nips nips2008 nips2008-195 knowledge-graph by maker-knowledge-mining

195 nips-2008-Regularized Policy Iteration

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Author: Amir M. Farahmand, Mohammad Ghavamzadeh, Shie Mannor, Csaba Szepesvári

Abstract: In this paper we consider approximate policy-iteration-based reinforcement learning algorithms. In order to implement a ﬂexible function approximation scheme we propose the use of non-parametric methods with regularization, providing a convenient way to control the complexity of the function approximator. We propose two novel regularized policy iteration algorithms by adding L2 -regularization to two widely-used policy evaluation methods: Bellman residual minimization (BRM) and least-squares temporal difference learning (LSTD). We derive efﬁcient implementation for our algorithms when the approximate value-functions belong to a reproducing kernel Hilbert space. We also provide ﬁnite-sample performance bounds for our algorithms and show that they are able to achieve optimal rates of convergence under the studied conditions. 1

Reference: text

Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 We propose two novel regularized policy iteration algorithms by adding L2 -regularization to two widely-used policy evaluation methods: Bellman residual minimization (BRM) and least-squares temporal difference learning (LSTD). [sent-3, score-1.497]

2 We derive efﬁcient implementation for our algorithms when the approximate value-functions belong to a reproducing kernel Hilbert space. [sent-4, score-0.1]

3 We also provide ﬁnite-sample performance bounds for our algorithms and show that they are able to achieve optimal rates of convergence under the studied conditions. [sent-5, score-0.145]

4 1 Introduction A key idea in reinforcement learning (RL) is to learn an action-value function which can then be used to derive a good control policy [15]. [sent-6, score-0.588]

5 When the state space is large or inﬁnite, value-function approximation techniques are necessary, and their quality has a major impact on the quality of the learned policy. [sent-7, score-0.061]

6 , Chapter 8 of [15]), kernel regression [12], regression tree methods [5], and neural networks (e. [sent-10, score-0.105]

7 If the problem is easier (than some other problem), the method should deliver better solution(s) with the same amount of data. [sent-17, score-0.062]

8 Smoothness is one of the most important regularities: In regression when the target function has smoothness of order p the optimal rate of convergence of the squared L2 -error is n−2p/(2p+d) , ∗ Csaba Szepesv´ ri is on leave from MTA SZTAKI. [sent-22, score-0.279]

9 Hence, the rate of convergence is higher for larger p’s. [sent-26, score-0.078]

10 Methods that achieve the optimal rate are more desirable, at least in the limit for large n, and seem to perform well in practice. [sent-27, score-0.099]

11 However, only a few methods in the regression literature are known to achieve the optimal rates. [sent-28, score-0.092]

12 In fact, it is known that tree methods with averaging in the leaves, linear methods with piecewise constant basis functions, and kernel estimates do not achieve the optimal rate, while neural networks and regularized least-squares estimators do [7]. [sent-29, score-0.175]

13 An advantage of using a regularized least-squares estimator compared to neural networks is that these estimators do not get stuck in local minima and therefore their training is more reliable. [sent-30, score-0.086]

14 The problem setting is to ﬁnd a good policy in a batch or active learning scenario for inﬁnite-horizon expected total discounted reward Markovian decision problems with continuous state and ﬁnite action spaces. [sent-32, score-0.575]

15 We propose two novel policy evaluation algorithms by adding L2 -regularization to two widely-used policy evaluation methods in RL: Bellman residual minimization (BRM) [16; 3] and least-squares temporal difference learning (LSTD) [4]. [sent-33, score-1.334]

16 We show how our algorithms can be implemented efﬁciently when the value-function approximator belongs to a reproducing kernel Hilbert space. [sent-34, score-0.136]

17 In particular, we show that they are able to achieve a rate that is as good as the corresponding regression rate when the value functions belong to a known smoothness class. [sent-36, score-0.264]

18 We further show that this rate of convergence carries through to the performance of a policy found by running policy iteration with our regularized policy evaluation methods. [sent-37, score-1.881]

19 The results indicate that from the point of view of convergence rates RL is not harder than regression estimation, answering an open question of Antos et al. [sent-38, score-0.071]

20 To our best knowledge this is the ﬁrst work that addresses ﬁnite-sample performance of a regularized RL algorithm. [sent-41, score-0.086]

21 While regularization in RL has not been thoroughly explored, there has been a few works that used regularization. [sent-42, score-0.055]

22 Jung and Polani [9] explored adding regularization to BRM, but this solution is restricted to deterministic problems. [sent-49, score-0.055]

23 For a measurable space with domain S, we let M(S) and B(S; L) denote the set of probability measures over S and the space of bounded measurable functions with domain S and bound 0 < L < ∞, respectively. [sent-54, score-0.158]

24 , aM } is the ﬁnite set of M actions, P : X × A → M(X ) is the transition probability kernel with P (·|x, a) deﬁning the next-state distribution upon taking action a in state x, S(·|x, a) gives the corresponding distribution of immediate rewards, and γ ∈ (0, 1) is a discount factor. [sent-60, score-0.09]

25 We make the following assumptions on MDP: Assumption A1 (MDP Regularity) The set of states X is a compact subspace of the d-dimensional Euclidean space and the expected immediate rewards r(x, a) = rS(dr|x, a) are bounded by Rmax . [sent-61, score-0.111]

26 A policy is deterministic if it is a mapping from states to actions π : X → A. [sent-63, score-0.536]

27 The value and the action-value functions of a policy π, denoted respectively by V π and Qπ , are deﬁned as the expected sum of discounted rewards that are encountered when the policy π is executed: ∞ ∞ V π (x) = Eπ γ t Rt X 0 = x , Qπ (x, a) = Eπ γ t Rt X0 = x, A0 = a . [sent-64, score-1.116]

28 t=0 t=0 Here Rt denotes the reward received at time step t; Rt ∼ S(·|Xt , At ), Xt evolves according to Xt+1 ∼ P (·|Xt , At ), and At is sampled from the policy At ∼ π(·|Xt ). [sent-65, score-0.512]

29 It is easy to see that for any policy π, the functions V π and Qπ are bounded by Vmax = Qmax = Rmax /(1−γ). [sent-66, score-0.542]

30 The action-value function of a policy is the unique ﬁxed-point of the Bellman operator T π : B(X × A) → B(X × A) deﬁned by (T π Q)(x, a) = r(x, a) + γ Q(y, π(y))P (dy|x, a). [sent-67, score-0.568]

31 Given an MDP, the goal is to ﬁnd a policy that attains the best possible values, V ∗ (x) = supπ V π (x), ∀x ∈ X . [sent-68, score-0.512]

32 We say that a deterministic policy π is greedy w. [sent-71, score-0.56]

33 Greedy policies are important because any greedy policy w. [sent-75, score-0.602]

34 In this paper we shall deal with a variant of the policy iteration algorithm [8]. [sent-80, score-0.641]

35 In the basic version of policy iteration an optimal policy is found by computing a series of policies, each being greedy w. [sent-81, score-1.227]

36 Throughout the paper we denote by F M ⊂ { f : X × A → R } some subset of real-valued functions over the state-action space X × A, and use it as the set of admissible functions in the optimization problems of our algorithms. [sent-85, score-0.165]

37 (1) to F M by f 2 ν = 1 M M j=1 fj f 2 ν 2 ν,n , and = 1 nM n M 2 I{At =at } fj (Xt ) = t=1 j=1 1 nM n f 2 (Xt , At ), (2) t=1 where I{·} is the indicator function: for an event E, I{E} = 1 if and only if E holds and I{E} = 0, otherwise. [sent-94, score-0.078]

38 3 Approximate Policy Evaluation The ability to evaluate a given policy is the core requirement to run policy iteration. [sent-95, score-1.024]

39 Loosely speaking, in policy evaluation the goal is to ﬁnd a “close enough” approximation V (or Q) of the value (or action-value) function of a ﬁxed target policy π, V π (or Qπ ). [sent-96, score-1.133]

40 Most often these values have to be inferred from the observed system dynamics, where the observations do not necessarily come from following the target policy π. [sent-100, score-0.539]

41 Many policy evaluation techniques have been developed in RL. [sent-103, score-0.564]

42 1 Bellman Residual Minimization The idea of Bellman residual minimization (BRM) goes back at least to the work of Schweitzer and Seidmann [14]. [sent-106, score-0.135]

43 The basic idea of BRM comes from writing the ﬁxed-point equation for the Bellman operator in the form Qπ − T π Qπ = 0. [sent-108, score-0.056]

44 The resulting quantity, Q − T π Q, is called the Bellman residual of Q. [sent-110, score-0.094]

45 It seems, however, more natural to use a weighted L2 -norm to measure the magnitude of the Bellman residual as it leads to an optimization problem with favorable characteristics and enables an easy connection to regression 2 function estimation [7]. [sent-115, score-0.16]

46 Hence, we deﬁne the loss function LBRM (Q; π) = Q − T π Q ν , where ν is the stationary distribution of states in the input data. [sent-116, score-0.081]

47 They showed that optimizing the new loss function still makes sense and the empirical version of this loss is unbiased. [sent-133, score-0.062]

48 (4) requires solving the following nested optimization problems: ˆ h∗ = argmin h − T π Q Q h∈F M 2 , ν ˆ QBRM = argmin Q∈F M ˆ Q − T πQ 2 ν ˆ − h∗ − T π Q Q 2 . [sent-135, score-0.323]

49 They called the resulting algorithm least-squares policy iteration (LSPI), which is an approximate policy iteration algorithm based on LSTD. [sent-139, score-1.282]

50 Unlike BRM that minimizes the distance of Q and T π Q, LSTD minimizes the distance of Q and ΠT π Q, the back-projection of the image of Q under the Bellman operator, T π Q, onto the space of admissible functions F M (see Figure 1). [sent-140, score-0.163]

51 Formally, this means that LSTD minimizes the loss 2 function LLST D (Q; π) = Q − ΠT π Q ν . [sent-141, score-0.06]

52 It can also be seen as ﬁnding a good approximation for π the ﬁxed-point of operator ΠT . [sent-142, score-0.086]

53 The projection operator Π : B(X × A) → B(X × A) is deﬁned 2 by Πf = argminh∈F M h − f ν . [sent-143, score-0.056]

54 Figure 1: This ﬁgure shows the loss functions minimized by BRM, modiﬁed BRM, and LSTD methods. [sent-146, score-0.086]

55 4 Regularized Policy Iteration Algorithms In this section, we introduce two regularized policy iteration algorithms. [sent-158, score-0.727]

56 These algorithms are instances of the generic policy- iteration method, whose pseudo- code is shown in Table 1. [sent-159, score-0.158]

57 By assumption, the training sample Di used at the ith (1 ≤ i ≤ N ) iteration of the algorithm is a ﬁnite trajectory {(Xt , At , Rt )}1≤t≤n generated by a policy π, thus, At = π(Xt ) FittedPolicyQ(N ,Q(−1) ,PEval) and Rt ∼ S(·|Xt , At ). [sent-160, score-0.641]

58 Examples // N : number of iterations of such policy π are πi plus some // Q(−1) : Initial action- value function exploration or some stochastic // PEval: Fitting procedure stationary policy πb . [sent-161, score-1.084]

59 The actionfor i = 0 to N − 1 do value function Q(−1) is used to ˆ πi (·) ← π (·; Q(i−1) ) // the greedy policy w. [sent-162, score-0.56]

60 The proceend for dure PEval takes a policy πi (here ˆ return Q(N −1) or πN (·) = π (·; Q(N −1) ) the greedy policy w. [sent-167, score-1.072]

61 the current (i−1) action- value function Q ) along with training sample Di , Table 1: The pseudo- code of policy- iteration algorithm and returns an approximation to the action- value function of policy πi . [sent-170, score-0.671]

62 In this paper, we propose two approaches, one based on regularized (modiﬁed) BRM (REG- BRM), and one based on regularized LSTD (REG- LSTD). [sent-172, score-0.172]

63 , norms), and λh,n , λQ,n > 0 are regularization coefﬁcients. [sent-175, score-0.055]

64 In REG- LSTD, the next iteration is computed by solving the following nested optimization problems: ‚2 i h‚ h ‚ ‚ ˆ h∗ (·; Q) = argmin ‚h − T πi Q‚ + λh,n J(h) , Q(i) = argmin Q − h∗ (·; Q) h∈F M n Q∈F M 2 n i + λQ,n J(Q) . [sent-176, score-0.452]

65 (8) It is important to note that unlike the non- regularized case described in Sections 3. [sent-177, score-0.086]

66 This is because, although the ﬁrst equations ˆ in (7) and (8) are the same, the projected vector h∗ (·; Q) − T πi Q is not necessarily perpendicular to the admissible function space F M . [sent-180, score-0.102]

67 This is due to the regularization term λh,n J(h). [sent-181, score-0.055]

68 As a result, the ‚ ‚2 ‚ ‚2 ˆ Pythagorean theorem does not hold: Q − h∗ (·; Q) 2 = ‚ − T πi Q‚ − ‚ ∗ (·; Q) − T πi Q‚ , and ‚h ‚ ‚Q ˆ ‚ therefore the objective functions of the second equations in (7) and (8) are not equal and they do not have the same solution. [sent-182, score-0.066]

69 We can obtain Q(i) by solving the regularization problems of Eqs. [sent-184, score-0.079]

70 (7) and (8) in a reproducing kernel Hilbert space (RKHS) deﬁned by a Mercer kernel K. [sent-185, score-0.135]

71 In this case, we let the regularization 2 2 terms J(h) and J(Q) be the RKHS norms of h and Q, h H and Q H , respectively. [sent-186, score-0.1]

72 This is not immediate, because the solutions of these procedures are deﬁned with nested optimization problems. [sent-188, score-0.087]

73 ˜ ˜ k(Zi , Z 5 Theoretical Analysis of the Algorithms In this section, we analyze the statistical properties of the policy iteration algorithms based on REGBRM and REG-LSTD. [sent-199, score-0.67]

74 We provide ﬁnite-sample convergence results for the error between QπN , the action-value function of policy πN , the policy resulted after N iterations of the algorithms, and Q∗ , the optimal action-value function. [sent-200, score-1.119]

75 using a ﬁxed distribution over states ν and a ﬁxed stochastic policy πb , i. [sent-205, score-0.536]

76 samples, where Zt = t=1 (Xt , At ), Zt = Xt , π(Xt ) , Xt ∼ ν ∈ M(X ), At ∼ πb (·|Xt ), Xt ∼ P (·|Xt , At ), and π is the policy being evaluated. [sent-210, score-0.512]

77 (2) The function space F used in the optimization problems (7) and (8) is a Sobolev space Wk (Rd ) with 2k > d. [sent-212, score-0.092]

78 This assumption is used mainly for simplifying the proofs and can be extended to the case where the training sample is a single trajectory generated by a ﬁxed policy with appropriate mixing conditions as was done in [2]. [sent-223, score-0.512]

79 (2) Using Sobolev space allows us to explicitly show the effect of smoothness k on the convergence rate of our algorithms and to make comparison with the regression learning settings. [sent-224, score-0.256]

80 , they will be looked as “simple”, while they would be viewed as “complex” functions in H¨ lder spaces. [sent-228, score-0.06]

81 This is a standard assumption when studying the rate of convergence in supervised learning [7]. [sent-233, score-0.106]

82 If the solutions of our optimization problems are not bounded, they must be truncated, and thus, truncation arguments must be used in the analysis. [sent-237, score-0.056]

83 We now ﬁrst derive an upper bound on the policy evaluation error in Theorem 2. [sent-239, score-0.564]

84 We then show how the policy evaluation error propagates through the iterations of policy iteration in Lemma 3. [sent-240, score-1.239]

85 Choosing λQ,n = ` log(n) ´ 2k c1 nJ 2 (Qπ ) 2k+d and λh,n = Θ(λQ,n ), for any policy π, the following holds with probability at k least 1 − δ, for c1 , c2 , c3 , c4 > 0. [sent-244, score-0.512]

86 ‚Q − T Q‚ ≤ c2 Jk (Q ) 2k+d n n ν Theorem 2 shows how the number of samples and the difﬁculty of the problem as characterized 2 ˆ by Jk (Qπ ) inﬂuence the policy evaluation error. [sent-246, score-0.588]

87 With a large number of samples, we expect ||Q − π ˆ 2 ˆ T Q||ν to be small with high probability, where π is the policy being evaluated and Q is its estimated action-value function using REG-BRM or REG-LSTD. [sent-247, score-0.512]

88 , N − 1 denote the estimated action-value function and the Bellman residual at the ith iteration of our algorithms. [sent-251, score-0.223]

89 Theorem 2 indicates that at each iteration 2 ˆ i, the optimization procedure ﬁnds a function Q(i) such that εi ν is small with high probability. [sent-252, score-0.159]

90 Lemma 3, which was stated as Lemma 12 in [2], bounds the ﬁnal error after N iterations as a ˆ function of the intermediate errors. [sent-253, score-0.059]

91 Theorem 4 shows the effect of number of samples n, degree of smoothness k, number of iterations N , and concentrability coefﬁcients on the quality of the policy induced by the estimated actionvalue function. [sent-259, score-0.697]

92 We then showed how these algorithms can be implemented efﬁciently when the valuefunction approximation belongs to a reproducing kernel Hilbert space (RKHS). [sent-262, score-0.161]

93 We also proved ﬁnite-sample performance bounds for REG-BRM and REG-LSTD, and the regularized policy iteration algorithms built on top of them. [sent-263, score-0.781]

94 Our theoretical results indicate that our methods are able to achieve the optimal rate of convergence under the studied conditions. [sent-264, score-0.134]

95 One of the remaining problems is how to ﬁnd the regularization parameters: λh,n and λQ,n . [sent-265, score-0.055]

96 Here we used L2 -regularization, however, the idea can be extended naturally to L1 regularization in the style of Lasso, opening up the possibility of procedures that can handle a high number of irrelevant features. [sent-268, score-0.055]

97 Here the interesting question is if it is possible to achieve better rates than the one currently available in the literature, which scales quite unfavorably with the dimension of the action space [1]. [sent-275, score-0.091]

98 Learning near-optimal policies with Bellman-residual minia mization based ﬁtted policy iteration and a single sample path. [sent-286, score-0.683]

99 Neural ﬁtted Q iteration – ﬁrst experiences with a data efﬁcient neural reinforcement learning method. [sent-350, score-0.205]

100 Tight performance bounds on greedy policies based on imperfect value functions. [sent-371, score-0.115]

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Prior to the current work, the possibility of instability could not be avoided whenever four individually desirable algorithmic features were combined: 1) off-policy updates, 2) temporal-difference learning, 3) linear function approximation, and 4) linear complexity in memory and per-time-step computation. If any one of these four is abandoned, then stable methods can be obtained relatively easily. But each feature brings value and practitioners are loath to give any of them up, as we discuss later in a penultimate related-work section. In this paper we present the ﬁrst algorithm to achieve all four desirable features and be stable and convergent for all ﬁnite Markov decision processes, all target and behavior policies, and all feature representations for the linear approximator. Moreover, our algorithm does not use importance sampling and can be expected to be much better conditioned and of lower variance than importance sampling methods. Our algorithm can be viewed as performing stochastic gradient-descent in a novel objective function whose optimum is the least-squares TD solution. Our algorithm is also incremental and suitable for online use just as are simple temporaldifference learning algorithms such as Q-learning and TD(λ) (Sutton 1988). Our algorithm can be broadly characterized as a gradient-descent version of TD(0), and accordingly we call it GTD(0). 2 Sub-sampling and i.i.d. formulations of temporal-difference learning In this section we formulate the off-policy policy-evaluation problem for one-step temporaldifference learning such that the data consists of independent, identically-distributed (i.i.d.) samples. We start by considering the standard reinforcement learning framework, in which a learning agent interacts with an environment consisting of a ﬁnite Markov decision process (MDP). At each of a sequence of discrete time steps, t = 1, 2, . . ., the environment is in a state st ∈ S, the agent chooses an action at ∈ A, and then the environment emits a reward rt ∈ R, and transitions to its next state st+1 ∈ S. The state and action sets are ﬁnite. State transitions are stochastic and dependent on the immediately preceding state and action. Rewards are stochastic and dependent on the preceding state and action, and on the next state. The agent process generating the actions is termed the behavior policy. To start, we assume a deterministic target policy π : S → A. The objective is to learn an approximation to its state-value function: ∞ V π (s) = Eπ γ t−1 rt |s1 = s , (1) t=1 where γ ∈ [0, 1) is the discount rate. The learning is to be done without knowledge of the process dynamics and from observations of a single continuous trajectory with no resets. In many problems of interest the state set is too large for it to be practical to approximate the value of each state individually. Here we consider linear function approximation, in which states are mapped to feature vectors with fewer components than the number of states. That is, for each state s ∈ S there is a corresponding feature vector φ(s) ∈ Rn , with n |S|. The approximation to the value function is then required to be linear in the feature vectors and a corresponding parameter vector θ ∈ Rn : V π (s) ≈ θ φ(s). (2) Further, we assume that the states st are not visible to the learning agent in any way other than through the feature vectors. Thus this function approximation formulation can include partialobservability formulations such as POMDPs as a special case. The environment and the behavior policy together generate a stream of states, actions and rewards, s1 , a1 , r1 , s2 , a2 , r2 , . . ., which we can break into causally related 4-tuples, (s1 , a1 , r1 , s1 ), 2 (s2 , a2 , r2 , s2 ), . . . , where st = st+1 . For some tuples, the action will match what the target policy would do in that state, and for others it will not. We can discard all of the latter as not relevant to the target policy. For the former, we can discard the action because it can be determined from the state via the target policy. With a slight abuse of notation, let sk denote the kth state in which an on-policy action was taken, and let rk and sk denote the associated reward and next state. The kth on-policy transition, denoted (sk , rk , sk ), is a triple consisting of the starting state of the transition, the reward on the transition, and the ending state of the transition. The corresponding data available to the learning algorithm is the triple (φ(sk ), rk , φ(sk )). The MDP under the behavior policy is assumed to be ergodic, so that it determines a stationary state-occupancy distribution µ(s) = limk→∞ P r{sk = s}. For any state s, the MDP and target policy together determine an N × N state-transition-probability matrix P , where pss = P r{sk = s |sk = s}, and an N × 1 expected-reward vector R, where Rs = E[rk |sk = s]. These two together completely characterize the statistics of on-policy transitions, and all the samples in the sequence of (φ(sk ), rk , φ(sk )) respect these statistics. The problem still has a Markov structure in that there are temporal dependencies between the sample transitions. In our analysis we ﬁrst consider a formulation without such dependencies, the i.i.d. case, and then prove that our results extend to the original case. In the i.i.d. formulation, the states sk are generated independently and identically distributed according to an arbitrary probability distribution µ. From each sk , a corresponding sk is generated according to the on-policy state-transition matrix, P , and a corresponding rk is generated according to an arbitrary bounded distribution with expected value Rsk . The ﬁnal i.i.d. data sequence, from which an approximate value function is to be learned, is then the sequence (φ(sk ), rk , φ(sk )), for k = 1, 2, . . . Further, because each sample is i.i.d., we can remove the indices and talk about a single tuple of random variables (φ, r, φ ) drawn from µ. It remains to deﬁne the objective of learning. The TD error for the linear setting is δ = r + γθ φ − θ φ. (3) Given this, we deﬁne the one-step linear TD solution as any value of θ at which 0 = E[δφ] = −Aθ + b, (4) where A = E φ(φ − γφ ) and b = E[rφ]. This is the parameter value to which the linear TD(0) algorithm (Sutton 1988) converges under on-policy training, as well as the value found by LSTD(0) (Bradtke & Barto 1996) under both on-policy and off-policy training. The TD solution is always a ﬁxed-point of the linear TD(0) algorithm, but under off-policy training it may not be stable; if θ does not exactly satisfy (4), then the TD(0) algorithm may cause it to move away in expected value and eventually diverge to inﬁnity. 3 The GTD(0) algorithm We next present the idea and gradient-descent derivation leading to the GTD(0) algorithm. As discussed above, the vector E[δφ] can be viewed as an error in the current solution θ. The vector should be zero, so its norm is a measure of how far we are away from the TD solution. A distinctive feature of our gradient-descent analysis of temporal-difference learning is that we use as our objective function the L2 norm of this vector: J(θ) = E[δφ] E[δφ] . (5) This objective function is quadratic and unimodal; it’s minimum value of 0 is achieved when E[δφ] = 0, which can always be achieved. The gradient of this objective function is θ J(θ) = 2( = 2E φ( θ E[δφ])E[δφ] θ δ) E[δφ] = −2E φ(φ − γφ ) E[δφ] . (6) This last equation is key to our analysis. We would like to take a stochastic gradient-descent approach, in which a small change is made on each sample in such a way that the expected update 3 is the direction opposite to the gradient. This is straightforward if the gradient can be written as a single expected value, but here we have a product of two expected values. One cannot sample both of them because the sample product will be biased by their correlation. However, one could store a long-term, quasi-stationary estimate of either of the expectations and then sample the other. The question is, which expectation should be estimated and stored, and which should be sampled? Both ways seem to lead to interesting learning algorithms. First let us consider the algorithm obtained by forming and storing a separate estimate of the ﬁrst expectation, that is, of the matrix A = E φ(φ − γφ ) . This matrix is straightforward to estimate from experience as a simple arithmetic average of all previously observed sample outer products φ(φ − γφ ) . Note that A is a stationary statistic in any ﬁxed-policy policy-evaluation problem; it does not depend on θ and would not need to be re-estimated if θ were to change. Let Ak be the estimate of A after observing the ﬁrst k samples, (φ1 , r1 , φ1 ), . . . , (φk , rk , φk ). Then this algorithm is deﬁned by k 1 Ak = φi (φi − γφi ) (7) k i=1 along with the gradient descent rule: θk+1 = θk + αk Ak δk φk , k ≥ 1, (8) where θ1 is arbitrary, δk = rk + γθk φk − θk φk , and αk > 0 is a series of step-size parameters, possibly decreasing over time. We call this algorithm A TD(0) because it is essentially conventional TD(0) preﬁxed by an estimate of the matrix A . Although we ﬁnd this algorithm interesting, we do not consider it further here because it requires O(n2 ) memory and computation per time step. The second path to a stochastic-approximation algorithm for estimating the gradient (6) is to form and store an estimate of the second expectation, the vector E[δφ], and to sample the ﬁrst expectation, E φ(φ − γφ ) . Let uk denote the estimate of E[δφ] after observing the ﬁrst k − 1 samples, with u1 = 0. The GTD(0) algorithm is deﬁned by uk+1 = uk + βk (δk φk − uk ) (9) and θk+1 = θk + αk (φk − γφk )φk uk , (10) where θ1 is arbitrary, δk is as in (3) using θk , and αk > 0 and βk > 0 are step-size parameters, possibly decreasing over time. Notice that if the product is formed right-to-left, then the entire computation is O(n) per time step. 4 Convergence The purpose of this section is to establish that GTD(0) converges with probability one to the TD solution in the i.i.d. problem formulation under standard assumptions. In particular, we have the following result: Theorem 4.1 (Convergence of GTD(0)). Consider the GTD(0) iteration (9,10) with step-size se∞ ∞ 2 quences αk and βk satisfying βk = ηαk , η > 0, αk , βk ∈ (0, 1], k=0 αk = ∞, k=0 αk < ∞. Further assume that (φk , rk , φk ) is an i.i.d. sequence with uniformly bounded second moments. Let A = E φk (φk − γφk ) and b = E[rk φk ] (note that A and b are well-deﬁned because the distribution of (φk , rk , φk ) does not depend on the sequence index k). Assume that A is non-singular. Then the parameter vector θk converges with probability one to the TD solution (4). Proof. We use the ordinary-differential-equation (ODE) approach (Borkar & Meyn 2000). First, we rewrite the algorithm’s two iterations as a single iteration in a combined parameter vector with √ 2n components ρk = (vk , θk ), where vk = uk / η, and a new reward-related vector with 2n components gk+1 = (rk φk , 0 ): √ ρk+1 = ρk + αk η (Gk+1 ρk + gk+1 ) , where Gk+1 = √ − ηI (φk − γφk )φk 4 φk (γφk − φk ) 0 . Let G = E[Gk ] and g = E[gk ]. Note that G and g are well-deﬁned as by the assumption the process {φk , rk , φk }k is i.i.d. In particular, √ − η I −A b G= , g= . 0 A 0 Further, note that (4) follows from Gρ + g = 0, (11) where ρ = (v , θ ). Now we apply Theorem 2.2 of Borkar & Meyn (2000). For this purpose we write ρk+1 = ρk + √ √ αk η(Gρk +g+(Gk+1 −G)ρk +(gk+1 −g)) = ρk +αk (h(ρk )+Mk+1 ), where αk = αk η, h(ρ) = g + Gρ and Mk+1 = (Gk+1 − G)ρk + gk+1 − g. Let Fk = σ(ρ1 , M1 , . . . , ρk−1 , Mk ). Theorem 2.2 requires the veriﬁcation of the following conditions: (i) The function h is Lipschitz and h∞ (ρ) = limr→∞ h(rρ)/r is well-deﬁned for every ρ ∈ R2n ; (ii-a) The sequence (Mk , Fk ) is a martingale difference sequence, and (ii-b) for some C0 > 0, E Mk+1 2 | Fk ≤ C0 (1 + ρk 2 ) holds for ∞ any initial parameter vector ρ1 ; (iii) The sequence αk satisﬁes 0 < αk ≤ 1, k=1 αk = ∞, ∞ 2 ˙ k=1 (αk ) < +∞; and (iv) The ODE ρ = h(ρ) has a globally asymptotically stable equilibrium. Clearly, h(ρ) is Lipschitz with coefﬁcient G and h∞ (ρ) = Gρ. By construction, (Mk , Fk ) satisﬁes E[Mk+1 |Fk ] = 0 and Mk ∈ Fk , i.e., it is a martingale difference sequence. Condition (ii-b) can be shown to hold by a simple application of the triangle inequality and the boundedness of the the second moments of (φk , rk , φk ). Condition (iii) is satisﬁed by our conditions on the step-size sequences αk , βk . Finally, the last condition (iv) will follow from the elementary theory of linear differential equations if we can show that the real parts of all the eigenvalues of G are negative. First, let us show that G is non-singular. Using the determinant rule for partitioned matrices1 we get det(G) = det(A A) = 0. This indicates that all the eigenvalues of G are non-zero. Now, let λ ∈ C, λ = 0 be an eigenvalue of G with corresponding normalized eigenvector x ∈ C2n ; 2 that is, x = x∗ x = 1, where x∗ is the complex conjugate of x. Hence x∗ Gx = λ. Let √ 2 x = (x1 , x2 ), where x1 , x2 ∈ Cn . Using the deﬁnition of G, λ = x∗ Gx = − η x1 + x∗ Ax2 − x∗ A x1 . Because A is real, A∗ = A , and it follows that (x∗ Ax2 )∗ = x∗ A x1 . Thus, 1 2 1 2 √ 2 Re(λ) = Re(x∗ Gx) = − η x1 ≤ 0. We are now done if we show that x1 cannot be zero. If x1 = 0, then from λ = x∗ Gx we get that λ = 0, which contradicts with λ = 0. The next result concerns the convergence of GTD(0) when (φk , rk , φk ) is obtained by the off-policy sub-sampling process described originally in Section 2. We make the following assumption: Assumption A1 The behavior policy πb (generator of the actions at ) selects all actions of the target policy π with positive probability in every state, and the target policy is deterministic. This assumption is needed to ensure that the sub-sampled process sk is well-deﬁned and that the obtained sample is of “high quality”. Under this assumption it holds that sk is again a Markov chain by the strong Markov property of Markov processes (as the times selected when actions correspond to those of the behavior policy form Markov times with respect to the ﬁltration deﬁned by the original process st ). The following theorem shows that the conclusion of the previous result continues to hold in this case: Theorem 4.2 (Convergence of GTD(0) with a sub-sampled process.). Assume A1. Let the parameters θk , uk be updated by (9,10). Further assume that (φk , rk , φk ) is such that E φk 2 |sk−1 , 2 E rk |sk−1 , E φk 2 |sk−1 are uniformly bounded. Assume that the Markov chain (sk ) is aperiodic and irreducible, so that limk→∞ P(sk = s |s0 = s) = µ(s ) exists and is unique. Let s be a state randomly drawn from µ, and let s be a state obtained by following π for one time step in the MDP from s. Further, let r(s, s ) be the reward incurred. Let A = E φ(s)(φ(s) − γφ(s )) and b = E[r(s, s )φ(s)]. Assume that A is non-singular. Then the parameter vector θk converges with probability one to the TD solution (4), provided that s1 ∼ µ. Proof. The proof of Theorem 4.1 goes through without any changes once we observe that G = E[Gk+1 |Fk ] and g = E[gk+1 | Fk ]. 1 R According to this rule, if A ∈ Rn×n , B ∈ Rn×m , C ∈ Rm×n , D ∈ Rm×m then for F = [A B; C D] ∈ , det(F ) = det(A) det(D − CA−1 B). (n+m)×(n+m) 5 The condition that (sk ) is aperiodic and irreducible guarantees the existence of the steady state distribution µ. Further, the aperiodicity and irreducibility of (sk ) follows from the same property of the original process (st ). For further discussion of these conditions cf. Section 6.3 of Bertsekas and Tsitsiklis (1996). With considerable more work the result can be extended to the case when s1 follows an arbitrary distribution. This requires an extension of Theorem 2.2 of Borkar and Meyn (2000) to processes of the form ρk+1 + ρk (h(ρk ) + Mk+1 + ek+1 ), where ek+1 is a fast decaying perturbation (see, e.g., the proof of Proposition 4.8 of Bertsekas and Tsitsiklis (1996)). 5 Extensions to action values, stochastic target policies, and other sample weightings The GTD algorithm extends immediately to the case of off-policy learning of action-value functions. For this assume that a behavior policy πb is followed that samples all actions in every state with positive probability. Let the target policy to be evaluated be π. In this case the basis functions are dependent on both the states and actions: φ : S × A → Rn . The learning equations are unchanged, except that φt and φt are redeﬁned as follows: φt = φ(st , at ), (12) φt = (13) π(st+1 , a)φ(st+1 , a). a (We use time indices t denoting physical time.) Here π(s, a) is the probability of selecting action a in state s under the target policy π. Let us call the resulting algorithm “one-step gradient-based Q-evaluation,” or GQE(0). Theorem 5.1 (Convergence of GQE(0)). Assume that st is a state sequence generated by following some stationary policy πb in a ﬁnite MDP. Let rt be the corresponding sequence of rewards and let φt , φt be given by the respective equations (12) and (13), and assume that E φt 2 |st−1 , 2 E rt |st−1 , E φt 2 |st−1 are uniformly bounded. Let the parameters θt , ut be updated by Equations (9) and (10). Assume that the Markov chain (st ) is aperiodic and irreducible, so that limt→∞ P(st = s |s0 = s) = µ(s ) exists and is unique. Let s be a state randomly drawn from µ, a be an action chosen by πb in s, let s be the next state obtained and let a = π(s ) be the action chosen by the target policy in state s . Further, let r(s, a, s ) be the reward incurred in this transition. Let A = E φ(s, a)(φ(s, a) − γφ(s , a )) and b = E[r(s, a, s )φ(s, a)]. Assume that A is non-singular. Then the parameter vector θt converges with probability one to a TD solution (4), provided that s1 is selected from the steady-state distribution µ. The proof is almost identical to that of Theorem 4.2, and hence it is omitted. Our main convergence results are also readily generalized to stochastic target policies by replacing the sub-sampling process described in Section 2 with a sample-weighting process. That is, instead of including or excluding transitions depending upon whether the action taken matches a deterministic policy, we include all transitions but give each a weight. For example, we might let the weight wt for time step t be equal to the probability π(st , at ) of taking the action actually taken under the target policy. We can consider the i.i.d. samples now to have four components (φk , rk , φk , wk ), with the update rules (9) and (10) replaced by uk+1 = uk + βk (δk φk − uk )wk , (14) θk+1 = θk + αk (φk − γφk )φk uk wk . (15) and Each sample is also weighted by wk in the expected values, such as that deﬁning the TD solution (4). With these changes, Theorems 4.1 and 4.2 go through immediately for stochastic policies. The reweighting is, in effect, an adjustment to the i.i.d. sampling distribution, µ, and thus our results hold because they hold for all µ. The choice wt = π(st , at ) is only one possibility, notable for its equivalence to our original case if the target policy is deterministic. Another natural weighting is wt = π(st , at )/πb (st , at ), where πb is the behavior policy. This weighting may result in the TD solution (4) better matching the target policy’s value function (1). 6 6 Related work There have been several prior attempts to attain the four desirable algorithmic features mentioned at the beginning this paper (off-policy stability, temporal-difference learning, linear function approximation, and O(n) complexity) but none has been completely successful. One idea for retaining all four desirable features is to use importance sampling techniques to reweight off-policy updates so that they are in the same direction as on-policy updates in expected value (Precup, Sutton & Dasgupta 2001; Precup, Sutton & Singh 2000). Convergence can sometimes then be assured by existing results on the convergence of on-policy methods (Tsitsiklis & Van Roy 1997; Tadic 2001). However, the importance sampling weights are cumulative products of (possibly many) target-to-behavior-policy likelihood ratios, and consequently they and the corresponding updates may be of very high variance. The use of “recognizers” to construct the target policy directly from the behavior policy (Precup, Sutton, Paduraru, Koop & Singh 2006) is one strategy for limiting the variance; another is careful choice of the target policies (see Precup, Sutton & Dasgupta 2001). However, it remains the case that for all of such methods to date there are always choices of problem, behavior policy, and target policy for which the variance is inﬁnite, and thus for which there is no guarantee of convergence. Residual gradient algorithms (Baird 1995) have also been proposed as a way of obtaining all four desirable features. These methods can be viewed as gradient descent in the expected squared TD error, E δ 2 ; thus they converge stably to the solution that minimizes this objective for arbitrary differentiable function approximators. However, this solution has always been found to be much inferior to the TD solution (exempliﬁed by (4) for the one-step linear case). In the literature (Baird 1995; Sutton & Barto 1998), it is often claimed that residual-gradient methods are guaranteed to ﬁnd the TD solution in two special cases: 1) systems with deterministic transitions and 2) systems in which two samples can be drawn for each next state (e.g., for which a simulation model is available). Our own analysis indicates that even these two special requirements are insufﬁcient to guarantee convergence to the TD solution.2 Gordon (1995) and others have questioned the need for linear function approximation. He has proposed replacing linear function approximation with a more restricted class of approximators, known as averagers, that never extrapolate outside the range of the observed data and thus cannot diverge. Rightly or wrongly, averagers have been seen as being too constraining and have not been used on large applications involving online learning. Linear methods, on the other hand, have been widely used (e.g., Baxter, Tridgell & Weaver 1998; Sturtevant & White 2006; Schaeffer, Hlynka & Jussila 2001). The need for linear complexity has also been questioned. Second-order methods for linear approximators, such as LSTD (Bradtke & Barto 1996; Boyan 2002) and LSPI (Lagoudakis & Parr 2003; see also Peters, Vijayakumar & Schaal 2005), can be effective on moderately sized problems. If the number of features in the linear approximator is n, then these methods require memory and per-timestep computation that is O(n2 ). Newer incremental methods such as iLSTD (Geramifard, Bowling & Sutton 2006) have reduced the per-time-complexity to O(n), but are still O(n2 ) in memory. Sparsiﬁcation methods may reduce the complexity further, they do not help in the general case, and may apply to O(n) methods as well to further reduce their complexity. Linear function approximation is most powerful when very large numbers of features are used, perhaps millions of features (e.g., as in Silver, Sutton & M¨ ller 2007). In such cases, O(n2 ) methods are not feasible. u 7 Conclusion GTD(0) is the ﬁrst off-policy TD algorithm to converge under general conditions with linear function approximation and linear complexity. As such, it breaks new ground in terms of important, 2 For a counterexample, consider that given in Dayan’s (1992) Figure 2, except now consider that state A is actually two states, A and A’, which share the same feature vector. The two states occur with 50-50 probability, and when one occurs the transition is always deterministically to B followed by the outcome 1, whereas when the other occurs the transition is always deterministically to the outcome 0. In this case V (A) and V (B) will converge under the residual-gradient algorithm to the wrong answers, 1/3 and 2/3, even though the system is deterministic, and even if multiple samples are drawn from each state (they will all be the same). 7 absolute abilities not previous available in existing algorithms. We have conducted empirical studies with the GTD(0) algorithm and have conﬁrmed that it converges reliably on standard off-policy counterexamples such as Baird’s (1995) “star” problem. On on-policy problems such as the n-state random walk (Sutton 1988; Sutton & Barto 1998), GTD(0) does not seem to learn as efﬁciently as classic TD(0), although we are still exploring different ways of setting the step-size parameters, and other variations on the algorithm. It is not clear that the GTD(0) algorithm in its current form will be a fully satisfactory solution to the off-policy learning problem, but it is clear that is breaks new ground and achieves important abilities that were previously unattainable. Acknowledgments The authors gratefully acknowledge insights and assistance they have received from David Silver, Eric Wiewiora, Mark Ring, Michael Bowling, and Alborz Geramifard. This research was supported by iCORE, NSERC and the Alberta Ingenuity Fund. References Baird, L. C. (1995). Residual algorithms: Reinforcement learning with function approximation. In Proceedings of the Twelfth International Conference on Machine Learning, pp. 30–37. Morgan Kaufmann. Baxter, J., Tridgell, A., Weaver, L. (1998). Experiments in parameter learning using temporal differences. International Computer Chess Association Journal, 21, 84–99. Bertsekas, D. P., Tsitsiklis. J. (1996). Neuro-Dynamic Programming. Athena Scientiﬁc, 1996. Borkar, V. S. and Meyn, S. P. (2000). The ODE method for convergence of stochastic approximation and reinforcement learning. SIAM Journal on Control And Optimization , 38(2):447–469. Boyan, J. (2002). Technical update: Least-squares temporal difference learning. Machine Learning, 49:233– 246. Bradtke, S., Barto, A. G. (1996). Linear least-squares algorithms for temporal difference learning. Machine Learning, 22:33–57. Dayan, P. (1992). The convergence of TD(λ) for general λ. Machine Learning, 8:341–362. Geramifard, A., Bowling, M., Sutton, R. S. (2006). Incremental least-square temporal difference learning. Proceedings of the National Conference on Artiﬁcial Intelligence, pp. 356–361. Gordon, G. J. (1995). Stable function approximation in dynamic programming. Proceedings of the Twelfth International Conference on Machine Learning, pp. 261–268. Morgan Kaufmann, San Francisco. Lagoudakis, M., Parr, R. (2003). Least squares policy iteration. Journal of Machine Learning Research, 4:1107-1149. Peters, J., Vijayakumar, S. and Schaal, S. (2005). Natural Actor-Critic. Proceedings of the 16th European Conference on Machine Learning, pp. 280–291. Precup, D., Sutton, R. S. and Dasgupta, S. (2001). Off-policy temporal-difference learning with function approximation. Proceedings of the 18th International Conference on Machine Learning, pp. 417–424. Precup, D., Sutton, R. S., Paduraru, C., Koop, A., Singh, S. (2006). Off-policy Learning with Recognizers. Advances in Neural Information Processing Systems 18. Precup, D., Sutton, R. S., Singh, S. (2000). Eligibility traces for off-policy policy evaluation. Proceedings of the 17th International Conference on Machine Learning, pp. 759–766. Morgan Kaufmann. Schaeffer, J., Hlynka, M., Jussila, V. (2001). Temporal difference learning applied to a high-performance gameplaying program. Proceedings of the International Joint Conference on Artiﬁcial Intelligence, pp. 529–534. Silver, D., Sutton, R. S., M¨ ller, M. (2007). Reinforcement learning of local shape in the game of Go. u Proceedings of the 20th International Joint Conference on Artiﬁcial Intelligence, pp. 1053–1058. Sturtevant, N. R., White, A. M. (2006). Feature construction for reinforcement learning in hearts. In Proceedings of the 5th International Conference on Computers and Games. Sutton, R. S. (1988). Learning to predict by the method of temporal differences. Machine Learning, 3:9–44. Sutton, R. S., Barto, A. G. (1998). Reinforcement Learning: An Introduction. MIT Press. Sutton, R.S., Precup D. and Singh, S (1999). Between MDPs and semi-MDPs: A framework for temporal abstraction in reinforcement learning. Artiﬁcial Intelligence, 112:181–211. Sutton, R. S., Rafols, E.J., and Koop, A. 2006. Temporal abstraction in temporal-difference networks. Advances in Neural Information Processing Systems 18. Tadic, V. (2001). On the convergence of temporal-difference learning with linear function approximation. In Machine Learning 42:241–267 Tsitsiklis, J. N., and Van Roy, B. (1997). An analysis of temporal-difference learning with function approximation. IEEE Transactions on Automatic Control, 42:674–690. Watkins, C. J. C. H. (1989). Learning from Delayed Rewards. Ph.D. thesis, Cambridge University. 8

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