nips nips2009 nips2009-113 knowledge-graph by maker-knowledge-mining

113 nips-2009-Improving Existing Fault Recovery Policies

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Author: Guy Shani, Christopher Meek

Abstract: An automated recovery system is a key component in a large data center. Such a system typically employs a hand-made controller created by an expert. While such controllers capture many important aspects of the recovery process, they are often not systematically optimized to reduce costs such as server downtime. In this paper we describe a passive policy learning approach for improving existing recovery policies without exploration. We explain how to use data gathered from the interactions of the hand-made controller with the system, to create an improved controller. We suggest learning an indeﬁnite horizon Partially Observable Markov Decision Process, a model for decision making under uncertainty, and solve it using a point-based algorithm. We describe the complete process, starting with data gathering, model learning, model checking procedures, and computing a policy. 1

Reference: text

Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 While such controllers capture many important aspects of the recovery process, they are often not systematically optimized to reduce costs such as server downtime. [sent-6, score-0.293]

2 In this paper we describe a passive policy learning approach for improving existing recovery policies without exploration. [sent-7, score-0.854]

3 In such large systems it is prohibitively costly to have a technician decide on a repair action for each observed problem. [sent-14, score-0.677]

4 Therefore, these systems often use some automatic repair policy or controller to choose appropriate repair actions. [sent-15, score-1.53]

5 These repair policies typically receive failure messages from the system. [sent-16, score-0.75]

6 The repair policy receives notiﬁcations and decides which actions can ﬁx the observed problems. [sent-19, score-1.08]

7 Given the POMDP parameters, we can compute a policy that optimizes repair costs, but learning the POMDP parameters may be difﬁcult. [sent-24, score-0.907]

8 Typically, the controller will not be designed to perform exploration, and we cannot expect such logs to contain sufﬁcient data to train standard RL techniques. [sent-31, score-0.288]

9 In this paper we introduce a passive policy learning approach, that uses only available information without exploration, to improve an existing repair policy. [sent-32, score-1.011]

10 We suggest a model-checking phase, providing supporting evidence for the quality of the learned model, which may be crucial to help the system administrators decide whether the learned model is appropriate. [sent-34, score-0.323]

11 We proceed to compute a policy for our learned model, that can then be used in the data center instead of the original hand-made controller. [sent-35, score-0.562]

12 We experiment with a synthetic, yet realistic, simulation of machine failures, showing how the policy of the learned POMDP performs close to optimal, and outperforms a set of simpler techniques that learn a policy directly in history space. [sent-36, score-1.104]

13 Many other real world applications, such as assembly lines, medical diagnosis systems, and failure detection and recovery systems, are also controlled by hand-made controllers. [sent-38, score-0.304]

14 Key aspects of the problem include the nature of repair actions and costs, machine failure, failure detection, and control policies. [sent-41, score-0.681]

15 Key aspects of repair actions include: (1) actions may succeed or fail stochastically. [sent-42, score-0.766]

16 We label actions using increasing levels, where problems ﬁxed by an action at level i, are also ﬁxed by any action of level j > i. [sent-44, score-0.458]

17 (3) Action costs are typically escalating, where lower level actions that ﬁx minor problems are relatively cheap, while higher level actions are more expensive. [sent-46, score-0.397]

18 In view of the escalating nature of actions and costs, a natural choice for a policy is an escalation policy. [sent-54, score-0.875]

19 Such policies choose a starting level based on the ﬁrst observation, and execute an action at that level. [sent-55, score-0.415]

20 In many cases, due to the non-deterministic success of repair actions, each action is tried several times. [sent-56, score-0.615]

21 After the controller decides that the action at the current level cannot ﬁx the problem, the controller escalates to the next action level. [sent-57, score-0.68]

22 For example, the number of retries of an action before an escalation occurs, and the entry level given an observation. [sent-59, score-0.4]

23 System administrators typically collect logs of the hand-made controller execution, for maintenance purposes. [sent-61, score-0.328]

24 We would like to use this knowledge to construct an improved policy that will perform better than the original policy. [sent-63, score-0.467]

25 Formally, we assume that we receive as input a log L of repair sessions. [sent-64, score-0.462]

26 Each repair session is a sequence l = o0 , a1 , o1 , . [sent-65, score-0.507]

27 , onl , starting with an error notiﬁcation, followed by a set of repair actions and observations until the problem is ﬁxed. [sent-68, score-0.673]

28 In a POMDP the true state is not directly observable and we thus maintain a belief state b ∈ B — a probability distribution over states, where b(s) is the probability that the system is at state s. [sent-93, score-0.315]

29 We assume that every repair session starts with an error observation, typically provided by one of the watchdogs. [sent-94, score-0.545]

30 It is convenient to deﬁne a policy for a POMDP as a mapping from belief states to actions π : B → A. [sent-98, score-0.682]

31 Our goal is to ﬁnd an optimal policy that brings the machine to the healthy state with the minimal cost. [sent-99, score-0.572]

32 One method for computing a policy is through a value function, V , assigning a value to each belief state b. [sent-100, score-0.541]

33 By associating an action a(α) which each vector, a policy π : B → A can be deﬁned through π(b) = a(αb ). [sent-108, score-0.598]

34 In this framework the POMDP has a single special action aT , available in any state, that terminates the repair session. [sent-114, score-0.615]

35 3 3 Learning Policies from System Logs In this section we propose two alternatives for computing a recovery policy given the logs. [sent-119, score-0.553]

36 We begin with a simple, model-free, history-based policy computation. [sent-120, score-0.445]

37 1 Model-Free Learning of Q-values The optimal policy for a POMDP can be expressed as a mapping from action-observation histories to actions. [sent-123, score-0.518]

38 Histories are directly observable, allowing us to use the standard Q function terminology, where Q(h, a) is the expected cost of executing action a with history h and continuing the session until it terminates. [sent-124, score-0.529]

39 The Q function is hence the average cost until repair of executing the action a in history h, under the policy that generated L. [sent-129, score-1.344]

40 An approach that generalizes better is to use a ﬁnite history window of size k, discarding all the observations and action occurring more than k steps ago. [sent-132, score-0.322]

41 In practice, the only parameters that are known are the set of possible repair actions and the set of possible observations, but even the number of possible errors is not initially known, let alone the probability of repair or observation. [sent-136, score-1.113]

42 Given the log of repair sessions, we can use a learning algorithm to learn the parameters of the POMDP. [sent-137, score-0.462]

43 Such evidence can be helpful for the system administrators in order to make a decision whether to replace the existing policy with a new policy. [sent-148, score-0.611]

44 Using an imperfect learner such as Baum-Welch does not guarantee that the resulting model indeed maximizes the likelihood of the observations given the policy, even for the same policy that was used to generate the training data. [sent-149, score-0.549]

45 We say that two POMDPs M1 =< S1 , A, tr1 , C1 , Ω, O1 > and M2 =< S2 , A, tr2 , C2 , Ω, O2 > are indistinguishable if for each policy π : H → A, E[ t Ct |M1 , π] = E[ t Ct |M2 , π]. [sent-158, score-0.47]

46 That is, the models are indistinguishable if any policy has the same expected accumulated cost when executed in both models. [sent-159, score-0.595]

47 Many policies cannot be evaluated on the real system because we cannot tolerate damaging policies. [sent-160, score-0.312]

48 We hence focus the model checking phase on comparing the expected cost of the hand-made policy predicted be the learned model to the true expected cost on the real system. [sent-162, score-0.989]

49 To estimate the expected cost in the real system, we use the average cost of the sessions in the logs. [sent-163, score-0.29]

50 To estimate the expected cost of the policy on the learned POMDP we execute a set of trials, each simulating a repair session, using the learned parameters of the POMDP to govern the trial advancement (observation emissions given the history and action). [sent-164, score-1.434]

51 For example, if the predicted cost of the policy over the learned POMDP is more than 20% away from the true expected cost, we may deduce that the learned mode does not properly capture the system dynamics. [sent-166, score-0.828]

52 While checking the models under a single policy cannot ensure that the models are identical, it can detect whether the model is defective. [sent-167, score-0.549]

53 If the learned model produces a substantially different expectation over the cost of a policy than the real system, we know that the model is corrupted prior to executing its optimal policy on the real system. [sent-168, score-1.294]

54 After ensuring that the original policy performs similarity on the real system and on the learned model, we can also evaluate the performance of the computed policy on the learned model. [sent-169, score-1.217]

55 Thus, we can compare the quality of the new policy to the existing one, helping us to understand the potential cost reduction of the new policy. [sent-170, score-0.55]

56 In the simulator we assume that a machine can be in one of n error states, or in healthy state, we also assume n possible repair actions, and m possible observations. [sent-173, score-0.696]

57 We assume that each action was designed to ﬁx a single error state, and set the number of errors to be the number of repair actions. [sent-174, score-0.69]

58 3 · j−i if j ≥ i and 0 otherwise, simulating the escalation power j of repair actions. [sent-177, score-0.706]

59 We set C(s, ai ) = 4i and C(sH , aT ) = 0, simulating the exponential growth of costs in the real AutoPilot system [6], and the zero downtime caused by terminating the session in the healthy state. [sent-178, score-0.409]

60 We execute a hand-made escalation policy with 3 retries (see Section 2) over the simulator and gather a log of repair sequences. [sent-180, score-1.327]

61 Each repair sequence begins with selecting an error uniformly, and executing the policy until the error is ﬁxed. [sent-181, score-1.07]

62 Then, we use the logs in order to learn a Q function 5 Table 1: Average cost of recovery policies in simulation, with increasing model size. [sent-182, score-0.535]

63 Results are averaged over 10 executions, and the worst standard error across all recovery policies is reported in the last column. [sent-183, score-0.343]

64 For the POMDP model, we initialize the number of states to the number of repair actions, initialize transition uniformly, and observation randomly, and execute the Baum-Welch algorithm. [sent-208, score-0.587]

65 For policies, we use πE for the escalation policy, πM for the policy computed by Perseus on the learned model, and πM ∗ for the Perseus policy over the ‘perfect learner’ model. [sent-212, score-1.201]

66 For the history-based Q functions, we use Q to denote the function computed for the complete history, and Qi denotes a policy over history sufﬁxes of length i. [sent-213, score-0.564]

67 A column header π, S denotes the estimated cost of executing π on the simulator, and π, M denotes the estimated cost of executing π on the model M . [sent-214, score-0.355]

68 1 Results We begin with showing the improvement of the policy of the learned models over the original escalation policies. [sent-217, score-0.778]

69 As Table 1 demonstrates, learning a POMDP model and computing its policy always result in a substantial reduction in costs. [sent-218, score-0.47]

70 In the smaller domains Q learning, especially with a history window of 3 (Q3 ) does fairly well, but in the larger domains all history-based policies do not perform well. [sent-223, score-0.345]

71 As we explained above, a model checking phase, comparing the expected cost of a policy on both the real system and the learned model, can provide evidence as to the validity of the learned model. [sent-225, score-0.979]

72 Indeed, as we see in Table 2, the learned models predict an expected cost that is within 3% of the real expected cost. [sent-226, score-0.314]

73 To further validate our learned models, we also compare the expected cost of the policies computed from the models (M and M ∗ ) over the model and the simulator. [sent-227, score-0.442]

74 As expected, the M ∗ predicted costs are within measurement error of the true costs of the policy on the real system. [sent-229, score-0.716]

75 When we use fewer observations, the quality of the learned model is reduced (further from M ∗ ), but the policy that is learned still signiﬁcantly outperforms the original escalation policy. [sent-233, score-0.898]

76 It is important to note that the hand-made escalation policy that we use is very natural for domains with actions that have escalating costs and effects. [sent-234, score-0.968]

77 Also, the number of actions and errors that we use is similar to these used by current controllers of repair services [6]. [sent-235, score-0.743]

78 As such, the improvements 6 Table 2: Comparing expected cost of policies on the learned model and the simulator for model checking. [sent-236, score-0.575]

79 Results are averaged over 10 executions, and the worst standard error across all recovery policies is reported in the last column. [sent-237, score-0.343]

80 Problem parameters |E| |O| |L| 2 2 10000 4 2 10000 4 4 10000 8 4 50000 8 8 50000 Escalation policy π E , M ∗ πE , M πE , S 21. [sent-238, score-0.445]

81 5 < 250 < 275 that we achieve over this hand-made policy hint that similar gains can be made over the real system. [sent-261, score-0.492]

82 As the input policy does not explore, the set of resulting histories does not provide enough coverage of the history space. [sent-266, score-0.658]

83 For example, if the current repair policy only escalates, the history-based approach will never observe a higher level action followed by a low level action, and cannot evaluate its expected cost. [sent-267, score-1.107]

84 Indeed, the ﬁnite history window methods (except for the reactive policy) improve upon the original escalation policy in some cases. [sent-270, score-0.877]

85 It is therefore safe to assume that action a will not ﬁx error e if we never observed a to ﬁx e in the logs, except for the terminating action aT . [sent-282, score-0.384]

86 [8] suggests the CSFR model which is similar to our POMDP formalization, except for a deterministic observation function, the escalation of actions, and the terminating action. [sent-290, score-0.313]

87 Also, as we argue above, in our case a Q function produces substantially inferior policies because of its lack of generalization power in partially observable domains. [sent-296, score-0.299]

88 They then suggest a one step lookahead repair strategy, and a multi-step lookahead, that uses a value function over a belief space similar to the Littman et al. [sent-304, score-0.546]

89 Bayer-Zubek and Dietterich [16] use a set of examples, similar to our logs, to learn a policy for disease diagnosis. [sent-306, score-0.445]

90 6 Conclusion We have presented an approach to improving imperfect repair policies through learning a POMDP model of the problem. [sent-310, score-0.72]

91 Our method takes as input a log of interaction of the existing controller with the system, learns a POMDP model, and computes a policy for the POMDP that can be used in the the real system. [sent-311, score-0.68]

92 The advantage of our method is that it does not require the existing controller to actively explore the effects of actions in all conditions, which may result in unacceptable costs in the real system. [sent-312, score-0.48]

93 We experiment with a synthetic, yet realistic, example of a hand-made escalation policy, where actions are ordered by increasing cost, and any action is repeated a number of times. [sent-314, score-0.521]

94 We show how the policy of the learned model signiﬁcantly improves the original escalation policy. [sent-315, score-0.803]

95 In the future we intend to use the improved policies to manage repairs in a real data center within the AutoPilot system [6]. [sent-316, score-0.312]

96 There are a number of interesting theoretical questions about our passive policy learning method and about passive policy learning in general. [sent-320, score-1.044]

97 First, for what families of initial policies and system dynamics would a passive policy learning method be expected to yields an improvement in expected costs. [sent-321, score-0.881]

98 Second, what families of initial policies and systems dynamics would a passive policy learning method be expected to yield the optimal policy. [sent-322, score-0.766]

99 Third, how would one characterize when iteratively applying a passive policy learning method would yield expected improvements in expected costs. [sent-323, score-0.616]

100 Finally, while this paper focuses on the important failure recovery problem, our methods may be applicable to a wide range of similar systems, such as assembly line management, and medical diagnosis systems, that currently employ hand-made imperfect controllers. [sent-324, score-0.293]

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P : S × S × A → [0, 1] is the transition function, where P (s , s, a) represents the probability of transiting to state s from state s, given action a. The function r : S × A → R is the reward function, and α : S → [0, 1] is the initial state distribution. The objective is to maximize the inﬁnite-horizon discounted cumulative reward. To shorten the notation, we assume an arbitrary ordering of the states: s1 , s2 , . . . , sn . Then, Pa and ra are used to denote the probabilistic transition matrix and reward for action a. The solution of an MDP is a policy π : S × A → [0, 1] from a set of possible policies Π, such that for all s ∈ S, a∈A π(s, a) = 1. We assume that the policies may be stochastic, but stationary [21]. A policy is deterministic when π(s, a) ∈ {0, 1} for all s ∈ S and a ∈ A. The transition and reward functions for a given policy are denoted by Pπ and rπ . The value function update for a policy π is denoted by Lπ , and the Bellman operator is denoted by L. That is: Lπ v = Pπ v + rπ Lv = max Lπ v. π∈Π The optimal value function, denoted v ∗ , satisﬁes v ∗ = Lv ∗ . We focus on linear value function approximation for discounted inﬁnite-horizon problems. In linear value function approximation, the value function is represented as a linear combination of nonlinear basis functions (vectors). For each state s, we deﬁne a row-vector φ(s) of features. The rows of the basis matrix M correspond to φ(s), and the approximation space is generated by the columns of the matrix. That is, the basis matrix M , and the value function v are represented as:   − φ(s1 ) −   M = − φ(s2 ) − v = M x. . . . Deﬁnition 1. A value function, v, is representable if v ∈ M ⊆ R|S| , where M = colspan (M ), and is transitive-feasible when v ≥ Lv. We denote the set of transitive-feasible value functions as: K = {v ∈ R|S| v ≥ Lv}. 2 Notice that the optimal value function v ∗ is transitive-feasible, and M is a linear space. Also, all the inequalities are element-wise. Because the new formulation is related to ALP, we introduce it ﬁrst. It is well known that an inﬁnite horizon discounted MDP problem may be formulated in terms of solving the following linear program: minimize v c(s)v(s) s∈S v(s) − γ s.t. P (s , s, a)v(s ) ≥ r(s, a) ∀(s, a) ∈ (S, A) (1) s ∈S We use A as a shorthand notation for the constraint matrix and b for the right-hand side. The value c represents a distribution over the states, usually a uniform one. That is, s∈S c(s) = 1. The linear program in Eq. (1) is often too large to be solved precisely, so it is approximated to get an approximate linear program by assuming that v ∈ M [8], as follows: minimize cT v x Av ≥ b s.t. (2) v∈M The constraint v ∈ M denotes the approximation. To actually solve this linear program, the value function is represented as v = M x. In the remainder of the paper, we assume that 1 ∈ M to guarantee the feasibility of the ALP, where 1 is a vector of all ones. The optimal solution of the ALP, v , satisﬁes that v ≥ v ∗ . Then, the objective of Eq. (2) represents the minimization of v − v ∗ 1,c , ˜ ˜ ˜ where · 1,c is a c-weighted L1 norm [7]. The ultimate goal of the optimization is not to obtain a good value function v , but a good policy. ˜ The quality of the policy, typically chosen to be greedy with respect to v , depends non-trivially on ˜ the approximate value function. The ABP formulation will minimize policy loss by minimizing L˜ − v ∞ , which bounds the policy loss as follows. v ˜ Theorem 2 (e.g. [25]). Let v be an arbitrary value function, and let v be the value of the greedy ˜ ˆ policy with respect to v . Then: ˜ 2 v∗ − v ∞ ≤ ˆ L˜ − v ∞ , v ˜ 1−γ In addition, if v ≥ L˜, the policy loss is smallest for the greedy policy. ˜ v Policies, like value functions, can be represented as vectors. Assume an arbitrary ordering of the state-action pairs, such that o(s, a) → N maps a state and an action to its position. The policies are represented as θ ∈ R|S|×|A| , and we use the shorthand notation θ(s, a) = θ(o(s, a)). Remark 3. The corresponding π and θ are denoted as π θ and θπ and satisfy: π θ (s, a) = θπ (s, a). We will also consider approximations of the policies in the policy-space, generated by columns of a matrix N . A policy is representable when π ∈ N , where N = colspan (N ). 3 Approximate Bilinear Programs This section shows how to formulate minv∈M Lv − v ∞ as a separable bilinear program. Bilinear programs are a generalization of linear programs with an additional bilinear term in the objective function. A separable bilinear program consists of two linear programs with independent constraints and are fairly easy to solve and analyze. Deﬁnition 4 (Separable Bilinear Program). A separable bilinear program in the normal form is deﬁned as follows: T T minimize f (w, x, y, z) = sT w + r1 x + xT Cy + r2 y + sT z 1 2 w,x y,z s.t. A1 x + B1 w = b1 A2 y + B2 z = b2 w, x ≥ 0 y, z ≥ 0 3 (3) We separate the variables using a vertical line and the constraints using different columns to emphasize the separable nature of the bilinear program. In this paper, we only use separable bilinear programs and refer to them simply as bilinear programs. An approximate bilinear program can now be formulated as follows. minimize θT λ + λ θ λ,λ ,v Bθ = 1 z = Av − b s.t. θ≥0 z≥0 (4) λ+λ1≥z λ≥0 θ∈N v∈M All variables are vectors except λ , which is a scalar. The symbol z is only used to simplify the notation and does not need to represent an optimization variable. The variable v is deﬁned for each state and represents the value function. Matrix A represents constraints that are identical to the constraints in Eq. (2). The variables λ correspond to all state-action pairs. These variables represent the Bellman residuals that are being minimized. The variables θ are deﬁned for all state-action pairs and represent policies in Remark 3. The matrix B represents the following constraints: θ(s, a) = 1 ∀s ∈ S. a∈A As with approximate linear programs, we initially assume that all the constraints on z are used. In realistic settings, however, the constraints would be sampled or somehow reduced. We defer the discussion of this issue until Section 6. Note that the constraints in our formulation correspond to elements of z and θ. Thus when constraints are omitted, also the corresponding elements of z and θ are omitted. To simplify the notation, the value function approximation in this problem is denoted only implicitly by v ∈ M, and the policy approximation is denoted by θ ∈ N . In an actual implementation, the optimization variables would be x, y using the relationships v = M x and θ = N y. We do not assume any approximation of the policy space, unless mentioned otherwise. We also use v or θ to refer to partial solutions of Eq. (4) with the other variables chosen appropriately to achieve feasibility. The ABP formulation is closely related to approximate linear programs, and we discuss the connection in Section 5. We ﬁrst analyze the properties of the optimal solutions of the bilinear program and then show and discuss the solution methods in Section 4. The following theorem states the main property of the bilinear formulation. ˜˜ ˜ ˜ Theorem 5. b Let (θ, v , λ, λ ) be an optimal solution of Eq. (4) and assume that 1 ∈ M. Then: ˜ ˜ ˜ θT λ + λ = L˜ − v v ˜ ∞ ≤ min v∈K∩M Lv − v ∞ ≤ 2 min Lv − v v∈M ∞ ≤ 2(1 + γ) min v − v ∗ v∈M ∞. ˜ In addition, π θ minimizes the Bellman residual with regard to v , and its value function v satisﬁes: ˜ ˆ 2 min Lv − v ∞ . v − v∗ ∞ ≤ ˆ 1 − γ v∈M The proof of the theorem can be found in [19]. It is important to note that, as Theorem 5 states, the ABP approach is equivalent to a minimization over all representable value functions, not only the transitive-feasible ones. Notice also the missing coefﬁcient 2 (2 instead of 4) in the last equation of Theorem 5. This follows by subtracting a constant vector 1 from v to balance the lower bounds ˜ on the Bellman residual error with the upper ones. This modiﬁed approximate value function will have 1/2 of the original Bellman residual but an identical greedy policy. Finally, note that whenever v ∗ ∈ M, both ABP and ALP will return the optimal value function. The ABP solution minimizes the L∞ norm of the Bellman residual due to: 1) the correspondence between θ and the policies, and 2) the dual representation with respect to variables λ and λ . The theorem then follows using techniques similar to those used for approximate linear programs [7]. 4 Algorithm 1: Iterative algorithm for solving Eq. (3) (x0 , w0 ) ← random ; (y0 , z0 ) ← arg miny,z f (w0 , x0 , y, z) ; i←1; while yi−1 = yi or xi−1 = xi do (yi , zi ) ← arg min{y,z A2 y+B2 z=b2 y,z≥0} f (wi−1 , xi−1 , y, z) ; (xi , wi ) ← arg min{x,w A1 x+B1 w=b1 x,w≥0} f (w, x, yi , zi ) ; i←i+1 return f (wi , xi , yi , zi ) 4 Solving Bilinear Programs In this section we describe simple methods for solving ABPs. We ﬁrst describe optimal methods, which have exponential complexity, and then discuss some approximation strategies. Solving a bilinear program is an NP-complete problem [3]. The membership in NP follows from the ﬁnite number of basic feasible solutions of the individual linear programs, each of which can be checked in polynomial time. The NP-hardness is shown by a reduction from the SAT problem [3]. The NP-completeness of ABP compares unfavorably with the polynomial complexity of ALP. However, most other ADP algorithms are not guaranteed to converge to a solution in ﬁnite time. The following theorem shows that the computational complexity of the ABP formulation is asymptotically the same as the complexity of the problem it solves. Theorem 6. b Determining minv∈K∩M Lv − v ∞ < is NP-complete for the full constraint representation, 0 < γ < 1, and a given > 0. In addition, the problem remains NP-complete when 1 ∈ M, and therefore minv∈M Lv − v ∞ < is also NP-complete. As the theorem states, the value function approximation does not become computationally simpler even when 1 ∈ M – a universal assumption in the paper. Notice that ALP can determine whether minv∈K∩M Lv − v ∞ = 0 in polynomial time. The proof of Theorem 6 is based on a reduction from SAT and can be found in [19]. The policy in the reduction determines the true literal in each clause, and the approximate value function corresponds to the truth value of the literals. The approximation basis forces literals that share the same variable to have consistent values. Bilinear programs are non-convex and are typically solved using global optimization techniques. The common solution methods are based on concave cuts [11] or branch-and-bound [6]. In ABP settings with a small number of features, the successive approximation algorithm [17] may be applied efﬁciently. We are, however, not aware of commercial solvers available for solving bilinear programs. Bilinear programs can be formulated as concave quadratic minimization problems [11], or mixed integer linear programs [11, 16], for which there are numerous commercial solvers available. Because we are interested in solving very large bilinear programs, we describe simple approximate algorithms next. Optimal scalable methods are beyond the scope of this paper. The most common approximate method for solving bilinear programs is shown in Algorithm 1. It is designed for the general formulation shown in Eq. (3), where f (w, x, y, z) represents the objective function. The minimizations in the algorithm are linear programs which can be easily solved. Interestingly, as we will show in Section 5, Algorithm 1 applied to ABP generalizes a version of API. While Algorithm 1 is not guaranteed to ﬁnd an optimal solution, its empirical performance is often remarkably good [13]. Its basic properties are summarized by the following proposition. Proposition 7 (e.g. [3]). Algorithm 1 is guaranteed to converge, assuming that the linear program solutions are in a vertex of the optimality simplex. In addition, the global optimum is a ﬁxed point of the algorithm, and the objective value monotonically improves during execution. 5 The proof is based on the ﬁnite count of the basic feasible solutions of the individual linear programs. Because the objective function does not increase in any iteration, the algorithm will eventually converge. In the context of MDPs, Algorithm 1 can be further reﬁned. For example, the constraint v ∈ M in Eq. (4) serves mostly to simplify the bilinear program and a value function that violates it may still be acceptable. The following proposition motivates the construction of a new value function from two transitive-feasible value functions. Proposition 8. Let v1 and v2 be feasible value functions in Eq. (4). Then the value function ˜ ˜ v (s) = min{˜1 (s), v2 (s)} is also feasible in Eq. (4). Therefore v ≥ v ∗ and v ∗ − v ∞ ≤ ˜ v ˜ ˜ ˜ min { v ∗ − v1 ∞ , v ∗ − v2 ∞ }. ˜ ˜ The proof of the proposition is based on Jensen’s inequality and can be found in [19]. Proposition 8 can be used to extend Algorithm 1 when solving ABPs. One option is to take the state-wise minimum of values from multiple random executions of Algorithm 1, which preserves the transitive feasibility of the value function. However, the increasing number of value functions used to obtain v also increases the potential sampling error. ˜ 5 Relationship to ALP and API In this section, we describe the important connections between ABP and the two closely related ADP methods: ALP, and API with L∞ minimization. Both of these methods are commonly used, for example to solve factored MDPs [10]. Our analysis sheds light on some of their observed properties and leads to a new convergent form of API. ABP addresses some important issues with ALP: 1) ALP provides value function bounds with respect to L1 norm, which does not guarantee small policy loss, 2) ALP’s solution quality depends signiﬁcantly on the heuristically-chosen objective function c in Eq. (2) [7], and 3) incomplete constraint samples in ALP easily lead to unbounded linear programs. The drawback of using ABP, however, is the higher computational complexity. Both the ﬁrst and the second issues in ALP can be addressed by choosing the right objective function [7]. Because this objective function depends on the optimal ALP solution, it cannot be practically computed. Instead, various heuristics are usually used. The heuristic objective functions may lead to signiﬁcant improvements in speciﬁc domains, but they do not provide any guarantees. ABP, on the other hand, has no such parameters that require adjustments. The third issue arises when the constraints of an ALP need to be sampled in some large domains. The ALP may become unbounded with incomplete samples because its objective value is deﬁned using the L1 norm on the states, and the constraints are deﬁned using the L∞ norm of the Bellman residual. In ABP, the Bellman residual is used in both the constraints and objective function. The objective function of ABP is then bounded below by 0 for an arbitrarily small number of samples. ABP can also improve on API with L∞ minimization (L∞ -API for short), which is a leading method for solving factored MDPs [10]. Minimizing the L∞ approximation error is theoretically preferable, since it is compatible with the existing bounds on policy loss [10]. In contrast, few practical bounds exist for API with the L2 norm minimization [14], such as LSPI [12]. L∞ -API is shown in Algorithm 2, where f (π) is calculated using the following program: minimize φ φ,v s.t. (I − γPπ )v + 1φ ≥ rπ −(I − γPπ )v + 1φ ≥ −rπ (5) v∈M Here I denotes the identity matrix. We are not aware of a convergence or a divergence proof of L∞ -API, and this analysis is beyond the scope of this paper. 6 Algorithm 2: Approximate policy iteration, where f (π) denotes a custom value function approximation for the policy π. π0 , k ← rand, 1 ; while πk = πk−1 do vk ← f (πk−1 ) ; ˜ πk (s) ← arg maxa∈A r(s, a) + γ s ∈S P (s , s, a)˜k (s) ∀s ∈ S ; v k ←k+1 We propose Optimistic Approximate Policy Iteration (OAPI), a modiﬁcation of API. OAPI is shown in Algorithm 2, where f (π) is calculated using the following program: minimize φ φ,v s.t. Av ≥ b (≡ (I − γPπ )v ≥ rπ ∀π ∈ Π) −(I − γPπ )v + 1φ ≥ −rπ (6) v∈M In fact, OAPI corresponds to Algorithm 1 applied to ABP because Eq. (6) corresponds to Eq. (4) with ﬁxed θ. Then, using Proposition 7, we get the following corollary. Corollary 9. Optimistic approximate policy iteration converges in ﬁnite time. In addition, the Bellman residual of the generated value functions monotonically decreases. OAPI differs from L∞ -API in two ways: 1) OAPI constrains the Bellman residuals by 0 from below and by φ from above, and then it minimizes φ. L∞ -API constrains the Bellman residuals by φ from both above and below. 2) OAPI, like API, uses only the current policy for the upper bound on the Bellman residual, but uses all the policies for the lower bound on the Bellman residual. L∞ -API cannot return an approximate value function that has a lower Bellman residual than ABP, given the optimality of ABP described in Theorem 5. However, even OAPI, an approximate ABP algorithm, performs comparably to L∞ -API, as the following theorem states. Theorem 10. b Assume that L∞ -API converges to a policy π and a value function v that both φ satisfy: φ = v − Lπ v ∞ = v − Lv ∞ . Then v = v + 1−γ 1 is feasible in Eq. (4), and it is a ﬁxed ˜ point of OAPI. In addition, the greedy policies with respect to v and v are identical. ˜ The proof is based on two facts. First, v is feasible with respect to the constraints in Eq. (4). The ˜ Bellman residual changes for all the policies identically, since a constant vector is added. Second, because Lπ is greedy with respect to v , we have that v ≥ Lπ v ≥ L˜. The value function v is ˜ ˜ ˜ v ˜ therefore transitive-feasible. The full proof can be found in [19]. To summarize, OAPI guarantees convergence, while matching the performance of L∞ -API. The convergence of OAPI is achieved because given a non-negative Bellman residual, the greedy policy also minimizes the Bellman residual. Because OAPI ensures that the Bellman residual is always non-negative, it can progressively reduce it. In comparison, the greedy policy in L∞ -API does not minimize the Bellman residual, and therefore L∞ -API does not always reduce it. Theorem 10 also explains why API provides better solutions than ALP, as observed in [10]. From the discussion above, ALP can be seen as an L1 -norm approximation of a single iteration of OAPI. L∞ -API, on the other hand, performs many such ALP-like iterations. 6 Empirical Evaluation As we showed in Theorem 10, even OAPI, the very simple approximate algorithm for ABP, can perform as well as existing state-of-the art methods on factored MDPs. However, a deeper understanding of the formulation and potential solution methods will be necessary in order to determine the full practical impact of the proposed methods. In this section, we validate the approach by applying it to the mountain car problem, a simple reinforcement learning benchmark problem. We have so far considered that all the constraints involving z are present in the ABP in Eq. (4). Because the constraints correspond to all state-action pairs, it is often impractical to even enumerate 7 (a) L∞ error of the Bellman residual Features 100 144 OAPI 0.21 (0.23) 0.13 (0.1) ALP 13. (13.) 3.6 (4.3) LSPI 9. (14.) 3.9 (7.7) API 0.46 (0.08) 0.86 (1.18) (b) L2 error of the Bellman residual Features 100 144 OAPI 0.2 (0.3) 0.1 (1.9) ALP 9.5 (18.) 0.3 (0.4) LSPI 1.2 (1.5) 0.9 (0.1) API 0.04 (0.01) 0.08 (0.08) Table 1: Bellman residual of the ﬁnal value function. The values are averages over 5 executions, with the standard deviations shown in parentheses. them. This issue can be addressed in at least two ways. First, a small randomly-selected subset of the constraints can be used in the ABP, a common approach in ALP [9, 5]. The ALP sampling bounds can be easily extended to ABP. Second, the structure of the MDP can be used to reduce the number of constraints. Such a reduction is possible, for example, in factored MDPs with L∞ -API and ALP [10], and can be easily extended to OAPI and ABP. In the mountain-car benchmark, an underpowered car needs to climb a hill [23]. To do so, it ﬁrst needs to back up to an opposite hill to gain sufﬁcient momentum. The car receives a reward of 1 when it climbs the hill. In the experiments we used a discount factor γ = 0.99. The experiments are designed to determine whether OAPI reliably minimizes the Bellman residual in comparison with API and ALP. We use a uniformly-spaced linear spline to approximate the value function. The constraints were based on 200 uniformly sampled states with all 3 actions per state. We evaluated the methods with the number of the approximation features 100 and 144, which corresponds to the number of linear segments. The results of ABP (in particular OAPI), ALP, API with L2 minimization, and LSPI are depicted in Table 1. The results are shown for both L∞ norm and uniformly-weighted L2 norm. The runtimes of all these methods are comparable, with ALP being the fastest. Since API (LSPI) is not guaranteed to converge, we ran it for at most 20 iterations, which was an upper bound on the number of iterations of OAPI. The results demonstrate that ABP minimizes the L∞ Bellman residual much more consistently than the other methods. Note, however, that all the considered algorithms would perform signiﬁcantly better given a ﬁner approximation. 7 Conclusion and Future Work We proposed and analyzed approximate bilinear programming, a new value-function approximation method, which provably minimizes the L∞ Bellman residual. ABP returns the optimal approximate value function with respect to the Bellman residual bounds, despite the formulation with regard to transitive-feasible value functions. We also showed that there is no asymptotically simpler formulation, since ﬁnding the closest value function and solving a bilinear program are both NP-complete problems. Finally, the formulation leads to the development of OAPI, a new convergent form of API which monotonically improves the objective value function. While we only discussed approximate solutions of the ABP, a deeper study of bilinear solvers may render optimal solution methods feasible. ABPs have a small number of essential variables (that determine the value function) and a large number of constraints, which can be leveraged by the solvers [15]. The L∞ error bound provides good theoretical guarantees, but it may be too conservative in practice. A similar formulation based on L2 norm minimization may be more practical. We believe that the proposed formulation will help to deepen the understanding of value function approximation and the characteristics of existing solution methods, and potentially lead to the development of more robust and widely-applicable reinforcement learning algorithms. Acknowledgements This work was supported by the Air Force Ofﬁce of Scientiﬁc Research under Grant No. FA955008-1-0171. We also thank the anonymous reviewers for their useful comments. 8 References [1] Pieter Abbeel, Varun Ganapathi, and Andrew Y. Ng. Learning vehicular dynamics, with application to modeling helicopters. In Advances in Neural Information Processing Systems, pages 1–8, 2006. [2] Daniel Adelman. A price-directed approach to stochastic inventory/routing. Operations Research, 52:499–514, 2004. [3] Kristin P. Bennett and O. L. Mangasarian. Bilinear separation of two sets in n-space. Technical report, Computer Science Department, University of Wisconsin, 1992. [4] Dimitri P. Bertsekas and Sergey Ioffe. Temporal differences-based policy iteration and applications in neuro-dynamic programming. Technical Report LIDS-P-2349, LIDS, 1997. [5] Guiuseppe Calaﬁore and M.C. Campi. Uncertain convex programs: Randomized solutions and conﬁdence levels. Mathematical Programming, Series A, 102:25–46, 2005. [6] Alberto Carpara and Michele Monaci. Bidimensional packing by bilinear programming. Mathematical Programming Series A, 118:75–108, 2009. [7] Daniela P. de Farias. The Linear Programming Approach to Approximate Dynamic Programming: Theory and Application. PhD thesis, Stanford University, 2002. [8] Daniela P. de Farias and Ben Van Roy. The linear programming approach to approximate dynamic programming. Operations Research, 51:850–856, 2003. [9] Daniela Pucci de Farias and Benjamin Van Roy. On constraint sampling in the linear programming approach to approximate dynamic programming. Mathematics of Operations Research, 29(3):462–478, 2004. [10] Carlos Guestrin, Daphne Koller, Ronald Parr, and Shobha Venkataraman. Efﬁcient solution algorithms for factored MDPs. Journal of Artiﬁcial Intelligence Research, 19:399–468, 2003. [11] Reiner Horst and Hoang Tuy. Global optimization: Deterministic approaches. Springer, 1996. [12] Michail G. Lagoudakis and Ronald Parr. Least-squares policy iteration. Journal of Machine Learning Research, 4:1107–1149, 2003. [13] O. L. Mangasarian. The linear complementarity problem as a separable bilinear program. Journal of Global Optimization, 12:1–7, 1995. [14] Remi Munos. Error bounds for approximate policy iteration. In International Conference on Machine Learning, pages 560–567, 2003. [15] Marek Petrik and Shlomo Zilberstein. Anytime coordination using separable bilinear programs. In Conference on Artiﬁcial Intelligence, pages 750–755, 2007. [16] Marek Petrik and Shlomo Zilberstein. Average reward decentralized Markov decision processes. In International Joint Conference on Artiﬁcial Intelligence, pages 1997–2002, 2007. [17] Marek Petrik and Shlomo Zilberstein. A bilinear programming approach for multiagent planning. Journal of Artiﬁcial Intelligence Research, 35:235–274, 2009. [18] Marek Petrik and Shlomo Zilberstein. Constraint relaxation in approximate linear programs. In International Conference on Machine Learning, pages 809–816, 2009. [19] Marek Petrik and Shlomo Zilberstein. 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