nips nips2010 nips2010-192 knowledge-graph by maker-knowledge-mining

192 nips-2010-Online Classification with Specificity Constraints

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Author: Andrey Bernstein, Shie Mannor, Nahum Shimkin

Abstract: We consider the online binary classiﬁcation problem, where we are given m classiﬁers. At each stage, the classiﬁers map the input to the probability that the input belongs to the positive class. An online classiﬁcation meta-algorithm is an algorithm that combines the outputs of the classiﬁers in order to attain a certain goal, without having prior knowledge on the form and statistics of the input, and without prior knowledge on the performance of the given classiﬁers. In this paper, we use sensitivity and speciﬁcity as the performance metrics of the meta-algorithm. In particular, our goal is to design an algorithm that satisﬁes the following two properties (asymptotically): (i) its average false positive rate (fp-rate) is under some given threshold; and (ii) its average true positive rate (tp-rate) is not worse than the tp-rate of the best convex combination of the m given classiﬁers that satisﬁes fprate constraint, in hindsight. We show that this problem is in fact a special case of the regret minimization problem with constraints, and therefore the above goal is not attainable. Hence, we pose a relaxed goal and propose a corresponding practical online learning meta-algorithm that attains it. In the case of two classiﬁers, we show that this algorithm takes a very simple form. To our best knowledge, this is the ﬁrst algorithm that addresses the problem of the average tp-rate maximization under average fp-rate constraints in the online setting. 1

Reference: text

Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 il Abstract We consider the online binary classiﬁcation problem, where we are given m classiﬁers. [sent-10, score-0.197]

2 An online classiﬁcation meta-algorithm is an algorithm that combines the outputs of the classiﬁers in order to attain a certain goal, without having prior knowledge on the form and statistics of the input, and without prior knowledge on the performance of the given classiﬁers. [sent-12, score-0.352]

3 We show that this problem is in fact a special case of the regret minimization problem with constraints, and therefore the above goal is not attainable. [sent-15, score-0.419]

4 Hence, we pose a relaxed goal and propose a corresponding practical online learning meta-algorithm that attains it. [sent-16, score-0.378]

5 To our best knowledge, this is the ﬁrst algorithm that addresses the problem of the average tp-rate maximization under average fp-rate constraints in the online setting. [sent-18, score-0.416]

6 1 In this paper we focus on the online classiﬁcation problem, where no training set is given in advance. [sent-29, score-0.197]

7 Therefore, it is convenient to formulate the online classiﬁcation problem as a repeated game between an agent and some abstract opponent that stands for the collective behavior of the classiﬁers and the realized labels. [sent-36, score-0.909]

8 We note that, in this formulation, we can identify the agent with a corresponding online classiﬁcation meta-algorithm. [sent-37, score-0.454]

9 There is a rich literature that deals with the online classiﬁcation problem, in the competitive ratio framework, such as [5, 1]. [sent-38, score-0.197]

10 Our performance metrics will be the average tp-rate and fp-rate of the meta-algorithm, while the performance guarantees will be expressed in the regret minimization framework. [sent-41, score-0.487]

11 In a seminal paper, ∗ Hannan [8] introduced the optimal reward-in-hindsight rn with respect to the empirical distribu∗ tion of opponent’s actions, as a performance goal of an online algorithm. [sent-42, score-0.345]

12 In our case, rn is in fact the maximal tp-rate the agent could get at time n by knowing the classiﬁcation probabilities and actual labels beforehand, using the best convex combination of the classiﬁers. [sent-43, score-0.425]

13 The regret is then ∗ deﬁned as the difference between rn and the actual average tp-rate obtained by the agent. [sent-44, score-0.447]

14 Hannan showed in [8] that there exist online algorithms whose √ regret converges to zero (or below) as time progresses, regardless of the opponent’s actions, at 1/ n rate. [sent-45, score-0.489]

15 These algorithms can be directly applied to the problem of online classiﬁcation when the goal is only to obtain no-regret with respect to the optimal tp-rate in hindsight. [sent-48, score-0.235]

16 In particular, it is reasonable to require (following the Neyman-Pearson approach) that, in the long term, the average fp-rate of the agent will be below some given threshold 0 < γ < 1. [sent-50, score-0.302]

17 In this case the tp-rate can be considered as the average reward obtained by the agent, while fp-rate – as the average cost. [sent-51, score-0.266]

18 This is in fact a special case of the regret minimization problem with constraints whose study was initiated by Mannor et al. [sent-52, score-0.451]

19 They deﬁned the constrained reward-in-hindsight with respect to the empirical distribution of opponent’s actions, as a performance goal of an online algorithm. [sent-54, score-0.363]

20 This quantity is the maximal average reward the agent could get in hindsight, had he known the opponent’s actions beforehand, by using any ﬁxed (mixed) action, while satisfying the average cost constraints. [sent-55, score-0.694]

21 The desired online algorithm then has to satisfy two requirements: (i) it should have a vanishing regret (with respect to the constrained reward-in-hindsight); and (ii) it should asymptotically satisfy the average cost constraints. [sent-56, score-0.902]

22 The positive result is that a relaxed goal, which is deﬁned in terms of the convex hull of the constrained reward-in-hindsight over an appropriate space, is attainable. [sent-58, score-0.406]

23 The two no-regret algorithms proposed in [11] explicitly involve either the convex hull or a calibrated forecast of the opponent’s actions. [sent-59, score-0.185]

24 Both of these algorithms may not be computationally feasible, since there are no efﬁcient (polynomial time) procedures for the computation of both the convex hull and a calibrated forecast. [sent-60, score-0.185]

25 Instead of examining the constrained tp-rate in hindsight (or its convex hull), our starting point is the “standard” regret with respect to the optimal (unconstrained) tp-rate, and we consider a certain relaxation thereof. [sent-62, score-0.618]

26 We then ﬁnd the minimal constant needed in order to have a vanishing regret (with respect to this relaxed goal) while asymptotically satisfying the average fp-rate constraint. [sent-64, score-0.553]

27 We know that if the constraints are always satisﬁed, then the optimal tp-rate in-hindsight is attainable (using relatively simple no-regret algorithms). [sent-66, score-0.217]

28 One of the main contributions of this paper is a computationally 2 feasible online algorithm, the Constrained Regret Matching (CRM) algorithm, that attains the posed performance goal. [sent-69, score-0.255]

29 We note that although we focus in this paper on the online classiﬁcation problem, our algorithm can be easily extended to the general case of regret minimization under average cost constraints. [sent-70, score-0.661]

30 In Section 2 we formally deﬁne the online classiﬁcation problem and the goal of the meta-algorithm. [sent-72, score-0.235]

31 In Section 3 we present the general problem of constrained regret minimization, and show that the online classiﬁcation problem is its special case. [sent-73, score-0.652]

32 In Section 4 we deﬁne our relaxed goal in terms of the unconstrained optimal tp-rate in-hindsight, propose the CRM algorithm, and show that it can be implemented efﬁciently. [sent-74, score-0.172]

33 2 Online Classiﬁcation We consider the online binary classiﬁcation problem from an abstract space to {1, −1}. [sent-77, score-0.197]

34 An online classiﬁcation metaalgorithm is an algorithm that combines the outputs of the given classiﬁers in order to attain a certain goal, without prior knowledge on the form and statistics of the input, and without prior knowledge on the performance of the given classiﬁers. [sent-83, score-0.352]

35 In the online setting, no training set is given in advance, and therefore these rates have to be updated online, using the obtained data at each stage. [sent-93, score-0.197]

36 Observe that this data is expressed in terms of the vector zn {fn (a)}a∈A , bn ∈ [0, 1]m × {−1, 1}. [sent-94, score-0.29]

37 We let rn = r(an , zn ) fn (an ) I {bn = 1} and cn = c(an , zn ) fn (an ) I {bn = 0} denote the reward and the cost of the agent at time n. [sent-95, score-1.03]

38 Note that rn is the probability that the instance with positive label at time n will be classiﬁed correctly by the agent, while cn is the probability that the instance n n ¯ with negative label will be classiﬁed incorrectly. [sent-96, score-0.264]

39 Then, βtp (n) k=1 rk / k=1 I {bn = 1} and n n ¯f p (n) β k=1 ck / k=1 I {bn = −1} are the average tp-rate and fp-rate of the agent at time n, respectively. [sent-97, score-0.302]

40 In fact, this problem is a special case of the regret minimization problem with constraints. [sent-102, score-0.381]

41 In the next section we thus present the general constrained regret minimization framework, and discuss its applicability to the case of online classiﬁcation. [sent-103, score-0.671]

42 1 Constrained Regret Minimization Model Deﬁnition We consider the problem of an agent facing an arbitrary varying environment. [sent-105, score-0.257]

43 We identify the environment with some abstract opponent, and therefore obtain a repeated game formulation between the agent and the opponent. [sent-106, score-0.347]

44 At time step n, the following events occur: (i) The agent chooses an action an , and the opponent chooses an action zn , simultaneously; (ii) the agent observes zn ; and (iii) the agent receives a reward rn = r(an , zn ) ∈ R and a cost cn = c(an , zn ) ∈ n n 1 1 ¯ R . [sent-116, score-2.261]

45 We let rn ¯ k=1 rk and cn k=1 ck denote the average reward and cost of the agent n n at time n, respectively. [sent-117, score-0.693]

46 At time n, the agent chooses an action an according to the decision rule πn : Hn → ∆(A), ∞ where ∆(A) is the set of probability distributions over the set A. [sent-119, score-0.396]

47 That is, at each time step, a strategy prescribes some mixed action p ∈ ∆(A), based on the observed history. [sent-121, score-0.256]

48 We denote the mixed action of the opponent by q ∈ ∆(Z), which is the probability density over Z. [sent-123, score-0.557]

49 In what follows, we will use the shorthand notation r(p, q) a∈A p(a) z∈Z q(z)r(a, z) for the expected reward under mixed actions p ∈ ∆(A) and q ∈ ∆(Z). [sent-124, score-0.35]

50 We make the following assumption that the agent can satisfy the constraints in expectation against any mixed action of the opponent. [sent-126, score-0.584]

51 1 is essential, since otherwise the opponent can violate the average-cost constraints simply by playing the corresponding stationary strategy q. [sent-131, score-0.499]

52 n Let qn (z) ¯ k=1 δ {z − zk } /n denote the empirical density of the opponent’s actions at time n, so that qn ∈ ∆(Z). [sent-132, score-0.296]

53 , zn ) 1 max n a∈A n k=1 1 r(a, zk ) = max r(a, z) a∈A z∈Z n n k=1 δ {z − zk } = max r(a, qn ), ¯ a∈A ∗ implying that rn = r∗ (¯n ). [sent-136, score-0.422]

54 The simplest reward envelope is the (unconstrained) bestresponse envelope (BE) ρ = r∗ . [sent-138, score-0.428]

55 The n-stage regret of the algorithm (with respect to the BE) is then r∗ (¯n ) − rn . [sent-139, score-0.431]

56 The no-regret algorithm must ensure that the regret vanishes as n → ∞ regardless q ¯ of the opponent’s actions. [sent-140, score-0.321]

57 Obviously, the BE need not be attainable in the presence of constraints, and therefore other reward envelopes should be considered. [sent-142, score-0.369]

58 Hence, we use the following deﬁnition (introduced in [11]) in order to assess the online performance of the agent. [sent-143, score-0.197]

59 A reward envelope ρ : ∆(Z) → R is Γ-attainable if there exists a strategy π for the agent such that, almost surely, (i) lim supn→∞ (ρ(¯n ) − rn ) ≤ 0 , q ¯ and (ii) limn→∞ d(¯n , Γ) = 0, for every strategy of the opponent. [sent-146, score-0.797]

60 Such a strategy π is called constrained no-regret strategy with respect to ρ. [sent-148, score-0.256]

61 (1) p∈∆(A) ∗ We refer to rΓ as the constrained best-response envelope (CBE). [sent-150, score-0.254]

62 The ﬁrst positive result that appeared in the literature was that of Shimkin [12], which showed that ∗ the value vΓ minq∈∆(Z) rΓ (q) of the constrained game is attainable by the agent. [sent-151, score-0.402]

63 The algorithm which attains the value is based on Blackwell’s approachability theory [3], and is computationally ∗ efﬁcient provided that vΓ can be computed ofﬂine. [sent-152, score-0.178]

64 Unfortunately, it was shown in [11] that rΓ (q) ∗ ∗ itself is not attainable in general. [sent-153, score-0.147]

65 However, the (lower) convex hull of rΓ (q), conv (rΓ ), is attainable1 . [sent-154, score-0.193]

66 To our best knowledge, 1 The (lower) convex hull of a function f : X → R is the largest convex function which is nowhere larger than f . [sent-156, score-0.214]

67 It should be noted that the problem that is considered here can not be formulated as an instance of online convex optimization [13, 9] – see [11] for a discussion on this issue. [sent-160, score-0.283]

68 The reward at time n is rn = r(an , zn ) = fn (an ) I {bn = 1} and the cost is cn = c(an , zn ) = fn (an ) I {bn = −1}. [sent-166, score-0.773]

69 Note that in this case, the mixed action of the opponent q ∈ ∆(Z) is q(f, b) = q(f |b)q(b), where q(f |b) is the conditional density of the predictions of the classiﬁers and q(b) is the probability of the label b. [sent-167, score-0.557]

70 We where βf p (q; a) f ¯ note that keeping the average fp-rate of the agent βf p (n) ≤ γ is equivalent to keeping n (1/n) k=1 cγ (ak , zk ) ≤ 0. [sent-171, score-0.365]

71 Now, using (1), ∗ ∗ (2), and (3) we have that rγ (q) = q(1)βγ (q), where ∗ βγ (q) max p∈∆(A) p(a)βtp (q; a) : so that a∈A a∈A p(a)βf p (q; a) ≤ γ , (5) is the optimal constrained tp-rate in hindsight under distribution q. [sent-186, score-0.214]

72 Finally, note that the value of the ∗ constrained game vγ minq∈∆(Z) rγ (q) = 0 in this case. [sent-187, score-0.218]

73 As a consequence of this formulation, the algorithms proposed in [11] can be in principle used in ∗ order to attain the convex hull of rγ . [sent-188, score-0.231]

74 However, given the implementation difﬁculties associated with these algorithms, we are motivated to examine more carefully the problem of regret minimization with constraints and provide more practical no-regret algorithms with formal guarantees. [sent-189, score-0.416]

75 5 4 Constrained Regret Matching We next deﬁne a relaxed reward envelope for the online classiﬁcation problem. [sent-192, score-0.584]

76 The proposed is in fact applicable to the problem of constrained regret minimization in general. [sent-193, score-0.474]

77 Our starting point here in deﬁning an attainable reward envelope will be the BE r∗ (q) = q(1)β ∗ (q). [sent-195, score-0.449]

78 Clearly, r∗ is in general not attainable in the presence of fp-constraints, and we thus consider a ∗ ∗ q(1)(β ∗ (q) − α). [sent-196, score-0.147]

79 Furthermore, recall ∗ that the value vγ of the constrained game is attainable by the agent. [sent-199, score-0.365]

80 (6) (7) SR We note that rα∗ (q) is strictly above 0 at some point, unless the game is in some sense trivial (see the supplementary material for a proof). [sent-204, score-0.165]

81 1, we are seeking for a strategy π that is: (i) an α-relaxed no-regret strategy for the average reward, and (ii) ensures that the cost constraints are asymptotically satisﬁed. [sent-206, score-0.331]

82 Let α Rk (a) [fk (a) − fk (ak ) − α] I {bk = 1} , a ∈ A, Lk cγ (ak , zk ), (8) denote the instantaneous α-regret and the instantaneous constraint violation (respectively) at time k. [sent-210, score-0.162]

83 We have that the average α-regret and constraints violation at time n are α ¯ ¯ R (a) = qn (1) βtp (¯n ; a) − βtp (n) − α , a ∈ A; Ln = qn (0)[βf p (n) − γ]. [sent-211, score-0.304]

84 We note that the mixed action required by the CRM algorithm always exists provided that α ≥ α∗ . [sent-222, score-0.221]

85 Note also that when the average constraints violation Ln−1 is non-positive, the minimum in (10) is obtained by α p = pα . [sent-224, score-0.16]

86 ﬁnd a mixed action p ∈ ∆(A) such that  α  f (a) − a ∈A p(a )f (a ) − α ≤ 0, a∈A Rn−1 (a) +  Ln−1 p(a )f (a ) − γ ≤ 0, + a ∈A ∀z = (f, 1) ∈ Z, ∀z = (f, −1) ∈ Z, (11) α where Rn (a) and Ln,i are given in (9). [sent-238, score-0.192]

87 In practice, it may be possible to attain rα with α < α∗ if the opponent is not entirely adversarial. [sent-241, score-0.44]

88 5 The Special Case of Two Classiﬁers If m = 2, we can obtain explicit expressions for the reward envelopes and for the algorithm. [sent-251, score-0.222]

89 We compared the performance of the CRM algorithm to a simple unconstrained no-regret algorithm that treats both the true-positive and false positive probabilities similarly, but with different weight. [sent-268, score-0.179]

90 In particular, the reward at stage n of this algorithm is gn (w) = fn (an ) I {bn = 1} − wfn (an ) I {bn = −1} for some weight parameter 7 tp-rate fp-rate w = 1. [sent-269, score-0.34]

91 6 Conclusion We studied regret minimization with average-cost constraints, with the focus on computationally feasible algorithm for the special case of online classiﬁcation problem with speciﬁcity constraints. [sent-290, score-0.607]

92 We deﬁned a relaxed version of the best-response reward envelope and showed that it can be attained by the agent while satisfying the constraints, provided that the relaxation parameter is above a certain threshold. [sent-291, score-0.736]

93 This algorithm generally solves a linear program at each time step, while in some special case the algorithm’s mixed action reduces to the simple α-regret matching strategy. [sent-293, score-0.29]

94 To the best of our knowledge, this is the ﬁrst algorithm that addresses the problem of the average tp-rate maximization under average fp-rate constraints in the online setting. [sent-294, score-0.416]

95 In addition, an adaptive scheme that adapts the relaxation parameter online was brieﬂy discussed. [sent-295, score-0.285]

96 Finally, the special case of two classiﬁers was discussed, and the experimental results for this case show that our algorithm outperforms a simple no-regret algorithm which takes as the reward function a weighted sum of the tp-rate and fp-rate. [sent-296, score-0.269]

97 First, the guaranteed convergence rate of the algorithm is of O(1/ n) since it is based on Blackwell’s approachability theorem4 . [sent-298, score-0.151]

98 Second, additional constraints can be easily incorporated in the presented framework, since the general regret minimization framework assumes a vector of constraints. [sent-299, score-0.416]

99 In particular, in case of a single constraint, the maximal attainable relaxed goal is the convex hull of the CBE (see [11]) but no polynomial algorithms are known that attain this goal. [sent-301, score-0.534]

100 We note that it is possible to improve the dependence to log(m) by using a potential based Blackwell’s approachability strategy (see for example [4], Chapter 7. [sent-306, score-0.155]

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PGRD has few parameters, improves the reward 1 function online during an agent’s lifetime, takes advantage of knowledge about the agent’s structure (through the gradient computation), and is linear in the number of reward function parameters. Notation. Formally, we consider discrete-time partially-observable environments with a ﬁnite number of hidden states s ∈ S, actions a ∈ A, and observations o ∈ O; these ﬁnite set assumptions are useful for our theorems, but our algorithm can handle inﬁnite sets in practice. Its dynamics are governed by a state-transition function P (s |s, a) that deﬁnes a distribution over next-states s conditioned on current state s and action a, and an observation function Ω(o|s) that deﬁnes a distribution over observations o conditioned on current state s. The agent designer’s goals are speciﬁed via the objective reward function RO . 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. Stochastic optimization of controlled partially observable Markov decision processes. In Proceedings of the 39th IEEE Conference on Decision and Control, 2000. [3] Jonathan Baxter, Peter L. Bartlett, and Lex Weaver. Experiments with Inﬁnite-Horizon, Policy-Gradient Estimation, 2001. [4] Shalabh Bhatnagar, Richard S. Sutton, M Ghavamzadeh, and Mark Lee. Natural actor-critic algorithms. Automatica, 2009. [5] Ronen I. Brafman and Moshe Tennenholtz. R-MAX - A General Polynomial Time Algorithm for NearOptimal Reinforcement Learning. Journal of Machine Learning Research, 3:213–231, 2001. [6] S. Elfwing, Eiji Uchibe, K. Doya, and H. I. Christensen. Co-evolution of Shaping Rewards and MetaParameters in Reinforcement Learning. Adaptive Behavior, 16(6):400–412, 2008. [7] J. Zico Kolter and Andrew Y. Ng. Near-Bayesian exploration in polynomial time. In Proceedings of the 26th International Conference on Machine Learning, pages 513–520, 2009. [8] Harold J. Kushner and G. George Yin. 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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