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37 nips-2010-Basis Construction from Power Series Expansions of Value Functions

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Author: Sridhar Mahadevan, Bo Liu

Abstract: This paper explores links between basis construction methods in Markov decision processes and power series expansions of value functions. This perspective provides a useful framework to analyze properties of existing bases, as well as provides insight into constructing more effective bases. Krylov and Bellman error bases are based on the Neumann series expansion. These bases incur very large initial Bellman errors, and can converge rather slowly as the discount factor approaches unity. The Laurent series expansion, which relates discounted and average-reward formulations, provides both an explanation for this slow convergence as well as suggests a way to construct more efﬁcient basis representations. The ﬁrst two terms in the Laurent series represent the scaled average-reward and the average-adjusted sum of rewards, and subsequent terms expand the discounted value function using powers of a generalized inverse called the Drazin (or group inverse) of a singular matrix derived from the transition matrix. Experiments show that Drazin bases converge considerably more quickly than several other bases, particularly for large values of the discount factor. An incremental variant of Drazin bases called Bellman average-reward bases (BARBs) is described, which provides some of the same beneﬁts at lower computational cost. 1

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Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 edu Abstract This paper explores links between basis construction methods in Markov decision processes and power series expansions of value functions. [sent-5, score-0.267]

2 Krylov and Bellman error bases are based on the Neumann series expansion. [sent-7, score-0.496]

3 These bases incur very large initial Bellman errors, and can converge rather slowly as the discount factor approaches unity. [sent-8, score-0.436]

4 The Laurent series expansion, which relates discounted and average-reward formulations, provides both an explanation for this slow convergence as well as suggests a way to construct more efﬁcient basis representations. [sent-9, score-0.268]

5 Experiments show that Drazin bases converge considerably more quickly than several other bases, particularly for large values of the discount factor. [sent-11, score-0.417]

6 An incremental variant of Drazin bases called Bellman average-reward bases (BARBs) is described, which provides some of the same beneﬁts at lower computational cost. [sent-12, score-0.798]

7 Recently, there has been growing interest in automatic basis construction methods for constructing a problem-speciﬁc low-dimensional representation of an MDP. [sent-14, score-0.177]

8 Functions on the original state space, such as the reward function or the value function, are “compressed” by projecting them onto a basis matrix Φ, whose column space spans a low-dimensional subspace of the function space on the states of the original MDP. [sent-15, score-0.383]

9 Among the various approaches proposed are reward-sensitive bases, such as Krylov bases [10] and an incremental variant called Bellman error basis functions (BEBFs) [9]. [sent-16, score-0.678]

10 These approaches construct bases by dilating the (sampled) reward function by geometric powers of the (sampled) transition matrix of a policy. [sent-17, score-0.653]

11 An alternative approach, called proto-value functions, constructs reward-invariant bases by ﬁnding the eigenvectors of the symmetric graph Laplacian matrix induced by the neighborhood topology of the state space under the given actions [7]. [sent-18, score-0.505]

12 A fundamental dilemma that is revealed by these prior studies is that neither reward-sensitive nor reward-invariant eigenvector bases by themselves appear to be fully satisfactory. [sent-19, score-0.372]

13 A Chebyshev polynomial bound for the error due to approximation using Krylov bases was derived in [10], extending a known similar result for general Krylov approximation [12]. [sent-20, score-0.483]

14 This bound shows that performance of Krylov bases (and BEBFs) tends to degrade as the discount factor γ → 1. [sent-21, score-0.403]

15 Intuitively, the initial basis vectors capture short-term transient behavior near rewarding regions, and tend to poorly ap1 proximate the value function over the entire state space until a sufﬁciently large time scale is reached. [sent-22, score-0.189]

16 A straightforward geometrical analysis of approximation errors using least-squares ﬁxed point approximation onto a basis shows that the Bellman error decomposes into the sum of two terms: a reward error and a second term involving the feature prediction error [1, 8] (see Figure 1). [sent-23, score-0.638]

17 This analysis helps reveal sources of error: Krylov bases and BEBFs tend to have low reward error (or zero in the non-sampled case), and hence a large component of the error in using these bases tends to be due to the feature prediction error. [sent-24, score-1.11]

18 In contrast, PVFs tend to have large reward error since typical spiky goal reward functions are poorly approximated by smooth low-order eigenvectors; however, their feature prediction error can be quite low as the eigenvectors often capture invariant subspaces of the model transition matrix. [sent-25, score-0.682]

19 A hybrid approach that combined low-order eigenvectors of the transition matrix (or PVFs) with higher-order Krylov bases was proposed in [10], which empirically resulted in a better approach. [sent-26, score-0.486]

20 This paper demonstrates a more principled approach to address this problem, by constructing new bases that emerge from investigating the links between basis construction methods and different power series expansions of value functions. [sent-27, score-0.641]

21 In particular, instead of using the eigenvectors of the transition matrix, the proposed approach uses the average-reward or gain as the ﬁrst basis vector, and dilates the reward function by powers of the average-adjusted transition matrix. [sent-28, score-0.512]

22 It turns out that the gain is an element of the space of eigenvectors associated with the eigenvalue λ = 1 of the transition matrix. [sent-29, score-0.112]

23 The relevance of power series expansion to approximations of value functions was hinted at in early work by Schwartz [13] on undiscounted optimization, although he did not discuss basis construction. [sent-30, score-0.32]

24 Krylov and Bellman error basis functions (BEBFs) [10, 9, 12], as well as proto-value functions [7], can be related to terms in the Neumann series expansion. [sent-31, score-0.332]

25 Ultimately, the performance of these bases is limited by the speed of convergence of the Neumann expansion, and of course, other errors arising due to reward and feature prediction error. [sent-32, score-0.591]

26 It is well-known that discounted value functions can be written in the form of a Laurent series expansion, where the ﬁrst two terms 1 correspond to the average-reward term (scaled by 1−γ ), and the average-adjusted sum of rewards (or bias). [sent-34, score-0.176]

27 Higher order terms involve powers of the Drazin (or group) inverse of a singular matrix related to the transition matrix. [sent-35, score-0.172]

28 This expansion provides a mathematical framework for analyzing the properties of basis construction methods and developing newer representations. [sent-36, score-0.215]

29 In particular, Krylov bases converge slowly for high discount factors since the value function is dominated by the scaled average-reward term, which is poorly approximated by the initial BEBF or Krylov basis vectors as it involves the long-term limiting matrix P ∗ . [sent-37, score-0.666]

30 The Laurent series expansion leads to a new type of basis called a Drazin basis [6]. [sent-38, score-0.403]

31 An approximation of Drazin bases called Bellman average-reward bases (BARBs) is described and compared with BEBFs, Krylov bases, and PVFs. [sent-39, score-0.777]

32 The value function V associated with a deterministic policy π : S → A is deﬁned as the long-term expected sum of rewards received starting from a state, and following the policy π indeﬁnitely. [sent-41, score-0.133]

33 For a ﬁxed policy π, the induced discounted Markov reward process is deﬁned as (P, R, γ). [sent-44, score-0.264]

34 ˆ A popular approach to approximating V is to use a linear combination of basis functions V ≈ V = Φw, where the basis matrix Φ is of size |S| × k, and k |S|. [sent-45, score-0.353]

35 The Bellman error for a given basis Φ, denoted BE(Φ), is deﬁned as the difference between the two sides of the Bellman equation, when 1 In what follows, we suppress the dependence of P , R, and V on the policy π to avoid clutter. [sent-46, score-0.27]

36 2 γP ΦwΦ R (I − ΠΦ )R ΠΦ R (I − ΠΦ )γP ΦwΦ ΠΦ γΦwΦ Φ Figure 1: The Bellman error due to a basis and its decomposition. [sent-47, score-0.226]

37 If the Markov chain deﬁned by P is irreducible and aperiodic, then ΠΦ = Φ(ΦT D∗ Φ)−1 ΦT D∗ , where D∗ is a diagonal matrix whose entries contain the stationary distribution of the Markov chain. [sent-52, score-0.096]

38 Krylov bases correspond to successive terms in the Neumann series [10, 12]. [sent-58, score-0.446]

39 An incremental variant of the Krylov-based approach is called Bellman error basis functions (BEBFs) [9]. [sent-68, score-0.306]

40 In particular, given a set of basis functions Φk (where the ﬁrst one is assumed to equal R), the next basis is deﬁned to be φk+1 = T (Φk wΦk ) − Φk wΦk . [sent-69, score-0.316]

41 In the model-free reinforcement learning setting, φk+1 ˆ ˆ can be approximated by the temporal-difference (TD) error φk+1 = r + γ Qk (s , πk (s )) − Qk (s, a), ˆ given a set of stored transitions in the form (s, a, r, s ). [sent-70, score-0.124]

42 Here, Qk is the ﬁxed-point least-squares approximation to the action-value function Q(s, a) on the basis Φk . [sent-71, score-0.16]

43 It can be easily shown that BEBFs and Krylov bases deﬁne the same space [8]. [sent-72, score-0.372]

44 1 Convergence Analysis A key issue in evaluating the effectiveness of Krylov bases (and BEBFs) is the speed of convergence of the Neumann series. [sent-74, score-0.387]

45 As γ → 1, Krylov bases and BEBFs converge rather slowly, owing to a large increase in the weighted feature error. [sent-75, score-0.472]

46 Petrik [10] derived a bound for the error due to Krylov approximation, which depends on the condition number of I − γP , and the ratio of two Chebyshev polynomials on the complex plane. [sent-79, score-0.097]

47 It has been shown that BEBFs reduce the Bellman error at a rate bounded by value iteration [9]. [sent-81, score-0.081]

48 For standard value iteration, A = I − γP , and consequently the natural decomposition is to set 2 The two components of the Bellman error may partially (or fully) cancel each other out: the Bellman error of V itself is 0, but it generates non-zero reward and feature prediction errors. [sent-84, score-0.366]

49 99 Figure 2: Condition number of I − γP as γ → 1, where P is the transition matrix of the optimal policy in a chain MDP of 50 states with rewards in states 10 and 41 [9]. [sent-87, score-0.213]

50 First, note that the overall Bellman error can be reduced to the weighted feature error, since the reward error is 0 as R is in the span of both BEBFs and Krylov bases: ˆ BE(Φ) 2 = T (ΦwΦ ) − ΦwΦ 2 = R − (I − γP )V 2 = (I − ΠΦ )γP ΦwΦ 2 . [sent-94, score-0.463]

51 Figure 2 shows empirically that one reason for the slow convergence of BEBFs and Krylov bases is that as γ → 1, the condition number of I − γP signiﬁcantly increases. [sent-101, score-0.401]

52 Figure 3 compares the weighted feature error of BEBF bases (the performance of Krylov bases is identical and not shown) on a 50 state chain domain with a single goal reward of 1 in state 25. [sent-102, score-1.201]

53 Notice as γ increases, the feature error increases dramatically. [sent-104, score-0.116]

54 4 Laurent Series Expansion and Drazin Bases A potential solution to the slow convergence of BEBFs and Krylov bases is suggested by a different power series called the Laurent expansion. [sent-105, score-0.478]

55 This connection uses the following Laurent series expansion of V in terms of the average reward ρ of the policy π, the average-adjusted sum of rewards h, and higher order terms that involve the generalized spectral inverse (Drazin or group inverse) of I − P . [sent-107, score-0.422]

56 As γ increases, the weighted feature error grows much larger as the value function becomes progressively smoother and less like the reward function. [sent-146, score-0.361]

57 As γ → 1, the ﬁrst term in the Laurent series expansion grows quite large causing the slow convergence of Krylov bases and BEBFs. [sent-148, score-0.507]

58 (I − P )D is a generalized inverse of the singular matrix I − P called the Drazin inverse [2, 11]. [sent-149, score-0.126]

59 The matrix I − P of a Markov chain has index k = 1. [sent-154, score-0.091]

60 For index 1 matrices, the Drazin inverse turns out to be the same as the group inverse, which is deﬁned as (I − P )D = (I − P + P ∗ )−1 − P ∗ , 1 where the long-term limiting matrix P ∗ = limn→∞ n k=0 P k = I − (I − P )(I − P )D . [sent-155, score-0.102]

61 In terms of the expansion above, y−1 is the gain of the policy, y0 is its bias, and so on. [sent-169, score-0.087]

62 Analogous to the Krylov bases, the successive terms of the Laurent series expansion can be viewed as basis vectors. [sent-174, score-0.271]

63 More formally, the Drazin basis is deﬁned as the space spanned by the vectors [6]: Dm = {P ∗ R, (I − P )D R, ((I − P )D )2 R, . [sent-175, score-0.166]

64 (2) The ﬁrst basis vector is the average-reward or gain g = P ∗ R of policy π. [sent-179, score-0.213]

65 The second basis vector is the bias, or average-adjusted sum of rewards h. [sent-180, score-0.19]

66 Subsequent basis vectors correspond to higher-order terms in the Laurent series. [sent-181, score-0.156]

67 Building on this insight, an approximate Drazin basis called Bellman average-reward bases (BARBs) can be deﬁned as follows. [sent-185, score-0.535]

68 First, the approximate Drazin basis is deﬁned as the space spanned by the vectors Am = {P ∗ R, (P − P ∗ )R, (P − P ∗ )2 R, . [sent-186, score-0.166]

69 BARBs are similar to Krylov bases, except that the reward function is being dilated by the averageadjusted transition matrix P − P ∗ , and the ﬁrst basis element is the gain. [sent-193, score-0.387]

70 Analogous to BEBFs, in the model-free reinforcement learning setting, BARBs can be computed using the average-adjusted TD error ˆ ˆ φk+1 (s) = r − ρk (s) + Qk (s , πk (s )) − Qk (s, a). [sent-197, score-0.106]

71 1 Expressivity Properties Some results concerning the expressivity of approximate Drazin bases and BARBs are now discussed. [sent-201, score-0.391]

72 Theorem 2 For any k > 1, the following hold: span {Ak (R)} ⊆ span {BARBk+1 (R)} . [sent-203, score-0.092]

73 For k = 1, both approximate Drazin bases and BARBs contain the gain ρ. [sent-208, score-0.396]

74 For general k > 2, the new basis vector in Ak is P k−1 R, which can be shown to be part of BARBk+1 (R). [sent-210, score-0.145]

75 There is a similar decomposition of the average-adjusted Bellman error into a component that depends on the average-adjusted reward error and an undiscounted weighted feature error. [sent-212, score-0.455]

76 Theorem 3 Given a basis Φ, for any average reward Markov reward process (P, R), the Bellman error can be decomposed as follows: ˆ ˆ T (V ) − V = R − ρ + P ΦwΦ − ΦwΦ = (I − ΠΦ )(R − ρ) + (I − ΠΦ )P ΦwΦ = (I − ΠΦ )R − (I − ΠΦ )ρ + (I − ΠΦ )P ΦwΦ . [sent-213, score-0.578]

77 Proof: The three terms represent the reward error, the average-reward error, and the undiscounted weighted feature error. [sent-214, score-0.304]

78 A more detailed convergence analysis of BARBs is given in [4], based on the relationship between the approximation error and the mixing rate of the Markov chain deﬁned by P . [sent-216, score-0.162]

79 5 0 Bellman Error 6 BARBs BEBF PVF−MP 20 30 # Basis Functions 20 0 0 0 10 20 30 # Basis Functions 40 50 0 10 20 30 # Basis Functions 40 50 0 0 10 20 30 # Basis Functions 40 50 Figure 4: Experimental comparison on a 50 state chain MDP with rewards in state 10 and 41. [sent-227, score-0.154]

80 6 Experimental Comparisons Figure 4 compares the performance of Bellman average-reward basis functions (BARBs) vs. [sent-234, score-0.183]

81 Bellman-error basis functions (BEBFs), and a variant of proto-value functions (PVF-MP) on a 50 state chain MDP. [sent-235, score-0.289]

82 The PVF-MP algorithm selects basis functions incrementally based upon the Bellman error, where basis function k+1 is the PVF that has the largest inner product with the Bellman error resulting from the previous k basis functions. [sent-243, score-0.542]

83 PVFs have a high reward error, since the reward function is a set of two delta functions that is poorly approximated by the eigenvectors of the combinatorial Laplacian on the chain graph. [sent-244, score-0.489]

84 The overall Bellman error remains large due to the high reward error. [sent-246, score-0.25]

85 The reward error for BEBFs is by deﬁnition 0 as R is a basis vector itself. [sent-247, score-0.395]

86 However, the weighted feature error for BEBFs grows quite large as γ increases from 0. [sent-248, score-0.178]

87 Drazin and Krylov bases in the two-room gridworld MDP [7]. [sent-256, score-0.372]

88 Drazin bases perform the best, followed by BARBs, and then Krylov bases. [sent-257, score-0.372]

89 BEBFs on a 10 × 10 grid world MDP with a reward placed in the upper left corner state. [sent-260, score-0.18]

90 The Neumann series, which underlies Bellman error and Krylov bases, tends to converge slowly as γ → 1. [sent-265, score-0.114]

91 Drazin and Krylov bases in a 100 state two-room MDP [7]. [sent-267, score-0.401]

92 The reward was set at +100 for reaching a corner goal state in one of the rooms. [sent-269, score-0.209]

93 2 0 0 0 5 10 15 20 25 30 0 5 10 15 20 25 30 0 0 5 10 15 20 25 30 35 Figure 6: Experimental comparison of BARBs and BEBFs on a 10 × 10 grid world MDP with a reward in the upper left corner. [sent-298, score-0.169]

94 An incremental version of Drazin bases called Bellman average-reward bases (BARBs) was investigated. [sent-306, score-0.786]

95 Numerical experiments on simple MDPs show superior performance of Drazin bases and BARBs to BEBFs, Krylov bases, and PVFs. [sent-307, score-0.372]

96 Scaling BARBs and Drazin bases to large MDPs requires addressing sampling issues, and exploiting structure in transition matrices, such as using factored representations. [sent-308, score-0.419]

97 Reinforcement learning methods to estimate the ﬁrst few terms of the Laurent series were proposed in [5], and can be adapted for basis construction. [sent-310, score-0.199]

98 The Schultz expansion provides a way of rewriting the Neumann series using a multiplicative series of dyadic powers of the transition matrix, which is useful for multiscale bases [6]. [sent-311, score-0.596]

99 An investigation of basis construction from power series expansions of value functions. [sent-332, score-0.255]

100 Sensitive-discount optimality: Unifying discounted and average reward reinforcement learning. [sent-335, score-0.245]

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Crucially, as a result, the sequence of environment-states {st }∞ is affected by t=0 the choice of reward function; therefore, the agent designer’s return is affected as well. The optimal reward problem arises from the fact that while the objective reward function is ﬁxed as part of the problem description, the reward function is a choice to be made by the designer. We capture this choice abstractly by letting the reward be parameterized by some vector of parameters θ chosen from space of parameters Θ. Each θ ∈ Θ speciﬁes a reward function R(· ; θ) which in turn produces a distribution over environment state sequences via whatever RL method the agent uses. The expected N 1 return obtained by the designer for choice θ is U(θ) = limN →∞ E N t=0 RO (st ) R(·; θ) . The optimal reward parameters are given by the solution to the optimal reward problem [16, 17, 18]: θ∗ = arg max U(θ) = arg max lim E θ∈Θ θ∈Θ N →∞ 1 N N RO (st ) R(·; θ) . (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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