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130 nips-2010-Interval Estimation for Reinforcement-Learning Algorithms in Continuous-State Domains

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Author: Martha White, Adam White

Abstract: The reinforcement learning community has explored many approaches to obtaining value estimates and models to guide decision making; these approaches, however, do not usually provide a measure of conﬁdence in the estimate. Accurate estimates of an agent’s conﬁdence are useful for many applications, such as biasing exploration and automatically adjusting parameters to reduce dependence on parameter-tuning. Computing conﬁdence intervals on reinforcement learning value estimates, however, is challenging because data generated by the agentenvironment interaction rarely satisﬁes traditional assumptions. Samples of valueestimates are dependent, likely non-normally distributed and often limited, particularly in early learning when conﬁdence estimates are pivotal. In this work, we investigate how to compute robust conﬁdences for value estimates in continuous Markov decision processes. We illustrate how to use bootstrapping to compute conﬁdence intervals online under a changing policy (previously not possible) and prove validity under a few reasonable assumptions. We demonstrate the applicability of our conﬁdence estimation algorithms with experiments on exploration, parameter estimation and tracking. 1

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Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 ca Abstract The reinforcement learning community has explored many approaches to obtaining value estimates and models to guide decision making; these approaches, however, do not usually provide a measure of conﬁdence in the estimate. [sent-5, score-0.36]

2 Accurate estimates of an agent’s conﬁdence are useful for many applications, such as biasing exploration and automatically adjusting parameters to reduce dependence on parameter-tuning. [sent-6, score-0.434]

3 Computing conﬁdence intervals on reinforcement learning value estimates, however, is challenging because data generated by the agentenvironment interaction rarely satisﬁes traditional assumptions. [sent-7, score-0.436]

4 We illustrate how to use bootstrapping to compute conﬁdence intervals online under a changing policy (previously not possible) and prove validity under a few reasonable assumptions. [sent-10, score-0.562]

5 1 Introduction In reinforcement learning, an agent interacts with the environment, learning through trial-and-error based on scalar reward signals. [sent-12, score-0.544]

6 Many reinforcement learning algorithms estimate values for states to enable selection of maximally rewarding actions. [sent-13, score-0.235]

7 Obtaining conﬁdence intervals on these estimates has been shown to be useful in practice, including directing exploration [17, 19] and deciding when to exploit learned models of the environment [3]. [sent-14, score-0.655]

8 Moreover, there are several potential applications using conﬁdence estimates, such as teaching interactive agents (using conﬁdence estimates as feedback), adjusting behaviour in non-stationary environments and controlling behaviour in a parallel multi-task reinforcement learning setting. [sent-15, score-0.598]

9 Computing conﬁdence intervals was ﬁrst studied by Kaelbling for ﬁnite-state Markov decision processes (MDPs) [11]. [sent-16, score-0.253]

10 In this work we focus on constructing conﬁdence intervals for online model-free reinforcement learning agents. [sent-19, score-0.461]

11 The agent-environment interaction in reinforcement learning does not satisfy classical assumptions typically used for computing conﬁdence intervals, making accurate conﬁdence estimation challenging. [sent-20, score-0.268]

12 In the discrete case, certain simplifying assumptions make classical normal intervals more appropriate; in the continuous setting, we will need a different approach. [sent-21, score-0.317]

13 The main contribution of this work is a method to robustly construct conﬁdence intervals for approximated value functions in continuous-state reinforcement learning setting. [sent-22, score-0.436]

14 We ﬁrst describe boot1 strapping, a non-parametric approach to estimating conﬁdence intervals from data. [sent-23, score-0.226]

15 Then, we discuss how to address complications in computing conﬁdence intervals for sparse or local linear representations, common in reinforcement learning, such as tile coding, radial basis functions, tree-based representations and sparse distributed memories. [sent-25, score-0.575]

16 Finally, we propose several potential applications of conﬁdence intervals in reinforcement learning and conclude with an empirical investigation of the practicality of our conﬁdence estimation algorithm for exploration, tuning the temporal credit parameter and tracking. [sent-26, score-0.552]

17 2 Related Work Kaelbling was the ﬁrst to employ conﬁdence interval estimation method for exploration in ﬁnitestate MDPs [11]. [sent-27, score-0.434]

18 The agent estimates the probability of receiving a reward of 1. [sent-28, score-0.426]

19 Interval estimation for model-based reinforcement learning with discrete state spaces has been quite extensively studied. [sent-31, score-0.273]

20 (2004) investigated conﬁdence estimates for the parameters of the learned transition and reward models, assuming Gaussian rewards [5, 16]. [sent-33, score-0.271]

21 For example, for reinforcement learning domains, KWIK-RMAX biases exploration toward states that the algorithm currently does not “know” an accurate estimate of the value [23]. [sent-39, score-0.501]

22 , GPTD [6]) provide a natural measure of conﬁdence: one can use the posterior distribution to form credible intervals for the mean value of a state-action pair. [sent-44, score-0.226]

23 Although this approach is promising, we are interested in computing classical frequentist conﬁdence intervals for agents, while not restricting the underlying learning algorithm to use a model or particular update mechanism. [sent-46, score-0.226]

24 Several papers have demonstrated the empirical beneﬁts of using heuristic conﬁdence estimates to bias exploration [14, 17, 19] and guide data collection in model learning [9, 18]. [sent-47, score-0.414]

25 In the remainder of this work, we provide the ﬁrst study of estimating conﬁdence intervals for model-free, online reinforcement learning value estimates in the continuous-state setting. [sent-50, score-0.553]

26 3 Background In this section, we will introduce the reinforcement learning model of sequential decision making and bootstrapping, a family of techniques used to compute conﬁdence intervals for means of dependent data from an unknown (likely non-normal) underlying distribution. [sent-51, score-0.541]

27 1 Reinforcement Learning In reinforcement learning, an agent interacts with its environment, receiving observations and selecting actions to maximize a scalar reward signal provided by the environment. [sent-53, score-0.601]

28 Many reinforcement learning algorithms maintain an state-action value function, Qπ (s, a), equal to the expected discounted sum of future rewards for a given state-action pair: Qπ (s, a) = ∞ k Eπ k=0 γ rt+k+1 |st = s, at = a , where γ ∈ [0 1] discounts the contribution of future rewards. [sent-61, score-0.21]

29 The optimal state-action value function, Q∗ (s, a), is the maximum achievable value given the agent starts in state s and selects action a. [sent-62, score-0.218]

30 During learning the agent must ˆ balance selecting actions to achieve high reward (according to Q(s, a)) or selecting actions to gain more information about the environment. [sent-64, score-0.448]

31 We refer the reader to the introductory text [25] for a more detailed discussion on reinforcement learning. [sent-69, score-0.21]

32 2 Bootstrapping a conﬁdence interval for dependent data Bootstrapping is a statistical procedure for estimating the distribution of a statistic (such as the sample mean), particularly when the underlying distribution is complicated or unknown, samples are dependent and power calculations (e. [sent-71, score-0.343]

33 This estimate can then be used to approximate a 1 − α conﬁdence interval around the statistic: an interval for which the probability of seeing the statistic outside of the interval is low (probability α). [sent-74, score-0.373]

34 ), n we can use bootstrapping to compute a conﬁdence interval around the mean, Tn = n−1 i=1 xn . [sent-78, score-0.252]

35 Samples are “drawn” from Pn ∗ to produce a bootstrap sample, x∗ , . [sent-80, score-0.39]

36 Bootstrapped intervals have been shown to have a lower coverage error than √ normal intervals for dependent, non-normal data. [sent-92, score-0.765]

37 A normal interval has a coverage error of O(1/ n), whereas bootstrapping has a coverage error of O(n−3/2 ) [29]. [sent-93, score-0.84]

38 The coverage error represents how quickly the estimated interval converges to the true interval: higher order coverage error indicates faster convergence1 . [sent-94, score-0.661]

39 With the bootstrapped samples, a percentile-t (studentized) interval is constructed by ∗ ∗ P (T ∈ (2Tn − T1−α/2 , 2Tn − Tα/2 )) ≥ 1 − α ∗ ∗ ∗ where Tβ is the β sample quantile of Tn,1 , . [sent-96, score-0.222]

40 The remaining question is how to bootstrap from the sequence of samples. [sent-101, score-0.39]

41 In the next section, we describe the block bootstrap, applicable to Markov processes, which we will show represents the structure of data for value estimates in reinforcement learning. [sent-102, score-0.385]

42 1 More theoretically, coverage error is the approximation error in the Edgeworth expansions used to approximate the distribution in bootstrap proofs. [sent-103, score-0.694]

43 1 Moving Block Bootstrap In the moving block bootstrap method, blocks of consecutive samples are drawn with replacement from a set of overlapping blocks, making the k-th block {xk−1+t : t = 1 . [sent-106, score-0.74]

44 The bootstrap resample is the concatenation of n/l blocks chosen randomly with replacement, making a time series of length n; B of these concatenated resamples are used in the bootstrap estimate. [sent-110, score-0.889]

45 The block bootstrap is appropriate for sequential processes because the blocks implicitly maintain a time-dependent structure. [sent-111, score-0.524]

46 The moving block bootstrap was designed for stationary, dependent data; however, our scenario involves nonstationary data. [sent-113, score-0.694]

47 Lahiri [12] proved a coverage error of o(n−1/2 ) when applying the moving block bootstrap to nonstationary, dependent data, better than the normal coverage error. [sent-114, score-1.179]

48 Note that there are other bootstrapping techniques applicable to sequential, dependent data with lower coverage error, such as the double bootstrap [13], block-block bootstrap [1] and Markov or Sieve bootstrap [28]. [sent-116, score-1.635]

49 In particular, the Markov bootstrap has been shown to have a lower coverage error for Markov data than the block bootstrap under certain restricted conditions [10]. [sent-117, score-1.138]

50 4 Conﬁdence intervals for continuous-state Markov decision processes In this section, we present a theoretically sound approach to constructing conﬁdence intervals for parametrized Q(s, a) using bootstrapping for dependent data. [sent-119, score-0.725]

51 Notice that Pa incorporates P and R, using st , θ t (giving the policy π) and R to determine the reward passed to the algorithm to then obtain θ t+1 . [sent-143, score-0.445]

52 Based on these assumptions (stated formally in the supplement), we can prove that the moving block bootstrap produces an interval with a coverage error of o(n−1/2 for the studentized interval on fn (s, a). [sent-149, score-1.101]

53 l increases with n), then the moving block bootstrap produces a one-sided conﬁdence interval that is consistent and has a coverage error of o(n−1/2 ) for the studentization of the mean of the process {f (θ t , s, a)}, where Qt (s, a) = f (θ t , s, a). [sent-152, score-0.928]

54 The proof for the above theorem follows Lahiri’s proof [12] for the coverage error of the moving block bootstrap for nonstationary data. [sent-153, score-0.891]

55 The general approach for coverage error proofs involve approximating the unknown distribution with an Edgeworth expansion (see [7]), with the coverage error dependent on the order of the expansion, similar to the the idea of a Taylor series expansion. [sent-154, score-0.628]

56 θ n }, α) l = block length, B = num bootstrap resamples last w weights and conﬁdence level α (= 0. [sent-170, score-0.546]

57 2 Bootstrapped Conﬁdence Intervals for Sparse Representations We have shown that bootstrapping is a principled approach for computing intervals for global representations; sparse representations, however, complicate the solution. [sent-193, score-0.392]

58 We propose that, for sparse linear representations, the samples for the weights can be treated independently and still produce a reasonable, though currently unproven, bootstrap interval. [sent-200, score-0.492]

59 5 Proving coverage error results for sparse representations will require analyzing the covariance between components of θ over time. [sent-213, score-0.35]

60 5 Applications of conﬁdence intervals for reinforcement learning The most obvious application of interval estimation is to bias exploration to select actions with high uncertainty. [sent-218, score-0.927]

61 Conﬁdence-based exploration should be comparable to optimistic initialization in domains where exhaustive search is required and ﬁnd better policies in domains where noisy rewards and noisy dynamics can cause the optimistic initialization to be prematurely decreased and inhibit exploration. [sent-219, score-0.595]

62 Furthermore, conﬁdence-based exploration reduces parameter tuning because the policy does not require knowledge of the reward range, as in softmax and optimistic initialization. [sent-220, score-0.798]

63 Conﬁdence-based exploration could be beneﬁcial in domains where the problem dynamics and reward function change over time. [sent-221, score-0.508]

64 In an extreme case, the agent may converge to a near-optimal policy before the goal is teleported to another portion of the space. [sent-222, score-0.365]

65 If the agent continues to act greedily with respect to its action-value estimates without re-exploring, it may act sub-optimally indeﬁnitely. [sent-223, score-0.247]

66 These tracking domains require that the agent “notice” that its predictions are incorrect and begin searching for a better policy. [sent-224, score-0.224]

67 AN example of a changing reward signals arises in interactive teaching. [sent-225, score-0.212]

68 In this scenario, the a human teaching shapes the agent by providing a drifting reward signal. [sent-226, score-0.37]

69 Even in stationary domains, tracking the optimal policy may be more effective than converging due to the non-stationarity introduced by imperfect function approximation [26]. [sent-227, score-0.228]

70 For example, many TD-based reinforcement learning algorithms use an eligibility parameter (λ) to address the credit assignment problem. [sent-229, score-0.289]

71 Conﬁdence estimates could be used to increase λ when the agent is uncertain, reﬂecting and decrease λ for conﬁdent value estimates [25]. [sent-232, score-0.339]

72 Conﬁdence estimates could also be used to guide the behaviour policy for a parallel multi-task reinforcement learning system. [sent-233, score-0.566]

73 The behaviour policy should explore to provide samples that generalize well between the various target policies, speeding overall convergence. [sent-235, score-0.244]

74 For example, if one-sided intervals are maintained for each target value functions, the behaviour policy could select an action corresponding to the maximal sum of those intervals. [sent-236, score-0.466]

75 In an online reinforcement learning setting, these methods freeze the representation for a ﬁxed window of time to accurately evaluate the candidate [20]. [sent-240, score-0.235]

76 We evaluate a naive implementation of the block bootstrap method for (1) exploration in a noisy reward domain, (2) automatically tuning λ in the Cartpole domain and (3) tracking a moving goal in a navigation task. [sent-243, score-1.173]

77 The ﬂashing goal yields a reward selected uniformly from {100, −100, 5, −5, 50}. [sent-251, score-0.22]

78 The agent’s observation is a continuous (x, y) position and actions move the agent {N,S,E,W} perturbed by uniform noise 10% of the time. [sent-255, score-0.239]

79 We also compare our algorithm to an exploration policy using normal (instead of bootstrapped) intervals to investigate the effectiveness of making simplifying assumptions on the data distribution. [sent-259, score-0.718]

80 We present the results for the best parameter setting for each exploration policy for clarity. [sent-260, score-0.428]

81 The optimistic policy slowly converges to the small goal for lower initializations and does not favour either goal for higher initializations. [sent-263, score-0.333]

82 The softmax policy navigates to the small goal on most runs and also convergences slowly. [sent-264, score-0.259]

83 The normal-interval exploration policy does prefer the ﬂashing goal but not as quickly as the bootstrap policy. [sent-265, score-0.859]

84 Finally, the bootstrap-interval exploration policy achieves highest cumulative reward and is the only policy that converges to the ﬂashing goal, despite the large variance in the reward signal. [sent-266, score-0.923]

85 The conﬁdence estimates were only used to adjust λ for clarity: exploration was performed using optimistic initialization. [sent-278, score-0.497]

86 We also tested adjusting λ using normal conﬁdence intervals, however, the normal conﬁdence intervals resulted in worse performance then any ﬁxed value of λ. [sent-282, score-0.353]

87 We studied the effects of this form of non-stationarity, where the agent learns to go to a goal and then another, better goal becomes available (near the ﬁrst goal to better guide it to the next goal). [sent-294, score-0.309]

88 In our domain, the agent receives -1 reward per step and +10 at termination in a goal region. [sent-295, score-0.409]

89 We used = 0 to enable exploration only with optimistic initialization. [sent-298, score-0.43]

90 7 Conclusion In this work, we investigated constructing conﬁdence intervals on value estimates in the continuousstate reinforcement learning setting. [sent-303, score-0.528]

91 We proved that our conﬁdence estimate has low coverage error under mild assumptions on the learning algorithm. [sent-305, score-0.301]

92 We have begun testing the conﬁdencebased exploration on a mobile robot platform. [sent-309, score-0.355]

93 Robotic tasks, however, are often more naturally formulated as continual learning tasks with a sparse reward signal, such as negative reward for bumping into objects, or a positive reward for reaching some goal. [sent-313, score-0.589]

94 We could potentially improve coverage error by extending other bootstrapping techniques, such as the Markov bootstrap, to non-stationary data. [sent-319, score-0.416]

95 We could also explore the theoretical work on exponential bounds, such as the Azuma-Hoeffding inequality, to obtain different conﬁdence estimates with low coverage error. [sent-320, score-0.338]

96 R-max - a general polynomial time algorithm for near-optimal reinforcement learning. [sent-343, score-0.21]

97 On blocking rules for the bootstrap with dependent data. [sent-367, score-0.468]

98 Generalized model learning for reinforcement learning in factored domains. [sent-370, score-0.21]

99 An empirical evaluation of interval estimation for markov decision processes. [sent-444, score-0.217]

100 Online linear regression and its application to model-based reinforcement learning. [sent-452, score-0.21]

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(1) t=0 Our previous research on solving the optimal reward problem has focused primarily on the properties of the optimal reward function and its correspondence to the agent architecture and the environment [16, 17, 18]. This work has used inefﬁcient exhaustive search methods for ﬁnding good approximations to θ∗ (though there is recent work on using genetic algorithms to do this [6, 9, 12]). Our primary contribution in this paper is a new convergent online stochastic gradient method for ﬁnding approximately optimal reward functions. To our knowledge, this is the ﬁrst algorithm that improves reward functions in an online setting—during a single agent’s lifetime. In Section 2, we present the PGRD algorithm, prove its convergence, and relate it to OLPOMDP [2], a policy gradient algorithm. In Section 3, we present experiments demonstrating PGRD’s ability to approximately solve the optimal reward problem online. 2 PGRD: Policy Gradient for Reward Design PGRD builds on the following insight: the agent’s planning algorithm procedurally converts the reward function into behavior; thus, the reward function can be viewed as a speciﬁc parameterization of the agent’s policy. Using this insight, PGRD updates the reward parameters by estimating the gradient of the objective return with respect to the reward parameters, θ U(θ), from experience, using standard policy gradient techniques. In fact, we show that PGRD can be viewed as an (independently interesting) generalization of the policy gradient method OLPOMDP [2]. Speciﬁcally, we show that OLPOMDP is special case of PGRD when the planning depth d is zero. In this section, we ﬁrst present the family of local planning agents for which PGRD improves the reward function. Next, we develop PGRD and prove its convergence. Finally, we show that PGRD generalizes OLPOMDP and discuss how adding planning to OLPOMDP affects the space of policies available to the optimization method. 2 1 2 3 4 5 Input: T , θ0 , {αt }∞ , β, γ t=0 o0 , i0 = initializeStart(); for t = 0, 1, 2, 3, . . . do ∀a Qt (a; θt ) = plan(it , ot , T, R(it , ·, ·; θt ), d,γ); at ∼ µ(a|it ; Qt ); rt+1 , ot+1 = takeAction(at ); µ(a |i ;Q ) 6 7 8 9 t zt+1 = βzt + θt t |itt ;Qt ) t ; µ(a θt+1 = θt + αt (rt+1 zt+1 − λθt ) ; it+1 = updateInternalState(it , at , ot+1 ); end Figure 1: PGRD (Policy Gradient for Reward Design) Algorithm A Family of Limited Agents with Internal State. Given a Markov model T deﬁned over the observation space O and action space A, denote T (o |o, a) the probability of next observation o given that the agent takes action a after observing o. Our agents use the model T to plan. We do not assume that the model T is an accurate model of the environment. The use of an incorrect model is one type of agent limitation we examine in our experiments. In general, agents can use non-Markov models deﬁned in terms of the history of observations and actions; we leave this for future work. The agent maintains an internal state feature vector it that is updated at each time step using it+1 = updateInternalState(it , at , ot+1 ). The internal state allows the agent to use reward functions T that depend on the agent’s history. We consider rewards of the form R(it , o, a; θt ) = θt φ(it , o, a), where θt is the reward parameter vector at time t, and φ(it , o, a) is a vector of features based on internal state it , planning state o, and action a. Note that if φ is a vector of binary indicator features, this representation allows for arbitrary reward functions and thus the representation is completely general. Many existing methods use reward functions that depend on history. Reward functions based on empirical counts of observations, as in PAC-MDP approaches [5, 20], provide some examples; see [14, 15, 13] for others. We present a concrete example in our empirical section. At each time step t, the agent’s planning algorithm, plan, performs depth-d planning using the model T and reward function R(it , o, a; θt ) with current internal state it and reward parameters θt . Speciﬁcally, the agent computes a d-step Q-value function Qd (it , ot , a; θt ) ∀a ∈ A, where Qd (it , o, a; θt ) = R(it , o, a; θt ) + γ o ∈O T (o |o, a) maxb∈A Qd−1 (it , o , b; θt ) and Q0 (it , o, a; θt ) = R(it , o, a; θt ). We emphasize that the internal state it and reward parameters θt are held invariant while planning. Note that the d-step Q-values are only computed for the current observation ot , in effect by building a depth-d tree rooted at ot . In the d = 0 special case, the planning procedure completely ignores the model T and returns Q0 (it , ot , a; θt ) = R(it , ot , a; θt ). Regardless of the value of d, we treat the end result of planning as providing a scoring function Qt (a; θt ) where the dependence on d, it and ot is dropped from the notation. To allow for gradient calculations, our agents act according to the τ Qt (a;θt ) def Boltzmann (soft-max) stochastic policy parameterized by Q: µ(a|it ; Qt ) = e eτ Qt (b;θt ) , where τ b is a temperature parameter that determines how stochastically the agent selects the action with the highest score. When the planning depth d is small due to computational limitations, the agent cannot account for events beyond the planning depth. We examine this limitation in our experiments. Gradient Ascent. To develop a gradient algorithm for improving the reward function, we need to compute the gradient of the objective return with respect to θ: θ U(θ). The main insight is to break the gradient calculation into the calculation of two gradients. The ﬁrst is the gradient of the objective return with respect to the policy µ, and the second is the gradient of the policy with respect to the reward function parameters θ. The ﬁrst gradient is exactly what is computed in standard policy gradient approaches [2]. The second gradient is challenging because the transformation from reward parameters to policy involves a model-based planning procedure. We draw from the work of Neu and Szepesv´ ri [10] which shows that this gradient computation resembles planning itself. We a develop PGRD, presented in Figure 1, explicitly as a generalization of OLPOMDP, a policy gradient algorithm developed by Bartlett and Baxter [2], because of its foundational simplicity relative to other policy-gradient algorithms such as those based on actor-critic methods (e.g., [4]). Notably, the reward parameters are the only parameters being learned in PGRD. 3 PGRD follows the form of OLPOMDP (Algorithm 1 in Bartlett and Baxter [2]) but generalizes it in three places. In Figure 1 line 3, the agent plans to compute the policy, rather than storing the policy directly. In line 6, the gradient of the policy with respect to the parameters accounts for the planning procedure. In line 8, the agent maintains a general notion of internal state that allows for richer parameterization of policies than typically considered (similar to Aberdeen and Baxter [1]). The algorithm takes as parameters a sequence of learning rates {αk }, a decaying-average parameter β, and regularization parameter λ > 0 which keeps the the reward parameters θ bounded throughout learning. Given a sequence of calculations of the gradient of the policy with respect to the parameters, θt µ(at |it ; Qt ), the remainder of the algorithm climbs the gradient of objective return θ U(θ) using OLPOMDP machinery. In the next subsection, we discuss how to compute θt µ(at |it ; Qt ). Computing the Gradient of the Policy with respect to Reward. For the Boltzmann distribution, the gradient of the policy with respect to the reward parameters is given by the equation θt µ(a|it ; Qt ) = τ · µ(a|Qt )[ θt Qt (a|it ; θt ) − θt Qt (b; θt )], where τ is the Boltzmann b∈A temperature (see [10]). Thus, computing θt µ(a|it ; Qt ) reduces to computing θt Qt (a; θt ). The value of Qt depends on the reward parameters θt , the model, and the planning depth. However, as we present below, the process of computing the gradient closely resembles the process of planning itself, and the two computations can be interleaved. Theorem 1 presented below is an adaptation of Proposition 4 from Neu and Szepesv´ ri [10]. It presents the gradient computation for depth-d a planning as well as for inﬁnite-depth discounted planning. We assume that the gradient of the reward function with respect to the parameters is bounded: supθ,o,i,a θ R(i, o, a, θ) < ∞. The proof of the theorem follows directly from Proposition 4 of Neu and Szepesv´ ri [10]. a Theorem 1. Except on a set of measure zero, for any depth d, the gradient θ Qd (o, a; θ) exists and is given by the recursion (where we have dropped the dependence on i for simplicity) d θ Q (o, a; θ) = θ R(o, a; θ) π d−1 (b|o ) T (o |o, a) +γ o ∈O d−1 (o θQ , b; θ), (2) b∈A where θ Q0 (o, a; θ) = θ R(o, a; θ) and π d (a|o) ∈ arg maxa Qd (o, a; θ) is any policy that is greedy with respect to Qd . The result also holds for θ Q∗ (o, a; θ) = θ limd→∞ Qd (o, a; θ). The Q-function will not be differentiable when there are multiple optimal policies. This is reﬂected in the arbitrary choice of π in the gradient calculation. However, it was shown by Neu and Szepesv´ ri [10] that even for values of θ which are not differentiable, the above computation produces a a valid calculation of a subgradient; we discuss this below in our proof of convergence of PGRD. Convergence of PGRD (Figure 1). Given a particular ﬁxed reward function R(·; θ), transition model T , and planning depth, there is a corresponding ﬁxed randomized policy µ(a|i; θ)—where we have explicitly represented the reward’s dependence on the internal state vector i in the policy parameterization and dropped Q from the notation as it is redundant given that everything else is ﬁxed. Denote the agent’s internal-state update as a (usually deterministic) distribution ψ(i |i, a, o). Given a ﬁxed reward parameter vector θ, the joint environment-state–internal-state transitions can be modeled as a Markov chain with a |S||I| × |S||I| transition matrix M (θ) whose entries are given by M s,i , s ,i (θ) = p( s , i | s, i ; θ) = o,a ψ(i |i, a, o)Ω(o|s )P (s |s, a)µ(a|i; θ). We make the following assumptions about the agent and the environment: Assumption 1. The transition matrix M (θ) of the joint environment-state–internal-state Markov chain has a unique stationary distribution π(θ) = [πs1 ,i1 (θ), πs2 ,i2 (θ), . . . , πs|S| ,i|I| (θ)] satisfying the balance equations π(θ)M (θ) = π(θ), for all θ ∈ Θ. Assumption 2. During its execution, PGRD (Figure 1) does not reach a value of it , and θt at which µ(at |it , Qt ) is not differentiable with respect to θt . It follows from Assumption 1 that the objective return, U(θ), is independent of the start state. The original OLPOMDP convergence proof [2] has a similar condition that only considers environment states. Intuitively, this condition allows PGRD to handle history-dependence of a reward function in the same manner that it handles partial observability in an environment. Assumption 2 accounts for the fact that a planning algorithm may not be fully differentiable everywhere. However, Theorem 1 showed that inﬁnite and bounded-depth planning is differentiable almost everywhere (in a measure theoretic sense). Furthermore, this assumption is perhaps stronger than necessary, as stochastic approximation algorithms, which provide the theory upon which OLPOMDP is based, have been shown to converge using subgradients [8]. 4 In order to state the convergence theorem, we must deﬁne the approximate gradient which OLPOMDP def T calculates. Let the approximate gradient estimate be β U(θ) = limT →∞ t=1 rt zt for a ﬁxed θ and θ PGRD parameter β, where zt (in Figure 1) represents a time-decaying average of the θt µ(at |it , Qt ) calculations. It was shown by Bartlett and Baxter [2] that β U(θ) is close to the true value θ U(θ) θ for large values of β. Theorem 2 proves that PGRD converges to a stable equilibrium point based on this approximate gradient measure. This equilibrium point will typically correspond to some local optimum in the return function U(θ). Given our development and assumptions, the theorem is a straightforward extension of Theorem 6 from Bartlett and Baxter [2] (proof omitted). ∞ Theorem 2. Given β ∈ [0, 1), λ > 0, and a sequence of step sizes αt satisfying t=0 αt = ∞ and ∞ 2 t=0 (αt ) < ∞, PGRD produces a sequence of reward parameters θt such that θt → L as t → ∞ a.s., where L is the set of stable equilibrium points of the differential equation ∂θ = β U(θ) − λθ. θ ∂t PGRD generalizes OLPOMDP. As stated above, OLPOMDP, when it uses a Boltzmann distribution in its policy representation (a common case), is a special case of PGRD when the planning depth is zero. First, notice that in the case of depth-0 planning, Q0 (i, o, a; θ) = R(i, o, a, θ), regardless of the transition model and reward parameterization. We can also see from Theorem 1 that 0 θ Q (i, o, a; θ) = θ R(i, o, a; θ). Because R(i, o, a; θ) can be parameterized arbitrarily, PGRD can be conﬁgured to match standard OLPOMDP with any policy parameterization that also computes a score function for the Boltzmann distribution. In our experiments, we demonstrate that choosing a planning depth d > 0 can be beneﬁcial over using OLPOMDP (d = 0). In the remainder of this section, we show theoretically that choosing d > 0 does not hurt in the sense that it does not reduce the space of policies available to the policy gradient method. Speciﬁcally, we show that when using an expressive enough reward parameterization, PGRD’s space of policies is not restricted relative to OLPOMDP’s space of policies. We prove the result for inﬁnite planning, but the extension to depth-limited planning is straightforward. Theorem 3. There exists a reward parameterization such that, for an arbitrary transition model T , the space of policies representable by PGRD with inﬁnite planning is identical to the space of policies representable by PGRD with depth 0 planning. Proof. Ignoring internal state for now (holding it constant), let C(o, a) be an arbitrary reward function used by PGRD with depth 0 planning. Let R(o, a; θ) be a reward function for PGRD with inﬁnite (d = ∞) planning. The depth-∞ agent uses the planning result Q∗ (o, a; θ) to act, while the depth-0 agent uses the function C(o, a) to act. Therefore, it sufﬁces to show that one can always choose θ such that the planning solution Q∗ (o, a; θ) equals C(o, a). For all o ∈ O, a ∈ A, set R(o, a; θ) = C(o, a) − γ o T (o |o, a) maxa C(o , a ). Substituting Q∗ for C, this is the Bellman optimality equation [22] for inﬁnite-horizon planning. Setting R(o, a; θ) as above is possible if it is parameterized by a table with an entry for each observation–action pair. Theorem 3 also shows that the effect of an arbitrarily poor model can be overcome with a good choice of reward function. This is because a Boltzmann distribution can, allowing for an arbitrary scoring function C, represent any policy. We demonstrate this ability of PGRD in our experiments. 3 Experiments The primary objective of our experiments is to demonstrate that PGRD is able to use experience online to improve the reward function parameters, thereby improving the agent’s obtained objective return. Speciﬁcally, we compare the objective return achieved by PGRD to the objective return achieved by PGRD with the reward adaptation turned off. In both cases, the reward function is initialized to the objective reward function. A secondary objective is to demonstrate that when a good model is available, adding the ability to plan—even for small depths—improves performance relative to the baseline algorithm of OLPOMDP (or equivalently PGRD with depth d = 0). Foraging Domain for Experiments 1 to 3: The foraging environment illustrated in Figure 2(a) is a 3 × 3 grid world with 3 dead-end corridors (rows) separated by impassable walls. The agent (bird) has four available actions corresponding to each cardinal direction. Movement in the intended direction fails with probability 0.1, resulting in movement in a random direction. If the resulting direction is 5 Objective Return 0.15 D=6, α=0 & D=6, α=5×10 −5 D=4, α=2×10 −4 D=0, α=5×10 −4 0.1 0.05 0 D=4, α=0 D=0, α=0 1000 2000 3000 4000 5000 Time Steps C) Objective Return B) A) 0.15 D=6, α=0 & D=6, α=5×10 −5 D=3, α=3×10 −3 D=1, α=3×10 −4 0.1 D=3, α=0 0.05 D=0, α=0.01 & D=1, α=0 0 1000 2000 3000 4000 5000 D=0, α=0 Time Steps Figure 2: A) Foraging Domain, B) Performance of PGRD with observation-action reward features, C) Performance of PGRD with recency reward features blocked by a wall or the boundary, the action results in no movement. There is a food source (worm) located in one of the three right-most locations at the end of each corridor. The agent has an eat action, which consumes the worm when the agent is at the worm’s location. After the agent consumes the worm, a new worm appears randomly in one of the other two potential worm locations. Objective Reward for the Foraging Domain: The designer’s goal is to maximize the average number of worms eaten per time step. Thus, the objective reward function RO provides a reward of 1.0 when the agent eats a worm, and a reward of 0 otherwise. The objective return is deﬁned as in Equation (1). Experimental Methodology: We tested PGRD for depth-limited planning agents of depths 0–6. Recall that PGRD for the agent with planning depth 0 is the OLPOMDP algorithm. For each depth, we jointly optimized over the PGRD algorithm parameters, α and β (we use a ﬁxed α throughout learning). We tested values for α on an approximate logarithmic scale in the range (10−6 , 10−2 ) as well as the special value of α = 0, which corresponds to an agent that does not adapt its reward function. We tested β values in the set 0, 0.4, 0.7, 0.9, 0.95, 0.99. Following common practice [3], we set the λ parameter to 0. We explicitly bound the reward parameters and capped the reward function output both to the range [−1, 1]. We used a Boltzmann temperature parameter of τ = 100 and planning discount factor γ = 0.95. Because we initialized θ so that the initial reward function was the objective reward function, PGRD with α = 0 was equivalent to a standard depth-limited planning agent. Experiment 1: A fully observable environment with a correct model learned online. In this experiment, we improve the reward function in an agent whose only limitation is planning depth, using (1) a general reward parameterization based on the current observation and (2) a more compact reward parameterization which also depends on the history of observations. Observation: The agent observes the full state, which is given by the pair o = (l, w), where l is the agent’s location and w is the worm’s location. Learning a Correct Model: Although the theorem of convergence of PGRD relies on the agent having a ﬁxed model, the algorithm itself is readily applied to the case of learning a model online. In this experiment, the agent’s model T is learned online based on empirical transition probabilities between observations (recall this is a fully observable environment). Let no,a,o be the number of times that o was reached after taking action a after observing o. The agent models the probability of seeing o as no,a,o T (o |o, a) = . n o o,a,o Reward Parameterizations: Recall that R(i, o, a; θ) = θT φ(i, o, a), for some φ(i, o, a). (1) In the observation-action parameterization, φ(i, o, a) is a binary feature vector with one binary feature for each observation-action pair—internal state is ignored. This is effectively a table representation over all reward functions indexed by (o, a). As shown in Theorem 3, the observation-action feature representation is capable of producing arbitrary policies over the observations. In large problems, such a parameterization would not be feasible. (2) The recency parameterization is a more compact representation which uses features that rely on the history of observations. The feature vector is φ(i, o, a) = [RO (o, a), 1, φcl (l, i), φcl,a (l, a, i)], where RO (o, a) is the objective reward function deﬁned as above. The feature φcl (l) = 1 − 1/c(l, i), where c(l, i) is the number of time steps since the agent has visited location l, as represented in the agent’s internal state i. Its value is normalized to the range [0, 1) and is high when the agent has not been to location l recently. The feature φcl,a (l, a, i) = 1 − 1/c(l, a, i) is similarly deﬁned with respect to the time since the agent has taken action a in location l. Features based on recency counts encourage persistent exploration [21, 18]. 6 Results & Discussion: Figure 2(b) and Figure 2(c) present results for agents that use the observationaction parameterization and the recency parameterization of the reward function respectively. The horizontal axis is the number of time steps of experience. The vertical axis is the objective return, i.e., the average objective reward per time step. Each curve is an average over 130 trials. The values of d and the associated optimal algorithm parameters for each curve are noted in the ﬁgures. First, note that with d = 6, the agent is unbounded, because food is never more than 6 steps away. Therefore, the agent does not beneﬁt from adapting the reward function parameters (given that we initialize to the objective reward function). Indeed, the d = 6, α = 0 agent performs as well as the best reward-optimizing agent. The performance for d = 6 improves with experience because the model improves with experience (and thus from the curves it is seen that the model gets quite accurate in about 1500 time steps). The largest objective return obtained for d = 6 is also the best objective return that can be obtained for any value of d. Several results can be observed in both Figures 2(b) and (c). 1) Each curve that uses α > 0 (solid lines) improves with experience. This is a demonstration of our primary contribution, that PGRD is able to effectively improve the reward function with experience. That the improvement over time is not just due to model learning is seen in the fact that for each value of d < 6 the curve for α > 0 (solid-line) which adapts the reward parameters does signiﬁcantly better than the corresponding curve for α = 0 (dashed-line); the α = 0 agents still learn the model. 2) For both α = 0 and α > 0 agents, the objective return obtained by agents with equivalent amounts of experience increases monotonically as d is increased (though to maintain readability we only show selected values of d in each ﬁgure). This demonstrates our secondary contribution, that the ability to plan in PGRD signiﬁcantly improves performance over standard OLPOMDP (PGRD with d = 0). There are also some interesting differences between the results for the two different reward function parameterizations. With the observation-action parameterization, we noted that there always exists a setting of θ for all d that will yield optimal objective return. This is seen in Figure 2(b) in that all solid-line curves approach optimal objective return. In contrast, the more compact recency reward parameterization does not afford this guarantee and indeed for small values of d (< 3), the solid-line curves in Figure 2(c) converge to less than optimal objective return. Notably, OLPOMDP (d = 0) does not perform well with this feature set. On the other hand, for planning depths 3 ≤ d < 6, the PGRD agents with the recency parameterization achieve optimal objective return faster than the corresponding PGRD agent with the observation-action parameterization. Finally, we note that this experiment validates our claim that PGRD can improve reward functions that depend on history. Experiment 2: A fully observable environment and poor given model. Our theoretical analysis showed that PGRD with an incorrect model and the observation–action reward parameterization should (modulo local maxima issues) do just as well asymptotically as it would with a correct model. Here we illustrate this theoretical result empirically on the same foraging domain and objective reward function used in Experiment 1. We also test our hypothesis that a poor model should slow down the rate of learning relative to a correct model. Poor Model: We gave the agents a ﬁxed incorrect model of the foraging environment that assumes there are no internal walls separating the 3 corridors. Reward Parameterization: We used the observation–action reward parameterization. With a poor model it is no longer interesting to initialize θ so that the initial reward function is the objective reward function because even for d = 6 such an agent would do poorly. Furthermore, we found that this initialization leads to excessively bad exploration and therefore poor learning of how to modify the reward. Thus, we initialize θ to uniform random values near 0, in the range (−10−3 , 10−3 ). Results: Figure 3(a) plots the objective return as a function of number of steps of experience. Each curve is an average over 36 trials. As hypothesized, the bad model slows learning by a factor of more than 10 (notice the difference in the x-axis scales from those in Figure 2). Here, deeper planning results in slower learning and indeed the d = 0 agent that does not use the model at all learns the fastest. However, also as hypothesized, because they used the expressive observation–action parameterization, agents of all planning depths mitigate the damage caused by the poor model and eventually converge to the optimal objective return. Experiment 3: Partially observable foraging world. Here we evaluate PGRD’s ability to learn in a partially observable version of the foraging domain. In addition, the agents learn a model under the erroneous (and computationally convenient) assumption that the domain is fully observable. 7 0.1 −4 D = 0, α = 2 ×10 D = 2, α = 3 ×10 −5 −5 D = 6, α = 2 ×10 0.05 D = 0&2&6, α = 0 0 1 2 3 Time Steps 4 5 x 10 4 0.06 D = 6, α = 7 ×10 D = 2, α = 7 ×10 −4 0.04 D = 1, α = 7 ×10 −4 D = 0, α = 5 ×10 −4 D = 0, α = 0 D = 1&2&6, α = 0 0.02 0 C) −4 1000 2000 3000 4000 5000 Time Steps Objective Return B) 0.08 0.15 Objective Return Objective Return A) 2.5 2 x 10 −3 D=6, α=3×10 −6 D=0, α=1×10 −5 1.5 D=0&6, α=0 1 0.5 1 2 3 Time Steps 4 5 x 10 4 Figure 3: A) Performance of PGRD with a poor model, B) Performance of PGRD in a partially observable world with recency reward features, C) Performance of PGRD in Acrobot Partial Observation: Instead of viewing the location of the worm at all times, the agent can now only see the worm when it is colocated with it: its observation is o = (l, f ), where f indicates whether the agent is colocated with the food. Learning an Incorrect Model: The model is learned just as in Experiment 1. Because of the erroneous full observability assumption, the model will hallucinate about worms at all the corridor ends based on the empirical frequency of having encountered them there. Reward Parameterization: We used the recency parameterization; due to the partial observability, agents with the observation–action feature set perform poorly in this environment. The parameters θ are initialized such that the initial reward function equals the objective reward function. Results & Discussion: Figure 3(b) plots the mean of 260 trials. As seen in the solid-line curves, PGRD improves the objective return at all depths (only a small amount for d = 0 and signiﬁcantly more for d > 0). In fact, agents which don’t adapt the reward are hurt by planning (relative to d = 0). This experiment demonstrates that the combination of planning and reward improvement can be beneﬁcial even when the model is erroneous. Because of the partial observability, optimal behavior in this environment achieves less objective return than in Experiment 1. Experiment 4: Acrobot. In this experiment we test PGRD in the Acrobot environment [22], a common benchmark task in the RL literature and one that has previously been used in the testing of policy gradient approaches [23]. This experiment demonstrates PGRD in an environment in which an agent must be limited due to the size of the state space and further demonstrates that adding model-based planning to policy gradient approaches can improve performance. Domain: The version of Acrobot we use is as speciﬁed by Sutton and Barto [22]. It is a two-link robot arm in which the position of one shoulder-joint is ﬁxed and the agent’s control is limited to 3 actions which apply torque to the elbow-joint. Observation: The fully-observable state space is 4 dimensional, with two joint angles ψ1 and ψ2 , and ˙ ˙ two joint velocities ψ1 and ψ2 . Objective Reward: The designer receives an objective reward of 1.0 when the tip is one arm’s length above the ﬁxed shoulder-joint, after which the bot is reset to its initial resting position. Model: We provide the agent with a perfect model of the environment. Because the environment is continuous, value iteration is intractable, and computational limitations prevent planning deep enough to compute the optimal action in any state. The feature vector contains 13 entries. One feature corresponds to the objective reward signal. For each action, there are 5 features corresponding to each of the state features plus an additional feature representing the height of the tip: φ(i, o, a) = ˙ ˙ [RO (o), {ψ1 (o), ψ2 (o), ψ1 (o), ψ2 (o), h(o)}a ]. The height feature has been used in previous work as an alternative deﬁnition of objective reward [23]. Results & Discussion: We plot the mean of 80 trials in Figure 3(c). Agents that use the ﬁxed (α = 0) objective reward function with bounded-depth planning perform according to the bottom two curves. Allowing PGRD and OLPOMDP to adapt the parameters θ leads to improved objective return, as seen in the top two curves in Figure 3(c). Finally, the PGRD d = 6 agent outperforms the standard OLPOMDP agent (PGRD with d = 0), further demonstrating that PGRD outperforms OLPOMDP. Overall Conclusion: We developed PGRD, a new method for approximately solving the optimal reward problem in bounded planning agents that can be applied in an online setting. We showed that PGRD is a generalization of OLPOMDP and demonstrated that it both improves reward functions in limited agents and outperforms the model-free OLPOMDP approach. 8 References [1] Douglas Aberdeen and Jonathan Baxter. Scalable Internal-State Policy-Gradient Methods for POMDPs. Proceedings of the Nineteenth International Conference on Machine Learning, 2002. [2] Peter L. Bartlett and Jonathan Baxter. 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