nips nips2008 nips2008-150 knowledge-graph by maker-knowledge-mining

150 nips-2008-Near-optimal Regret Bounds for Reinforcement Learning

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Author: Peter Auer, Thomas Jaksch, Ronald Ortner

Abstract: For undiscounted reinforcement learning in Markov decision processes (MDPs) we consider the total regret of a learning algorithm with respect to an optimal policy. In order to describe the transition structure of an MDP we propose a new parameter: An MDP has diameter D if for any pair of states s, s there is a policy which moves from s to s in at most D steps (on average). We present a rein√ ˜ forcement learning algorithm with total regret O(DS AT ) after T steps for any unknown MDP with S states, A actions per state, and diameter D. This bound holds with high probability. We also present a corresponding lower bound of √ Ω( DSAT ) on the total regret of any learning algorithm. 1

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Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 at Abstract For undiscounted reinforcement learning in Markov decision processes (MDPs) we consider the total regret of a learning algorithm with respect to an optimal policy. [sent-3, score-0.518]

2 In order to describe the transition structure of an MDP we propose a new parameter: An MDP has diameter D if for any pair of states s, s there is a policy which moves from s to s in at most D steps (on average). [sent-4, score-0.686]

3 We present a rein√ ˜ forcement learning algorithm with total regret O(DS AT ) after T steps for any unknown MDP with S states, A actions per state, and diameter D. [sent-5, score-0.753]

4 We also present a corresponding lower bound of √ Ω( DSAT ) on the total regret of any learning algorithm. [sent-7, score-0.371]

5 1 Introduction In a Markov decision process (MDP) M with ﬁnite state space S and ﬁnite action space A, a learner in state s ∈ S needs to choose an action a ∈ A. [sent-8, score-0.464]

6 When executing action a in state s, the learner receives a random reward r with mean r(s, a) according to some distribution on [0, 1]. [sent-9, score-0.407]

7 Further, ¯ according to the transition probabilities p (s |s, a), a random transition to a state s ∈ S occurs. [sent-10, score-0.306]

8 In contrast to important other work on reinforcement learning — where the performance of the learned policy is considered (see e. [sent-12, score-0.32]

9 For that, we compare the rewards collected by the algorithm during learning with the rewards of an optimal policy. [sent-15, score-0.37]

10 The accumulated reward of an algorithm A after T steps in an MDP M is deﬁned as T R(M, A, s, T ) := t=1 rt , where s is the initial state and rt are the rewards received during the execution of algorithm A. [sent-17, score-0.632]

11 The average reward 1 ρ(M, A, s) := lim T E [R(M, A, s, T )] T →∞ can be maximized by an appropriate stationary policy π : S → A which deﬁnes an optimal action for each state [4]. [sent-18, score-0.698]

12 In order to measure this transition structure we propose a new parameter, the diameter D of an MDP. [sent-20, score-0.341]

13 The diameter D is the time it takes to move from any state s to any other state s , using an appropriate policy for this pair of states s and s : Deﬁnition 1. [sent-21, score-0.711]

14 Let T (s |M, π, s) be the ﬁrst (random) time step in which state s is reached when policy π is executed on MDP M with initial state s. [sent-22, score-0.486]

15 Then the diameter of M is given by D(M ) := max min E [T (s |M, π, s)] . [sent-23, score-0.283]

16 s,s ∈S π:S→A A ﬁnite diameter seems necessary for interesting bounds on the regret of any algorithm with respect to an optimal policy. [sent-24, score-0.682]

17 Hence, the learner may suffer regret D for such exploration, and it is very plausible that the diameter appears in the regret bound. [sent-26, score-0.941]

18 For MDPs with ﬁnite diameter (which usually are called communicating, see e. [sent-27, score-0.247]

19 [4]) the optimal average reward ρ∗ does not depend on the initial state (cf. [sent-29, score-0.346]

20 π The optimal average reward is the natural benchmark for a learning algorithm A, and we deﬁne the total regret of A after T steps as1 ∆(M, A, s, T ) := T ρ∗ (M ) − R(M, A, s, T ). [sent-33, score-0.633]

21 In the following, we present our reinforcement learning algorithm U CRL 2 (a variant of the UCRL algorithm of [5]) which uses upper conﬁdence bounds to choose an optimistic policy. [sent-34, score-0.35]

22 We show ˜ that the total regret of U CRL 2 after T steps is O(D|S| |A|T ). [sent-35, score-0.383]

23 A corresponding lower bound of Ω( D|S||A|T ) on the total regret of any learning algorithm is given as well. [sent-36, score-0.388]

24 These results establish the diameter as an important parameter of an MDP. [sent-37, score-0.247]

25 Further, the diameter seems to be more natural than other parameters that have been proposed for various PAC and regret bounds, such as the mixing time [3, 6] or the hitting time of an optimal policy [7] (cf. [sent-38, score-0.852]

26 These algorithms achieve ε-optimal average reward with probability 1 − δ after time polynomial in 1 , 1 , |S|, |A|, and the mixδ ε mix ing time Tε (see below). [sent-42, score-0.281]

27 As the polynomial dependence on ε is of order 1/ε3 , the PAC bounds translate into T 2/3 regret bounds at the best. [sent-43, score-0.485]

28 Moreover, both algorithms need the ε-return mixing mix mix time Tε of an optimal policy π ∗ as input parameter. [sent-44, score-0.429]

29 This parameter Tε is the number of steps ∗ mix until the average reward of π over these Tε steps is ε-close to the optimal average reward ρ∗ . [sent-45, score-0.613]

30 This additional dependency on ε It is easy to construct MDPs of diameter D with Tε further increases the exponent in the above mentioned regret bounds for E3 and R-max. [sent-47, score-0.628]

31 The MBIE algorithm of Strehl and Littman [9, 10] — similarly to our approach — applies conﬁdence bounds to compute an optimistic policy. [sent-49, score-0.258]

32 However, Strehl and Littman consider only a discounted reward setting, which seems to be less natural when dealing with regret. [sent-50, score-0.165]

33 Their deﬁnition of regret measures the difference between the rewards2 of an optimal policy and the rewards of the learning algorithm along the trajectory taken by the learning algorithm. [sent-51, score-0.757]

34 In contrast, we are interested in the regret of the learning algorithm in respect to the rewards of the optimal policy along the trajectory of the optimal policy. [sent-52, score-0.794]

35 These index policies choose actions optimistically by using conﬁdence bounds only for the estimates in the current state. [sent-54, score-0.27]

36 The regret bounds for the index policies of [11] and the OLP algorithm of [7] are asymptotically logarithmic in T . [sent-55, score-0.478]

37 However, unlike our bounds, these bounds depend on the gap between the “quality” of the best and the second best action, and these asymptotic bounds also hide an additive term which is exponential in the number of states. [sent-56, score-0.178]

38 This bound holds uniformly over time and under weaker assumptions: While [7] and [11] consider only ergodic MDPs in which any policy will reach every state after a sufﬁcient number of steps, we make only the more natural assumption of a ﬁnite diameter. [sent-59, score-0.384]

39 We assume an unknown MDP M to be learned, with S := |S| states, A := |A| actions, and ﬁnite diameter D := D(M ). [sent-63, score-0.247]

40 With probability 1 − δ it holds that for any initial state s ∈ S and any T > 1, the regret of U CRL 2 is bounded by ∆(M, U CRL 2, s, T ) ≤ c1 · DS T A log T δ , for a constant c1 which is independent of M , T , and δ. [sent-66, score-0.449]

41 With probability 1 − δ the average per-step regret is at most ε for any T ≥ c2 D2 S 2 A log ε2 DSA δε steps, where c2 is a constant independent of M . [sent-69, score-0.348]

42 For some c4 > 0, any algorithm A, and any natural numbers S, A ≥ 10, D ≥ 20 logA S, and T ≥ DSA, there is an MDP 3 M with S states, A actions, and diameter D, such that for any initial state s ∈ S the expected regret of A after T steps is √ E [∆(M, A, s, T )] ≥ c4 · DSAT . [sent-75, score-0.738]

43 its transition probabilities and reward distributions) is allowed to change times up to step T , such that the diameter is always at most D (we assume an initial change at time t = 1). [sent-80, score-0.576]

44 In this model we measure regret as the sum of missed rewards compared to the policies which are optimal after the changes of the MDP. [sent-81, score-0.572]

45 , this regret is upper bounded by c5 · 1 3 2 T 3 DS A log T δ with probability 1 − 2δ. [sent-85, score-0.336]

46 There, the transition probabilities are assumed to be ﬁxed and known to the learner, but the rewards are allowed √ to change in every step. [sent-87, score-0.28]

47 A best possible upper bound of O( T ) on the regret against an optimal stationary policy, given all the reward changes in advance, is derived. [sent-88, score-0.554]

48 3 The diameter of any MDP with S states and A actions is at least logA S. [sent-89, score-0.392]

49 For all (s, a) in S × A initialize the state-action counts for episode k, vk (s, a) := 0. [sent-98, score-0.414]

50 Further, set the state-action counts prior to episode k, Nk (s, a) := # {τ < tk : sτ = s, aτ = a} . [sent-99, score-0.449]

51 1) to ﬁnd a policy πk and an optimistic ˜ ˜ MDP Mk ∈ Mk such that ˜ ˜ ρk := min ρ(Mk , πk , s) ≥ ˜ s max M ∈Mk ,π,s 1 ρ(M , π, s ) − √ . [sent-106, score-0.441]

52 While vk (st , πk (st )) < max{1, Nk (st , πk (st ))} do ˜ ˜ (a) Choose action at := πk (st ), obtain reward rt , and observe next state st+1 . [sent-108, score-0.565]

53 As such, it deﬁnes a set M of statistically ˜ plausible MDPs given the observations so far, and chooses an optimistic MDP M (with respect to the achievable average reward) among these plausible MDPs. [sent-114, score-0.318]

54 Then it executes a policy π which is ˜ ˜ (nearly) optimal for the optimistic MDP M . [sent-115, score-0.442]

55 More precisely, U CRL 2 (Figure 1) proceeds in episodes and computes a new policy πk only at the ˜ beginning of each episode k. [sent-116, score-0.489]

56 In Steps 2–3, U CRL 2 computes estimates pk (s |s, a) and rk (s, a) for the ˆ ˆ transition probabilities and mean rewards from the observations made before episode k. [sent-118, score-0.663]

57 In Step 4, a set Mk of plausible MDPs is deﬁned in terms of conﬁdence regions around the estimated mean rewards rk (s, a) and transition probabilities pk (s |s, a). [sent-119, score-0.504]

58 In Step 5, extended value iteration (see below) is used to choose a near˜ optimal policy πk on an optimistic MDP Mk ∈ Mk . [sent-121, score-0.548]

59 This policy πk is executed throughout episode ˜ ˜ k (Step 6). [sent-122, score-0.474]

60 Episode k ends when a state s is visited in which the action a = πk (s) induced by the ˜ current policy has been chosen in episode k equally often as before episode k. [sent-123, score-0.864]

61 The counts vk (s, a) keep track of these occurrences in episode k. [sent-125, score-0.445]

62 1 Extended Value Iteration In Step 5 of the U CRL 2 algorithm we need to ﬁnd a near-optimal policy πk for an optimistic MDP. [sent-127, score-0.422]

63 ˜ While value iteration typically calculates a policy for a ﬁxed MDP, we also need to select an op˜ timistic MDP Mk which gives almost maximal reward among all plausible MDPs. [sent-128, score-0.529]

64 Formally, this can be seen as undiscounted value iteration [4] on an MDP with extended action set. [sent-130, score-0.259]

65 ˜ While (4) may look like a step of value iteration with an inﬁnite action space, maxp p · ui is actually a linear optimization problem over the convex polytope P(s, a). [sent-132, score-0.452]

66 5 The value iteration is stopped when 1 max ui+1 (s) − ui (s) − min ui+1 (s) − ui (s) < √ , s∈S s∈S tk (5) which means that the change of the state values is almost uniform and actually close to the average reward of the optimal policy. [sent-134, score-1.045]

67 It can be shown that the actions, rewards, and transition probabilities ˜ chosen in (4) for this i-th iteration deﬁne an optimistic MDP Mk and a policy πk which satisfy ˜ condition (3) of algorithm U CRL 2. [sent-135, score-0.612]

68 We also make the assumption that the rewards r(s, a) are deterministic and known to the learner. [sent-138, score-0.174]

69 Considering unknown stochastic rewards adds little to the proof and only lower order terms to the regret bounds. [sent-140, score-0.505]

70 We start by considering the regret in a single episode k. [sent-143, score-0.51]

71 Since the optimistic average reward ρk ˜ of the optimistically chosen policy πk is essentially larger than the true optimal average reward ρ∗ , ˜ it is sufﬁcient to calculate by how much the optimistic average reward ρk overestimates the actual ˜ √ ˜ rewards of policy πk . [sent-144, score-1.625]

72 By the choice of πk and Mk in Step 5 of U CRL 2, ρk ≥ ρ∗ − 1/ tk . [sent-145, score-0.196]

73 Thus the ˜ ˜ ˜ 4 Since the policy πk is ﬁxed for episode k, vk (s, a) = 0 only for a = πk (s). [sent-146, score-0.616]

74 Nevertheless, we ﬁnd it ˜ ˜ convenient to use a notation which explicitly includes the action a in vk (s, a). [sent-147, score-0.27]

75 5 Because of the special structure of the polytope P(s, a), the linear program in (4) can be solved very efﬁciently in O(S) steps after sorting the state values ui (s ). [sent-148, score-0.391]

76 regret ∆k during episode k is bounded as tk+1 −1 tk+1 −1 ∗ (ρ − rt ) ≤ ∆k := t=tk (˜k − rt ) + ρ t=tk tk+1 − tk √ . [sent-151, score-0.805]

77 tk √ The sum over k of the second term on the right hand side is O( T ) and will not be considered further in this proof sketch. [sent-152, score-0.241]

78 The ﬁrst term on the right hand side can be rewritten using the known deterministic rewards r(s, a) and the occurrences of state action pairs (s, a) in episode k, tk+1 −1 (˜k − rt ) = ρ ∆k t=tk 4. [sent-153, score-0.67]

79 As an important observation for our analysis, we ﬁnd that for any iteration i the range of the state values is bounded by the diameter of the MDP M , max ui (s) − min ui (s) ≤ D. [sent-157, score-0.866]

80 (7) s s To see this, observe that ui (s) is the total expected reward after i steps of an optimal non-stationary i-step policy starting in state s, on the MDP with extended action set as considered for the extended value iteration. [sent-158, score-1.008]

81 The diameter of this extended MDP is at most D as it contains the actions of the true MDP M . [sent-159, score-0.391]

82 If there were states with ui (s1 ) − ui (s0 ) > D, then an improved value for ui (s0 ) could be achieved by the following policy: First follow a policy which moves from s0 to s1 most quickly, which takes at most D steps on average. [sent-160, score-0.954]

83 Since only D of the i rewards of the policy for s1 are missed, this policy gives ui (s0 ) ≥ ui (s1 ) − D, proving (7). [sent-162, score-1.054]

84 For the convergence criterion (5) it can be shown that at the corresponding iteration 1 |ui+1 (s) − ui (s) − ρk | ≤ √ ˜ tk for all s ∈ S, where ρk is the average reward of the policy πk chosen in this iteration on the ˜ ˜ ˜ optimistic MDP Mk . [sent-163, score-1.136]

85 7 Expanding ui+1 (s) according to (4), we get pk (s |s, πk (s)) · ui (s ) ˜ ˜ ui+1 (s) = r(s, πk (s)) + ˜ s and hence ρk − r(s, πk (s)) − ˜ ˜ pk (s |s, πk (s)) · ui (s ) − ui (s) ˜ ˜ s 1 ≤√ . [sent-164, score-0.791]

86 Since the rows of P k sum to 1, we can replace ui by wk ˜ with wk (s) = ui (s) − mins ui (s) (we again use the subscript k to reference the episode). [sent-166, score-0.833]

87 The last term on the right hand side of (8) is of lower order, and by (7) we have ∆k wk ∞ ˜ v k P k − I wk , ≤ D. [sent-167, score-0.258]

88 We expect to receive average reward ρk per step, such that the difference of the state ˜ values after i + 1 and i steps should be about ρk . [sent-169, score-0.347]

89 2 Completing the Proof ˜ ˜ Replacing the transition matrix P k of the policy πk in the optimistic MDP Mk by the transition ˜ matrix P k of πk in the true MDP M , we get ˜ ∆k ˜ ˜ v k P k − I wk = v k P k − P k + P k − I wk ˜ = v k P k − P k wk + v k P k − I wk . [sent-171, score-1.041]

90 (11) The intuition about the second term in (11) is that the counts of the state visits v k are relatively close to the stationary distribution of the transition matrix P k , such that v k P k −I should be small. [sent-172, score-0.264]

91 This yields v k P k − I wk = O D T log k T δ with high probability, which gives a lower order term in our regret bound. [sent-174, score-0.463]

92 Thus, our regret bound is ˜ mainly determined by the ﬁrst term in (11). [sent-175, score-0.347]

93 The average reward of a policy which chooses action a∗ in state s0 is 2δ+ε > 1 , 2 1 while the average reward of any other policy is 2 . [sent-180, score-1.081]

94 Thus the regret suffered by a suboptimal action in state s0 is Ω(ε/δ). [sent-181, score-0.526]

95 The main observation for the proof of the lower bound is that any algorithm needs to probe Ω(A ) actions in state s0 for Ω δ/ε2 times on average, to detect the “good” action a∗ reliably. [sent-182, score-0.4]

96 Considering k := S/2 copies of this MDP where only one of the copies has such a “good” action a∗ , we ﬁnd that Ω(kA ) actions in the s0 -states of the copies need to be probed for Ω δ/ε2 times to detect the “good” action. [sent-183, score-0.386]

97 This can be done by introducing A + 1 additional deterministic actions per state, which do not leave the s1 -states but connect the s0 -states of the k copies by inducing an A -ary tree structure on the s0 -states (1 action for going toward the root, A actions to go toward the leaves). [sent-186, score-0.414]

98 The diameter of the resulting MDP is at most 2(D/10 + logA k ) which is twice the time to travel to or from the root for any state in the MDP. [sent-187, score-0.337]

99 Thus we have constructed an MDP with ≤ S states, ≤ A actions, and diameter ≤ D which forces regret √ Ω( DSAT ) on any algorithm. [sent-188, score-0.547]

100 Optimistic linear programming gives logarithmic regret for irreducible mdps. [sent-229, score-0.329]

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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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