nips nips2007 nips2007-34 knowledge-graph by maker-knowledge-mining

34 nips-2007-Bayesian Policy Learning with Trans-Dimensional MCMC

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Author: Matthew Hoffman, Arnaud Doucet, Nando D. Freitas, Ajay Jasra

Abstract: A recently proposed formulation of the stochastic planning and control problem as one of parameter estimation for suitable artiﬁcial statistical models has led to the adoption of inference algorithms for this notoriously hard problem. At the algorithmic level, the focus has been on developing Expectation-Maximization (EM) algorithms. In this paper, we begin by making the crucial observation that the stochastic control problem can be reinterpreted as one of trans-dimensional inference. With this new interpretation, we are able to propose a novel reversible jump Markov chain Monte Carlo (MCMC) algorithm that is more efﬁcient than its EM counterparts. Moreover, it enables us to implement full Bayesian policy search, without the need for gradients and with one single Markov chain. The new approach involves sampling directly from a distribution that is proportional to the reward and, consequently, performs better than classic simulations methods in situations where the reward is a rare event.

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Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 uk Abstract A recently proposed formulation of the stochastic planning and control problem as one of parameter estimation for suitable artiﬁcial statistical models has led to the adoption of inference algorithms for this notoriously hard problem. [sent-14, score-0.257]

2 In this paper, we begin by making the crucial observation that the stochastic control problem can be reinterpreted as one of trans-dimensional inference. [sent-16, score-0.138]

3 With this new interpretation, we are able to propose a novel reversible jump Markov chain Monte Carlo (MCMC) algorithm that is more efﬁcient than its EM counterparts. [sent-17, score-0.592]

4 Moreover, it enables us to implement full Bayesian policy search, without the need for gradients and with one single Markov chain. [sent-18, score-0.379]

5 The new approach involves sampling directly from a distribution that is proportional to the reward and, consequently, performs better than classic simulations methods in situations where the reward is a rare event. [sent-19, score-0.555]

6 A large number of methods have been proposed in the literature relying on value function approximation and policy search; including [3, 10, 14, 16, 18]. [sent-22, score-0.388]

7 In this paper, we follow the policy learning approach because of its promise and remarkable success in complex domains; see for example [13, 15]. [sent-23, score-0.355]

8 Our work is strongly motivated by a recent formulation of stochastic planning and control problems as inference problems. [sent-24, score-0.193]

9 This line of work appears to have been initiated in [5], where the authors used EM as an alternative to standard stochastic gradient algorithms to maximize an expected cost. [sent-25, score-0.066]

10 In [2], a planning problem under uncertainty was solved using a Viterbi algorithm. [sent-26, score-0.083]

11 In these works, the standard discounted reward control problem was expressed in terms of an inﬁnite mixture of MDPs. [sent-30, score-0.272]

12 To make the problem tractable, the authors proposed to truncate the inﬁnite horizon time. [sent-31, score-0.138]

13 Here, we make the observation that, in this probabilistic interpretation of stochastic control, the objective function can be written as the expectation of a positive function with respect to a transdimensional probability distribution, i. [sent-32, score-0.092]

14 In this paper, we propose a full Bayesian policy search alternative to the EM algorithm. [sent-38, score-0.443]

15 In this approach, we set a prior distribution on the set of policy parameters and derive an artiﬁcial posterior distribution which is proportional to the prior times the expected reward. [sent-39, score-0.439]

16 We sample from the resulting artiﬁcial posterior distribution using a single transdimensional MCMC algorithm, which only involves a simple modiﬁcation of the MCMC algorithm developed to implement the EM. [sent-42, score-0.125]

17 Although the Bayesian policy search approach can beneﬁt from gradient information, it does not require gradients. [sent-43, score-0.433]

18 Moreover, since the target is proportional to the expected reward, the simulation is guided to areas of high reward automatically. [sent-44, score-0.257]

19 In the ﬁxed policy case, the value function is often computed using importance sampling. [sent-45, score-0.4]

20 In this context, our algorithm could be reinterpreted as an MCMC algorithm sampling from the optimal importance distribution. [sent-46, score-0.149]

22 is a discrete-time index, µ(·) is the initial state distribution, {Xn } is the X −valued state process, {An } is the A−valued action process, {Rn } is a positive real-valued reward process, fa denotes the transition density, ga the reward density and πθ is a randomized policy. [sent-50, score-0.639]

23 If we have a deterministic policy then πθ ( a| x) = δϕθ (x) (a). [sent-51, score-0.389]

24 In this case, the transition model fa ( ·| x) assumes the parametrization fθ ( ·| x). [sent-52, score-0.089]

25 The reward model could also be parameterized as gθ ( ·| x). [sent-53, score-0.23]

26 We are here interested in maximizing with respect to the parameters of the policy θ the expected future reward ∞ π Vµ (θ) = E γ n−1 Rn , n=1 where 0 < γ < 1 is a discount factor and the expectation is with respect to the probabilistic model deﬁned in (1). [sent-55, score-0.556]

27 As shown in [20], it is possible to re-write this objective of optimizing an inﬁnite horizon discounted reward MDP (where the reward happens at each step) as one of optimizing an inﬁnite mixture of ﬁnite horizon MDPs (where the reward only happens at the last time step). [sent-56, score-0.834]

28 Speciﬁcally we have: ∞ π Vµ (θ) = (1 − γ) −1 Epθ [RK ] = (1 − γ)−1 rk pθ (k, x1:k , a1:k , rk ) dx1:k da1:k drk k=1 2 (3) for a randomized policy. [sent-58, score-0.566]

29 Similarly, for a deterministic policy, the representation (3) also holds for the trans-dimensional probability distribution deﬁned on {k} × X k × R+ given by k pθ (k, x1:k , rk ) = (1 − γ) γ k−1 µ (x1 ) gθ ( rk | xk ) fθ ( xn | xn−1 ) . [sent-59, score-0.852]

30 pθ (K, X1:K , A1:K , RK ))] , e rk pθ (k, x1:k , a1:k , rk ) . [sent-62, score-0.53]

31 The standard Monte Carlo EM approach consists of sampling from pθ (k, x1:k , a1:k , rk ) using MCMC to obtain a Monte Carlo estimate of the Q function. [sent-65, score-0.29]

32 As pθ (k, x1:k , a1:k , rk ) is proportional to the reward, the samples will consequently be drawn in regions of high reward. [sent-66, score-0.348]

33 This is a particularly interesting feature in situations where the reward function is concentrated in a region of low probability mass under pθ (k, x1:k , rk ), which is often the case in high-dimensional control settings. [sent-67, score-0.537]

34 Note that if we wanted to estimate π Vµ (θ) using importance sampling, then the distribution pθ (k, x1:k , a1:k , rk ) corresponds to the optimal zero-variance importance distribution. [sent-68, score-0.445]

35 pθ (k, x1:k , a1:k , rk ) = Alternatively, instead of sampling from pθ (k, x1:k , a1:k , rk ) using MCMC, we could proceed as in [20] to derive forward-backward algorithms to implement the E-step which can be implemented here using Sequential Monte Carlo (SMC) techniques. [sent-69, score-0.579]

36 Finally, we remark that for a deterministic policy, we can introduce the trans-dimensional distribution: rk pθ (k, x1:k , rk ) pθ (k, x1:k , rk ) = Epθ [RK ] . [sent-73, score-0.829]

37 In addition, and for ease of presentation only, we focus the discussion on deterministic policies and reward functions gθ ( rn | xn ) = δr(xn ) (rn ) ; the extension of our algorithms to the randomized case is straightforward. [sent-74, score-0.404]

38 3 Bayesian policy exploration The EM algorithm is particularly sensitive to initialization and might get trapped in a severe loπ cal maximum of Vµ (θ). [sent-75, score-0.378]

39 Moreover, in the general state-space setting that we are considering, the particle smoothers in the E-step can be very expensive computationally. [sent-76, score-0.058]

40 The idea consists of introducing a vague prior distribution p (θ) on the parameters of the policy θ. [sent-79, score-0.426]

41 By construction, this target distribution admits the following marginal in θ π p (θ) ∝ Vµ (θ) p (θ) and we can select an improper prior distribution p (θ) ∝ 1 if 3 Θ π Vµ (θ) dθ < ∞. [sent-81, score-0.052]

42 If we could sample from p (θ), then the generated samples θ(i) would concentrate themselves π in regions where Vµ (θ) is large. [sent-82, score-0.078]

43 We cannot sample from p (θ) directly but we can developed a trans-dimensional MCMC algorithm which will generate asymptotically samples from p (θ, k, x1:k ), hence samples from p (θ). [sent-83, score-0.157]

44 Assume the current state of the Markov chain targeting p (θ, k, x1:k ) is (θ, k, x1:k ). [sent-85, score-0.167]

45 We propose ﬁrst to update the components (k, x1:k ) conditional upon θ using a combination of birth, death and update moves using the reversible jump MCMC algorithm [7, 8, 17]. [sent-86, score-0.849]

46 Then we propose to update θ conditional upon the current value of (k, x1:k ). [sent-87, score-0.083]

47 The details of the reversible jump algorithm are presented in the following section. [sent-91, score-0.459]

48 • If (u ≤ bk ) – then carry out a “birth” move: Increase the horizon length of the MDP, say k(i) = k(i−1) + 1 and insert a new state. [sent-96, score-0.291]

49 – else if (u ≤ bk + dk ) then carry out a “death” move: decrease the horizon length of the MDP, say k(i) = k(i−1) − 1 and an existing state. [sent-97, score-0.459]

50 (i) – else let k(i) = k(i−1) and generate samples x1:k(i) of the MDP states. [sent-98, score-0.051]

51 (i) • Sample the policy parameters θ(i) conditional on the samples (x1:k(i) , k(i) ). [sent-100, score-0.406]

52 Figure 1: Generic reversible jump MCMC for Bayesian policy learning. [sent-101, score-0.814]

53 We note that for a given θ the samples of the states and horizon generated by this Markov chain will also be distributed (asymptotically) according to the trans-dimensional distribution pθ (k, x1:k ). [sent-102, score-0.277]

54 The trans-dimensional simulation approach has the advantage that the samples will concentrate themselves automatically in regions where pθ (k) has high probability masses. [sent-105, score-0.102]

55 4 Trans-Dimensional Markov chain Monte Carlo We present a simple reversible jump method composed of two reversible moves (birth and death) and several update moves. [sent-107, score-0.931]

56 Assume the current state of the Markov chain targeting pθ (k, x1:k ) is (k, x1:k ). [sent-108, score-0.167]

57 With probability1 bk , we propose a birth move; that is we sample a location uniformly in the interval {1, . [sent-109, score-0.451]

58 , k + 1}, and propose the candidate (k + 1, x1:j−1 , x∗ , xj:k ) where X ∗ ∼ qθ ( ·| xj−1:j ). [sent-117, score-0.069]

59 This candidate is accepted with probability Abirth = min{1, αbirth } where we have for j ∈ {2, . [sent-118, score-0.085]

60 r (xk ) bk qθ ( x∗ | xk ) With probability dk , we propose a death move; that is J ∼ U {1, . [sent-123, score-0.792]

61 , k} and we propose the candidate (k − 1, x1:j−1 , xj+1:k ) which is accepted with probability Adeath = min{1, αdeath } where for j ∈ {2, . [sent-126, score-0.121]

62 γr (xk ) fθ ( xk | xk−1 ) dk The αbirth and αdeath terms derived above can be thought of as ratios between the distribution over the newly proposed state of the chain (i. [sent-130, score-0.489]

63 These terms must also ensure reversibility and the dimension-matching requirement for reversible jump MCMC. [sent-133, score-0.459]

64 αdeath = Finally with probability uk = 1 − bk − dk , we propose a standard (ﬁxed dimensional) move where we update all or a subset of the components x1:k using say Metropolis-Hastings or Gibbs moves. [sent-135, score-0.483]

65 If one adopts a simple one-at-a time Metropolis-Hastings scheme with proposals qθ ( x∗ | xj−1:j+1 ) to update the j-th term, then the candidate is accepted with probability Aupdate = min{1, αupdate } where for j ∈ {2, . [sent-138, score-0.132]

66 r (xk ) fθ ( xk | xk−1 ) qθ ( x∗ | xk−1:k ) (i) Under weak assumptions on the model, the Markov chain {K (i) , X1:K } generated by this transition kernel will be irreducible and aperiodic and hence will generate asymptotically samples from the target distribution pθ (k, x1:k ). [sent-142, score-0.416]

67 Hence the probability of having the reversible moves accepted will be reasonable. [sent-144, score-0.38]

68 Standard Bayesian applications of reversible jump MCMC usually do not enjoy this property and it makes it more difﬁcult to design fast mixing algorithms. [sent-145, score-0.496]

69 Ten sample points are shown distributed according to µ, the initial distribution, and the contour plot corresponds to the reward function r. [sent-164, score-0.229]

70 The red line denotes the policy parameterized by some angle θ, while a path is drawn in blue sampled from this policy. [sent-165, score-0.384]

71 We will consider state- and action-spaces X = A = R2 such that each state x ∈ X is a 2d position and each action a ∈ A is a vector corresponding to a change in position. [sent-167, score-0.052]

72 Finally we will let µ be a normal distribution about the origin, and consider a reward (as in [20]) given by 1 an unnormalized Gaussian about some point m, i. [sent-169, score-0.227]

73 For these experiments we chose a simple, stochastic policy parameterized by θ ∈ [0, 2π]. [sent-173, score-0.424]

74 Intuitively, this policy corresponds to choosing a direction θ in which the agent will walk. [sent-175, score-0.355]

75 For a state-space with initial distribution and reward function deﬁned as in Figure 2 the optimal policy corresponds to θ = π/4. [sent-177, score-0.609]

76 We ﬁrst implemented a simple SMC-based extension of the EM algorithm described in [20], wherein a particle ﬁlter was used for the forwards/backwards ﬁlters. [sent-178, score-0.058]

77 The ﬁrst thing of note is the terrible performance of the SMC-based algorithm—in fact we had to make the reward broader and closer to the initial position in order to ensure that the algorithm converges in a reasonable amount 2 of time. [sent-181, score-0.201]

78 This comes as no surprise considering the O(N 2 kmax ) time complexity necessary for computing the importance weights. [sent-182, score-0.136]

79 While there do exist methods [9] for reducing this complexity to 2 O(N log N kmax ), the discrepancy between this and the reversible jump MCMC method suggests that the MCMC approach may be more adapted to this class of problems. [sent-183, score-0.55]

80 In the ﬁnite/discrete case 2 it is also possible, as shown by Toussaint et al (2006), to reduce the kmax term to kmax by calculating updates only using messages from the backwards recursion. [sent-184, score-0.212]

81 The SMC method might further be improved by better choices for the artiﬁcial distribution ηn (xn ) in the backwards ﬁlter. [sent-185, score-0.056]

82 Also shown in ﬁgure 3 is the performance of a Monte Carlo EM algorithm using reversible jump MCMC in the E-step. [sent-188, score-0.459]

83 7 0 500 1000 1500 2000 2500 cpu time (in seconds) 0. [sent-203, score-0.058]

84 7 0 200 400 600 800 1000 cpu time (in seconds) Figure 3: The left ﬁgure shows estimates for the policy parameter θ as a function of the CPU time used to calculate that value. [sent-204, score-0.413]

85 In both plots a red line denotes the known optimal policy parameter of π/4. [sent-208, score-0.405]

86 Finally, we also compared the proposed Bayesian policy exploration method to the PEGASUS [14] approach using a local search method. [sent-211, score-0.407]

87 6 Discussion We believe that formulating stochastic control as a trans-dimensional inference problem is fruitful. [sent-215, score-0.111]

88 This formulation relies on minimal assumptions and allows us to apply modern inference algorithms to solve control problems. [sent-216, score-0.095]

89 We have focused here on Monte Carlo methods and have presented— to the best of our knowledge—the ﬁrst application of reversible jump MCMC to policy search. [sent-217, score-0.814]

90 Our results, on an illustrative example, showed that this trans-dimensional MCMC algorithm is more effective that standard policy search methods and alternative Monte Carlo methods relying on particle ﬁlters. [sent-218, score-0.498]

91 For such scenarios, we expect that it will be necessary to develop more efﬁcient MCMC strategies to explore the policy space efﬁciently. [sent-220, score-0.355]

92 7 Evolution of policy parameters against transition-model samples policy parameter (theta) 0. [sent-250, score-0.761]

93 20 1000 2000 3000 4000 5000 6000 7000 8000 number of samples taken from transition-model 9000 Figure 4: Convergence of PEGASUS and our Bayesian policy search algorithm when started from θ = 0 and converging to the optimum of θ∗ = π/4. [sent-257, score-0.458]

94 For our algorithm we plot samples taken directly from the MCMC algorithm itself: plotting the empirical average would produce an estimate whose convergence is almost immediate, but we also wanted to show the “burn-in” period. [sent-259, score-0.116]

95 For both algorithms lines denoting one standard deviation are shown and performance is plotted against the number of samples taken from the transition model. [sent-260, score-0.088]

96 On solving integral equations using Markov chain Monte Carlo methods. [sent-265, score-0.097]

97 Reversible jump Markov chain Monte Carlo computation and Bayesian model determination. [sent-270, score-0.261]

98 Fast particle smoothing: If i had a million particles. [sent-283, score-0.058]

99 Optimal Bayesian design by inhomogeneous Markov chain simuu o lation. [sent-299, score-0.134]

100 PEGASUS: A policy search method for large MDPs and POMDPs. [sent-320, score-0.407]

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Here V ∗ is called the optimal value function and a policy π ∗ that satisﬁes V π (x) = ∗ ∗ ∗ V (x) for all x ∈ X is called optimal. The optimal action-value function Q (x, a) is Q (x, a) = supπ Qπ (x, a). We say that a (deterministic stationary) policy π is greedy w.r.t. an action-value function Q ∈ B(X × A), and we write π = π (·; Q), if, for all x ∈ X , π(x) ∈ argmaxa∈A Q(x, a). ˆ Under mild technical assumptions, such a greedy policy always exists. Any greedy policy w.r.t. Q∗ is optimal. For π : X → A we deﬁne its evaluation operator, T π : B(X × A) → B(X × A), by (T π Q)(x, a) = r(x, a) + γ X Q(y, π(y)) P (dy|x, a). It is known that Qπ = T π Qπ . Further, if we let the Bellman operator, T : B(X × A) → B(X × A), deﬁned by (T Q)(x, a) = r(x, a) + γ X supb∈A Q(y, b) P (dy|x, a) then Q∗ = T Q∗ . It is known that V π and Qπ are bounded by Rmax /(1 − γ), just like Q∗ and V ∗ . For π : X → A, the operator E π : B(X × A) → B(X ) is deﬁned by (E π Q)(x) = Q(x, π(x)), while E : B(X × A) → B(X ) is deﬁned by (EQ)(x) = supa∈A Q(x, a). Throughout the paper F ⊂ {f : X × A → R} will denote a subset of real-valued functions over the state-action space X × A and Π ⊂ AX will be a set of policies. For ν ∈ M (X ) and f : X → R p measurable, we let (for p ≥ 1) f p,ν = X |f (x)|p ν(dx). We simply write f ν for f 2,ν . 2 Further, we extend · ν to F by f ν = A X |f |2 (x, a) dν(x) dλA (a), where λA is the uniform distribution over A. We shall use the shorthand notation νf to denote the integral f (x)ν(dx). We denote the space of bounded measurable functions with domain X by B(X ). Further, the space of measurable functions bounded by 0 < K < ∞ shall be denoted by B(X ; K). We let · ∞ denote the supremum norm. 2 Fitted Q-iteration with approximate policy maximization We assume that we are given a ﬁnite trajectory, {(Xt , At , Rt )}1≤t≤N , generated by some stochastic stationary policy πb , called the behavior policy: At ∼ πb (·|Xt ), Xt+1 ∼ P (·|Xt , At ), Rt ∼ def S(·|Xt , At ), where πb (·|x) is a density with π0 = inf (x,a)∈X ×A πb (a|x) > 0. The generic recipe for ﬁtted Q-iteration (FQI) [5] is Qk+1 = Regress(Dk (Qk )), (1) where Regress is an appropriate regression procedure and Dk (Qk ) is a dataset deﬁning a regression problem in the form of a list of data-point pairs: Dk (Qk ) = (Xt , At ), Rt + γ max Qk (Xt+1 , b) b∈A 1≤t≤N .1 Fitted Q-iteration can be viewed as approximate value iteration applied to action-value functions. To see this note that value iteration would assign the value (T Qk )(x, a) = r(x, a) + γ maxb∈A Qk (y, b) P (dy|x, a) to Qk+1 (x, a) [6]. Now, remember that the regression function for the jointly distributed random variables (Z, Y ) is deﬁned by the conditional expectation of Y given Z: m(Z) = E [Y |Z]. Since for any ﬁxed function Q, E [Rt + γ maxb∈A Q(Xt+1 , b)|Xt , At ] = (T Q)(Xt , At ), the regression function corresponding to the data Dk (Q) is indeed T Q and hence if FQI solved the regression problem deﬁned by Qk exactly, it would simulate value iteration exactly. However, this argument itself does not directly lead to a rigorous analysis of FQI: Since Qk is obtained based on the data, it is itself a random function. Hence, after the ﬁrst iteration, the “target” function in FQI becomes random. Furthermore, this function depends on the same data that is used to deﬁne the regression problem. Will FQI still work despite these issues? To illustrate the potential difﬁculties consider a dataset where X1 , . . . , XN is a sequence of independent random variables, which are all distributed uniformly at random in [0, 1]. Further, let M be a random integer greater than N which is independent of the dataset (Xt )N . Let U be another random variable, uniformly t=1 distributed in [0, 1]. Now deﬁne the regression problem by Yt = fM,U (Xt ), where fM,U (x) = sgn(sin(2M 2π(x + U ))). Then it is not hard to see that no matter how big N is, no procedure can 1 Since the designer controls Qk , we may assume that it is continuous, hence the maximum exists. 2 estimate the regression function fM,U with a small error (in expectation, or with high probability), even if the procedure could exploit the knowledge of the speciﬁc form of fM,U . On the other hand, if we restricted M to a ﬁnite range then the estimation problem could be solved successfully. The example shows that if the complexity of the random functions deﬁning the regression problem is uncontrolled then successful estimation might be impossible. Amongst the many regression methods in this paper we have chosen to work with least-squares methods. In this case Equation (1) takes the form N Qk+1 = argmin Q∈F t=1 1 πb (At |Xt ) 2 Q(Xt , At ) − Rt + γ max Qk (Xt+1 , b) b∈A . (2) We call this method the least-squares ﬁtted Q-iteration (LSFQI) method. Here we introduced the weighting 1/πb (At |Xt ) since we do not want to give more weight to those actions that are preferred by the behavior policy. Besides this weighting, the only parameter of the method is the function set F. This function set should be chosen carefully, to keep a balance between the representation power and the number of samples. As a speciﬁc example for F consider neural networks with some ﬁxed architecture. In this case the function set is generated by assigning weights in all possible ways to the neural net. Then the above minimization becomes the problem of tuning the weights. Another example is to use linearly parameterized function approximation methods with appropriately selected basis functions. In this case the weight tuning problem would be less demanding. Yet another possibility is to let F be an appropriate restriction of a Reproducing Kernel Hilbert Space (e.g., in a ball). In this case the training procedure becomes similar to LS-SVM training [7]. As indicated above, the analysis of this algorithm is complicated by the fact that the new dataset is deﬁned in terms of the previous iterate, which is already a function of the dataset. Another complication is that the samples in a trajectory are in general correlated and that the bias introduced by the imperfections of the approximation architecture may yield to an explosion of the error of the procedure, as documented in a number of cases in, e.g., [8]. Nevertheless, at least for ﬁnite action sets, the tools developed in [1, 3, 2] look suitable to show that under appropriate conditions these problems can be overcome if the function set is chosen in a judicious way. However, the results of these works would become essentially useless in the case of an inﬁnite number of actions since these previous bounds grow to inﬁnity with the number of actions. Actually, we believe that this is not an artifact of the proof techniques of these works, as suggested by the counterexample that involved random targets. The following result elaborates this point further: Proposition 2.1. Let F ⊂ B(X × A). Then even if the pseudo-dimension of F is ﬁnite, the fatshattering function of ∨ Fmax = VQ : VQ (·) = max Q(·, a), Q ∈ F a∈A 2 can be inﬁnite over (0, 1/2). Without going into further details, let us just note that the ﬁniteness of the fat-shattering function is a sufﬁcient and necessary condition for learnability and the ﬁniteness of the fat-shattering function is implied by the ﬁniteness of the pseudo-dimension [9].The above proposition thus shows that without imposing further special conditions on F, the learning problem may become infeasible. One possibility is of course to discretize the action space, e.g., by using a uniform grid. However, if the action space has a really high dimensionality, this approach becomes unfeasible (even enumerating 2dA points could be impossible when dA is large). Therefore we prefer alternate solutions. Another possibility is to make the functions in F, e.g., uniformly Lipschitz in their state coordinates. ∨ Then the same property will hold for functions in Fmax and hence by a classical result we can bound the capacity of this set (cf. pp. 353–357 of [10]). One potential problem with this approach is that this way it might be difﬁcult to get a ﬁne control of the capacity of the resulting set. 2 The proof of this and the other results are given in the appendix, available in the extended version of this paper, downloadable from http://hal.inria.fr/inria-00185311/en/. 3 In the approach explored here we modify the ﬁtted Q-iteration algorithm by introducing a policy set Π and a search over this set for an approximately greedy policy in a sense that will be made precise in a minute. Our algorithm thus has four parameters: F, Π, K, Q0 . Here F is as before, Π is a user-chosen set of policies (mappings from X to A), K is the number of iterations and Q0 is an initial value function (a typical choice is Q0 ≡ 0). The algorithm computes a sequence of iterates (Qk , πk ), k = 0, . . . , K, deﬁned by the following equations: ˆ N π0 ˆ = argmax π∈Π Q0 (Xt , π(Xt )), t=1 N Qk+1 = argmin Q∈F t=1 1 Q(Xt , At ) − Rt + γQk (Xt+1 , πk (Xt+1 )) ˆ πb (At |Xt ) 2 , (3) N πk+1 ˆ = argmax π∈Π Qk+1 (Xt , π(Xt )). (4) t=1 Thus, (3) is similar to (2), while (4) deﬁnes the policy search problem. The policy search will generally be solved by a gradient procedure or some other appropriate method. The cost of this step will be primarily determined by how well-behaving the iterates Qk+1 are in their action arguments. For example, if they were quadratic and if π was linear then the problem would be a quadratic optimization problem. However, except for special cases3 the action value functions will be more complicated, in which case this step can be expensive. Still, this cost could be similar to that of searching for the maximizing actions for each t = 1, . . . , N if the approximately maximizing actions are similar across similar states. This algorithm, which we could also call a ﬁtted actor-critic algorithm, will be shown to overcome the above mentioned complexity control problem provided that the complexity of Π is controlled appropriately. Indeed, in this case the set of possible regression problems is determined by the set ∨ FΠ = { V : V (·) = Q(·, π(·)), Q ∈ F, π ∈ Π } , ∨ and the proof will rely on controlling the complexity of FΠ by selecting F and Π appropriately. 3 3.1 The main theoretical result Outline of the analysis In order to gain some insight into the behavior of the algorithm, we provide a brief summary of its error analysis. The main result will be presented subsequently. For f ,Q ∈ F and a policy π, we deﬁne the tth TD-error as follows: dt (f ; Q, π) = Rt + γQ(Xt+1 , π(Xt+1 )) − f (Xt , At ). Further, we deﬁne the empirical loss function by 1 ˆ LN (f ; Q, π) = N N t=1 d2 (f ; Q, π) t , λ(A)πb (At |Xt ) where the normalization with λ(A) is introduced for mathematical convenience. Then (3) can be ˆ written compactly as Qk+1 = argminf ∈F LN (f ; Qk , πk ). ˆ ˆ The algorithm can then be motivated by the observation that for any f ,Q, and π, LN (f ; Q, π) is an unbiased estimate of def 2 L(f ; Q, π) = f − T π Q ν + L∗ (Q, π), (5) where the ﬁrst term is the error we are interested in and the second term captures the variance of the random samples: L∗ (Q, π) = E [Var [R1 + γQ(X2 , π(X2 ))|X1 , A1 = a]] dλA (a). A 3 Linear quadratic regulation is such a nice case. It is interesting to note that in this special case the obvious choices for F and Π yield zero error in the limit, as can be proven based on the main result of this paper. 4 ˆ This result is stated formally by E LN (f ; Q, π) = L(f ; Q, π). Since the variance term in (5) is independent of f , argminf ∈F L(f ; Q, π) = 2 π argminf ∈F f − T Q ν . Thus, if πk were greedy w.r.t. Qk then argminf ∈F L(f ; Qk , πk ) = ˆ ˆ 2 argminf ∈F f − T Qk ν . Hence we can still think of the procedure as approximate value iteration over the space of action-value functions, projecting T Qk using empirical risk minimization on the space F w.r.t. · ν distances in an approximate manner. Since πk is only approximately greedy, we ˆ will have to deal with both the error coming from the approximate projection and the error coming from the choice of πk . To make this clear, we write the iteration in the form ˆ ˆ ˆ Qk+1 = T πk Qk + εk = T Qk + εk + (T πk Qk − T Qk ) = T Qk + εk , def ˆ ˆ where εk is the error committed while computing T πk Qk , εk = T πk Qk − T Qk is the error committed because the greedy policy is computed approximately and εk = εk + εk is the total error of step k. Hence, in order to show that the procedure is well behaved, one needs to show that both errors are controlled and that when the errors are propagated through these equations, the resulting error stays controlled, too. Since we are ultimately interested in the performance of the policy obtained, we will also need to show that small action-value approximation errors yield small performance losses. For these we need a number of assumptions that concern either the training data, the MDP, or the function sets used for learning. 3.2 Assumptions 3.2.1 Assumptions on the training data We shall assume that the data is rich, is in a steady state, and is fast-mixing, where, informally, mixing means that future depends weakly on the past. Assumption A2 (Sample Path Properties) Assume that {(Xt , At , Rt )}t=1,...,N is the sample path of πb , a stochastic stationary policy. Further, assume that {Xt } is strictly stationary (Xt ∼ ν ∈ M (X )) and exponentially β-mixing with the actual rate given by the parameters (β, b, κ).4 We further assume that the sampling policy πb satisﬁes π0 = inf (x,a)∈X ×A πb (a|x) > 0. The β-mixing property will be used to establish tail inequalities for certain empirical processes.5 Note that the mixing coefﬁcients do not need to be known. In the case when no mixing condition is satisﬁed, learning might be impossible. To see this just consider the case when X1 = X2 = . . . = XN . Thus, in this case the learner has many copies of the same random variable and successful generalization is thus impossible. We believe that the assumption that the process is in a steady state is not essential for our result, as when the process reaches its steady state quickly then (at the price of a more involved proof) the result would still hold. 3.2.2 Assumptions on the MDP In order to prevent the uncontrolled growth of the errors as they are propagated through the updates, we shall need some assumptions on the MDP. A convenient assumption is the following one [11]: Assumption A3 (Uniformly stochastic transitions) For all x ∈ X and a ∈ A, assume that P (·|x, a) is absolutely continuous w.r.t. ν and the Radon-Nikodym derivative of P w.r.t. ν is bounded def < +∞. uniformly with bound Cν : Cν = supx∈X ,a∈A dP (·|x,a) dν ∞ Note that by the deﬁnition of measure differentiation, Assumption A3 means that P (·|x, a) ≤ Cν ν(·). This assumption essentially requires the transitions to be noisy. We will also prove (weaker) results under the following, weaker assumption: 4 For the deﬁnition of β-mixing, see e.g. [2]. We say “empirical process” and “empirical measure”, but note that in this work these are based on dependent (mixing) samples. 5 5 Assumption A4 (Discounted-average concentrability of future-state distributions) Given ρ, ν, m ≥ 1 and an arbitrary sequence of stationary policies {πm }m≥1 , assume that the futuredef state distribution ρP π1 P π2 . . . P πm is absolutely continuous w.r.t. ν. Assume that c(m) = π1 π2 πm def satisﬁes m≥1 mγ m−1 c(m) < +∞. We shall call Cρ,ν = supπ1 ,...,πm d(ρP Pdν ...P ) ∞ max (1 − γ)2 m≥1 mγ m−1 c(m), (1 − γ) m≥1 γ m c(m) the discounted-average concentrability coefﬁcient of the future-state distributions. The number c(m) measures how much ρ can get ampliﬁed in m steps as compared to the reference distribution ν. Hence, in general we expect c(m) to grow with m. In fact, the condition that Cρ,µ is ﬁnite is a growth rate condition on c(m). Thanks to discounting, Cρ,µ is ﬁnite for a reasonably large class of systems (see the discussion in [11]). A related assumption is needed in the error analysis of the approximate greedy step of the algorithm: Assumption A5 (The random policy “makes no peak-states”) Consider the distribution µ = (ν × λA )P which is the distribution of a state that results from sampling an initial state according to ν and then executing an action which is selected uniformly at random.6 Then Γν = dµ/dν ∞ < +∞. Note that under Assumption A3 we have Γν ≤ Cν . This (very mild) assumption means that after one step, starting from ν and executing this random policy, the probability of the next state being in a set is upper bounded by Γν -times the probability of the starting state being in the same set. def Besides, we assume that A has the following regularity property: Let Py(a, h, ρ) = (a , v) ∈ RdA +1 : a − a 1 ≤ ρ, 0 ≤ v/h ≤ 1 − a − a 1 /ρ denote the pyramid with hight h and base given by the 1 def -ball B(a, ρ) = a ∈ RdA : a − a 1 ≤ρ centered at a. Assumption A6 (Regularity of the action space) We assume that there exists α > 0, such that for all a ∈ A, for all ρ > 0, λ(Py(a, 1, ρ) ∩ (A × R)) λ(A) ≥ min α, λ(Py(a, 1, ρ)) λ(B(a, ρ)) For example, if A is an 1 . -ball itself, then this assumption will be satisﬁed with α = 2−dA . Without assuming any smoothness of the MDP, learning in inﬁnite MDPs looks hard (see, e.g., [12, 13]). Here we employ the following extra condition: Assumption A7 (Lipschitzness of the MDP in the actions) Assume that the transition probabilities and rewards are Lipschitz w.r.t. their action variable, i.e., there exists LP , Lr > 0 such that for all (x, a, a ) ∈ X × A × A and measurable set B of X , |P (B|x, a) − P (B|x, a )| ≤ LP a − a 1 , |r(x, a) − r(x, a )| ≤ Lr a − a 1 . Note that previously Lipschitzness w.r.t. the state variables was used, e.g., in [11] to construct consistent planning algorithms. 3.2.3 Assumptions on the function sets used by the algorithm These assumptions are less demanding since they are under the control of the user of the algorithm. However, the choice of these function sets will greatly inﬂuence the performance of the algorithm, as we shall see it from the bounds. The ﬁrst assumption concerns the class F: Assumption A8 (Lipschitzness of candidate action-value functions) Assume F ⊂ B(X × A) and that any elements of F is uniformly Lipschitz in its action-argument in the sense that |Q(x, a) − Q(x, a )| ≤ LA a − a 1 holds for any x ∈ X , a,a ∈ A, and Q ∈ F . 6 Remember that λA denotes the uniform distribution over the action set A. 6 We shall also need to control the capacity of our function sets. We assume that the reader is familiar with the concept of VC-dimension.7 Here we use the pseudo-dimension of function sets that builds upon the concept of VC-dimension: Deﬁnition 3.1 (Pseudo-dimension). The pseudo-dimension VF + of F is deﬁned as the VCdimension of the subgraphs of functions in F (hence it is also called the VC-subgraph dimension of F). Since A is multidimensional, we deﬁne VΠ+ to be the sum of the pseudo-dimensions of the coordinate projection spaces, Πk of Π: dA V Π+ = VΠ + , k=1 k Πk = { πk : X → R : π = (π1 , . . . , πk , . . . , πdA ) ∈ Π } . Now we are ready to state our assumptions on our function sets: Assumption A9 (Capacity of the function and policy sets) Assume that F ⊂ B(X × A; Qmax ) for Qmax > 0 and VF + < +∞. Also, A ⊂ [−A∞ , A∞ ]dA and VΠ+ < +∞. Besides their capacity, one shall also control the approximation power of the function sets involved. Let us ﬁrst consider the policy set Π. Introduce e∗ (F, Π) = sup inf ν(EQ − E π Q). Q∈F π∈Π Note that inf π∈Π ν(EQ − E π Q) measures the quality of approximating νEQ by νE π Q. Hence, e∗ (F, Π) measures the worst-case approximation error of νEQ as Q is changed within F. This can be made small by choosing Π large. Another related quantity is the one-step Bellman-error of F w.r.t. Π. This is deﬁned as follows: For a ﬁxed policy π, the one-step Bellman-error of F w.r.t. T π is deﬁned as E1 (F; π) = sup inf Q∈F Q ∈F Q − T πQ ν . Taking again a pessimistic approach, the one-step Bellman-error of F is deﬁned as E1 (F, Π) = sup E1 (F; π). π∈Π Typically by increasing F, E1 (F, Π) can be made smaller (this is discussed at some length in [3]). However, it also holds for both Π and F that making them bigger will increase their capacity (pseudo-dimensions) which leads to an increase of the estimation errors. Hence, F and Π must be selected to balance the approximation and estimation errors, just like in supervised learning. 3.3 The main result Theorem 3.2. Let πK be a greedy policy w.r.t. QK , i.e. πK (x) ∈ argmaxa∈A QK (x, a). Then under Assumptions A1, A2, and A5–A9, for all δ > 0 we have with probability at least 1 − δ: given Assumption A3 (respectively A4), V ∗ − V πK ∞ (resp. V ∗ − V πK 1,ρ ), is bounded by    d 1+1 κ+1   A   4κ (log N + log(K/δ))  + γK , C E1 (F, Π) + e∗ (F, Π) + 1/4   N   where C depends on dA , VF + , (VΠ+ )dA , γ, κ, b, β, Cν (resp. Cρ,ν ), Γν , LA , LP ,Lr , α, λ(A), π0 , k=1 k κ+1 ˆ Qmax , Rmax , Rmax , and A∞ . In particular, C scales with V 4κ(dA +1) , where V = 2VF + + VΠ+ plays the role of the “combined effective” dimension of F and Π. 7 Readers not familiar with VC-dimension are suggested to consult a book, such as the one by Anthony and Bartlett [14]. 7 4 Discussion We have presented what we believe is the ﬁrst ﬁnite-time bounds for continuous-state and actionspace RL that uses value functions. Further, this is the ﬁrst analysis of ﬁtted Q-iteration, an algorithm that has proved to be useful in a number of cases, even when used with non-averagers for which no previous theoretical analysis existed (e.g., [15, 16]). In fact, our main motivation was to show that there is a systematic way of making these algorithms work and to point at possible problem sources the same time. We discussed why it can be difﬁcult to make these algorithms work in practice. We suggested that either the set of action-value candidates has to be carefully controlled (e.g., assuming uniform Lipschitzness w.r.t. the state variables), or a policy search step is needed, just like in actorcritic algorithms. The bound in this paper is similar in many respects to a previous bound of a Bellman-residual minimization algorithm [2]. It looks that the techniques developed here can be used to obtain results for that algorithm when it is applied to continuous action spaces. Finally, although we have not explored them here, consistency results for FQI can be obtained from our results using standard methods, like the methods of sieves. We believe that the methods developed here will eventually lead to algorithms where the function approximation methods are chosen based on the data (similar to adaptive regression methods) so as to optimize performance, which in our opinion is one of the biggest open questions in RL. Currently we are exploring this possibility. Acknowledgments Andr´ s Antos would like to acknowledge support for this project from the Hungarian Academy of Sciences a (Bolyai Fellowship). Csaba Szepesv´ ri greatly acknowledges the support received from the Alberta Ingenuity a Fund, NSERC, the Computer and Automation Research Institute of the Hungarian Academy of Sciences. References [1] A. Antos, Cs. Szepesv´ ri, and R. Munos. Learning near-optimal policies with Bellman-residual minia mization based ﬁtted policy iteration and a single sample path. In COLT-19, pages 574–588, 2006. [2] A. Antos, Cs. Szepesv´ ri, and R. Munos. Learning near-optimal policies with Bellman-residual minia mization based ﬁtted policy iteration and a single sample path. Machine Learning, 2007. (accepted). [3] A. Antos, Cs. Szepesv´ ri, and R. Munos. Value-iteration based ﬁtted policy iteration: learning with a a single trajectory. In IEEE ADPRL, pages 330–337, 2007. [4] D. P. Bertsekas and S.E. Shreve. Stochastic Optimal Control (The Discrete Time Case). Academic Press, New York, 1978. [5] D. Ernst, P. Geurts, and L. Wehenkel. Tree-based batch mode reinforcement learning. Journal of Machine Learning Research, 6:503–556, 2005. [6] R.S. Sutton and A.G. Barto. Reinforcement Learning: An Introduction. Bradford Book. MIT Press, 1998. [7] N. Cristianini and J. Shawe-Taylor. An introduction to support vector machines (and other kernel-based learning methods). Cambridge University Press, 2000. [8] J.A. Boyan and A.W. Moore. Generalization in reinforcement learning: Safely approximating the value function. In NIPS-7, pages 369–376, 1995. [9] P.L. Bartlett, P.M. Long, and R.C. Williamson. Fat-shattering and the learnability of real-valued functions. Journal of Computer and System Sciences, 52:434–452, 1996. [10] A.N. Kolmogorov and V.M. Tihomirov. -entropy and -capacity of sets in functional space. American Mathematical Society Translations, 17(2):277–364, 1961. [11] R. Munos and Cs. Szepesv´ ri. Finite time bounds for sampling based ﬁtted value iteration. Technical a report, Computer and Automation Research Institute of the Hungarian Academy of Sciences, Kende u. 13-17, Budapest 1111, Hungary, 2006. [12] A.Y. Ng and M. Jordan. PEGASUS: A policy search method for large MDPs and POMDPs. In Proceedings of the 16th Conference in Uncertainty in Artiﬁcial Intelligence, pages 406–415, 2000. [13] P.L. Bartlett and A. Tewari. Sample complexity of policy search with known dynamics. In NIPS-19. MIT Press, 2007. [14] M. Anthony and P. L. Bartlett. Neural Network Learning: Theoretical Foundations. Cambridge University Press, 1999. [15] M. Riedmiller. Neural ﬁtted Q iteration – ﬁrst experiences with a data efﬁcient neural reinforcement learning method. In 16th European Conference on Machine Learning, pages 317–328, 2005. [16] S. Kalyanakrishnan and P. Stone. Batch reinforcement learning in a complex domain. In AAMAS-07, 2007. 8

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