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4 nips-2010-A Computational Decision Theory for Interactive Assistants

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Author: Alan Fern, Prasad Tadepalli

Abstract: We study several classes of interactive assistants from the points of view of decision theory and computational complexity. We ﬁrst introduce a class of POMDPs called hidden-goal MDPs (HGMDPs), which formalize the problem of interactively assisting an agent whose goal is hidden and whose actions are observable. In spite of its restricted nature, we show that optimal action selection in ﬁnite horizon HGMDPs is PSPACE-complete even in domains with deterministic dynamics. We then introduce a more restricted model called helper action MDPs (HAMDPs), where the assistant’s action is accepted by the agent when it is helpful, and can be easily ignored by the agent otherwise. We show classes of HAMDPs that are complete for PSPACE and NP along with a polynomial time class. Furthermore, we show that for general HAMDPs a simple myopic policy achieves a regret, compared to an omniscient assistant, that is bounded by the entropy of the initial goal distribution. A variation of this policy is shown to achieve worst-case regret that is logarithmic in the number of goals for any goal distribution. 1

Reference: text

Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 We ﬁrst introduce a class of POMDPs called hidden-goal MDPs (HGMDPs), which formalize the problem of interactively assisting an agent whose goal is hidden and whose actions are observable. [sent-6, score-0.57]

2 We then introduce a more restricted model called helper action MDPs (HAMDPs), where the assistant’s action is accepted by the agent when it is helpful, and can be easily ignored by the agent otherwise. [sent-8, score-1.154]

3 Furthermore, we show that for general HAMDPs a simple myopic policy achieves a regret, compared to an omniscient assistant, that is bounded by the entropy of the initial goal distribution. [sent-10, score-0.466]

4 A variation of this policy is shown to achieve worst-case regret that is logarithmic in the number of goals for any goal distribution. [sent-11, score-0.48]

5 The assistant needs to correctly reason about the relative merits of taking different actions in the presence of signiﬁcant uncertainty about the goals of the human agent. [sent-16, score-0.788]

6 In this paper, we formulate and study several classes of interactive assistant problems from the points of view of decision theory and computational complexity. [sent-23, score-0.588]

7 In a HGMDP, a (human) agent and a (computer) assistant take actions in turns. [sent-27, score-1.057]

8 The objective for the assistant is to ﬁnd a history-dependent policy that maximizes the expected reward of the agent given the agent’s goal-based policy and its goal distribution. [sent-29, score-1.45]

9 This motivates a more restricted model called Helper Action MDP (HAMDP), where the assistant executes a helper action at each step. [sent-31, score-0.849]

10 The agent is obliged to accept the helper action if it is helpful for its goal and receives a reward bonus (or cost reduction) for doing so. [sent-32, score-0.861]

11 Otherwise, the agent can continue with its own preferred action without any reward or penalty to the assistant. [sent-33, score-0.583]

12 The main positive result of the paper is to give a simple myopic policy for general stochastic HAMDPs that has a regret which is upper bounded by the entropy of the goal distribution. [sent-37, score-0.558]

13 Furthermore we give a variant of this policy that is able to achieve worst-case and expected regret that is logarithmic in the number of goals without any prior knowledge of the goal distribution. [sent-38, score-0.504]

14 While the current HAMDP results are conﬁned to unobtrusively assisting a competent agent, they provide a strong foundation for analyzing more complex classes of assistant problems, possibly including direct communication, coordination, partial observability, and irrationality of users. [sent-40, score-0.596]

15 Our objective is to select actions for the assistant in order to help the agent maximize its reward. [sent-42, score-1.071]

16 An HGMDP describes the dynamics and reward structure of the environment via a ﬁrst-order Markov model, where it is assumed that the state is fully observable to both the agent and assistant. [sent-46, score-0.586]

17 In addition, an HGMDP describes the possible goals of the agent and the behavior of the agent when pursuing those goals. [sent-47, score-0.823]

18 For example, it can be convenient to model basic communication actions of the agent as changing aspects of the state, and the result of such actions will often be goal dependent. [sent-52, score-0.694]

19 We consider a ﬁnite-horizon episodic problem setting where the agent begins each episode in a state drawn from IS with a goal drawn from IG . [sent-53, score-0.514]

20 The process then alternates between the agent and assistant executing actions (including noops) in the environment until the horizon is reached. [sent-55, score-1.12]

21 The agent is assumed to select actions according to π. [sent-56, score-0.496]

22 The reward for the episode is equal to the sum of the rewards of the actions executed by the agent and assistant during the episode. [sent-58, score-1.231]

23 The objective of the assistant is to reason about the HGMDP and observed state-action history in order to select actions that maximize the expected (or worst-case) total reward of an episode. [sent-59, score-0.855]

24 The assistant can reduce the effort for the agent by opening the relevant doors for the agent. [sent-62, score-0.959]

25 The assistant can select actions that offer the agent a small number of shortcuts through the folder structure. [sent-64, score-1.091]

26 Given knowledge of the agent’s goal g in an HGMDP, the assistant’s problem reduces to solving an MDP over assistant actions. [sent-65, score-0.613]

27 The MDP transition function captures both the state change due to the assistant action and also the ensuing state change due to the agent action selected according to the policy π given g. [sent-66, score-1.412]

28 Likewise the reward function on a transition captures the reward due to the assistant action and the ensuing agent action conditioned on g. [sent-67, score-1.394]

29 The optimal policy for this MDP corresponds to an optimal assistant policy for g. [sent-68, score-0.931]

30 However, since the real assistant will often have uncertainty about the agent’s goal, it is unlikely that this optimal performance will be achieved. [sent-69, score-0.576]

31 We can view an HGMDP as a collection of |G| MDPs that share the same state space, where the assistant is placed in one of the MDPs at the beginning of each episode, but cannot observe which one. [sent-71, score-0.594]

32 Given an HGMDP M , a horizon m = O(|M |) where |M | is the size of the encoding of M , and a reward target r∗ , the short-term reward maximization problem asks whether there exists a historydependent assistant policy that achieves an expected ﬁnite horizon reward of at least r∗ . [sent-78, score-1.159]

33 This result shows that any POMDP can be encoded as an HGMDP with deterministic dynamics, where the stochastic dynamics of the POMDP are captured via the stochastic agent policy in the HGMDP. [sent-84, score-0.605]

34 HAMDPs will address the ﬁrst issue by assuming that the agent is competent at (approximately) maximizing reward without the assistant. [sent-89, score-0.486]

35 The last two issues will be addressed by assuming that the agent will always “detect and exploit” helpful actions and that the assistant actions do not hurt the agent. [sent-90, score-1.189]

36 Informally, the HAMDP provides the assistant with a helper action for each of the agent’s actions. [sent-91, score-0.849]

37 Whenever a helper action h is executed directly before the corresponding agent action a, the agent receives a bonus reward of 1. [sent-92, score-1.261]

38 However, the agent will only accept the helper action h (by taking a) and hence receive the bonus, if a is an action that the agent considers to be good for achieving the goal without the assistant. [sent-93, score-1.2]

39 Thus, the primary objective of the assistant in an HAMDP is to maximize the number of helper actions that get accepted by the agent. [sent-94, score-0.899]

40 While simple, this model captures much of the essence of assistance domains where assistant actions cause minimal harm and the agent is able to detect and accept good assistance when it arises. [sent-95, score-1.186]

41 An HAMDP is an HGMDP S, G, A, A , T, R, π, IS , IG with the following constraints: 3 • The agent and the assistant actions sets are A = {a1 , . [sent-96, score-1.057]

42 , hn }, so that for each ai there is a corresponding helper action hi . [sent-102, score-0.469]

43 States in W × A encode the current world state and the previous assistant action. [sent-104, score-0.594]

44 • The reward function R is 0 for all assistant actions. [sent-105, score-0.67]

45 For agent actions the reward is zero unless the agent selects the action ai in state (s, hi ) which gives a reward of 1. [sent-106, score-1.396]

46 That is, the agent receives a bonus of 1 whenever its action corresponds to the preceding helper action. [sent-107, score-0.69]

47 • The assistant always acts from states in W , and T is such that taking hi in s deterministically transitions to (s, hi ). [sent-108, score-0.776]

48 • The agent always acts from states in S ×A , resulting in states in S according to a transition function that does not depend on hi , i. [sent-109, score-0.513]

49 • Finally, for the agent policy, let Π(s, g) be a function that returns a set of actions and P (s, g) be a distribution over those actions. [sent-112, score-0.496]

50 We will view Π(s, g) as the set of actions that the agent considers acceptable (or equally good) in state s for goal g. [sent-113, score-0.598]

51 The agent policy π always selects ai after its helper action hi whenever ai is acceptable. [sent-114, score-1.075]

52 Otherwise the agent draws an action according to P (s, g). [sent-116, score-0.474]

53 In a HAMDP, the primary impact of an assistant action is to inﬂuence the reward of the following agent action. [sent-117, score-1.144]

54 The only rewards in HAMDPS are the bonuses received whenever the agent accepts a helper action. [sent-118, score-0.577]

55 Any additional environmental reward is assumed to be already captured by the agent policy via Π(s, g) that contains actions that approximately optimize this reward. [sent-119, score-0.775]

56 4 Regret Analysis for HAMDPs Given an assistant policy π , the regret of a particular episode is the extra reward that an omniscient assistant with knowledge of the goal would achieve over π . [sent-131, score-1.712]

57 For HAMDPs the omniscient assistant can always achieve a reward equal to the ﬁnite horizon m, because it can always select a helper action that will be accepted by the agent. [sent-132, score-1.067]

58 Thus, the regret of an execution of π in a HAMDP is equal to the number of helper actions that are not accepted by the agent, which we will call mispredictions. [sent-133, score-0.481]

59 We now show that a simple myopic policy is able to achieve regret bounds that are logarithmic in the number of goals. [sent-135, score-0.493]

60 Intuitively, our myopic assistant policy π will select an action that has the highest ˆ probability of being accepted with respect to a “coarsened” version of the posterior distribution over goals. [sent-137, score-1.029]

61 The myopic policy in state s given history H is based on the consistent goal set C(H), which is the set of goals that have non-zero probability with respect to history H. [sent-138, score-0.568]

62 The myopic policy is deﬁned as: π (s, H) = arg max IG (C(H) ∩ G(s, a)) ˆ a where G(s, a) = {g | a ∈ Π(s, g)} is the set of goals for which the agent considers a to be an acceptable action in state s. [sent-140, score-0.949]

63 For any HAMDP the expected regret of the myopic policy is bounded above by the entropy of the goal distribution H(IG ). [sent-143, score-0.582]

64 Consider a misprediction step where the myopic policy selects helper action hi in state s given history H, but the agent does not accept the action and instead selects a∗ = ai . [sent-148, score-1.402]

65 By the deﬁnition of the myopic policy we know that IG (C(H) ∩ G(s, ai )) ≥ IG (C(H) ∩ G(s, a∗ )), since otherwise the assistant would not have chosen hi . [sent-149, score-1.052]

66 Since H(IG ) ≤ log(|G|), this result implies that for HAMDPs the expected regret of the myopic policy is no more than logarithmic in the number of goals. [sent-161, score-0.517]

67 There exists a HAMDP such that for any assistant policy the expected regret is at least log(|G|)/2. [sent-165, score-0.898]

68 Since each episode is of length log(|G|), the expected regret of an episode for any policy is log(|G|)/2. [sent-171, score-0.467]

69 Resolving the gap between the myopic policy bound and this regret lower bound is an open problem. [sent-172, score-0.473]

70 Suppose that the assistant uses an approximate goal distribution IG instead of the true underlying goal distribution IG when computing the myopic policy. [sent-174, score-0.825]

71 That is, the assistant selects actions that maximize IG (C(H) ∩ G(s, a)), which we will refer to as the myopic policy relative to IG . [sent-175, score-1.037]

72 For any HAMDP with goal distribution IG , the expected regret of the myopic policy with respect to distribution IG is bounded above by H(IG ) + KL(IG IG ). [sent-178, score-0.582]

73 A consequence of Theorem 5 is that the myopic policy with respect to the uniform goal distribution has expected regret bounded by log(|G|) for any HAMDP, showing that logarithmic regret can be achieved without knowledge of IG . [sent-181, score-0.745]

74 For any HAMDP, the worst case and hence expected regret of the myopic policy with respect to the uniform goal distribution is bounded above by log(|G|). [sent-184, score-0.605]

75 In our case IG = 1/|G| which shows a worst case regret bound of log(|G|), which also bounds the expected regret of the uniform myopic policy. [sent-187, score-0.493]

76 The myopic policy achieves the optimal expected reward for HAMDPs with deterministic agent policies. [sent-193, score-0.895]

77 We now consider the case where both the agent policy and the environment are deterministic, and attempt to minimize the worst possible regret compared to an omniscient assistant who knows the agent’s goal. [sent-195, score-1.345]

78 Since the agent policy and the environment are both deterministic, there is at most one trajectory per goal in the tree. [sent-203, score-0.644]

79 The minimum worst-case regret of any policy for an HAMDP for deterministic environments and deterministic agent policies is equal to the tree rank of its optimal trajectory tree. [sent-207, score-0.94]

80 If the agent policy is deterministic, the problem of minimizing the maximum regret in HAMDPs in deterministic environments is in P. [sent-209, score-0.754]

81 The assumption of deterministic agent policy may be too restrictive in many domains. [sent-213, score-0.587]

82 We now consider HAMDPs in which the agent policies have a constant bound on the number of possible actions in Π(s, g) for each state-goal pair. [sent-214, score-0.514]

83 The branching factor of a HAMDP is the largest number of possible actions in Π(s, g) by the agent in any state for any goal and any assistant’s action. [sent-217, score-0.602]

84 Indeed, in many domains, it is reasonable to constrain the assistant so that the agent has the ﬁnal say on approving the actions proposed by the assistant. [sent-233, score-1.057]

85 These scenarios range from the ubiquitous auto-complete functions and Microsoft’s infamous Paperclip to more sophisticated adaptive programs such as SmartEdit [7] and TaskTracer [3] that learn assistant policies from users’ long-term behaviors. [sent-234, score-0.579]

86 Many open problems remain including generalization of these and other results to more general assistant frameworks, including partially observable and adversarial settings, learning assistants, and multi-agent assistance. [sent-236, score-0.59]

87 To show PSPACEhardness, we reduce the QSAT problem to the problem of the existence of a history-dependent assistant policy of expected reward ≥ r. [sent-239, score-0.864]

88 The agent chooses a goal uniformly randomly from the set of goals formed from φ and hides it from the assistant. [sent-260, score-0.511]

89 The actions of the assistant are to set the existentially quantiﬁed variables. [sent-262, score-0.693]

90 The agent simulates setting the universally quantiﬁed variables by choosing actions from the set {0, 1} with equal probability. [sent-263, score-0.496]

91 The episode terminates when all the variables are set, and the assistant gets a reward of 1 if the value of the clause is 1 at the end and a reward of 0 otherwise. [sent-264, score-0.904]

92 Note that the assistant does not get any useful feedback from the agent until it is too late and it either makes a mistake or solves the goal. [sent-265, score-0.939]

93 The best the assistant can do is to ﬁnd an optimal historydependent policy that maximizes the expected reward over the goals in Φ. [sent-266, score-0.991]

94 If Φ is satisﬁable, then there is an assistant policy that leads to a reward of 1 over all goals and all agent actions, and hence has an expected value of 1 over the goal distribution. [sent-267, score-1.375]

95 The agent accepts all actions until the last one and sets the variable as suggested by the assistant. [sent-272, score-0.519]

96 The last action is accepted by the agent if the goal clause evaluates to 1, otherwise not. [sent-274, score-0.614]

97 When the agent policy is deterministic, the initial goal distribution IG and the history of agent actions and states H fully capture the belief state of the agent. [sent-281, score-1.156]

98 Since the assistant does not get any useful information until it makes the clause true or fails to do so, its optimal policy is to choose the assignment that maximizes the number of satisﬁed clauses so that the mistakes are minimized. [sent-306, score-0.806]

99 The assistant makes a single prediction mistake on the last literal of each clause that is not satisﬁed by the assignment. [sent-307, score-0.621]

100 An intelligent personal assistant for task and time management. [sent-374, score-0.561]

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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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