sometimes called an improper prior, is used as an\nuninformative prior, signifying that we have no prior knowledge of \u2713, i.e., all values are equally likely.\nBased on Proposition 2.1 we can de\ufb01ne a high-probability credibility region by:\n\n\u2326 Inx.\n\nwhere c = 2\nn2\n\nx+nxnu\n\n\u21e5e(Dn) := {\u2713 : (\u2713 \u00b5\u2713)>\u2303\u27131(\u2713 \u00b5\u2713) \uf8ff c},\n() for 0 << 1. Then, \u2713tr = vec ([Atr Btr]) 2 \u21e5e(Dn) w.p. 1 .\n\n(2)\n\n+ , and is denoted K = {K, \u2303}. Let {ti}N\n\nPolicies Though not necessarily optimal, we will restrict our attention to static-gain policies of\nthe form ut = Kxt +\u2303 1/2et, where et \u21e0N (0, I) represent random excitations for the purpose of\nlearning. Static-gain policies are popular in practice, due to simplicity of synthesis and implementation\n[12, 13, 11], and encompass many common control strategies, e.g., proportional-derivative (PD)\ncontrol. A policy comprises K 2 Rp\u21e5n and \u2303 2 Snu\ni=0 2 N,\nwith 0 = t0 \uf8ff t1 \uf8ff . . . ,\uf8ff tN = T , partition the time horizon T into N intervals. The ith interval\nis of length Ti := ti ti1. We will then design N policies, {Ki}N\ni=1, such that Ki = {Ki, \u2303i} is\ndeployed during the ith interval, t 2 [ti1, ti]. For convenience, we de\ufb01ne the function I : R+ 7! N\ngiven by I(t) := arg mini2N{i : t \uf8ff tj}, which maps time t to the index i = I(t) of the policy to\nbe deployed. We also make use of the notation ut = K(xt) as shorthand for ut = Kxt +\u2303 1/2et.\nWorst-case dynamics We are now in a position to de\ufb01ne the optimization problem that we wish to\nsolve in this paper. In the absence of knowledge of the true dynamics, {Atr, Btr}, given initial data\nD0, we wish to \ufb01nd a sequence of policies {Ki}N\nt=1 c(xt, ut),\nassuming that, at time t, the system evolves according to the worst-case dynamics within the high-\nprobability credibility region \u21e5e(Dt), i.e.,\n\ni=0 that minimize the expected costPT\n\nc(xt, ut)# , s.t. xt+1 = Atxt + Btut + wt, ut = KI(t)(xt),\n\n(3)\n\nwhere the expectation is w.r.t. wt \u21e0N 0, 2\n\nwInx and et \u21e0N (0, Inu). We choose to optimize for\n\nthe worst-case dynamics so as to bound, with high probability, the cost of applying the policies to the\nunknown true system. In principle, problems such as (3) can be solved via dynamic programing (DP)\n[17]. However, such DP-based solutions require gridding to obtain \ufb01nite state-action spaces, and\nare hence computationally intractable for systems of even modest dimension [6]; cf. also [31, \u00a7IV]\nfor a discussion of RL methods for \ufb01nite state-action spaces. In what follows, we will present an\napproximate solution to this problem, which we refer to as a \u2018robust reinforcement learning\u2019 (RRL)\nproblem, that retains the continuous sate-action space formulation and is based on convex optimization.\nTo facilitate such a solution, we require a re\ufb01ned means of quantifying system uncertainty, which we\npresent next.\n\nE\" TXt=0\n\nmin\n{Ki}N\n\ni=1\n\nsup\n\n{At,Bt}2\u21e5e(Dt)\n\n3\n\n\f3 Modeling uncertainty for robust control\n\nIn this paper, we adopt a model-based approach to control, in which quantifying uncertainty in\nthe estimates of the system dynamics is of central importance. From Proposition 2.1 the posterior\ndistribution over parameters is Gaussian, which allows us to construct an \u2018ellipsoidal\u2019 credibility\nregion \u21e5e, centered about the ordinary least squares estimates of the model parameters, as in (2).\nTo allow for an exact convex formulation of the control problem involving the worst-case dynamics,\ncf. \u00a74.2, it is desirable to work with a credibility region that bounds uncertainty in terms of the\nspectral properties of the parameter error matrix [ \u02c6A Atr, \u02c6B Btr], where { \u02c6A, \u02c6B} are the ordinary\nleast squares estimates, i.e. vec\u21e3[ \u02c6A \u02c6B]\u2318 = \u00b5\u2713, cf. Proposition 2.1. To this end, we will work with\nmodels of the form M(D) := { \u02c6A, \u02c6B, D} where D 2 Snx+nu speci\ufb01es the following region, in\nparameter space, centered about { \u02c6A, \u02c6B}:\n\n\u21e5m(M) := {A, B : X>DX I, X = [ \u02c6A A, \u02c6B B]>}.\n\n(4)\n\nx+nxnu\n\nThe following lemma, cf. \u00a7A.1.2 for proof, suggests a speci\ufb01c means of constructing D, so as to\nensure that \u21e5m de\ufb01nes a high-probability credibility region:\n\n(). Then [Atr, Btr] 2 \u21e5m(M) w.p. 1 .\n\nLemma 3.1. Given data Dn from (1), and 0 << 1, let D = 1\nc = 2\nn2\n\nut \uf8ff xt\nut >\nFor convenience, we will make use of the following shorthand notation: M(Dti) = { \u02c6Ai, \u02c6Bi, Di}.\nCredibility regions of the form (4), i.e. bounds on the spectral properties of the estimation error, have\nappeared in recent works on data-driven and adaptive control, cf. e.g., [11, Proposition 2.4] which\nmakes use of results from high-dimensional statistics [40]. The construction in [11, Proposition\n2.4] requires {xt+1, xt, ut} to be independent, and as such is not directly applicable to time series\ndata, without subsampling to attain uncorrelated samples (though more complicated extensions to\ncircumvent this limitation have been suggested [38]). Lemma 3.1 is directly applicable to correlated\ntime series data, and provides a credibility region that is well suited to the RRL problem, cf. \u00a74.3.\n\nt=1 \uf8ff xt\n\nwcPn1\n\n2\n\n, with\n\n4 Convex approximation to robust reinforcement learning problem\n\nEquipped with the high-probability bound on the spectral properties of the parameter estimation\nerror presented in Lemma 3.1, we now proceed with the main contribution of this paper: a convex\napproximation to the \u2018robust reinforcement learning\u2019 (RRL) problem in (3).\n\n4.1 Steady-state approximation of cost\n\nIn pursuit of a more tractable formulation, we \ufb01rst introduce the following approximation of (3),\n\nNXi=1\n\nsup\n\n{A,B}2\n\n\u21e5m(M(Dti ))\n\nE24\n\ntiXt=ti1\n\nc(xt, ut)35 , s.t. xt+1 = Axt + But + wt, ut = Ki(xt).\n\n(5)\n\nObserve that (5) has introduced two approximations to (3). First, in (5) we only update the \u2018worst-\ncase\u2019 model at the beginning of each epoch, when we deploy a new policy, rather than at each time\nstep as in (3). This introduces some conservatism, as model uncertainty will generally decrease\nas more data is collected, but results in a simpler control synthesis problem. Second, we select\nthe worst-case model from the \u2018spectral\u2019 credibility region \u21e5m as de\ufb01ned in (4), rather than the\n\u2018ellipsoidal\u2019 region \u21e5e de\ufb01ned in (2). Again, this introduces some conservatism as \u21e5e \u2713 \u21e5m, cf.\n\u00a7A.1.2, but permits convex optimization of the worst-case cost, cf. \u00a74.2. For convenience, we denote\n\nJ\u2327 (x1,K, \u21e5m(M)) := sup\n{A,B}2\n\n\u21e5m(M)X\u2327\n\nt=1\n\nc(xt, ut), s.t. xt+1 = Axt + But + wt, ut = K(xt).\n\n4\n\n\fNext, we approximate the cost between epochs with the in\ufb01nite-horizon cost, scaled appropriately for\nthe epoch duration, i.e., between the i 1th and ith epoch we approximate the cost as\nJTi(xti,Ki, \u21e5m(M(Dti1))) \u21e1 Ti\u21e5{J1(Ki, \u21e5m(M(Dti1))) := lim\n\u2327!1\n(6)\nThis approximation is accurate when the epoch duration Ti is suf\ufb01ciently long relative to the time\nrequired for the state to reach the stationary distribution. Substituting (6) into (5), the cost function\nthat we seek to minimize becomes\n\n1\n\u2327\n\nJ\u2327 (0,Ki, \u21e5m(M(Dti1)))}.\n\nE\uf8ffXN\n\ni=1\n\nTi \u21e5 J1Ki, \u21e5m(M(Dti1)) .\n\n(7)\n\nThe expectation in (7) is w.r.t. to wt and et, as Dti depends on the random variables x1:ti and u1:ti,\nwhich evolve according to the worst-case dynamics in (5).\n\n4.2 Optimization of worst-case cost\n\n1\n\n1\n\u2327\n\n1\n\u2327\n\n(8)\n\n\u2327!1\n\nlim\n\u2327!1\n\n0 R lim\n\nThe previous subsection introduced an approximation of our \u2018ideal\u2019 problem (3), based on the worst-\ncase in\ufb01nite horizon cost, cf. (7). In this subsection we present a convex approach to the optimization\nof J1(K, \u21e5m(M)) w.r.t. K, given M. The in\ufb01nite horizon cost can be expressed as\nut \uf8ff xt\n\n\u2327 E\" \u2327Xt=1\nx>t Qxt + u>t Rut# = tr \uf8ff Q 0\nE\"\uf8ff xt\nut >#! .\n\u2327Xt=1\nUnder the feedback policy K, the covariance appearing on the RHS of (8) can be expressed as\nE\" lim\nKxt +\u2303 1/2et ># =\uf8ff W\nKxt +\u2303 1/2et \uf8ff\n\u2327Xt=1\uf8ff\nKW KW K> +\u2303 ,\nwhere W = E\u21e5xtx>t\u21e4 denotes the stationary state covariance. For known A and B, W is given by\n(10)\ni.e., arg minW tr W s.t. (10). To optimize J1(K, \u21e5m(M)) via convex optimization, there are two\nchallenges to overcome: i. non-convexity of jointly searching for K and W , satisfying (10) and\nminimizing (8), ii. computing W for worst-case {A, B}2 \u21e5m(M), rather than known {A, B}.\nLet us begin with the \ufb01rst challenge: nonconvexity. To facilitate a convex formulation of the RRL\nproblem (7) we write (10) as\n\nW \u232b (A + BK)W (A + BK)> + B\u2303B> + 2\n\nthe (minimum trace) solution to the Lyapunov inequality\n\nW K>\n\nwInx,\n\n\u2327!1\n\n(9)\n\nxt\n\nxt\n\nW \u232b [A B]\uf8ff W\n\nKW KW K> +\u2303 [A B]> + 2\n\nW K>\n\nwInx,\n\n(11)\n\nand introduce the change of variables Z = W K> and Y = KW K> +\u2303 , collated in the variable\n\n\u2305= \uf8ff W Z\nZ> Y . With this change of variables, minimizing (8) subject to (11) is a convex program.\nNow, we turn to the second challenge: computation of the stationary state covariance under the\nworst-case dynamics. As a suf\ufb01cient condition, we require (11) to hold for all {A, B}2 \u21e5m(M). In\nparticular, we de\ufb01ne the following approximation of J1(K,M)\n\u02dcJ1(K,M) := min\nW2Snx\n\nKW KW K> +\u2303 \u25c6 , s.t. (11) holds 8{ A, B}2 \u21e5m(M).\n\n0 R \uf8ff W\n\ntr\u2713\uf8ff Q 0\n\n(12)\nLemma 4.1. Consider the worst-case cost J1(K,M), cf. (6), and the approximation \u02dcJ1(K,M),\ncf. (12). \u02dcJ1(K,M) J1(K,M).\nProof: cf. \u00a7A.1.3. To optimize \u02dcJ1(K,M), as de\ufb01ned in (12), we make use of the following result\nfrom [28]:\n\nW K>\n\n++\n\n5\n\n\fH\n\nF\nB\n\nG\nB>\n\nA + P\n\nF + GX\n\n35 \u232b 0,\n\nH\nF> C I\nG>\n\n(F + GX)> C + X>B + B>X + X>AX \u232b 0, iff 24\n\nTheorem 4.1. The data matrices (A,B,C,P,F,G,H) satisfy, for all X with I X>PX \u232b 0, the\nrobust fractional quadratic matrix inequality\n\uf8ff\nfor some 0.\nTo put (11) in a form to which Theorem 4.1 is applicable, we make use of of the nominal parameters\n\u02c6A and \u02c6B. With X de\ufb01ned as in (4), such that [A B] = [ \u02c6A \u02c6B] X0, we can express (11) as\nwI\n := W[ \u02c6A \u02c6B]\u2305[ \u02c6A \u02c6B]>+X>\u2305[ \u02c6A \u02c6B]>+[ \u02c6A \u02c6B]\u2305XX>\u2305X \u232b 2\nwhere the \u2018iff\u2019 follows from the Schur complement. Given this equivalent representation, by Theorem\n4.1, (11) holds for all X>DM I (i.e. all {A, B}2 \u21e5m(M)) iff\nI\nwI W [ \u02c6A \u02c6B]\u2305[ \u02c6A \u02c6B]> I\n0\u2305\n\nwInx () \uf8ff\n\n\u02c6A \u02c6B]>\nwhich is simply (13) with the substitutions A = \u2305, B =\u2305[\nF = wI, G = 0, and P = D. We now have the following result, cf. \u00a7A.1.4 for proof.\n\u02dcJ1(K, \u21e5m(M)), cf. (12), is given by the SDP:\nTheorem 4.2. The solution to minK\n\nS(, \u2305, \u02c6A, \u02c6B, D) :=24\n\n\u02c6A \u02c6B]>, C = W [ \u02c6A \u02c6B]\u2305[ \u02c6A \u02c6B]>,\n\n35 \u232b 0,\n\n[ \u02c6A \u02c6B]\u2305>\nD \u2305\n\nwI\n\n0\n\nI\nwI\n\n \u232b 0,\n\n[\n\n(13)\n\n(14)\n\n(15)\n\nmin\n,\u2305\n\ntr (blkdiag(Q, R)\u2305) , s.t. S(, \u2305, \u02c6A, \u02c6B, D) \u232b 0, 0,\n\n\u02dcJ1(K, \u21e5m(M)) is purely an \u2018exploitation\u2019 problem \u2303 ! 0 in the above SDP; in\n\nwith the optimal policy given by K = {Z>W 1, Y Z>W 1Z}.\nNote that as minK\ngeneral, \u2303 6= 0 in the RRL setting (i.e. (7)) where exploration is bene\ufb01cial.\n4.3 Approximate uncertainty propagation\nLet us now return to the RRL problem (7). Given a model M, \u00a74.2 furnished us with a convex\nmethod to minimize the worst-case cost. However, at time t = 0, we have access only to data D0,\nand therefore, only M(D0). To optimize (7) we need to approximate the models {M(Dti)}N1\ni=1\nbased on the future data, {Dti}N1\ni=1 , that we expect to see. To this end, we denote the approximate\nmodel, at time t = tj given data Dti, by \u02dcMj(Dti) := { \u02dcAj|i, \u02dcBj|i, \u02dcDj|i}\u21e1 E\u21e5M(Dtj )|Dti\u21e4. We\nnow describe speci\ufb01c choices for \u02dcAj|i, \u02dcBj|i, and \u02dcDj|i, beginning with the latter.\nRecall that the uncertainty matrix D at the ith epoch is denoted Di. The uncertainty matrix at the\nut \uf8ff xt\nut >\n\ni + 1th epoch is then given by Di+1 = Di + 1\n. We approximate the\n2\nempirical covariance matrix in this expression with the worst-case state covariance Wi as follows:\n\nt=ti\uf8ff xt\n\nut ># \u21e1 Ti+1\uf8ff Wi\n\nThis approximation makes use of the same calculation appearing in (9). The equality makes use of\nthe change of variables introduced in \u00a74.2. Note that in proof of Theorem 4.2, cf. \u00a7A.1.4, it was\n\nut \uf8ff xt\nK>W KW K> +\u2303 , when \u2305 is the solution of (15).\n\nE\"ti+1Xt=ti\uf8ff xt\nshown that \u2305= \uf8ff W\nNext, we turn our attention to approximating the effect of future data on the nominal parameter\nestimates { \u02c6A, \u02c6B}. Updating these (ordinary least squares) estimates based on the expected value\nof future observations involves dif\ufb01cult integrals that must be approximated numerically [27, \u00a75].\nTo preserve convexity in our formulation, we approximate future nominal parameter estimates with\nthe current estimates, i.e., given data Dti we set \u02dcAj|i = \u02c6Ai and \u02dcBj|i = \u02c6Bi. To summarize, our\napproximate model at epoch j is given by \u02dcMj(Dti) = { \u02c6Ai, \u02c6Bi, Di + 1\n\nk=i+1 Tk+1\u2305k}.\n\nwcPti+1\nK>i Wi KiWiK>i +\u2303 i = Ti+1\u2305i.\n\nWiK>i\n\nW K>\n\n(16)\n\n2\n\nwcPj\n\n6\n\n\f4.4 Final convex program and receding horizon application\nWe are now in a position to present a convex approximation to our original problem (3). By\nsubstituting \u02dcJ1(\u00b7,\u00b7) for J1(\u00b7,\u00b7), and \u02dcMi(D0) for M(Dti) in (7), we attain the cost function:\ni=1Ti \u21e5 \u02dcJ1\u21e3Ki, \u21e5m( \u02dcMi1(D0))\u2318 . Consider the ith term in this sum, which can be optimized\nPN\nvia the SDP (15), with D = \u02dcDi1|0. Notice two important facts: 1. for \ufb01xed multiplier , the\nuncertainty \u02dcDi1|0 enters linearly in the constraint S(\u00b7) \u232b 0, cf. (15); 2. \u02dcDi1|0 is linear in the\ndecision variables {\u2305k}i1\nk=1, cf. end of \u00a74.3. Therefore, the constraint S(\u00b7) \u232b 0 remains linear in the\ndecision variables, which means that the cost function derived by substituting \u02dcMi(D0) into (7) can\nbe optimized as an SDP, cf. (17) below.\nHitherto, we have considered the problem of minimizing the expected cost over time horizon T\ngiven initial data D0. In practical applications, we employ a receding horizon strategy, i.e., at the\nith epoch, given data Dti1, we \ufb01nd a sequence of policies {Kj}i+h\nj=i that minimize the approximate\nh-step-ahead expected cost\ni+hXj=i+1\n\n\u02c6J(i, h,{Kj}i+h\nand then apply Ki during the ith epoch. At the beginning of the i + 1th epoch, we repeat the process;\ncf. Algorithm1. The problem min{Kj}N\n\nj=i ,Dti1) := Ti \u02dcJ1(Ki, \u21e5m(M(Dti1))) +\n\nj=i Xi+h\nS j, \u2305k, \u02c6Ai, \u02c6Bi, Di +\n\nj=i ,Dti1) can be solved as the SDP:\ns.t. S(i, \u2305i, \u02c6Ai, \u02c6Bi, Di) \u232b 0, \u2305j \u232b 0 8j,\nTk+1\u2305k! \u232b 0 for j = i + 1, . . . , i + h. (17b)\njXk=i+1\nSelecting multipliers For optimization of \u02dcJ1(K,M) given a model M, i.e., (15), the simultaneous\nsearch for the policy K and multiplier is convex, as D is \ufb01xed. However, in the RRL setting,\n\u2018D\u2019 is a function of the decision variables \u2305i, cf.\nj=i+1 2\nRh1\n+ must be speci\ufb01ed in advance. We propose the following method of selecting the multipliers:\n\u02dcJ1(K, \u21e5m(M(Dti1))) via the SDP (15). Then, compute the\ngiven Dti1, solve \u00afK = arg minK\ncost \u02c6J(i, h,{ \u00afK}i+h\nj=i ,Dti1) by solving (17), but with the policies \ufb01xed to \u00afK, and the multipliers\n{j}i+h\n+ as free decision variables. In other words, approximate the worst-case cost of\ndeploying the \u00afK, h epochs into the future. Then, use the multipliers found during the calculation of\nthis cost for control policy synthesis at the ith epoch.\n\n(17b), and so the multipliers {j}i+h\n\nTj \u02dcJ1(Kj, \u21e5m( \u02dcM(Dti1))),\n\ntr (blkdiag(Q, R)\u2305 j) ,\n\n\u02c6J(i, h,{Kj}i+h\n\nj=i 2 Rh\n\n1\n2\nwc\n\n(17a)\n\nj=i\n\nmin\n\ni0,{\u2305j}i+h\n\nj=i\n\nComputational complexity The proposed method can be implemented via semide\ufb01nite program-\ning (SDP) for which computational complexity is well-understood. In particular, the cost of solving the\nSDP (15) scales as O(max{m3, mn3, m2n2}) [26], where m = (1/2)nx(nx + 1) + (1/2)nu(nu +\n1) + nxnu + 1 denotes the dimensionality of the decision variables, and n = 3nx + nu is the\ndimensionality of the LMI S \u232b 0. The cost of solving the SDP (17) is then given, approximately, by\nthe cost of (15) multiplied by the horizon h.\n\n5 Experimental results\n\nNumerical simulations\n\nAtr =\" 1.1\n\n0.5\n0\n0.9\n0 0.2\n\nIn this section, we consider the RRL problem with parameters\n0\n0.1\n\n2 # , Q = I, R = blkdiag(0.1, 1), w = 0.5.\n\n0.8 # , Btr =\" 0\n\n0.1\n0\n\n1\n0\n\nWe partition the time horizon T = 103 into N = 10 equally spaced intervals, each of length Ti = 100.\nFor robustness, we set = 0.05. Each experimental trial consists of the following procedure. Initial\n\n7\n\n\fAlgorithm 1 Receding horizon application to true system\n1: Input: initial data D0, con\ufb01dence , LQR cost matrices Q and R, epochs {ti}N\ni=1.\n2: for i = 1 : N do\n3:\n4:\n5:\n6:\n\nCompute/update nominal model M(Dti1).\nSolve convex program (17).\nRecover policy Ki: Ki = Z>i W 1\nApply policy to true system for ti1 < t \uf8ff ti, which evolves according to (1) with ut =\nForm Dti = Dti1 [{ xti1:ti, uti1:ti} based on newly observed data.\n\nKixt +\u2303 1/2\n\nand \u2303i = Yi Z>i W 1\n\ni Zi.\n\ni\n\net.\n\ni\n\n7:\n8: end for\n\ndata D0 is obtained by driving the system forward 6 time steps, excited by \u02dcut \u21e0N (0, I). This\nopen-loop experiment is repeated 100 times, such that D0 = {\u02dcxi\ni=1. We then apply three\nmethods: i. rrl - the method proposed in \u00a74.4, with look-ahead horizon h = 10; ii. nom - applying\n\u02dcJ1(K, \u21e5m(M(Dti))), i.e., a pure greedy exploitation\nthe \u2018nominal\u2019 robust policy Ki = arg minK\npolicy, with no explicit exploration; iii. greedy - \ufb01rst obtaining a nominal robustly stabilizing policy\nas with nom, but then optimizing (i.e., increasing, if possible) the exploration variance \u2303 until the\ngreedy policy and the rrl policy have the same theoretical worst-case cost at the current epoch. This\nis a greedy exploration policy; iv. ts - Thompson sampling [4]; v. rbst - the robust adaptive-control\nsynthesis method proposed in [13]. We perform 100 of these trials and plot the results in Figure 2.\n\n1:6, \u02dcu1:6}100\n\n(a)\n\n(b)\n\n(c)\n\nFigure 2: Results for the experiments described in \u00a75. (a) cost and \u2018information\u2019 (a scalar measure of\nuncertainty de\ufb01ned in \u00a75) when the system evolves according to the worst-case dynamics. The trace\ndenotes the median, and the shaded region spans from the 10th to the 90th percentile. (b) cost and\ninformation when policies are applied to the true system. (c) sum of costs (over all time steps) for\nthe worst-case dynamics (left) and the true system (right). Note that greedy is abbreviated as grdy.\n\ni\n\nIn Figure 2(a) we plot the cost at each epoch when the system evolves according to the worst-case\ndynamics. we also plot the information, de\ufb01ned as 1/max(D1\n), at the ith epoch, which is the\n(inverse) of the 2-norm of parameter error, cf. (4). This is a scalar measure of uncertainty: the\nlarger the information, the more certain our estimate of the system (in an absolute sense). ts is\nomitted from these results as the closed-loop behavior diverges (i.e. attains in\ufb01nite worst-case cost)\nin 96% of the trials conducted. In Figure 2(b) we plot the cost, and information, at each epoch\nwhen the policies are applied to the true system. Figure 2(c) plots the total cost (sum of costs\nover all epochs). We make the following observations. Concerning worst-case performance, nom\nattains the lowest cost at the initial epoch, as it does no explicit exploration. However, methods that\nincorporate exploration achieve better performance at subsequent epochs (and in terms of total cost)\ndue to greater reduction in uncertainty. Of these methods, the proposed rrl performs best, optimally\nbalancing exploration with exploitation; we emphasize that this balance of exploration/exploitation\noccurs automatically. Furthermore, observe that greedy actually achieves higher information (lower\nabsolute uncertainty) relative to rrl, yet attains higher cost. This suggests that rrl is reducing the\n\n8\n\n\funcertainty in a structured way, targeting uncertainty reduction in the parameters that \u2018matter most for\ncontrol\u2019. Results on the true system are qualitatively similar, with rrl attaining better performance\nthan all other methods except ts. We note that Thompson sampling performs well when the policy\nhappens to stabilize the system; however, as stability is not a consideration during ts synthesis, this\ncannot be guaranteed.\n\nHardware-in-the-loop experiment\nIn this section, we consider the RRL problem for a hardware-\nin-the-loop simulation comprised of the interconnection of a physical servo mechanism (Quanser\nQUBE 2) and a synthetic (simulated) LTI dynamical system. Control of servomechanisms is a\nubiquitous task in practice (e.g. robotics); furthermore, the planar servo can be modeled reasonably\nwell (globally) as a linear system. As such, this setup represents a good compromise between the\ncomplexities of a real world system (backlash, friction, unmodeled dynamics, disturbances, etc) and a\nsystem that approximately satis\ufb01es the assumptions of the method; cf. Appendix A.2 for full details\nof the experimental setup. An experimental trial consisted of the following procedure. Initial data was\nobtained by simulating the system for 0.5 seconds, under closed-loop feedback control (cf. Appendix\nA.2) with data sampled at 500Hz, to give 250 initial data points. We then applied methods rrl (with\nhorizon h = 5) and greedy as described in \u00a75. The total control horizon was T = 1250 (2.5 seconds\nat 500Hz) and was divided into N = 5 intervals, each of duration 0.5 seconds. We performed 5\nof these experimental trials and plot the results in Figure3. In Figure 3(b) and (d) we plot the total\ncost (the sum of the costs at each epoch), and the cost at each epoch, respectively, for each method,\nand observe signi\ufb01cantly better performance from rrl in both cases. Additional plots decomposing\nthe cost into that associated with the physical and synthetic system are available in Appendix A.2.\nWe also applied ts, and observed that the resulting policy was unable to stabilize the system; cf.\nFigure 3(c) which demonstrates divergence of the angular position of the servomotor under ts.\n\n4\n\n2\n\n0\n\n-2\n\n-4\n\n0\n\n3\n\n4\n\n5\n\n1\n\n2\n\n(c)\n\n(a)\n\n(b)\n\nFigure 3: Results for the hardware-in-the-loop experiment in \u00a75. (a) median costs at each epoch; the\nshaded region covers the best/worst costs at each epoch. (b) total costs (sum of costs at each epoch).\n(c) angular position of the servo motor (x1) under feedback control with the proposed method and\nThompson sampling; the latter results in divergence (uncontrolled revolutions of the servo motor).\n\n6 Conclusion\n\nWe have presented an algorithm for robust, targeted exploration in RL for linear systems with quadratic\nrewards. Policies are robust in the sense that stability of the closed loop system is guaranteed, with\nhigh probability, during learning, and targeted, in the sense that uncertainty is reduced so as to\nimprove performance of the controller on the speci\ufb01c task at hand. Roughly speaking, the policy\nprioritizes uncertainty reduction in the parameters that \u2018matter most for control\u2019. The search for a\npolicy is formulated as a convex program; solving to global optimality then automatically gives the\noptimal tradeoff between exploration and exploitation, in the worst-case setting.\n\nAcknowledgments\nThis research was \ufb01nancially supported by the project NewLEADS - New Directions in Learning\nDynamical Systems (contract number: 621-2016-06079), funded by the Swedish Research Council\n\n9\n\n\fand by the project ASSEMBLE (contract number: RIT15-0012), funded by the Swedish Foundation\nfor Strategic Research (SSF).\n\nReferences\n[1] Y. Abbasi-Yadkori and C. Szepesv\u00e1ri. Regret bounds for the adaptive control of linear quadratic systems.\n\nIn Proceedings of the 24th Annual Conference on Learning Theory, pages 1\u201326, 2011.\n\n[2] P. Abbeel and A. Y. Ng. Exploration and apprenticeship learning in reinforcement learning. In Proceedings\n\nof the 22nd international conference on Machine learning, pages 1\u20138. ACM, 2005.\n\n[3] M. Abeille and A. Lazaric. 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