nips nips2010 nips2010-93 knowledge-graph by maker-knowledge-mining

93 nips-2010-Feature Construction for Inverse Reinforcement Learning

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Author: Sergey Levine, Zoran Popovic, Vladlen Koltun

Abstract: The goal of inverse reinforcement learning is to ﬁnd a reward function for a Markov decision process, given example traces from its optimal policy. Current IRL techniques generally rely on user-supplied features that form a concise basis for the reward. We present an algorithm that instead constructs reward features from a large collection of component features, by building logical conjunctions of those component features that are relevant to the example policy. Given example traces, the algorithm returns a reward function as well as the constructed features. The reward function can be used to recover a full, deterministic, stationary policy, and the features can be used to transplant the reward function into any novel environment on which the component features are well deﬁned. 1

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Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 edu Abstract The goal of inverse reinforcement learning is to ﬁnd a reward function for a Markov decision process, given example traces from its optimal policy. [sent-7, score-0.467]

2 We present an algorithm that instead constructs reward features from a large collection of component features, by building logical conjunctions of those component features that are relevant to the example policy. [sent-9, score-0.928]

3 Given example traces, the algorithm returns a reward function as well as the constructed features. [sent-10, score-0.326]

4 The reward function can be used to recover a full, deterministic, stationary policy, and the features can be used to transplant the reward function into any novel environment on which the component features are well deﬁned. [sent-11, score-1.046]

5 1 Introduction Inverse reinforcement learning aims to ﬁnd a reward function for a Markov decision process, given only example traces from its optimal policy. [sent-12, score-0.437]

6 IRL solves the general problem of apprenticeship learning, in which the goal is to learn the policy from which the examples were taken. [sent-13, score-0.217]

7 The MDP formalism provides a compact method for specifying a task in terms of a reward function, and IRL further simpliﬁes task speciﬁcation by requiring only a demonstration of the task being performed. [sent-14, score-0.326]

8 However, current IRL methods generally require not just expert demonstrations, but also a set of features or basis functions that concisely capture the structure of the reward function [1, 7, 9, 10]. [sent-15, score-0.605]

9 It is often easier to enumerate all potentially relevant component features (“components”) than to manually specify a set of features that is both complete and fully relevant. [sent-17, score-0.335]

10 For example, when emulating a human driver, it is easier to list all known aspects of the environment than to construct a complete and fully relevant reward basis. [sent-18, score-0.455]

11 The difﬁculty of performing IRL given only such components is that many of them may have important logical relationships that make it impossible to represent the reward function as their linear combination, while enumerating all possible relationships is intractable. [sent-19, score-0.464]

12 Others, like the car’s speed and the presence of police, might have an important logical relationship for a driver who prefers to speed. [sent-21, score-0.258]

13 We present an IRL algorithm that constructs reward features out of a large collection of component features, many of which may be irrelevant for the expert’s policy. [sent-22, score-0.565]

14 The Feature construction for Inverse Reinforcement Learning (FIRL) algorithm constructs features as logical conjunctions of the components that are most relevant for the observed examples, thus capturing their logical relationships. [sent-23, score-0.557]

15 At the same time, it ﬁnds a reward function for which the optimal policy matches 1 the examples. [sent-24, score-0.481]

16 The reward function can be used to recover a deterministic, stationary policy for the expert, and the features can be used to transplant the reward to any novel environment on which the component features are well deﬁned. [sent-25, score-1.201]

17 In this way, the features act as a portable explanation for the expert’s policy, enabling the expert’s behavior to be predicted in unfamiliar surroundings. [sent-26, score-0.192]

18 2 Algorithm Overview We deﬁne a Markov decision process as M = {S, A, θ, γ, R}, where S is a state space, A is a set of actions, θsas is the probability of a transition from s ∈ S to s ∈ S under action a ∈ A, γ ∈ [0, 1) is a discount factor, and R(s, a) is a reward function. [sent-27, score-0.399]

19 The optimal policy π ∗ is the policy ∞ that maximizes the expected discounted sum of rewards E [ t=0 γ t R(st , at )|π ∗ , θ]. [sent-28, score-0.362]

20 FIRL also accepts a set of component features of the form δ : S → Z, which are used to construct a set of relevant features for representing R. [sent-33, score-0.359]

21 The algorithm iteratively constructs both the features and the reward function. [sent-34, score-0.493]

22 The optimization step of the ith iteration computes a reward function R(i) using the current set of features Φ(i−1) , and the following ﬁtting step determines a new set of features Φ(i) . [sent-37, score-0.657]

23 The objective of the optimization step is to ﬁnd a reward function R(i) that best ﬁts the last feature hypothesis Φ(i−1) while remaining consistent with the examples D. [sent-38, score-0.506]

24 In contrast, the FIRL optimization step aims to discover regions where the current features are insufﬁcient, and must be able to step outside of the constraints of the these features. [sent-41, score-0.265]

25 To this end, the reward function R(i) is found by solving a quadratic program, with constraints that keep R(i) consistent with D, and an objective that penalizes the deviation of R(i) from its projection onto the linear basis formed by the features Φ(i−1) . [sent-42, score-0.51]

26 The ﬁtting step analyzes the reward function R(i) to generate a new feature hypothesis Φ(i) that better captures the variation in the reward function. [sent-43, score-0.754]

27 Intuitively, the regions where R(i) is poorly represented by Φ(i−1) correspond to features that must be reﬁned further, while regions where different features take on similar rewards are indicative of redundant features that should be merged. [sent-44, score-0.512]

28 A simple explanation of the reward function is often more likely to be the correct one [7], so we prefer the smallest tree that produces a sufﬁciently rich feature set to represent a reward function consistent with the examples. [sent-48, score-0.838]

29 To obtain such a tree, we stop subdividing a node when setting the reward for all states in φ to their average induces an optimal policy consistent with the examples. [sent-49, score-0.572]

30 Since the optimization is constrained to be consistent with D, if the current set of features is insufﬁcient to represent a consistent reward function, R(i) will not be well-represented by the features Φ(i−1) . [sent-51, score-0.661]

31 This intra-feature reward variance is detected in the ﬁtting step, and the features that were insufﬁciently reﬁned are subdivided further, while redundant features that have little variance between them are merged. [sent-52, score-0.586]

32 3 Optimization Step During the ith optimization step, we compute a reward function R(i) using the examples D and the current feature set Φ(i−1) . [sent-53, score-0.429]

33 This reward function is chosen so that the optimal policy under the reward is consistent with the examples D and so that it minimizes the sum of squared errors between R(i) and its projection onto the linear basis of features Φ(i−1) . [sent-54, score-0.99]

34 Thus, TΦ→R TR→Φ R is a vector where 2 the reward in each state is the average over all rewards in the feature that state belongs to. [sent-56, score-0.5]

35 Letting π R denote the optimal policy under R, the reward optimization problem can be expressed as: min R − TΦ→R TR→Φ R 2 R π R (s) = a s. [sent-57, score-0.506]

36 In practice, we prefer a reward function under which the examples are not just part of an optimal policy, but are part of the unique optimal policy [7]. [sent-70, score-0.537]

37 To prevent rewards under which example actions “tie” for the optimal choice, we require that ai,t be better than all other actions in state si,t by some margin ε, which we accomplish by adding ε to all inequality constraints for state si,t . [sent-71, score-0.221]

38 The precise value of ε is not important, since changing it only scales the reward function by a constant. [sent-72, score-0.326]

39 Recall that the ﬁtting step must determine not only which features must be reﬁned further, but also which features can be merged. [sent-75, score-0.283]

40 Since the purpose of the added term is to allow superﬂuous features to be merged because they take on similar values, we prefer for a feature to be very similar to one of its neighbors, rather than to have minimal distance to all of them. [sent-80, score-0.254]

41 4 Fitting Step Once the reward function R(i) for the current feature set Φ(i−1) is computed, we formulate a new feature hypothesis Φ(i) that is better able to represent this reward function. [sent-89, score-0.781]

42 The objective of this step is to construct a set of features that gives greater resolution in regions where the old features are too coarse, and lower resolution in regions where the old features are unnecessarily ﬁne. [sent-90, score-0.507]

43 The tree is rooted at the node t0 , and each node of the tree is deﬁned as tj = {δj , φj , tj− , tj+ }. [sent-92, score-0.223]

44 tj− and tj+ are the left and right subtrees, φj ⊆ S is the set of states belonging to node j (initialized as φ0 = S), and δj is the component feature that acts as the splitting test at node j. [sent-93, score-0.224]

45 In our implementation, all component features are binary, though the generalization to multivariate components and non-binary trees is straightforward. [sent-95, score-0.206]

46 The new set of features consists of indicators for each of the leaf clusters φ (where t is a leaf node), and can be equivalently expressed as a conjunction of components: letting j0 , . [sent-96, score-0.248]

47 As discussed in Section 2, we prefer the smallest tree that produces a rich enough feature set to represent a reward function consistent with the examples D. [sent-106, score-0.514]

48 This is done by constructing a new reward function R(i) for (i) ˆ (i) (s, a) = |φ |−1 ˆ (i) (s, a) = R(i) (s, a) otherwise. [sent-108, score-0.326]

49 The which R s∈φ R (s, a) if s ∈ φ , and R ˆ (i) is determined with value iteration and, if the policy is consistent with the optimal policy under R ˆ examples D, t becomes a leaf and R(i) is updated to be equal to R(i) . [sent-109, score-0.403]

50 Recall that ∆(φk1 , φk2 ) is a linear penalty on the difference between the average rewards of states in φk1 and φk2 , and can be used to drive the rewards in these features closer together so that they can be merged in a subsequent iteration. [sent-113, score-0.346]

51 These complex features are more likely to be the result of overﬁtting, and can be merged to form smaller trees. [sent-115, score-0.176]

52 A gridworld consists of an N×N grid of states, with ﬁve actions possible in each state, corresponding to movement in each of the compass directions and standing in place. [sent-157, score-0.166]

53 The expert’s policy is the optimal policy under this reward. [sent-161, score-0.31]

54 The example set D is generated by randomly sampling states and following the expert’s policy for 100 steps. [sent-162, score-0.188]

55 Since the prior algorithms do not perform feature construction, they were tested either with indicators for each of the 64 regions (referred to as “perfect” features), or with indicators for each state (the “primitive” features). [sent-163, score-0.197]

56 FIRL was instead provided with 2N component features corresponding to splits on the x and y axes, so that δx,i (sx,y ) = 1 if x ≥ i, and δy,i (sx,y ) = 1 if y ≥ i. [sent-164, score-0.178]

57 We ﬁrst compare the running times of the algorithms (using perfect features for prior methods) on gridworlds of varying sizes, shown in Table 1. [sent-166, score-0.23]

58 For the mixed policies produced by Abbeel & Ng, we computed the metrics for each policy and mixed them using the policy weights λ [1]. [sent-176, score-0.385]

59 For the non-deterministic policies of LPAL, percent misprediction is 5 40% 30% 20% 10% 0% 2 4 8 16 32 64 128 256 512 examples non-deterministic 60% percent misprediction feature expectation dist percent misprediction 50% deterministic 0. [sent-177, score-0.814]

60 To evaluate reward transfer and see how the method performs with more realistic component features, we populated a world with objects. [sent-199, score-0.453]

61 The expert received a reward of −2 for being within 3 units of an object with inner color 1, otherwise a reward of −1 for being within 2 units of a wall, otherwise a reward of 1 for being within 1 unit of an object with inner color 2, and 0 otherwise. [sent-203, score-1.101]

62 All other colors acted as distractors, allowing us to evaluate the robustness of feature construction to irrelevant components. [sent-204, score-0.172]

63 For each trial, the learned reward tree was used to test accuracy on 10 more random environments, by specifying a reward for each state according to the regression tree. [sent-205, score-0.773]

64 In Figure 2(a), we evaluate how FIRL performs with varying numbers of iterations on both the training and transfer environments, as well as on the gridworld from the previous section. [sent-208, score-0.239]

65 The results indicate that FIRL converged to a stable hypothesis more quickly than in the gridworld, since the square regions in the gridworld required many more partitions than the objectrelative features. [sent-209, score-0.196]

66 convergence analysis 10% 60% 50% 40% 30% 20% FIRL with objects FIRL transfer 10% FIRL gridworld 0% constraint violation 9% 70% 2 4 6 8 10 12 14 16 18 20 iterations percent violation percent misprediction 80% 8% 7% 6% 5% 4% 3% 2% 1% 0% 0. [sent-211, score-0.605]

67 measure the percent of examples that are violated with varying amounts of non-determinism, by varying the probability β with which an action moves the agent to the desired state. [sent-218, score-0.211]

68 LPAL was not used in the comparison because it does not return a reward function, and therefore cannot transfer 6 40% 30% A&N; MMP FIRL A&N; transfer 10% MMP transfer FIRL transfer 0% 2 8 14 20 non-deterministic 0. [sent-223, score-0.642]

69 05 0 2 colors 8 14 50% 40% 30% 20% 10% 0% 20 colors 2 8 14 colors non-deterministic 0. [sent-228, score-0.165]

70 3 60% percent misprediction feature expectation dist percent misprediction 50% 20% deterministic 0. [sent-229, score-0.523]

71 05 0 2 8 14 20 colors Figure 3: Comparison of FIRL and Abbeel & Ng on training environments and randomly generated transfer environments, with increasing numbers of component features. [sent-235, score-0.257]

72 FIRL maintained higher transfer accuracy in the presence of distractors by constructing features out of relevant components. [sent-236, score-0.272]

73 The results in Figure 3 indicate that accuracy on the training environment remained largely stable, while transfer accuracy gradually decreased with more colors due to the ambiguity caused by large numbers of distractors. [sent-242, score-0.188]

74 Prior algorithms were more affected by distractors on the training environments, and their inability to construct features prevented them from capturing a portable “explanation” of the expert’s reward. [sent-243, score-0.226]

75 They therefore could not transfer the learned policy to other environments with comparable accuracy. [sent-244, score-0.337]

76 In contrast to the gridworld experiments, the expert’s reward function in these environments was encoded in terms of logical relationships between the component features, which standard IRL algorithms cannot capture. [sent-245, score-0.691]

77 The agent can drive at speeds 1 through 4, and can move one lane left or one lane right. [sent-251, score-0.208]

78 The component features take the form “is a vehicle of type X at most n car-lengths in front/behind me,” where X can be either all vehicles, cars, motorcycles, police, or civilian, and n is in the range from 0 to 5 car-lengths. [sent-253, score-0.204]

79 There are equivalent features for checking for cars in front or behind in the lanes to the left and to the right of the agent’s, as well as a feature for each of the four speeds and each lane the agent can occupy. [sent-254, score-0.358]

80 For this experiment, we implemented expert policies for two behaviors: a “lawful” driver and an “outlaw” driver. [sent-256, score-0.262]

81 The lawful driver prefers to drive fast, but does not exceed speed 2 in the right lane, or speed 3 in the middle lane. [sent-257, score-0.294]

82 The outlaw driver also prefers to drive fast, but slows down to speed 2 or below when within 2 car-lengths of a police vehicle (to avoid arrest). [sent-258, score-0.365]

83 In Table 2, we compare the policies learned from traces of the two experts by FIRL, MMP, and Abbeel & Ng’s algorithm. [sent-259, score-0.165]

84 As can be seen in the supplemental videos, the policy learned by FIRL closely matched that of the expert, maintaining a high speed whenever possible but not driving fast in the wrong lane or near police vehicles. [sent-262, score-0.432]

85 The policies learned by Abbeel & Ng’s algorithm and MMP drove at the minimum speed when trained on either the lawful or outlaw expert traces. [sent-263, score-0.434]

86 Because prior methods only represented the reward as a linear combination of the provided features, they were unable to determine the logical connection between speed and the other features. [sent-264, score-0.493]

87 The policies learned by these methods found the nearest “optimal” position with respect to their learned feature weights, accepting the cost of violating the speed expectation in exchange for best matching the expectation of all other (largely irrelevant) features. [sent-265, score-0.238]

88 FIRL, on the other hand, correctly established 7 “Lawful” policies percent mis- feature expectprediction ation distance Expert FIRL MMP A&N; Random 0. [sent-266, score-0.242]

89 220 “Outlaw” policies average percent mis- feature expectspeed prediction ation distance average speed 2. [sent-276, score-0.299]

90 htm the logical connection between speed and police vehicles or lanes, and drove fast when appropriate, as indicated by the average speed in Table 2. [sent-301, score-0.407]

91 As a baseline, the table also shows the performance of a random policy generated by picking weights for the component features uniformly at random. [sent-302, score-0.333]

92 6 Discussion and Future Work This paper presents an IRL algorithm that constructs reward features, represented as a regression tree, out of a large collection of component features. [sent-303, score-0.411]

93 By combining relevant components into logical conjunctions, the FIRL algorithm is able to discover logical precedence relationships that would not otherwise be apparent. [sent-304, score-0.275]

94 The learned regression tree concisely captures the structure of the reward function and acts as a portable “explanation” of the observed behavior in terms of the provided components, allowing the learned reward function to be transplanted onto different environments. [sent-305, score-0.827]

95 While it can be difﬁcult to manually construct a complete list of relevant reward features for such an agent, it is comparatively easier to list all aspects of the environment that a human or animal is aware of. [sent-308, score-0.609]

96 With FIRL, such a list can be used to form hypotheses about reward features, possibly leading to increased understanding of the agent’s motivations. [sent-309, score-0.35]

97 Approximate linear programming has been used to solve MDPs with continuous state spaces [5], and a similar approach could be used to construct a tractable set of constraints for the optimization step, making it possible to perform feature construction on continuous or extremely large state spaces. [sent-312, score-0.243]

98 A more intelligent hypothesis proposal scheme, perhaps with a Bayesian approach, could more readily incorporate priors on potential features to penalize excessively deep trees or prevent improbable conjunctions of components. [sent-320, score-0.231]

99 Furthermore, while regression trees provide a principled method for constructing logical conjunctions of component features, if the desired features are not readily expressible as conjunctions of simple components, other regression methods may be used in the ﬁtting step. [sent-321, score-0.432]

100 For example, the algorithm could be modiﬁed to perform feature adaptation by using the ﬁtting step to adapt a set of continuously-parameterized features to best ﬁt the reward function. [sent-322, score-0.529]

similar papers computed by tfidf model

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At each time step, the designer receives reward RO (st ) ∈ [0, 1] based on the current state st of the environment, where the subscript denotes time. The designer’s objective return is the expected mean objective reward N 1 obtained over an inﬁnite horizon, i.e., limN →∞ E N t=0 RO (st ) . In the standard view of RL, the agent uses the same reward function as the designer to align the interests of the agent and the designer. Here we allow for a separate agent reward function R(· ). An agent’s reward function can in general be deﬁned in terms of the history of actions and observations, but is often more pragmatically deﬁned in terms of some abstraction of history. We deﬁne the agent’s reward function precisely in Section 2. Optimal Reward Problem. An RL agent attempts to act so as to maximize its own cumulative reward, or return. 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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Stochastic Approximation and Recursive Algorithms and Applications. Springer, 2nd edition, 2010. [9] Cetin Mericli, Tekin Mericli, and H. Levent Akin. A Reward Function Generation Method Using Genetic ¸ ¸ ¸ Algorithms : A Robot Soccer Case Study (Extended Abstract). In Proc. of the 9th Int. Conf. on Autonomous Agents and Multiagent Systems (AAMAS 2010), number 2, pages 1513–1514, 2010. [10] Gergely Neu and Csaba Szepesv´ ri. Apprenticeship learning using inverse reinforcement learning and a gradient methods. In Proceedings of the 23rd Conference on Uncertainty in Artiﬁcial Intelligence, pages 295–302, 2007. [11] Andrew Y. Ng, Stuart J. Russell, and D. Harada. Policy invariance under reward transformations: Theory and application to reward shaping. In Proceedings of the 16th International Conference on Machine Learning, pages 278–287, 1999. [12] Scott Niekum, Andrew G. Barto, and Lee Spector. Genetic Programming for Reward Function Search. IEEE Transactions on Autonomous Mental Development, 2(2):83–90, 2010. [13] Pierre-Yves Oudeyer, Frederic Kaplan, and Verena V. Hafner. Intrinsic Motivation Systems for Autonomous Mental Development. IEEE Transactions on Evolutionary Computation, 11(2):265–286, April 2007. [14] J¨ rgen Schmidhuber. Curious model-building control systems. In IEEE International Joint Conference on u Neural Networks, pages 1458–1463, 1991. [15] Satinder Singh, Andrew G. Barto, and Nuttapong Chentanez. Intrinsically Motivated Reinforcement Learning. In Proceedings of Advances in Neural Information Processing Systems 17 (NIPS), pages 1281–1288, 2005. [16] Satinder Singh, Richard L. Lewis, and Andrew G. Barto. Where Do Rewards Come From? In Proceedings of the Annual Conference of the Cognitive Science Society, pages 2601–2606, 2009. [17] Satinder Singh, Richard L. Lewis, Andrew G. Barto, and Jonathan Sorg. Intrinsically Motivated Reinforcement Learning: An Evolutionary Perspective. 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