nips nips2009 nips2009-60 knowledge-graph by maker-knowledge-mining

60 nips-2009-Convergent Temporal-Difference Learning with Arbitrary Smooth Function Approximation

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Author: Shalabh Bhatnagar, Doina Precup, David Silver, Richard S. Sutton, Hamid R. Maei, Csaba Szepesvári

Abstract: We introduce the ﬁrst temporal-difference learning algorithms that converge with smooth value function approximators, such as neural networks. Conventional temporal-difference (TD) methods, such as TD(λ), Q-learning and Sarsa have been used successfully with function approximation in many applications. However, it is well known that off-policy sampling, as well as nonlinear function approximation, can cause these algorithms to become unstable (i.e., the parameters of the approximator may diverge). Sutton et al. (2009a, 2009b) solved the problem of off-policy learning with linear TD algorithms by introducing a new objective function, related to the Bellman error, and algorithms that perform stochastic gradient-descent on this function. These methods can be viewed as natural generalizations to previous TD methods, as they converge to the same limit points when used with linear function approximation methods. We generalize this work to nonlinear function approximation. We present a Bellman error objective function and two gradient-descent TD algorithms that optimize it. We prove the asymptotic almost-sure convergence of both algorithms, for any ﬁnite Markov decision process and any smooth value function approximator, to a locally optimal solution. The algorithms are incremental and the computational complexity per time step scales linearly with the number of parameters of the approximator. Empirical results obtained in the game of Go demonstrate the algorithms’ effectiveness. 1

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Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 Sutton University of Alberta, Edmonton, AB, Canada Abstract We introduce the ﬁrst temporal-difference learning algorithms that converge with smooth value function approximators, such as neural networks. [sent-3, score-0.187]

2 However, it is well known that off-policy sampling, as well as nonlinear function approximation, can cause these algorithms to become unstable (i. [sent-5, score-0.165]

3 (2009a, 2009b) solved the problem of off-policy learning with linear TD algorithms by introducing a new objective function, related to the Bellman error, and algorithms that perform stochastic gradient-descent on this function. [sent-9, score-0.199]

4 These methods can be viewed as natural generalizations to previous TD methods, as they converge to the same limit points when used with linear function approximation methods. [sent-10, score-0.118]

5 We present a Bellman error objective function and two gradient-descent TD algorithms that optimize it. [sent-12, score-0.136]

6 We prove the asymptotic almost-sure convergence of both algorithms, for any ﬁnite Markov decision process and any smooth value function approximator, to a locally optimal solution. [sent-13, score-0.09]

7 The algorithms are incremental and the computational complexity per time step scales linearly with the number of parameters of the approximator. [sent-14, score-0.089]

8 1 Introduction We consider the problem of estimating the value function of a given stationary policy of a Markov Decision Process (MDP). [sent-16, score-0.131]

9 This problem arises as a subroutine of generalized policy iteration and is generally thought to be an important step in developing algorithms that can learn good control policies in reinforcement learning (e. [sent-17, score-0.28]

10 A key property of the TD(λ) algorithm is that it can be combined with function approximators in order to generalize the observed data to unseen states. [sent-21, score-0.102]

11 TD(λ) is known to converge when used with linear function approximators, if states are sampled according to the policy being evaluated – a scenario called onpolicy learning (Tsitsiklis & Van Roy, 1997). [sent-25, score-0.21]

12 However, the absence of either of these requirements can cause the parameters of the function approximator to diverge when trained with TD methods (e. [sent-26, score-0.17]

13 The question of whether it is possible to create TD-style algorithms that are guaranteed to converge when used with nonlinear function approximation has remained open until now. [sent-29, score-0.283]

14 Residual gradient algorithms (Baird, 1995) ∗ On leave from MTA SZTAKI, Hungary. [sent-30, score-0.116]

15 However, unlike TD, these algorithms usually require two independent samples from each state. [sent-32, score-0.059]

16 Moreover, even if two samples are provided, the solution to which they converge may not be desirable (Sutton et al. [sent-33, score-0.115]

17 In this paper we deﬁne the ﬁrst TD algorithms that are stable when used with smooth nonlinear function approximators (such as neural networks). [sent-35, score-0.316]

18 Our starting point is the family of TD-style algorithms introduced recently by Sutton et al. [sent-36, score-0.095]

19 Their goal was to address the instability of TD learning with linear function approximation, when the policy whose value function is sought differs from the policy used to generate the samples (a scenario called off-policy learning). [sent-38, score-0.296]

20 These algorithms were designed to approximately follow the gradient of an objective function whose unique optimum is the ﬁxed point of the original TD(0) algorithm. [sent-39, score-0.194]

21 Here, we extend the ideas underlying this family of algorithms to design TD-like algorithms which converge, under mild assumptions, almost surely, with smooth nonlinear approximators. [sent-40, score-0.273]

22 Under some technical conditions, the limit points of the new algorithms correspond to the limit points of the original (not necessarily convergent) nonlinear TD algorithm. [sent-41, score-0.165]

23 The algorithms are incremental, and the cost of each update is linear in the number of parameters of the function approximator, as in the original TD algorithm. [sent-42, score-0.084]

24 First, we extend the objective function of Sutton et al. [sent-44, score-0.086]

25 (2009b), in a natural way, to the nonlinear function approximation case. [sent-45, score-0.145]

26 To overcome potential instability issues, we introduce a projection step in the weight update. [sent-49, score-0.1]

27 The almost sure convergence of the algorithm then follows from standard two-time-scale stochastic approximation arguments. [sent-50, score-0.111]

28 In the rest of the paper, we ﬁrst introduce the setting and our notation (Section 2), review previous relevant work (Section 3), introduce the algorithms (Section 4), analyze them (Section 5) and illustrate the algorithms’ performance (Section 6). [sent-51, score-0.059]

29 2 Notation and Background We consider policy evaluation in ﬁnite state and action Markov Decision Processes (MDPs). [sent-52, score-0.159]

30 The policy to be evaluated is a mapping π : S × A → [0, 1]. [sent-54, score-0.131]

31 The value function of π, V π : S → R, maps each state s to a number representing the inﬁnite-horizon expected discounted return obtained if policy π is followed from state s. [sent-55, score-0.187]

32 To simplify the notation, from now on we will drop the superscript π everywhere, since the policy to be evaluated will be kept ﬁxed. [sent-62, score-0.131]

33 Assume that the policy to be evaluated is followed and it gives rise to the trajectory (s0 , a0 , r1 , s1 , a1 , r2 , s2 , . [sent-63, score-0.131]

34 More generally, we may assume that we are given an inﬁnite sequence of 3-tuples, (sk , rk , s ), k that satisﬁes the following: Assumption A1 (sk )k≥0 is an S-valued stationary Markov process, sk ∼ d(·), rk = R(sk ) and s ∼ P (sk , ·). [sent-68, score-0.286]

35 When the state space is large (or inﬁnite) a function approximation method can be used to facilitate the generalization of observed transitions to unvisited or rarely visited states. [sent-74, score-0.14]

36 We denote by Vθ (s) the value of state s ∈ S returned by the function approximator with parameters θ. [sent-76, score-0.147]

37 The goal of policy evaluation becomes to ﬁnd θ such that Vθ ≈ V . [sent-77, score-0.131]

38 3 TD Algorithms with function approximation The classical TD(0) algorithm with function approximation (Sutton, 1988; Sutton & Barto, 1998) starts with an arbitrary value of the parameters, θ0 . [sent-78, score-0.078]

39 Upon observing the k th transition, it computes the scalar-valued temporal-difference error, δk = rk + γVθk (s ) − Vθk (sk ), k which is then used to update the parameter vector as follows: θk+1 ← θk + αk δk ∇Vθk (sk ). [sent-79, score-0.085]

40 However, it is well known that TD(0) may not converge; the stability of the algorithm is affected both by the actual function approximator Vθ and by the way in which transitions are sampled. [sent-86, score-0.161]

41 Sutton et al (2009a, 2009b) tackled this problem in the case of linear function approximation, in which Vθ (s) = θ φ(s), where φ : S → Rn , but where transitions may be sampled in an off-policy manner. [sent-87, score-0.078]

42 (2009b) rely on an error function, called mean-square projected Bellman error (MSPBE)2 , which has the same unique optimum as Equation (2). [sent-90, score-0.082]

43 This function, which we denote J, projects the Bellman error measure, T Vθ − Vθ onto the linear space M = {Vθ | θ ∈ Rn } with respect to the metric · D . [sent-91, score-0.103]

44 More precisely: D (3) J(θ) = Π(T Vθ − Vθ ) 2 = Π T Vθ − Vθ 2 = E[δφ] E[φφ ]−1 E[δφ], D D 2 2 where V D is the weighted quadratic norm deﬁned by V D = s∈S d(s)V (s) , and the scalar TD(0) error for a given transition (s, r, s ) is δ = r + γθ φ − θ φ. [sent-93, score-0.068]

45 The negative gradient of the MSPBE objective function is: 1 (4) − ∇J(θ) = E (φ − γφ )φ w = E[δφ] − γE φ φ w, 2 where w = E[φφ ]−1 E[δφ]. [sent-94, score-0.107]

46 In order to develop an efﬁcient (O(n)) stochastic gradient algorithm, Sutton et al. [sent-96, score-0.124]

47 They introduce a new set of weights, wk , whose purpose is to estimate w for a ﬁxed value of the θ parameter. [sent-98, score-0.476]

48 These weights are updated on a “fast” timescale, as follows: wk+1 = wk + βk (δk − φ wk )φk . [sent-99, score-0.976]

49 Two update rules can be obtained, based on two slightly different calculations: θk+1 = θk + αk (φk − γφ )(φ wk ) (an algorithm called GTD2), or (6) k k θk+1 = θk + αk δk φk − αk γφ (φ wk ) (an algorithm called TDC). [sent-101, score-0.977]

50 k k (7) 2 This error function was also described in (Antos et al. [sent-102, score-0.063]

51 3 4 Nonlinear Temporal Difference Learning Our goal is to generalize this approach to the case in which Vθ is a smooth, nonlinear function approximator. [sent-106, score-0.106]

52 The ﬁrst step is to ﬁnd a good objective function on which to do gradient descent. [sent-107, score-0.107]

53 In the linear case, MSPBE was chosen as a projection of the Bellman error on a natural hyperplane–the subspace to which Vθ is restricted. [sent-108, score-0.093]

54 However, in the nonlinear case, the value function is no longer restricted to a plane, but can move on a nonlinear surface. [sent-109, score-0.212]

55 More precisely, assuming that Vθ is a differentiable function of θ, M = {Vθ ∈ R|S| | θ ∈ Rn } becomes a differentiable submanifold of R|S| . [sent-110, score-0.082]

56 The tangent plane P Mθ of M at θ is the hyperplane of R|S| that (i) passes through Vθ and (ii) is orthogonal to the normal of M at θ. [sent-113, score-0.196]

57 The tangent space T Mθ is the translation of P Mθ to the ∂ origin. [sent-114, score-0.097]

58 Let Πθ be the projection that projects vectors of (R|S| , · D ) to T Mθ . [sent-116, score-0.105]

59 D (9) θ V θT Υ TV Υ T Vθ θ This is a natural generalization of the objective function deﬁned by (3), as the plane on which we project is parallel to the tangent plane at θ. [sent-119, score-0.293]

60 More precisely, let Υθ be the projection to P Mθ and let Πθ be the projection to T Mθ . [sent-120, score-0.132]

61 In other words, projecting onto the tangent space gives exactly the same distance as projecting onto the tangent plane, while being mathematically more convenient. [sent-122, score-0.33]

62 ∗ θ T J( θ) Vθ V θ∗ Figure 1: The MSPBE objective for nonlinear function approximation at two points in the value function space. [sent-125, score-0.195]

63 Υθ∗ T Vθ∗ = Vθ∗ Tangent plane TD(0) solution We now show that J(θ) can be re-written in the same way as done in (Sutton et al. [sent-127, score-0.109]

64 Assume Vθ (s0 ) is continuously differentiable as a function of θ, for any s0 ∈ S s. [sent-130, score-0.084]

65 Note that the assumption that E[ ∇Vθ (s)∇Vθ (s) ]−1 is non-singular is akin to the assumption that the feature vectors are independent in the linear function approximation case. [sent-139, score-0.089]

66 4 This is an important corollary, because it shows that the global optima of the proposed objective function will not modify the set of solutions that the usual TD(0) algorithm would ﬁnd (if it would indeed converge). [sent-143, score-0.079]

67 Assume that (i) Vθ (s0 ) is twice continuously differentiable in θ for any s0 ∈ ˆ S s. [sent-146, score-0.084]

68 From its d deﬁnition and the assumptions, W (u) is a symmetric, positive deﬁnite matrix, so du (W −1 )|u=θ = d d − W −1 (θ) ( du W |u=θ ) W −1 (θ), where we use the assumption that du W exists at θ and W −1 exists in a small neighborhood of θ. [sent-156, score-0.139]

69 Weight wk is updated as before on a “faster” timescale: wk+1 = wk + βk (δk − φ wk )φk . [sent-165, score-1.452]

70 (16) k Besides hk , the only new ingredient compared to the linear case is Γ : Rn → Rn , a mapping that projects its argument into an appropriately chosen compact set C with a smooth boundary. [sent-167, score-0.192]

71 The purpose of this projection is to prevent the parameters to diverge in the initial phase of the algorithm, which could happen due to the presence of the nonlinearities in the algorithm. [sent-168, score-0.117]

72 Projection is a common technique for stabilizing the transient behavior of stochastic approximation algorithms (see, e. [sent-169, score-0.129]

73 We expect this to happen frequently in practice: the main reason for the projection is to facilitate convergence analysis. [sent-173, score-0.138]

74 Assume that Vθ (s) and its gradient can each be computed in O(n) time, the usual case for approximators of interest (e. [sent-175, score-0.159]

75 The key is to note that ∇2 Vθk (sk )wk = ∇(∇Vθk (s) wk ), because wk does not depend on θk . [sent-180, score-0.952]

76 The scalar term ∇Vθk (s) wk can be computed in O(n) and its gradient, which is a vector, can also be computed in O(n). [sent-181, score-0.476]

77 Hence, the computation time per update for the proposed algorithms is linear in the number of parameters of the function approximator (just like in TD(0)). [sent-182, score-0.203]

78 Given ˆ projection Γ onto C, let operator Γ : C(C) → C(Rn ) be ˆ (θ) = lim Γ θ + ε v(θ) − θ . [sent-184, score-0.103]

79 ˆ (θ) = v(θ) when θ ∈ C ◦ , otherwise, if θ ∈ ∂C, Γv (θ) is the projection of v(θ) to ˆ In particular, Γv the tangent space of ∂C at θ. [sent-186, score-0.163]

80 The next theorem shows that under some technical conditions, the iterates produced by nonlinear GTD2 converge to K with probability one. [sent-191, score-0.238]

81 Let (sk , rk , s )k≥0 be a sequence of transitions that k satisﬁes A1. [sent-193, score-0.102]

82 (2009b)), it follows that θk converges almost surely to ˙ ˆ the set of asymptotically stable equilibria of θ = ΓF (θ), (θ(0) ∈ C), where F (θ) = g(θ, wθ ). [sent-209, score-0.082]

83 , Borkar & Meyn, 2000), wk (θ) converges to wθ , which by assumption stays bounded if θ comes from a bounded set. [sent-218, score-0.541]

84 6 Empirical results To illustrate the convergence properties of the algorithms, we applied them to the “spiral” counterexample of Tsitsikilis & Van Roy (1997), originally used to show the divergence of TD(0) with nonlinear function approximation. [sent-223, score-0.147]

85 The value function has sin= a ˆ ˆ gle parameter, θ, and takes the nonlinear spiral form Vθ (s) = a(s) cos (λθ) − b(s) sin (λθ) eθ . [sent-227, score-0.137]

86 Hence, we run our algorithms without a projection step. [sent-236, score-0.125]

87 We also use constant learning rates, in order to facilitate gradient descent through an error surface which is essentially ﬂat. [sent-237, score-0.115]

88 GTD2 and TDC converge to the correct solution, while TD(0) diverges. [sent-246, score-0.079]

89 To assess the performance of the new algorithms on a large scale problem, we used them to learn an evaluation function in 9x9 computer Go. [sent-248, score-0.059]

90 All rewards were zero, except upon winning the game, when the reward was 1. [sent-253, score-0.065]

91 We applied four algorithms to this problem: TD(0), the proposed algorithms (GTD2 and TDC) and residual gradient (RG). [sent-254, score-0.218]

92 7 Conclusions and future work In this paper, we solved a long-standing open problem in reinforcement learning, by establishing a family of temporal-difference learning algorithms that converge with arbitrary differentiable function approximators (including neural networks). [sent-285, score-0.371]

93 The algorithms perform gradient descent on a natural objective function, the projected Bellman error. [sent-286, score-0.166]

94 Of course, TD(0) need not converge with non-linear function approximation. [sent-288, score-0.079]

95 Our algorithms are on-line, incremental and their computational cost per update is linear in the number of parameters. [sent-289, score-0.114]

96 Local optimality is the best one can hope for, since nonlinear function approximation creates non-convex optimization problems. [sent-291, score-0.145]

97 However, more practical experience with these algorithms is needed. [sent-293, score-0.059]

98 We are currently working on extensions of these algorithms using eligibility traces, and on using them for solving control problems. [sent-294, score-0.059]

99 Learning near-optimal policies with Bellmana residual minimization based ﬁtted policy iteration and a single sample path. [sent-304, score-0.174]

100 The ODE method for convergence of stochastic approximation and reinforcement learning. [sent-318, score-0.201]

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While there exist some L2 -based bounds [14], they require values that are difﬁcult to obtain. Approximate linear programming (ALP) uses a linear program to compute the approximate value function in a particular vector space [7]. ALP has been previously used in a wide variety of settings [2, 9, 10]. Although ALP often does not perform as well as ADP, there have been some recent 1 efforts to close the gap [18]. ALP has better theoretical properties than ADP and policy search. It is guaranteed to converge and return the closest L1 -norm approximation v of the optimal value func˜ tion v ∗ up to a multiplicative factor. However, the L1 norm must be properly weighted to guarantee a small policy loss, and there is no reliable method for selecting appropriate weights [7]. To summarize, the existing reinforcement learning techniques often provide good solutions, but typically require signiﬁcant domain knowledge [20]. The domain knowledge is needed partly because useful a priori error bounds are not available, as mentioned above. Our goal is to develop a more robust method that is guaranteed to minimize an actual bound on the policy loss. We present a new formulation of value function approximation that provably minimizes a bound on the policy loss. Unlike in some other algorithms, the bound in this case does not rely on values that are hard to obtain. The new method uniﬁes policy search and value-function search methods to minimize the L∞ norm of the Bellman residual, which bounds the policy loss. We start with a description of the framework and notation in Section 2. Then, in Section 3, we describe the proposed Approximate Bilinear Programming (ABP) formulation. A drawback of this formulation is its computational complexity, which may be exponential. We show in Section 4 that this is unavoidable, because minimizing the approximation error bound is in fact NP-hard. Although our focus is on the formulation and its properties, we also discuss some simple algorithms for solving bilinear programs. Section 5 shows that ABP can be seen as an improvement of ALP and Approximate Policy Iteration (API). Section 6 demonstrates the applicability of ABP using a common reinforcement learning benchmark problem. A complete discussion of sampling strategies–an essential component for achieving robustness–is beyond the scope of this paper, but the issue is brieﬂy discussed in Section 6. Complete proofs of the theorems can be found in [19]. 2 Solving MDPs using ALP In this section, we formally deﬁne MDPs, their ALP formulation, and the approximation errors involved. These notions serve as a basis for developing the ABP formulation. A Markov Decision Process is a tuple (S, A, P, r, α), where S is the ﬁnite set of states, A is the ﬁnite set of actions. 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. 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