nips nips2009 nips2009-16 knowledge-graph by maker-knowledge-mining

16 nips-2009-A Smoothed Approximate Linear Program

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Author: Vijay Desai, Vivek Farias, Ciamac C. Moallemi

Abstract: We present a novel linear program for the approximation of the dynamic programming cost-to-go function in high-dimensional stochastic control problems. LP approaches to approximate DP naturally restrict attention to approximations that are lower bounds to the optimal cost-to-go function. Our program – the ‘smoothed approximate linear program’ – relaxes this restriction in an appropriate fashion while remaining computationally tractable. Doing so appears to have several advantages: First, we demonstrate superior bounds on the quality of approximation to the optimal cost-to-go function aﬀorded by our approach. Second, experiments with our approach on a challenging problem (the game of Tetris) show that the approach outperforms the existing LP approach (which has previously been shown to be competitive with several ADP algorithms) by an order of magnitude. 1

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Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 edu Abstract We present a novel linear program for the approximation of the dynamic programming cost-to-go function in high-dimensional stochastic control problems. [sent-8, score-0.173]

2 LP approaches to approximate DP naturally restrict attention to approximations that are lower bounds to the optimal cost-to-go function. [sent-9, score-0.118]

3 Our program – the ‘smoothed approximate linear program’ – relaxes this restriction in an appropriate fashion while remaining computationally tractable. [sent-10, score-0.148]

4 Doing so appears to have several advantages: First, we demonstrate superior bounds on the quality of approximation to the optimal cost-to-go function aﬀorded by our approach. [sent-11, score-0.117]

5 1 Introduction Many dynamic optimization problems can be cast as Markov decision problems (MDPs) and solved, in principle, via dynamic programming. [sent-13, score-0.076]

6 ADP algorithms seek to compute good approximations to the dynamic programing optimal cost-to-go function within the span of some pre-speciﬁed set of basis functions. [sent-16, score-0.174]

7 The program employed in the ALP approach is identical to the LP used for exact computation of the optimal cost-to-go function, with further constraints limiting solutions to the low-dimensional subspace spanned by the basis functions used. [sent-18, score-0.197]

8 The resulting low dimensional LP implicitly restricts attention to approximations that are lower bounds on the optimal cost-to-go function. [sent-19, score-0.101]

9 While the structure of this program appears crucial in establishing approximation guarantees for the approach, the restriction to lower bounds leads one to ask whether the ALP is the ‘right’ LP. [sent-20, score-0.239]

10 In particular, could an appropriate relaxation of the feasible region of the ALP allow for better approximations to the cost-to-go function while remaining computationally tractable? [sent-21, score-0.132]

11 Motivated by this question, the present paper presents a new linear program for ADP we call the ‘smoothed’ ALP (or SALP). [sent-22, score-0.081]

12 The SALP may be viewed as a relaxation of the ALP wherein one is allowed to violate the ALP constraints for any given state. [sent-23, score-0.063]

13 A user deﬁned ‘violation budget’ parameter controls the ‘expected’ violation across states; a budget of 0 thus yields the original ALP. [sent-24, score-0.269]

14 We specify a choice of this violation budget that yields a relaxation with attractive properties. [sent-25, score-0.318]

15 In particular, we are able to establish strong approximation guarantees for the SALP; these guarantees are substantially stronger than the corresponding guarantees for the ALP. [sent-26, score-0.156]

16 The number of constraints and variables in the SALP scale with the size of the MDP state space. [sent-27, score-0.069]

17 We nonetheless establish sample complexity bounds that demonstrate that an 1 appropriate ‘sampled’ SALP provides a good approximation to the SALP solution with a tractable number of sampled MDP states. [sent-28, score-0.246]

18 This sampled program is no more complex than the ‘sampled’ ALP and, as such, we demonstrate that the SALP is essentially no harder to solve than the ALP. [sent-29, score-0.153]

19 In detailed comparisons with the ALP, we estimate that the SALP provides an order of magnitude improvement over controllers designed via that approach for the game of Tetris. [sent-32, score-0.06]

20 2 Problem Formulation Our setting is that of a discrete-time, discounted inﬁnite-horizon, cost-minimizing MDP with a ﬁnite state space X and ﬁnite action space A. [sent-33, score-0.073]

21 Given the state and action at time t, xt and at , a per-stage cost g(xt , at ) is incurred. [sent-34, score-0.076]

22 A stationary policy µ : X → A is a mapping that determines the action at each time as a function of the state. [sent-36, score-0.099]

23 Given each initial state x0 = x, the expected discounted cost (cost-to-go function) of the policy µ is given by Jµ (x) Eµ ∞ αt g(xt , µ(xt )) x0 = x . [sent-37, score-0.144]

24 Denote by Pµ ∈ RX ×X the transition probability matrix for the policy µ, whose (x, x )th entry is Pµ(x) (x, x ). [sent-39, score-0.085]

25 Then, the cost-to-go function Jµ is the unique solution to the equation Tµ J = J, where the operator Tµ is deﬁned by Tµ J = gµ + αPµ J. [sent-41, score-0.059]

26 It is readily shown that the optimal cost-to-go function J ∗ is the unique solution to Bellman’s equation and that a corresponding optimal policy µ∗ is greedy with respect to J ∗ ; i. [sent-44, score-0.179]

27 The above program is indeed an LP since the constraint J(x) ≤ (T J)(x) is equivalent to the set of linear constraints J(x) ≤ g(x, a) + α x ∈X Pa (x, x )J(x ), ∀ a ∈ A. [sent-49, score-0.137]

28 Then, every feasible point for (1) is a component-wise lower bound to J ∗ , and J ∗ is the unique optimal solution to the exact LP (1). [sent-52, score-0.138]

29 For problems where X is prohibitively large, an ADP algorithm seeks to ﬁnd a good approximation to J ∗ . [sent-53, score-0.061]

30 Speciﬁcally, one considers a collection of basis functions {φ1 , . [sent-54, score-0.058]

31 φK ] to be a matrix with columns consisting of basis functions, one seeks an approximation of the form Jr = Φr, with the hope that Jr ∼ J ∗ . [sent-61, score-0.106]

32 In other words, the approximate cost-to-go function is necessarily a point-wise lower bound to the true cost-to-go function in the span of Φ. [sent-67, score-0.083]

33 Figure 1: A cartoon illustrating the feasible set and optimal solution for the ALP and SALP, in the case of a two-state MDP. [sent-70, score-0.126]

34 A careful relaxation from the feasible set of the ALP to that of the SALP can yield an improved approximation. [sent-72, score-0.066]

35 3 The Smoothed ALP The J ≤ T J constraints in the exact LP, which carry over to the ALP, impose a strong restriction on the cost-to-go function approximation: in particular they restrict us to approximations that are lower bounds to J ∗ at every point in the state space. [sent-73, score-0.16]

36 In the case where the state space is very large, and the number of basis functions is (relatively) small, it may be the case that constraints arising from rarely visited or pathological states are binding and inﬂuence the optimal solution. [sent-74, score-0.174]

37 In many cases, our ultimate goal is not to ﬁnd a lower bound on the optimal cost-to-go function, but rather a good approximation to J ∗ . [sent-75, score-0.109]

38 In these instances, it may be the case that relaxing the constraints in the ALP so as not to require a uniform lower bound may allow for better overall approximations to the optimal cost-to-go function. [sent-76, score-0.147]

39 Relaxing the feasible region of the ALP in Figure 1(b) to the light gray region in Figure 1(b) would yield the point ΦrSALP as an optimal solution. [sent-78, score-0.1]

40 The relaxation in this case is clearly beneﬁcial; it allows us to compute a better approximation to J ∗ than the point ΦrSALP . [sent-79, score-0.079]

41 The smoothed approximate linear program (SALP) is given by: (3) maximize r,s ν Φr subject to Φr ≤ T Φr + s, π s ≤ θ, s ≥ 0. [sent-81, score-0.179]

42 For each state x, s(x) is a non-negative decision variable (a slack) that allows for violation of the corresponding ALP constraint. [sent-83, score-0.223]

43 The parameter π ∈ RX is a probability distribution known as the constraint violation distribution. [sent-85, score-0.192]

44 The parameter θ is thus a violation budget: the expected violation of the Φr ≤ T Φr constraint, under the distribution π, must be less than θ. [sent-86, score-0.334]

45 Our analysis will present two types of results: First, we prove approximation guarantees (Sections 4. [sent-94, score-0.078]

46 2) that will indicate that the SALP computes approximations that are of comparable quality to the projection of J ∗ on the linear span of Φ. [sent-96, score-0.075]

47 3) that an implementable ‘sampled’ version of the SALP may be used to approximate the SALP with a tractable number of samples. [sent-98, score-0.058]

48 In the case of the ALP, one either assumes the ability to solve a linear program with as many constraints as there are states, or absent that, knowledge of a certain idealized sampling distribution, so that one can then proceed with solving a ‘sampled’ version of the ALP. [sent-101, score-0.17]

49 Our analysis of the SALP in this section is predicated on the knowledge of an idealized constraint violation distribution, which is this same idealized sampling distribution. [sent-102, score-0.308]

50 The distribution πµ∗ ,ν may be interpreted as yielding the discounted expected frequency of visits to a given state when the initial state is distributed according to ν and the system runs under the optimal policy µ∗ . [sent-105, score-0.208]

51 We note that the ‘sampled’ ALP introduced by de Farias and Van Roy [2] requires access to states sampled according to precisely this distribution. [sent-106, score-0.109]

52 1 A Simple Approximation Guarantee We present a ﬁrst, simple approximation guarantee for the following specialization of the SALP in (3): maximize r,s (4) ν Φr subject to Φr ≤ T Φr + s, πµ∗ ,ν s ≤ θ, s ≥ 0. [sent-108, score-0.129]

53 Before we proceed to state our result, we deﬁne a useful function: (r, θ) (5) minimize γ s,γ subject to Φr − T Φr ≤ s + γ1, πµ∗ ,ν s ≤ θ, s ≥ 0. [sent-109, score-0.062]

54 Armed with this deﬁnition, we are now in a position to state our ﬁrst, crude approximation guarantee: 4 Theorem 1. [sent-116, score-0.085]

55 1−α ,θ)+2θ as the approximation error associated The above theorem allows us to interpret (r 1−α with the SALP solution r. [sent-121, score-0.098]

56 This is precisely Theorem 2 in de Farias and Van Roy [1]; we recover their approximation guarantee for the ALP. [sent-124, score-0.097]

57 In the event that Φr∗ (x) − T Φr∗ (x) is large for only a small number of states (that is, the Bellman error of the approximation produced by r∗ is large for only a small number of states), we thus expect to have a choice of θ for which l(r∗ , θ) + 2θ l(r∗ , 0). [sent-126, score-0.098]

58 Thus, Theorem 1 reinforces the intuition (shown via Figure 1) that the SALP will permit closer approximations to J ∗ than the ALP. [sent-127, score-0.059]

59 Since it is unlikely that the basis functions Φ will provide a uniformly good approximation over the entire state space, the right hand side of our bound could be quite large. [sent-130, score-0.168]

60 While we do not show this here, this choice allows us to choose regions of the state space where we would like a better approximation of J ∗ . [sent-133, score-0.102]

61 Our guarantee does not suggest a concrete choice of the violation budget, θ. [sent-136, score-0.217]

62 2 A Better Approximation Guarantee With the intent of deriving stronger approximation guarantees, we begin this section by introducing a ‘nicer’ measure of the quality of approximation aﬀorded by Φ. [sent-139, score-0.107]

63 The weighting function ψ allows us to weight approximation error in a non-uniform fashion across the state space and in this manner potentially ignore approximation quality in regions of the state space that ‘don’t matter’. [sent-141, score-0.218]

64 In addition to specifying the constraint violation distribution π as we did for our previous bound, we will specify (implicitly) a particular choice of the violation budget θ. [sent-142, score-0.478]

65 For every ψ ∈ Ψ, let β(ψ) = maxµ Then, for an optimal solution (rSALP , s) to (6), we have: ¯ J ∗ − ΦrSALP 1,ν ≤ inf r,ψ∈Ψ J ∗ − Φr ∞,1/ψ 5 ν ψ+ 2(πµ∗ ,ν ψ + 1)(αβ(ψ) + 1) 1−α Pµ ψ ψ . [sent-147, score-0.071]

66 Thus, the approximation guarantee of Theorem 2 allows us to view the SALP as automating the critical procedure of identifying a good Lyapunov function for a given problem. [sent-154, score-0.08]

67 3 Sample Complexity Our analysis thus far has assumed we have the ability to solve the SALP, a program with a potentially intractable number of constraints and variables. [sent-156, score-0.112]

68 As it turns out, a solution to the SALP is well approximated by the solution to a certain ‘sampled’ program which we now ˆ describe: Let X = {x1 , x2 , . [sent-157, score-0.139]

69 Let us consider solving the following program which we call the sampled SALP: maximize r,s (7) ν Φr − 2 (1−α)S ˆ x∈X s(x) ˆ subject to (Φr)(x) ≤ (T Φr)(x) + s(x), ∀ x ∈ X , r ∈ N , s ≥ 0. [sent-161, score-0.188]

70 Notice that (7) is a linear program with S variables and S|A| constraints. [sent-163, score-0.081]

71 Under the conditions of Theorem 2, let rSALP be an optimal solution to the SALP (6), and let rSALP be an optimal solution to the sampled SALP (7). [sent-169, score-0.168]

72 Further, given ∈ (0, B] and δ ∈ (0, 1/2], suppose that the number of sampled states S satisﬁes 16eB 8 64B 2 2(K + 2) log + log . [sent-171, score-0.092]

73 1−α Theorem 3 establishes that the sampled SALP provides a close approximation to the solution of the SALP, in the sense that the approximation guarantees we established for the SALP are approximately valid for the solution to the sampled version with high probability. [sent-173, score-0.313]

74 This number depends linearly on the number of basis functions and the diameter of the 1 This includes those ψ that do not satisfy the Lyapunov condition β(ψ) ≤ 1/α. [sent-175, score-0.061]

75 6 feasible region, but is otherwise independent of the size of the state space for the MDP under consideration. [sent-176, score-0.072]

76 Exploiting the fact that the ALP has a small number of variables, de Farias and Van Roy [2] establish a sample complexity bound for a sampled version of the ALP analogous (7). [sent-179, score-0.13]

77 The number of samples required for this sampled ALP to produce a good approximation to the ALP can be shown to depend on the same problem parameters we have identiﬁed here, viz. [sent-180, score-0.105]

78 The sample complexity in that case is identical to the sample complexity bound established here up to constants and an additional multiplicative factor of B/ (for the sampled SALP). [sent-182, score-0.16]

79 Thus, the two sample complexity bounds are within polynomial terms of each other and we have established that the SALP is essentially no harder to solve than the ALP. [sent-183, score-0.091]

80 This section places the SALP on solid theoretical ground by establishing strong approximation guarantees for the SALP that represent a substantial improvement over those available for the ALP and sample complexity results that indicated that the SALP was implementable via sampling. [sent-184, score-0.17]

81 Tetris represents precisely the kind of large and unstructured MDP for which it is diﬃcult to design heuristic controllers, and hence policies designed by ADP algorithms are particularly relevant. [sent-188, score-0.057]

82 Given a sample size S, a collection X ⊂ X of S states was sampled. [sent-194, score-0.063]

83 Given the collection X of sampled states, an increasing sequence of choices of the violation budget θ ≥ 0 is considered. [sent-197, score-0.34]

84 For each choice of θ, the optimization program maximize r,s (8) 1 S ˆ x∈X (Φr)(x) ˆ subject to Φr(x) ≤ T Φr(x) + s(x), ∀ x ∈ X , 1 ˆ s(x) ≤ θ, x∈X S ˆ s(x) ≥ 0, ∀ x ∈ X, was solved. [sent-198, score-0.147]

85 This program is a version of the original SALP (3), but with sampled empirical distributions in place of the state-relevance weights ν and the constraint violation distribution π. [sent-199, score-0.331]

86 Given a vector of weights obtained by solving (8), the performance of the corresponding policy is evaluated via Monte Carlo simulation over 3, 000 games of Tetris. [sent-203, score-0.085]

87 For each pair (S, θ), the resulting average performance (averaged over 10 diﬀerent sets of sampled states) is shown in Figure 2. [sent-205, score-0.058]

88 It provides experimental evidence for the intuition expressed in Section 3 and the analytic result of Theorem 1: Relaxing the constraints of the ALP by allowing for a violation budget allows for better policy performance. [sent-206, score-0.4]

89 As the violation budget θ is increased from 0, performance dramatically improves. [sent-207, score-0.269]

90 2 Our baseline policy had an average performance of 113 points. [sent-210, score-0.085]

91 00256 θ = 0 (ALP) 4 2 0 50 100 150 200 250 300 ×103 Sample Size S Figure 2: Average performance of SALP for diﬀerent values of the number of sampled states S and the violation budget θ. [sent-216, score-0.361]

92 Algorithm Best Performance CPU Time ALP TD-Learning [7] ALP with bootstrapping [3] TD-Learning [8] Policy gradient [9] SALP 897 3,183 4,274 4,471 5,500 10,775 hours minutes hours minutes days hours Table 1: Comparison of the performance of the best policy found with various ADP methods. [sent-221, score-0.142]

93 Note that signiﬁcantly better policies are possible with this basis function architecture than any of the ADP algorithms in Table 1 discover. [sent-222, score-0.085]

94 In addition, the approach employs a number of rather arbitrary Tetris speciﬁc ‘modiﬁcations’ that are ultimately seen to be critical to performance - in the absence of these modiﬁcations, the method is unable to ﬁnd a policy for Tetris that scores above a few hundred points. [sent-225, score-0.085]

95 2 are approximation guarantees, which provide bounds on the approximation error given by the SALP approach versus the best approximation possible with the particular set of basis functions. [sent-229, score-0.217]

96 These provide bounds on the performance of the resulting SALP policies, as a function of the basis architecture. [sent-231, score-0.076]

97 Rather than solving a large linear program, such an algorithm would optimize a policy in an online fashion along a single system trajectory. [sent-233, score-0.105]

98 However, a sample path SALP variation would inherit all of the theoretical bounds developed here. [sent-235, score-0.059]

99 On constraint sampling in the linear programming approach to approximate dynamic programming. [sent-247, score-0.087]

100 Temporal diﬀerences–based policy iteration and applications in neuro–dynamic programming. [sent-275, score-0.085]

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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. 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