nips nips2008 nips2008-181 knowledge-graph by maker-knowledge-mining

181 nips-2008-Policy Search for Motor Primitives in Robotics

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Author: Jens Kober, Jan R. Peters

Abstract: Many motor skills in humanoid robotics can be learned using parametrized motor primitives as done in imitation learning. However, most interesting motor learning problems are high-dimensional reinforcement learning problems often beyond the reach of current methods. In this paper, we extend previous work on policy learning from the immediate reward case to episodic reinforcement learning. We show that this results in a general, common framework also connected to policy gradient methods and yielding a novel algorithm for policy learning that is particularly well-suited for dynamic motor primitives. The resulting algorithm is an EM-inspired algorithm applicable to complex motor learning tasks. We compare this algorithm to several well-known parametrized policy search methods and show that it outperforms them. We apply it in the context of motor learning and show that it can learn a complex Ball-in-a-Cup task using a real Barrett WAMTM robot arm. 1

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Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 de Abstract Many motor skills in humanoid robotics can be learned using parametrized motor primitives as done in imitation learning. [sent-6, score-1.302]

2 However, most interesting motor learning problems are high-dimensional reinforcement learning problems often beyond the reach of current methods. [sent-7, score-0.637]

3 In this paper, we extend previous work on policy learning from the immediate reward case to episodic reinforcement learning. [sent-8, score-1.115]

4 We show that this results in a general, common framework also connected to policy gradient methods and yielding a novel algorithm for policy learning that is particularly well-suited for dynamic motor primitives. [sent-9, score-1.518]

5 The resulting algorithm is an EM-inspired algorithm applicable to complex motor learning tasks. [sent-10, score-0.413]

6 We compare this algorithm to several well-known parametrized policy search methods and show that it outperforms them. [sent-11, score-0.668]

7 We apply it in the context of motor learning and show that it can learn a complex Ball-in-a-Cup task using a real Barrett WAMTM robot arm. [sent-12, score-0.654]

8 1 Introduction Policy search, also known as policy learning, has become an accepted alternative of value functionbased reinforcement learning [2]. [sent-13, score-0.742]

9 In this paper, we will extend the previous work in [17, 18] from the immediate reward case to episodic reinforcement learning and show how it relates to policy gradient methods [7, 8, 11, 10]. [sent-15, score-1.16]

10 Despite that many real-world motor learning tasks are essentially episodic [14], episodic reinforcement learning [1] is a largely undersubscribed topic. [sent-16, score-1.208]

11 The resulting framework allows us to derive a new algorithm called Policy Learning by Weighting Exploration with the Returns (PoWER) which is particularly well-suited for learning of trial-based tasks in motor control. [sent-17, score-0.444]

12 We are especially interested in a particular kind of motor control policies also known as dynamic motor primitives [22, 23]. [sent-18, score-1.044]

13 , we have a special kind of parametrized policy which is well-suited for robotics problems. [sent-21, score-0.671]

14 We show that the presented algorithm works well when employed in the context of learning dynamic motor primitives in four different settings, i. [sent-22, score-0.654]

15 Looking at these tasks from a human motor learning perspective, we have a human acting as teacher presenting an example for imitation learning and, subsequently, the policy will be improved by reinforcement learning. [sent-27, score-1.42]

16 Since such tasks are inherently single-stroke movements, we focus on the special class of episodic reinforcement learning. [sent-28, score-0.492]

17 In our experiments, we show how a presented movement is recorded using kinesthetic teach-in and, subsequently, how a Barrett WAMTM robot arm is learning the behavior by a combination of imitation and reinforcement learning. [sent-29, score-0.72]

18 1 2 Policy Search for Parameterized Motor Primitives Our goal is to ﬁnd reinforcement learning techniques that can be applied to a special kind of prestructured parametrized policies called motor primitives [22, 23], in the context of learning highdimensional motor control tasks. [sent-30, score-1.39]

19 3 and show how the general framework is related to policy gradients methods in 2. [sent-34, score-0.575]

20 [12] extends the [17] algorithm to episodic reinforcement learning for discrete states; we use continuous states. [sent-36, score-0.494]

21 Subsequently, we discuss how we can turn the parametrized motor primitives [22, 23] into explorative [19], stochastic policies. [sent-37, score-0.728]

22 1 Problem Statement & Notation In this paper, we treat motor primitive learning problems in the framework of reinforcement learning with a strong focus on the episodic case [1]. [sent-39, score-1.007]

23 We assume that at time t there is an actor in a state st and chooses an appropriate action at according to a stochastic policy π(at |st , t). [sent-40, score-0.836]

24 Such a policy is a probability distribution over actions given the current state. [sent-41, score-0.545]

25 The stochastic formulation allows a natural incorporation of exploration and, in the case of hidden state variables, the optimal timeinvariant policy has been shown to be stochastic [8]. [sent-42, score-0.721]

26 As we are interested in learning complex motor tasks consisting of a single stroke [23], we focus on ﬁnite horizons of length T with episodic restarts [1] and learn the optimal parametrized, stochastic policy for such reinforcement learning problems. [sent-44, score-1.575]

27 We assume an explorative version of the dynamic motor primitives [22, 23] as parametrized policy π with parameters θ ∈ Rn . [sent-45, score-1.244]

28 While episodic Reinforcement Learning (RL) problems with ﬁnite horizons are common in motor control, few methods exist in the RL literature, e. [sent-56, score-0.698]

29 The motor primitives based on dynamical systems [22, 23] are a particular type of time-variant policy representation as they have an internal phase which corresponds to a clock with additional ﬂexibility (e. [sent-60, score-1.139]

30 2 Episodic Policy Learning In this section, we discuss episodic reinforcement learning in policy space which we will refer to as Episodic Policy Learning. [sent-66, score-1.012]

31 For doing so, we ﬁrst discuss the lower bound on the expected return suggested in [17] for guaranteeing that policy update steps are improvements. [sent-67, score-0.572]

32 The reasons for this preference apply in policy learning: if the lower bound also becomes an equality for the sampling policy, 2 we can guarantee that the policy will be improved by optimizing the lower bound. [sent-74, score-1.036]

33 , generate rollouts τ using the current policy with parameters θ which we weight with the returns R (τ ) and subsequently match it with a new policy parametrized by θ . [sent-78, score-1.46]

34 The policy improvement step is equivalent to maximizing the lower bound on the expected return Lθ (θ ) and we show how it relates to previous policy learning methods. [sent-84, score-1.123]

35 2 Resulting Policy Updates In the following part, we will discuss three different policy updates which directly result from Section 2. [sent-87, score-0.518]

36 First, we show that policy gradients [7, 8, 11, 10] can be derived from the lower bound Lθ (θ ) (as was to be expected from supervised learning, see [13]). [sent-90, score-0.575]

37 Subsequently, we show that natural policy gradients can be seen as an additional constraint regularizing the change in the path distribution resulting from a policy update when improving the policy incrementally. [sent-91, score-1.648]

38 Finally, we will show how expectation-maximization (EM) algorithms for policy learning can be generated. [sent-92, score-0.551]

39 As this log-derivative only depends on the policy, we can estimate a gradient from rollouts without having a model by simply replacing the expectation by a sum; when θ is close to θ, we have the policy gradient estimator which is widely known as Episodic REINFORCE [7], i. [sent-95, score-0.8]

40 Equation (6) is equivalent to the policy gradient theorem [8] for θ → θ in the inﬁnite horizon case where the dependence on time t can be dropped. [sent-101, score-0.563]

41 Previously, similar ideas have been explored in immediate reinforcement learning [17, 18]. [sent-109, score-0.269]

42 In general, an EMalgorithm would choose the next policy parameters θ n+1 such that θ n+1 = argmaxθ Lθ (θ ). [sent-110, score-0.518]

43 In the case where π(at |st , t) belongs to the exponential family, the next policy can be determined analytically by setting Equation (6) to zero, i. [sent-111, score-0.518]

44 3 Policy learning by Weighting Exploration with the Returns (PoWER) ¯ In most learning control problems, we attempt to have a deterministic mean policy a = θ T φ(s, t) with parameters θ and basis functions φ. [sent-124, score-0.625]

45 In Section 3, we will introduce the basis functions of the motor primitives. [sent-125, score-0.38]

46 When learning motor primitives, we turn this deterministic mean policy ¯ a = θ T φ(s, t) into a stochastic policy using additive exploration ε(s, t) in order to make modelfree reinforcement learning possible, i. [sent-126, score-1.83]

47 , we always intend to have a policy π(at |st , t) which can be brought into the form a = θ T φ(s, t) + (φ(s, t)). [sent-128, score-0.518]

48 It is straightforward to obtain the Reward-Weighted Regression for episodic RL by solving Equation (7) for θ which naturally yields a weighted regression method with the state-action values Qπ (s, a, t) as weights. [sent-132, score-0.301]

49 This form of exploration has resulted into various applications in robotics such as T-Ball batting, Peg-In-Hole, humanoid robot locomotion, constrained reaching movements and operational space control, see [4, 10, 18] for both reviews and their own applications. [sent-133, score-0.529]

50 As a result, all methods relying on this state-independent exploration have proven too fragile for learning the Ballin-a-Cup task on a real robot system. [sent-135, score-0.364]

51 Alternatively, as introduced by [19], one could generate a form 2 of structured, state-dependent exploration (φ(s, t)) = εT φ(s, t) with [εt ]ij ∼ N (0, σij ), where t 2 σij are meta-parameters of the exploration that can also be optimized. [sent-136, score-0.262]

52 This argument results into ˆ the policy a ∼ π(at |st , t) = N (a|θ T φ(s, t), Σ(s, t)). [sent-137, score-0.518]

53 Inserting the resulting policy into Equation (7), we obtain the optimality condition in the sense of Equation (7) and can derive the update rule θ =θ+E T π t=1 Q (s, a, t)W(s, t) T T −1 E T π t=1 Q (s, a, t)W(s, t)εt (8) with W(s, t) = φ(s, t)φ(s, t) /(φ(s, t) φ(s, t)). [sent-138, score-0.518]

54 Note that for our motor primitives W reduces to a diagonal, constant matrix and cancels out. [sent-139, score-0.577]

55 In order to reduce the number of rollouts in this on-policy scenario, we reuse the rollouts through importance sampling as described in the context of reinforcement learning in [1]. [sent-141, score-0.628]

56 3 in the context of motor primitive learning for robotics. [sent-147, score-0.533]

57 For doing so, we will ﬁrst give a quick overview how the motor primitives work and how the algorithm can be used to adapt them. [sent-148, score-0.577]

58 As a signiﬁcantly more complex motor learning task, we show how the robot can learn a high-speed Ball-in-a-Cup [24] movement with motor primitives for all seven degrees of freedom of our Barrett WAMTM robot arm. [sent-154, score-1.459]

59 1 Using the Motor Primitives in Policy Search The motor primitive framework [22, 23] can be described as two coupled differential equations, i. [sent-156, score-0.48]

60 Both ¨ ˙ dynamical systems are chosen to be stable and to have the right properties so that they are useful for the desired class of motor control problems. [sent-159, score-0.442]

61 In this paper, we focus on single stroke movements as they frequently appear in human motor control [14, 23] and, thus, we will always choose the point attractor version of the motor primitives exactly as presented in [23] and not the older one in [22]. [sent-160, score-1.122]

62 The biggest advantage of the motor primitive framework of [22, 23] is that the function g is linear in the policy parameters θ and, thus, well-suited for imitation learning as well as for our presented reinforcement learning algorithm. [sent-161, score-1.423]

63 For example, if we would have to learn only a motor primitive for ¯ a single degree of freedom qi , then we could use a motor primitive in the form qi = g(qi , qi , y, θ) = ¨ ˙ φ(s)T θ where s = [qi , qi , y] is the state and where time is implicitly embedded in y. [sent-162, score-1.249]

64 We use the ˙ ¯ output of qi = φ(s)T θ = a as the policy mean. [sent-163, score-0.58]

65 (a) minimum motor command (b) passing through a point −250 1 average return average return In Sections 3. [sent-166, score-0.488]

66 −500 −10 2 −10 −1000 2 3 10 10 number of rollouts 2 3 10 10 number of rollouts FDG VPG eNAC RWR PoWER 3. [sent-171, score-0.384]

67 two tasks commonly studied in motor control literature for which the analytic solutions are known, i. [sent-175, score-0.452]

68 , a reaching task where a goal has to be reached at a certain time while the used motor commands have to be minimized and a reaching task of the same style with an additional via-point. [sent-177, score-0.444]

69 For both tasks, we use the same rewards as in [10] but we use the newer form of the motor primitives from [23]. [sent-179, score-0.577]

70 The episodic Reward-Weighted Regression (RWR) is outperformed by all other algorithms suggesting that this algorithm does not generalize well from the immediate reward case. [sent-182, score-0.373]

71 5 Figure 2: This ﬁgure shows the time series of the Underactuated Swing-Up where only a single joint of the robot is moved with a torque limit ensured by limiting the maximal motor current of that joint. [sent-183, score-0.675]

72 The resulting motion requires the robot to (i) ﬁrst move away from the target to limit the maximal required torque during the swing-up in (ii-iv) and subsequent stabilization (v). [sent-184, score-0.397]

73 7 PoWER dom is represented by the motor primitive as described eNAC in Section 3. [sent-196, score-0.48]

74 6 pendulum to an upright position and stabilize it there 50 100 150 200 number of rollouts in minimum time and with minimal motor torques. [sent-199, score-0.572]

75 Under these torque performance averaged over 20 learning limits, the robot needs to (i) ﬁrst move away from the runs with the error bars indicating the stantarget to limit the maximal required torque during the dard deviation. [sent-201, score-0.462]

76 The motor primitive with nine shape parameters and one goal parameter was initialized by imitation learning from a kinesthetic teach-in. [sent-207, score-0.761]

77 4 Ball-in-a-Cup on a Barrett WAMTM The most challenging application in this paper is the children’s game Ball-in-a-Cup [24] where a small cup is attached at the robot’s end-effector and this cup has a small wooden ball hanging down from the cup on a 40cm string. [sent-214, score-0.548]

78 The robot needs to move fast in order to induce a motion at the ball through the string, swing it up and catch it with the cup, a possible movement is illustrated in Figure 4 (top row). [sent-216, score-0.402]

79 The actions are the joint space accelerations where each of the seven joints is represented by a motor primitive. [sent-218, score-0.431]

80 The task is quite complex as the reward is not modiﬁed solely by the movements of the cup but foremost by the movements of the ball and the movements of the ball are very sensitive to changes in the movement. [sent-220, score-0.527]

81 6 Figure 4: This ﬁgure shows schematic drawings of the Ball-in-a-Cup motion, the ﬁnal learned robot motion as well as a kinesthetic teach-in. [sent-222, score-0.329]

82 The human cup motion was taught to the robot by imitation learning with 31 parameters per joint for an approximately 3 seconds long trajectory. [sent-224, score-0.612]

83 The robot manages to reproduce the imitated motion quite accurately, but the ball misses the cup by several centimeters. [sent-225, score-0.487]

84 75 iterations of our Policy learning by Weighting Exploration with the Returns (PoWER) algorithm the robot has improved its motion so that the ball goes in the cup. [sent-227, score-0.359]

85 average return Due to the complexity of the task, Ball-in-a-Cup is 1 even a hard motor learning task for children who usu0. [sent-229, score-0.494]

86 4 we ﬁrst initialize the motor primitives by imitation and, 0. [sent-233, score-0.745]

87 We recorded the motions of a human player by 0 0 20 40 60 80 100 kinesthetic teach-in in order to obtain an example for number of rollouts imitation as shown in Figure 4 (middle row). [sent-235, score-0.461]

88 From the Figure 5: This ﬁgure shows the expected imitation, it can be determined by cross-validation that return of the learned policy in the Ball-in31 parameters per motor primitive are needed. [sent-236, score-1.052]

89 pected, the robot fails to reproduce the the presented behavior and reinforcement learning is needed for self-improvement. [sent-238, score-0.444]

90 Figure 5 shows the expected return over the number of rollouts where convergence to a maximum is clearly recognizable. [sent-239, score-0.246]

91 The robot regularly succeeds at bringing the ball into the cup after approximately 75 iterations. [sent-240, score-0.44]

92 4 Conclusion In this paper, we have presented a new perspective on policy learning methods and an application to a highly complex motor learning task on a real Barrett WAMTM robot arm. [sent-241, score-1.164]

93 We have generalized the previous work in [17, 18] from the immediate reward case to the episodic case. [sent-242, score-0.373]

94 In the process, we could show that policy gradient methods are a special case of this more general framework. [sent-243, score-0.563]

95 During initial experiments, we realized that the form of exploration highly inﬂuences the speed of the policy learning method. [sent-244, score-0.682]

96 This empirical insight resulted in a novel policy learning algorithm, Policy learning by Weighting Exploration with the Returns (PoWER), an EM-inspired algorithm that outperforms several other policy search methods both on standard benchmarks as well as on a simulated Underactuated Swing-Up. [sent-245, score-1.159]

97 Instead, we mimic the way children learn Ball-in-a-Cup and ﬁrst present an example for imitation learning which is recorded using kinesthetic teach-in. [sent-250, score-0.329]

98 Subsequently, our reinforcement learning algorithm takes over and learns how to move the ball into 7 the cup reliably. [sent-251, score-0.47]

99 PEGASUS: A policy search method for large MDPs and POMDPs. [sent-268, score-0.546]

100 Policy gradient methods for reinforcement learning with function approximation. [sent-300, score-0.269]

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At each of a sequence of discrete time steps, t = 1, 2, . . ., the environment is in a state st ∈ S, the agent chooses an action at ∈ A, and then the environment emits a reward rt ∈ R, and transitions to its next state st+1 ∈ S. The state and action sets are ﬁnite. State transitions are stochastic and dependent on the immediately preceding state and action. Rewards are stochastic and dependent on the preceding state and action, and on the next state. The agent process generating the actions is termed the behavior policy. To start, we assume a deterministic target policy π : S → A. The objective is to learn an approximation to its state-value function: ∞ V π (s) = Eπ γ t−1 rt |s1 = s , (1) t=1 where γ ∈ [0, 1) is the discount rate. The learning is to be done without knowledge of the process dynamics and from observations of a single continuous trajectory with no resets. In many problems of interest the state set is too large for it to be practical to approximate the value of each state individually. Here we consider linear function approximation, in which states are mapped to feature vectors with fewer components than the number of states. That is, for each state s ∈ S there is a corresponding feature vector φ(s) ∈ Rn , with n |S|. The approximation to the value function is then required to be linear in the feature vectors and a corresponding parameter vector θ ∈ Rn : V π (s) ≈ θ φ(s). (2) Further, we assume that the states st are not visible to the learning agent in any way other than through the feature vectors. Thus this function approximation formulation can include partialobservability formulations such as POMDPs as a special case. The environment and the behavior policy together generate a stream of states, actions and rewards, s1 , a1 , r1 , s2 , a2 , r2 , . . ., which we can break into causally related 4-tuples, (s1 , a1 , r1 , s1 ), 2 (s2 , a2 , r2 , s2 ), . . . , where st = st+1 . For some tuples, the action will match what the target policy would do in that state, and for others it will not. We can discard all of the latter as not relevant to the target policy. For the former, we can discard the action because it can be determined from the state via the target policy. With a slight abuse of notation, let sk denote the kth state in which an on-policy action was taken, and let rk and sk denote the associated reward and next state. The kth on-policy transition, denoted (sk , rk , sk ), is a triple consisting of the starting state of the transition, the reward on the transition, and the ending state of the transition. The corresponding data available to the learning algorithm is the triple (φ(sk ), rk , φ(sk )). The MDP under the behavior policy is assumed to be ergodic, so that it determines a stationary state-occupancy distribution µ(s) = limk→∞ P r{sk = s}. For any state s, the MDP and target policy together determine an N × N state-transition-probability matrix P , where pss = P r{sk = s |sk = s}, and an N × 1 expected-reward vector R, where Rs = E[rk |sk = s]. These two together completely characterize the statistics of on-policy transitions, and all the samples in the sequence of (φ(sk ), rk , φ(sk )) respect these statistics. The problem still has a Markov structure in that there are temporal dependencies between the sample transitions. In our analysis we ﬁrst consider a formulation without such dependencies, the i.i.d. case, and then prove that our results extend to the original case. In the i.i.d. formulation, the states sk are generated independently and identically distributed according to an arbitrary probability distribution µ. From each sk , a corresponding sk is generated according to the on-policy state-transition matrix, P , and a corresponding rk is generated according to an arbitrary bounded distribution with expected value Rsk . The ﬁnal i.i.d. data sequence, from which an approximate value function is to be learned, is then the sequence (φ(sk ), rk , φ(sk )), for k = 1, 2, . . . Further, because each sample is i.i.d., we can remove the indices and talk about a single tuple of random variables (φ, r, φ ) drawn from µ. It remains to deﬁne the objective of learning. The TD error for the linear setting is δ = r + γθ φ − θ φ. (3) Given this, we deﬁne the one-step linear TD solution as any value of θ at which 0 = E[δφ] = −Aθ + b, (4) where A = E φ(φ − γφ ) and b = E[rφ]. This is the parameter value to which the linear TD(0) algorithm (Sutton 1988) converges under on-policy training, as well as the value found by LSTD(0) (Bradtke & Barto 1996) under both on-policy and off-policy training. The TD solution is always a ﬁxed-point of the linear TD(0) algorithm, but under off-policy training it may not be stable; if θ does not exactly satisfy (4), then the TD(0) algorithm may cause it to move away in expected value and eventually diverge to inﬁnity. 3 The GTD(0) algorithm We next present the idea and gradient-descent derivation leading to the GTD(0) algorithm. As discussed above, the vector E[δφ] can be viewed as an error in the current solution θ. The vector should be zero, so its norm is a measure of how far we are away from the TD solution. A distinctive feature of our gradient-descent analysis of temporal-difference learning is that we use as our objective function the L2 norm of this vector: J(θ) = E[δφ] E[δφ] . (5) This objective function is quadratic and unimodal; it’s minimum value of 0 is achieved when E[δφ] = 0, which can always be achieved. The gradient of this objective function is θ J(θ) = 2( = 2E φ( θ E[δφ])E[δφ] θ δ) E[δφ] = −2E φ(φ − γφ ) E[δφ] . (6) This last equation is key to our analysis. We would like to take a stochastic gradient-descent approach, in which a small change is made on each sample in such a way that the expected update 3 is the direction opposite to the gradient. This is straightforward if the gradient can be written as a single expected value, but here we have a product of two expected values. One cannot sample both of them because the sample product will be biased by their correlation. However, one could store a long-term, quasi-stationary estimate of either of the expectations and then sample the other. The question is, which expectation should be estimated and stored, and which should be sampled? Both ways seem to lead to interesting learning algorithms. First let us consider the algorithm obtained by forming and storing a separate estimate of the ﬁrst expectation, that is, of the matrix A = E φ(φ − γφ ) . This matrix is straightforward to estimate from experience as a simple arithmetic average of all previously observed sample outer products φ(φ − γφ ) . Note that A is a stationary statistic in any ﬁxed-policy policy-evaluation problem; it does not depend on θ and would not need to be re-estimated if θ were to change. Let Ak be the estimate of A after observing the ﬁrst k samples, (φ1 , r1 , φ1 ), . . . , (φk , rk , φk ). Then this algorithm is deﬁned by k 1 Ak = φi (φi − γφi ) (7) k i=1 along with the gradient descent rule: θk+1 = θk + αk Ak δk φk , k ≥ 1, (8) where θ1 is arbitrary, δk = rk + γθk φk − θk φk , and αk > 0 is a series of step-size parameters, possibly decreasing over time. We call this algorithm A TD(0) because it is essentially conventional TD(0) preﬁxed by an estimate of the matrix A . Although we ﬁnd this algorithm interesting, we do not consider it further here because it requires O(n2 ) memory and computation per time step. The second path to a stochastic-approximation algorithm for estimating the gradient (6) is to form and store an estimate of the second expectation, the vector E[δφ], and to sample the ﬁrst expectation, E φ(φ − γφ ) . Let uk denote the estimate of E[δφ] after observing the ﬁrst k − 1 samples, with u1 = 0. The GTD(0) algorithm is deﬁned by uk+1 = uk + βk (δk φk − uk ) (9) and θk+1 = θk + αk (φk − γφk )φk uk , (10) where θ1 is arbitrary, δk is as in (3) using θk , and αk > 0 and βk > 0 are step-size parameters, possibly decreasing over time. Notice that if the product is formed right-to-left, then the entire computation is O(n) per time step. 4 Convergence The purpose of this section is to establish that GTD(0) converges with probability one to the TD solution in the i.i.d. problem formulation under standard assumptions. In particular, we have the following result: Theorem 4.1 (Convergence of GTD(0)). Consider the GTD(0) iteration (9,10) with step-size se∞ ∞ 2 quences αk and βk satisfying βk = ηαk , η > 0, αk , βk ∈ (0, 1], k=0 αk = ∞, k=0 αk < ∞. Further assume that (φk , rk , φk ) is an i.i.d. sequence with uniformly bounded second moments. Let A = E φk (φk − γφk ) and b = E[rk φk ] (note that A and b are well-deﬁned because the distribution of (φk , rk , φk ) does not depend on the sequence index k). Assume that A is non-singular. Then the parameter vector θk converges with probability one to the TD solution (4). Proof. We use the ordinary-differential-equation (ODE) approach (Borkar & Meyn 2000). First, we rewrite the algorithm’s two iterations as a single iteration in a combined parameter vector with √ 2n components ρk = (vk , θk ), where vk = uk / η, and a new reward-related vector with 2n components gk+1 = (rk φk , 0 ): √ ρk+1 = ρk + αk η (Gk+1 ρk + gk+1 ) , where Gk+1 = √ − ηI (φk − γφk )φk 4 φk (γφk − φk ) 0 . Let G = E[Gk ] and g = E[gk ]. Note that G and g are well-deﬁned as by the assumption the process {φk , rk , φk }k is i.i.d. In particular, √ − η I −A b G= , g= . 0 A 0 Further, note that (4) follows from Gρ + g = 0, (11) where ρ = (v , θ ). Now we apply Theorem 2.2 of Borkar & Meyn (2000). For this purpose we write ρk+1 = ρk + √ √ αk η(Gρk +g+(Gk+1 −G)ρk +(gk+1 −g)) = ρk +αk (h(ρk )+Mk+1 ), where αk = αk η, h(ρ) = g + Gρ and Mk+1 = (Gk+1 − G)ρk + gk+1 − g. Let Fk = σ(ρ1 , M1 , . . . , ρk−1 , Mk ). Theorem 2.2 requires the veriﬁcation of the following conditions: (i) The function h is Lipschitz and h∞ (ρ) = limr→∞ h(rρ)/r is well-deﬁned for every ρ ∈ R2n ; (ii-a) The sequence (Mk , Fk ) is a martingale difference sequence, and (ii-b) for some C0 > 0, E Mk+1 2 | Fk ≤ C0 (1 + ρk 2 ) holds for ∞ any initial parameter vector ρ1 ; (iii) The sequence αk satisﬁes 0 < αk ≤ 1, k=1 αk = ∞, ∞ 2 ˙ k=1 (αk ) < +∞; and (iv) The ODE ρ = h(ρ) has a globally asymptotically stable equilibrium. Clearly, h(ρ) is Lipschitz with coefﬁcient G and h∞ (ρ) = Gρ. By construction, (Mk , Fk ) satisﬁes E[Mk+1 |Fk ] = 0 and Mk ∈ Fk , i.e., it is a martingale difference sequence. Condition (ii-b) can be shown to hold by a simple application of the triangle inequality and the boundedness of the the second moments of (φk , rk , φk ). Condition (iii) is satisﬁed by our conditions on the step-size sequences αk , βk . Finally, the last condition (iv) will follow from the elementary theory of linear differential equations if we can show that the real parts of all the eigenvalues of G are negative. First, let us show that G is non-singular. Using the determinant rule for partitioned matrices1 we get det(G) = det(A A) = 0. This indicates that all the eigenvalues of G are non-zero. Now, let λ ∈ C, λ = 0 be an eigenvalue of G with corresponding normalized eigenvector x ∈ C2n ; 2 that is, x = x∗ x = 1, where x∗ is the complex conjugate of x. Hence x∗ Gx = λ. Let √ 2 x = (x1 , x2 ), where x1 , x2 ∈ Cn . Using the deﬁnition of G, λ = x∗ Gx = − η x1 + x∗ Ax2 − x∗ A x1 . Because A is real, A∗ = A , and it follows that (x∗ Ax2 )∗ = x∗ A x1 . Thus, 1 2 1 2 √ 2 Re(λ) = Re(x∗ Gx) = − η x1 ≤ 0. We are now done if we show that x1 cannot be zero. If x1 = 0, then from λ = x∗ Gx we get that λ = 0, which contradicts with λ = 0. The next result concerns the convergence of GTD(0) when (φk , rk , φk ) is obtained by the off-policy sub-sampling process described originally in Section 2. We make the following assumption: Assumption A1 The behavior policy πb (generator of the actions at ) selects all actions of the target policy π with positive probability in every state, and the target policy is deterministic. This assumption is needed to ensure that the sub-sampled process sk is well-deﬁned and that the obtained sample is of “high quality”. Under this assumption it holds that sk is again a Markov chain by the strong Markov property of Markov processes (as the times selected when actions correspond to those of the behavior policy form Markov times with respect to the ﬁltration deﬁned by the original process st ). The following theorem shows that the conclusion of the previous result continues to hold in this case: Theorem 4.2 (Convergence of GTD(0) with a sub-sampled process.). Assume A1. Let the parameters θk , uk be updated by (9,10). Further assume that (φk , rk , φk ) is such that E φk 2 |sk−1 , 2 E rk |sk−1 , E φk 2 |sk−1 are uniformly bounded. Assume that the Markov chain (sk ) is aperiodic and irreducible, so that limk→∞ P(sk = s |s0 = s) = µ(s ) exists and is unique. Let s be a state randomly drawn from µ, and let s be a state obtained by following π for one time step in the MDP from s. Further, let r(s, s ) be the reward incurred. Let A = E φ(s)(φ(s) − γφ(s )) and b = E[r(s, s )φ(s)]. Assume that A is non-singular. Then the parameter vector θk converges with probability one to the TD solution (4), provided that s1 ∼ µ. Proof. The proof of Theorem 4.1 goes through without any changes once we observe that G = E[Gk+1 |Fk ] and g = E[gk+1 | Fk ]. 1 R According to this rule, if A ∈ Rn×n , B ∈ Rn×m , C ∈ Rm×n , D ∈ Rm×m then for F = [A B; C D] ∈ , det(F ) = det(A) det(D − CA−1 B). (n+m)×(n+m) 5 The condition that (sk ) is aperiodic and irreducible guarantees the existence of the steady state distribution µ. Further, the aperiodicity and irreducibility of (sk ) follows from the same property of the original process (st ). For further discussion of these conditions cf. Section 6.3 of Bertsekas and Tsitsiklis (1996). With considerable more work the result can be extended to the case when s1 follows an arbitrary distribution. This requires an extension of Theorem 2.2 of Borkar and Meyn (2000) to processes of the form ρk+1 + ρk (h(ρk ) + Mk+1 + ek+1 ), where ek+1 is a fast decaying perturbation (see, e.g., the proof of Proposition 4.8 of Bertsekas and Tsitsiklis (1996)). 5 Extensions to action values, stochastic target policies, and other sample weightings The GTD algorithm extends immediately to the case of off-policy learning of action-value functions. For this assume that a behavior policy πb is followed that samples all actions in every state with positive probability. Let the target policy to be evaluated be π. In this case the basis functions are dependent on both the states and actions: φ : S × A → Rn . The learning equations are unchanged, except that φt and φt are redeﬁned as follows: φt = φ(st , at ), (12) φt = (13) π(st+1 , a)φ(st+1 , a). a (We use time indices t denoting physical time.) Here π(s, a) is the probability of selecting action a in state s under the target policy π. Let us call the resulting algorithm “one-step gradient-based Q-evaluation,” or GQE(0). Theorem 5.1 (Convergence of GQE(0)). Assume that st is a state sequence generated by following some stationary policy πb in a ﬁnite MDP. Let rt be the corresponding sequence of rewards and let φt , φt be given by the respective equations (12) and (13), and assume that E φt 2 |st−1 , 2 E rt |st−1 , E φt 2 |st−1 are uniformly bounded. Let the parameters θt , ut be updated by Equations (9) and (10). Assume that the Markov chain (st ) is aperiodic and irreducible, so that limt→∞ P(st = s |s0 = s) = µ(s ) exists and is unique. Let s be a state randomly drawn from µ, a be an action chosen by πb in s, let s be the next state obtained and let a = π(s ) be the action chosen by the target policy in state s . Further, let r(s, a, s ) be the reward incurred in this transition. Let A = E φ(s, a)(φ(s, a) − γφ(s , a )) and b = E[r(s, a, s )φ(s, a)]. Assume that A is non-singular. Then the parameter vector θt converges with probability one to a TD solution (4), provided that s1 is selected from the steady-state distribution µ. The proof is almost identical to that of Theorem 4.2, and hence it is omitted. Our main convergence results are also readily generalized to stochastic target policies by replacing the sub-sampling process described in Section 2 with a sample-weighting process. That is, instead of including or excluding transitions depending upon whether the action taken matches a deterministic policy, we include all transitions but give each a weight. For example, we might let the weight wt for time step t be equal to the probability π(st , at ) of taking the action actually taken under the target policy. We can consider the i.i.d. samples now to have four components (φk , rk , φk , wk ), with the update rules (9) and (10) replaced by uk+1 = uk + βk (δk φk − uk )wk , (14) θk+1 = θk + αk (φk − γφk )φk uk wk . (15) and Each sample is also weighted by wk in the expected values, such as that deﬁning the TD solution (4). With these changes, Theorems 4.1 and 4.2 go through immediately for stochastic policies. The reweighting is, in effect, an adjustment to the i.i.d. sampling distribution, µ, and thus our results hold because they hold for all µ. The choice wt = π(st , at ) is only one possibility, notable for its equivalence to our original case if the target policy is deterministic. Another natural weighting is wt = π(st , at )/πb (st , at ), where πb is the behavior policy. This weighting may result in the TD solution (4) better matching the target policy’s value function (1). 6 6 Related work There have been several prior attempts to attain the four desirable algorithmic features mentioned at the beginning this paper (off-policy stability, temporal-difference learning, linear function approximation, and O(n) complexity) but none has been completely successful. One idea for retaining all four desirable features is to use importance sampling techniques to reweight off-policy updates so that they are in the same direction as on-policy updates in expected value (Precup, Sutton & Dasgupta 2001; Precup, Sutton & Singh 2000). Convergence can sometimes then be assured by existing results on the convergence of on-policy methods (Tsitsiklis & Van Roy 1997; Tadic 2001). However, the importance sampling weights are cumulative products of (possibly many) target-to-behavior-policy likelihood ratios, and consequently they and the corresponding updates may be of very high variance. The use of “recognizers” to construct the target policy directly from the behavior policy (Precup, Sutton, Paduraru, Koop & Singh 2006) is one strategy for limiting the variance; another is careful choice of the target policies (see Precup, Sutton & Dasgupta 2001). However, it remains the case that for all of such methods to date there are always choices of problem, behavior policy, and target policy for which the variance is inﬁnite, and thus for which there is no guarantee of convergence. Residual gradient algorithms (Baird 1995) have also been proposed as a way of obtaining all four desirable features. These methods can be viewed as gradient descent in the expected squared TD error, E δ 2 ; thus they converge stably to the solution that minimizes this objective for arbitrary differentiable function approximators. However, this solution has always been found to be much inferior to the TD solution (exempliﬁed by (4) for the one-step linear case). In the literature (Baird 1995; Sutton & Barto 1998), it is often claimed that residual-gradient methods are guaranteed to ﬁnd the TD solution in two special cases: 1) systems with deterministic transitions and 2) systems in which two samples can be drawn for each next state (e.g., for which a simulation model is available). Our own analysis indicates that even these two special requirements are insufﬁcient to guarantee convergence to the TD solution.2 Gordon (1995) and others have questioned the need for linear function approximation. He has proposed replacing linear function approximation with a more restricted class of approximators, known as averagers, that never extrapolate outside the range of the observed data and thus cannot diverge. Rightly or wrongly, averagers have been seen as being too constraining and have not been used on large applications involving online learning. Linear methods, on the other hand, have been widely used (e.g., Baxter, Tridgell & Weaver 1998; Sturtevant & White 2006; Schaeffer, Hlynka & Jussila 2001). The need for linear complexity has also been questioned. Second-order methods for linear approximators, such as LSTD (Bradtke & Barto 1996; Boyan 2002) and LSPI (Lagoudakis & Parr 2003; see also Peters, Vijayakumar & Schaal 2005), can be effective on moderately sized problems. If the number of features in the linear approximator is n, then these methods require memory and per-timestep computation that is O(n2 ). Newer incremental methods such as iLSTD (Geramifard, Bowling & Sutton 2006) have reduced the per-time-complexity to O(n), but are still O(n2 ) in memory. Sparsiﬁcation methods may reduce the complexity further, they do not help in the general case, and may apply to O(n) methods as well to further reduce their complexity. Linear function approximation is most powerful when very large numbers of features are used, perhaps millions of features (e.g., as in Silver, Sutton & M¨ ller 2007). In such cases, O(n2 ) methods are not feasible. u 7 Conclusion GTD(0) is the ﬁrst off-policy TD algorithm to converge under general conditions with linear function approximation and linear complexity. As such, it breaks new ground in terms of important, 2 For a counterexample, consider that given in Dayan’s (1992) Figure 2, except now consider that state A is actually two states, A and A’, which share the same feature vector. The two states occur with 50-50 probability, and when one occurs the transition is always deterministically to B followed by the outcome 1, whereas when the other occurs the transition is always deterministically to the outcome 0. In this case V (A) and V (B) will converge under the residual-gradient algorithm to the wrong answers, 1/3 and 2/3, even though the system is deterministic, and even if multiple samples are drawn from each state (they will all be the same). 7 absolute abilities not previous available in existing algorithms. We have conducted empirical studies with the GTD(0) algorithm and have conﬁrmed that it converges reliably on standard off-policy counterexamples such as Baird’s (1995) “star” problem. On on-policy problems such as the n-state random walk (Sutton 1988; Sutton & Barto 1998), GTD(0) does not seem to learn as efﬁciently as classic TD(0), although we are still exploring different ways of setting the step-size parameters, and other variations on the algorithm. It is not clear that the GTD(0) algorithm in its current form will be a fully satisfactory solution to the off-policy learning problem, but it is clear that is breaks new ground and achieves important abilities that were previously unattainable. Acknowledgments The authors gratefully acknowledge insights and assistance they have received from David Silver, Eric Wiewiora, Mark Ring, Michael Bowling, and Alborz Geramifard. This research was supported by iCORE, NSERC and the Alberta Ingenuity Fund. References Baird, L. C. (1995). Residual algorithms: Reinforcement learning with function approximation. In Proceedings of the Twelfth International Conference on Machine Learning, pp. 30–37. Morgan Kaufmann. Baxter, J., Tridgell, A., Weaver, L. (1998). Experiments in parameter learning using temporal differences. International Computer Chess Association Journal, 21, 84–99. Bertsekas, D. P., Tsitsiklis. J. (1996). Neuro-Dynamic Programming. Athena Scientiﬁc, 1996. Borkar, V. S. and Meyn, S. P. (2000). The ODE method for convergence of stochastic approximation and reinforcement learning. SIAM Journal on Control And Optimization , 38(2):447–469. Boyan, J. (2002). Technical update: Least-squares temporal difference learning. Machine Learning, 49:233– 246. Bradtke, S., Barto, A. G. (1996). Linear least-squares algorithms for temporal difference learning. Machine Learning, 22:33–57. Dayan, P. (1992). The convergence of TD(λ) for general λ. Machine Learning, 8:341–362. Geramifard, A., Bowling, M., Sutton, R. S. (2006). Incremental least-square temporal difference learning. Proceedings of the National Conference on Artiﬁcial Intelligence, pp. 356–361. Gordon, G. J. (1995). Stable function approximation in dynamic programming. Proceedings of the Twelfth International Conference on Machine Learning, pp. 261–268. Morgan Kaufmann, San Francisco. Lagoudakis, M., Parr, R. (2003). Least squares policy iteration. Journal of Machine Learning Research, 4:1107-1149. Peters, J., Vijayakumar, S. and Schaal, S. (2005). Natural Actor-Critic. Proceedings of the 16th European Conference on Machine Learning, pp. 280–291. Precup, D., Sutton, R. S. and Dasgupta, S. (2001). Off-policy temporal-difference learning with function approximation. Proceedings of the 18th International Conference on Machine Learning, pp. 417–424. Precup, D., Sutton, R. S., Paduraru, C., Koop, A., Singh, S. (2006). Off-policy Learning with Recognizers. Advances in Neural Information Processing Systems 18. Precup, D., Sutton, R. S., Singh, S. (2000). Eligibility traces for off-policy policy evaluation. Proceedings of the 17th International Conference on Machine Learning, pp. 759–766. Morgan Kaufmann. Schaeffer, J., Hlynka, M., Jussila, V. (2001). Temporal difference learning applied to a high-performance gameplaying program. Proceedings of the International Joint Conference on Artiﬁcial Intelligence, pp. 529–534. Silver, D., Sutton, R. S., M¨ ller, M. (2007). Reinforcement learning of local shape in the game of Go. u Proceedings of the 20th International Joint Conference on Artiﬁcial Intelligence, pp. 1053–1058. Sturtevant, N. R., White, A. M. (2006). Feature construction for reinforcement learning in hearts. In Proceedings of the 5th International Conference on Computers and Games. Sutton, R. S. (1988). Learning to predict by the method of temporal differences. Machine Learning, 3:9–44. Sutton, R. S., Barto, A. G. (1998). Reinforcement Learning: An Introduction. MIT Press. Sutton, R.S., Precup D. and Singh, S (1999). Between MDPs and semi-MDPs: A framework for temporal abstraction in reinforcement learning. Artiﬁcial Intelligence, 112:181–211. Sutton, R. S., Rafols, E.J., and Koop, A. 2006. Temporal abstraction in temporal-difference networks. Advances in Neural Information Processing Systems 18. Tadic, V. (2001). On the convergence of temporal-difference learning with linear function approximation. In Machine Learning 42:241–267 Tsitsiklis, J. N., and Van Roy, B. (1997). An analysis of temporal-difference learning with function approximation. IEEE Transactions on Automatic Control, 42:674–690. Watkins, C. J. C. H. (1989). Learning from Delayed Rewards. Ph.D. thesis, Cambridge University. 8

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Abstract: Motor primitives or motion templates have become an important concept for both modeling human motor control as well as generating robot behaviors using imitation learning. Recent impressive results range from humanoid robot movement generation to timing models of human motions. The automatic generation of skill libraries containing multiple motion templates is an important step in robot learning. Such a skill learning system needs to cluster similar movements together and represent each resulting motion template as a generative model which is subsequently used for the execution of the behavior by a robot system. In this paper, we show how human trajectories captured as multi-dimensional time-series can be clustered using Bayesian mixtures of linear Gaussian state-space models based on the similarity of their dynamics. The appropriate number of templates is automatically determined by enforcing a parsimonious parametrization. As the resulting model is intractable, we introduce a novel approximation method based on variational Bayes, which is especially designed to enable the use of efﬁcient inference algorithms. On recorded human Balero movements, this method is not only capable of ﬁnding reasonable motion templates but also yields a generative model which works well in the execution of this complex task on a simulated anthropomorphic SARCOS arm.

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