nips nips2005 nips2005-78 knowledge-graph by maker-knowledge-mining

78 nips-2005-From Weighted Classification to Policy Search

Source: pdf

Author: Doron Blatt, Alfred O. Hero

Abstract: This paper proposes an algorithm to convert a T -stage stochastic decision problem with a continuous state space to a sequence of supervised learning problems. The optimization problem associated with the trajectory tree and random trajectory methods of Kearns, Mansour, and Ng, 2000, is solved using the Gauss-Seidel method. The algorithm breaks a multistage reinforcement learning problem into a sequence of single-stage reinforcement learning subproblems, each of which is solved via an exact reduction to a weighted-classiﬁcation problem that can be solved using off-the-self methods. Thus the algorithm converts a reinforcement learning problem into simpler supervised learning subproblems. It is shown that the method converges in a ﬁnite number of steps to a solution that cannot be further improved by componentwise optimization. The implication of the proposed algorithm is that a plethora of classiﬁcation methods can be applied to ﬁnd policies in the reinforcement learning problem. 1

Reference: text

Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 edu Abstract This paper proposes an algorithm to convert a T -stage stochastic decision problem with a continuous state space to a sequence of supervised learning problems. [sent-8, score-0.388]

2 The optimization problem associated with the trajectory tree and random trajectory methods of Kearns, Mansour, and Ng, 2000, is solved using the Gauss-Seidel method. [sent-9, score-0.448]

3 The algorithm breaks a multistage reinforcement learning problem into a sequence of single-stage reinforcement learning subproblems, each of which is solved via an exact reduction to a weighted-classiﬁcation problem that can be solved using off-the-self methods. [sent-10, score-0.777]

4 Thus the algorithm converts a reinforcement learning problem into simpler supervised learning subproblems. [sent-11, score-0.398]

5 It is shown that the method converges in a ﬁnite number of steps to a solution that cannot be further improved by componentwise optimization. [sent-12, score-0.093]

6 The implication of the proposed algorithm is that a plethora of classiﬁcation methods can be applied to ﬁnd policies in the reinforcement learning problem. [sent-13, score-0.588]

7 1 Introduction There has been increased interest in applying tools from supervised learning to problems in reinforcement learning. [sent-14, score-0.301]

8 The goal is to leverage techniques and theoretical results from supervised learning for solving the more complex problem of reinforcement learning [3]. [sent-15, score-0.362]

9 In [6] and [4], classiﬁcation was incorporated into approximate policy iterations. [sent-16, score-0.435]

10 Bounds on the performance of a policy which is built from a sequence of classiﬁers were derived in [8] and [9]. [sent-18, score-0.485]

11 Similar to [8], we adopt the generative model assumption of [5] and tackle the problem of ﬁnding good policies within an inﬁnite class of policies, where performance is evaluated in terms of empirical averages over a set of trajectory trees. [sent-19, score-0.565]

12 In [8] the T-step reinforcement learning problem was converted to a set of weighted classiﬁcation problems by trying to ﬁt the classiﬁers to the maximal path on the trajectory tree of the decision process. [sent-20, score-0.796]

13 We show that while the task of ﬁnding the global optimum within a class of non-stationary policies may be overwhelming, the componentwise search leads to single step reinforcement learning problems which can be reduced to a sequence of weighted classiﬁcation problems. [sent-22, score-0.9]

14 The weighted classiﬁcation problems can be solved by applying weights-sensitive classiﬁers or by further reducing the weighted classiﬁcation problem to a standard classiﬁcation problem using re-sampling methods (see [7], [1], and references therein for a description of both approaches). [sent-24, score-0.345]

15 Based on this observation, an algorithm that converts the policy search problem into a sequence of weighted classiﬁcation problems is given. [sent-25, score-0.671]

16 It is shown that the algorithm converges in a ﬁnite number of steps to a solution, which cannot be further improved by changing the control of a single stage while holding the rest of the policy ﬁxed. [sent-26, score-0.692]

17 Consider a T-step MDP M = {S, A, D, Ps,a }, where S is a (possibly continuous) state space, A = {0, . [sent-28, score-0.17]

18 , L − 1} is a ﬁnite set of possible actions, D is the distribution of the initial state, and Ps,a is the distribution of the next state given that the current state is s and the action taken is a. [sent-31, score-0.602]

19 The reward granted when taking action a at state s and making a transition to state s′ is assumed to be a known deterministic and bounded function of s′ denoted by r : S → [−M, M ]. [sent-32, score-0.676]

20 No generality is lost in specifying a known deterministic reward since it is possible to augment the state variable by an additional random component whose distribution depends on the previous state and action, and specify the function r to extract this random component. [sent-33, score-0.607]

21 A non-stationary deterministic policy π = (π0 , π1 , . [sent-38, score-0.507]

22 The control πt speciﬁes the action taken at time t as a function of the state at time t. [sent-42, score-0.345]

23 The expected sum of rewards of a non-stationary deterministic policy π is given by T V (π) = Eπ r (St ) , (1) t=1 where the expectation is taken with respect to the distribution over the random state variables induced by the policy π. [sent-43, score-1.388]

24 Non-stationary deterministic policies are considered since the optimal policy for a ﬁnite horizon MDP is non-stationary and deterministic [10]. [sent-45, score-0.882]

25 Usually the optimal policy is deﬁned as the policy that maximizes the value conditioned on the initial state, i. [sent-46, score-0.996]

26 , T Vπ (s) = Eπ R (St ) |S0 = s , (2) t=1 for any realization s of S0 [10]. [sent-48, score-0.217]

27 The policy that maximizes the conditional value given each realization of the initial state also maximizes the value averaged over the initial state, and it is the unique maximizer if the distribution of the initial state D is positive over S. [sent-49, score-1.395]

28 When optimizing (1) over a restricted class of policies, which does not contain the optimal policy, the distribution over the initial state speciﬁes the importance of different regions of the state space in terms of the approximation error. [sent-51, score-0.525]

29 For example, assigning high probability to a certain region of S will favor policies that well approximate the optimal policy over that region. [sent-52, score-0.738]

30 Alternatively, maximizing (1) when D is a point mass at state s is equivalent to maximizing (2). [sent-53, score-0.232]

31 Following the generative model assumption of [5], the initial distribution D and the conditional distribution Ps,a are unknown but it is possible to generate realization of the initial state according to D and the next state according to Ps,a for arbitrary state-action pairs (s, a). [sent-54, score-0.847]

32 Given the generative model, n trajectory trees are constructed in the following manner. [sent-55, score-0.238]

33 The root of each tree is a realization of S0 generated according to the distribution D. [sent-56, score-0.44]

34 Given the realization of the initial state, realizations of the next state S1 given the L possible actions, denoted by S1 |a, a ∈ A, are generated. [sent-57, score-0.596]

35 Note that this notation omits the dependence on the value of the initial state. [sent-58, score-0.079]

36 Each of the L realizations of S1 is now the root of the subtree. [sent-59, score-0.207]

37 , it−1 the random variable generated at the node that follows the sequence of actions i0 , i1 , . [sent-64, score-0.162]

38 Hence, each tree is constructed using a single call to the initial state generator and LT − 2 calls to the next state generator. [sent-68, score-0.538]

39 It is possible to estimate the value of any policy in the class from the set of trajectory trees by simply averaging the sum of rewards on each tree along the path that agrees with the policy [5]. [sent-73, score-1.48]

40 Denote by V i (π) the observed value on the i’th tree along the path that corresponds to the policy π. [sent-74, score-0.577]

41 Then the value of the policy π is estimated by n Vn (π) = n−1 V i (π). [sent-75, score-0.435]

42 (3) i=1 In [5], the authors show that with high probability (over the data set) Vn (π) converges uniformly to V (π) (1) with rates that depend on the VC-dimension of the policy class. [sent-76, score-0.473]

43 This result motivates the use of policies π with high Vn (π), since with high probability these policies have high values of V (π). [sent-77, score-0.606]

44 In this paper, we consider the problem of ﬁnding policies that obtain high values of Vn (π). [sent-78, score-0.332]

45 3 A Reduction From a Single Step Reinforcement Learning Problem to Weighted Classiﬁcation The building block of the proposed algorithm is an exact reduction from a single step reinforcement learning to a weighted classiﬁcation problem. [sent-79, score-0.509]

46 An initial state S0 generated according to the distribution D is followed by one of L possible actions A ∈ {0, 1, . [sent-81, score-0.415]

47 , L − 1}, which leads to a transition to state S1 whose conditional distribution given the initial state is s and the action is a is given by Ps,a . [sent-84, score-0.546]

48 Given a class of policies Π, where policy in Π is a map from S to A, the goal is to ﬁnd π ∈ arg max Vn (π). [sent-85, score-0.982]

49 (4) π∈Π In this single step problem the data are n realization of the random element {S0 , S1 |0, S1 |1, . [sent-86, score-0.316]

50 Denote the i’th realization by {si , si |0, si |1, . [sent-90, score-0.799]

51 , S1 |L − 1)} is its empirical expectation n n−1 i=1 f (si , si |0, si |1, . [sent-97, score-0.622]

52 , si |L − 1), and I(·) is the indicator function taking a value 0 1 1 1 of one when its argument is true and zero otherwise. [sent-100, score-0.319]

53 The following proposition shows that the problem of maximizing the empirical reward (5) is equivalent to a weighted classiﬁcation problem. [sent-101, score-0.443]

54 Proposition 1 Given a class of policies Π and a set of n trajectory trees, L−1 r(S1 |l)I(π(S0 ) = l) arg max En π∈Π l=0 L−1 = arg min En max r(S1 |k) − r(S1 |l) I(π(S0 ) = l) . [sent-102, score-0.88]

55 , max r(si |k) − r(si |L − 1) 1 1 1 1 1 1 k k k is the realization of the L costs of classifying example i to each of the possible labels. [sent-106, score-0.37]

56 Note that the realizations of the costs are always non-negative and the cost of the correct classiﬁcation (arg maxk r(si |k)) is always zero. [sent-107, score-0.298]

57 The solution to the weighted classiﬁcation 1 problem is a map from S to A which minimizes the empirical weighted misclassiﬁcation error (6). [sent-108, score-0.311]

58 The proposition asserts that this mapping is also the control which maximizes the empirical reward (5). [sent-109, score-0.359]

59 In addition, L−1 En r(S1 |l)I(π(S0 ) = l) l=0 = L−1 En I(arg max r(S1 |k) = 0) k r(S1 |l)I(π(S0 ) = l) + r(S1 |l)I(π(S0 ) = l) + . [sent-117, score-0.093]

60 + l=0 L−1 En I(arg max r(S1 |k) = 1) k l=0 L−1 En I(arg max r(S1 |k) = L − 1) k r(S1 |l)I(π(S0 ) = l) . [sent-120, score-0.186]

61 l=0 Substituting (7) we obtain L−1 En r(S1 |l)I(π(S0 ) = l) = l=0 L−1 En {I(arg max r(S1 |k) = j)[r(S1 |j) − k j=0 (max r(S1 |k) − r(S1 |0))I(π(S0 ) = 0) − k (max r(S1 |k) − r(S1 |1))I(π(S0 ) = 1) − . [sent-121, score-0.093]

62 − k (max r(S1 |k) − r(S1 |L − 1))I(π(S0 ) = L − 1)]} = k L−1 En I(arg max r(S1 |k) = j)r(S1 |j) − k j=0 L−1 En max R(S1 |k) − R(S1 |l) I(π(S0 ) = l) l=0 k The term in the second to last line is independent of π(s) and the result follows. [sent-124, score-0.186]

63 In the binary case, the optimization problem is arg min En |r(S1 |0) − r(S1 |1)|I(π(S0 ) = arg max r(S1 |k)) , π∈Π k i. [sent-125, score-0.392]

64 When applying the reduction in [8] to our single step problem the costs are taken to be maxk∈{0,1} r(si |k) rather than |r(si |0) − r(si |1)|. [sent-129, score-0.243]

65 It is easy to set an example of a simple MDP and a restricted class of policies, which do not include the optimal policy, in which the classiﬁer that minimizes the weighted misclassiﬁcation problem with costs maxk∈{0,1} r(si |k) is 1 not equivalent to the optimal policy. [sent-131, score-0.25]

66 On the other hand, in [8] the choice maxk∈{0,1} r(si |k) led to a bound on the perfor1 mance of the policy in terms of the performance of the classiﬁer. [sent-133, score-0.435]

67 We do not pursue this type of bounds here since given the classiﬁer, the performance of the resulting policy can be directly estimated from (5). [sent-134, score-0.435]

68 Given a sequence of classiﬁers, the value of the induced sequence of controls (or policy) can be estimated directly by (3) with generalization guarantees provided by the bounds in [5]. [sent-135, score-0.236]

69 In [2], a certain single step binary reinforcement learning problem is converted to weighted classiﬁcation by averaging multiple realizations of the rewards under the two possible actions for each state. [sent-136, score-0.944]

70 As seen here, this Monte Carlo approach is not necessary; it is sufﬁcient to sample the rewards once for each state. [sent-137, score-0.15]

71 4 Finding Good Policies for a T -Step Markov Decision Processes By Solving a Sequence of Weighted Classiﬁcation Problems Given the class of policies Π, the algorithm updates the controls π0 , . [sent-138, score-0.409]

72 , πT −1 one at a time in a cyclic manner while holding the rest constant. [sent-141, score-0.077]

73 Each update is formulated as a single step reinforcement learning problem which is then converted to a weighted classiﬁcation problem. [sent-142, score-0.528]

74 In practice, if the weighted classiﬁcation problem is only approximately solved, then the new control is accepted only if it leads to higher value of V . [sent-143, score-0.2]

75 When updating πt , the trees are pruned from the root to stage t by keeping only the branch which agrees with the controls π0 , π1 , . [sent-144, score-0.492]

76 Note that when updating πt , each tree contributes one realization of the state at time t. [sent-153, score-0.527]

77 A result of the pruning process is that the ensemble of state realization are drawn from the distribution induced by the policy up to time t − 1. [sent-154, score-0.966]

78 In other words, the algorithm relaxes the requirement in [2] to have access to a baseline distribution - a distribution over the states that is induced by a good policy. [sent-155, score-0.128]

79 Our algorithm automatically generates samples from distributions that are induced by a sequence of monotonically improving policies. [sent-156, score-0.12]

80 In the example: pruning down according to π0 (S0 ) = 0, propagating rewards up according to π2 (S2 |00) = 1, and π2 (S2 |01) = 0. [sent-158, score-0.245]

81 Proposition 2 The algorithm converges after a ﬁnite number of iterations to a policy that cannot be further improved by changing one of the controls and holding the rest ﬁxed. [sent-159, score-0.64]

82 Proof 2 Writing the empirical average sum of rewards Vn (π) explicitly as   Vn (π) = En I(π0 (S 0 ) = i0 )I(π1 (S 1 |i0 ) = i1 ) . [sent-160, score-0.19]

83 , it−1 ) , t=1 it can be seen that the algorithm is a Gauss-Seidel algorithm for maximizing Vn (π), where, at each iteration, optimization of πt is carried out at one of the stages t while keeping πt′ , t′ = t ﬁxed. [sent-172, score-0.079]

84 A cycle with no improvements implies that we cannot increase the empirical average sum of rewards by updating one of the πt ’s. [sent-176, score-0.283]

85 5 Initialization There are two possible initial policies that can be extracted from the set of trajectory trees. [sent-177, score-0.511]

86 One possible initial policy is the myopic policy which is computed from the root of the tree downwards. [sent-178, score-1.156]

87 Staring from the root, π0 is found by solving the single stage reinforcement learning resulting from taking into account only the immediate reward at the next state. [sent-179, score-0.545]

88 Once the weighted classiﬁcation problem is solved the trees are pruned by following the action which agrees with π0 . [sent-180, score-0.474]

89 The remaining realizations of state S1 follow the distribution induced by the myopic control of the ﬁrst stage. [sent-181, score-0.487]

90 The second possible initial policy is computed from the leaves backward to the root. [sent-183, score-0.514]

91 Note that the distribution of the state at a leaf that is chosen at random is the distribution of the state when a randomized policy is used. [sent-184, score-0.833]

92 Given the classiﬁer, we use the equivalent control πT −1 to propagated the rewards up to the previous stage and solve the resulting weighted classiﬁcation problem. [sent-186, score-0.386]

93 This is carried out recursively up to the root of the tree. [sent-187, score-0.077]

94 In particular, when the state space, action space, and the reward function depend on time, and the distribution over the next state depends on all past states and actions, we will be dealing with non-stationary deterministic policies π = (π0 , π1 , . [sent-189, score-1.008]

95 POMDPs can be dealt with in terms of the belief states as a continuous state space MDP or as a non-Markovian process in which policies depend directly on all past observations. [sent-199, score-0.502]

96 While we focused on the trajectory tree method, the algorithm can be easily modiﬁed to solve the optimization problem associated with the random trajectory method [5] by adjusting the single step reinforcement learning reduction and the pruning method presented here. [sent-200, score-0.836]

97 The simulated system is a two-step MDP, with continuous state space S = [0, 1] and a binary action space A = {0, 1}. [sent-202, score-0.321]

98 Given state s and action a the next state s′ is generated by s′ = mod(s + 0. [sent-204, score-0.438]

99 We consider a class of policies parameterized by a continuous parameter: Π = {π(·; θ)|θ = (θ0 , θ1 ) ∈ [0, 2]2 }, where πi (s; θi ) = 1 when θi ≤ 1 and s > θi or when θi > 1 and s < θi − 1 and zero otherwise, i = 0, 1. [sent-208, score-0.372]

100 The path taken by the algorithm supperimposed on the contour plot of Vn (π(θ)) is also presented. [sent-210, score-0.077]

similar papers computed by tfidf model

tfidf for this paper:

wordName wordTfidf (topN-words)

[('policy', 0.435), ('policies', 0.303), ('si', 0.291), ('vn', 0.254), ('reinforcement', 0.229), ('realization', 0.217), ('en', 0.21), ('state', 0.17), ('reward', 0.166), ('rewards', 0.15), ('classi', 0.144), ('realizations', 0.13), ('trajectory', 0.129), ('weighted', 0.121), ('actions', 0.112), ('arg', 0.111), ('maxk', 0.108), ('action', 0.098), ('max', 0.093), ('tree', 0.092), ('cation', 0.088), ('trees', 0.085), ('initial', 0.079), ('mdp', 0.078), ('root', 0.077), ('deterministic', 0.072), ('induced', 0.07), ('st', 0.069), ('controls', 0.066), ('stage', 0.065), ('agrees', 0.064), ('pomdps', 0.061), ('costs', 0.06), ('reduction', 0.057), ('proposition', 0.056), ('componentwise', 0.055), ('langford', 0.052), ('holding', 0.052), ('control', 0.05), ('path', 0.05), ('sequence', 0.05), ('ers', 0.048), ('updating', 0.048), ('hero', 0.048), ('maximizes', 0.047), ('converted', 0.047), ('pruning', 0.045), ('cycle', 0.045), ('solved', 0.045), ('maximizer', 0.043), ('mansour', 0.043), ('inc', 0.043), ('step', 0.043), ('empirical', 0.04), ('supervised', 0.04), ('class', 0.04), ('converges', 0.038), ('michigan', 0.038), ('myopic', 0.038), ('ann', 0.038), ('barto', 0.038), ('arbor', 0.038), ('decision', 0.038), ('regions', 0.037), ('converts', 0.036), ('kearns', 0.036), ('nite', 0.036), ('mod', 0.035), ('pruned', 0.032), ('learning', 0.032), ('branch', 0.031), ('sons', 0.031), ('er', 0.031), ('maximizing', 0.031), ('misclassi', 0.03), ('continuous', 0.029), ('distribution', 0.029), ('maximal', 0.029), ('problem', 0.029), ('mi', 0.028), ('argument', 0.028), ('dynamic', 0.028), ('single', 0.027), ('taken', 0.027), ('immediate', 0.026), ('mappings', 0.026), ('rest', 0.025), ('according', 0.025), ('keeping', 0.024), ('generative', 0.024), ('binary', 0.024), ('optimization', 0.024), ('zadrozny', 0.024), ('ent', 0.024), ('plethora', 0.024), ('subproblems', 0.024), ('tenth', 0.024), ('ting', 0.024), ('iterations', 0.024), ('iteration', 0.024)]

similar papers list:

simIndex simValue paperId paperTitle

same-paper 1 1.0000004 78 nips-2005-From Weighted Classification to Policy Search

Author: Doron Blatt, Alfred O. Hero

2 0.33907196 145 nips-2005-On Local Rewards and Scaling Distributed Reinforcement Learning

Author: Drew Bagnell, Andrew Y. Ng

Abstract: We consider the scaling of the number of examples necessary to achieve good performance in distributed, cooperative, multi-agent reinforcement learning, as a function of the the number of agents n. We prove a worstcase lower bound showing that algorithms that rely solely on a global reward signal to learn policies confront a fundamental limit: They require a number of real-world examples that scales roughly linearly in the number of agents. For settings of interest with a very large number of agents, this is impractical. We demonstrate, however, that there is a class of algorithms that, by taking advantage of local reward signals in large distributed Markov Decision Processes, are able to ensure good performance with a number of samples that scales as O(log n). This makes them applicable even in settings with a very large number of agents n. 1

3 0.27866536 144 nips-2005-Off-policy Learning with Options and Recognizers

Author: Doina Precup, Cosmin Paduraru, Anna Koop, Richard S. Sutton, Satinder P. Singh

Abstract: We introduce a new algorithm for off-policy temporal-difference learning with function approximation that has lower variance and requires less knowledge of the behavior policy than prior methods. We develop the notion of a recognizer, a ﬁlter on actions that distorts the behavior policy to produce a related target policy with low-variance importance-sampling corrections. We also consider target policies that are deviations from the state distribution of the behavior policy, such as potential temporally abstract options, which further reduces variance. This paper introduces recognizers and their potential advantages, then develops a full algorithm for linear function approximation and proves that its updates are in the same direction as on-policy TD updates, which implies asymptotic convergence. Even though our algorithm is based on importance sampling, we prove that it requires absolutely no knowledge of the behavior policy for the case of state-aggregation function approximators. Off-policy learning is learning about one way of behaving while actually behaving in another way. For example, Q-learning is an off- policy learning method because it learns about the optimal policy while taking actions in a more exploratory fashion, e.g., according to an ε-greedy policy. Off-policy learning is of interest because only one way of selecting actions can be used at any time, but we would like to learn about many different ways of behaving from the single resultant stream of experience. For example, the options framework for temporal abstraction involves considering a variety of different ways of selecting actions. For each such option one would like to learn a model of its possible outcomes suitable for planning and other uses. Such option models have been proposed as fundamental building blocks of grounded world knowledge (Sutton, Precup & Singh, 1999; Sutton, Rafols & Koop, 2005). Using off-policy learning, one would be able to learn predictive models for many options at the same time from a single stream of experience. Unfortunately, off-policy learning using temporal-difference methods has proven problematic when used in conjunction with function approximation. Function approximation is essential in order to handle the large state spaces that are inherent in many problem do- mains. Q-learning, for example, has been proven to converge to an optimal policy in the tabular case, but is unsound and may diverge in the case of linear function approximation (Baird, 1996). Precup, Sutton, and Dasgupta (2001) introduced and proved convergence for the ﬁrst off-policy learning algorithm with linear function approximation. They addressed the problem of learning the expected value of a target policy based on experience generated using a different behavior policy. They used importance sampling techniques to reduce the off-policy case to the on-policy case, where existing convergence theorems apply (Tsitsiklis & Van Roy, 1997; Tadic, 2001). There are two important difﬁculties with that approach. First, the behavior policy needs to be stationary and known, because it is needed to compute the importance sampling corrections. Second, the importance sampling weights are often ill-conditioned. In the worst case, the variance could be inﬁnite and convergence would not occur. The conditions required to prevent this were somewhat awkward and, even when they applied and asymptotic convergence was assured, the variance could still be high and convergence could be slow. In this paper we address both of these problems in the context of off-policy learning for options. We introduce the notion of a recognizer. Rather than specifying an explicit target policy (for instance, the policy of an option), about which we want to make predictions, a recognizer speciﬁes a condition on the actions that are selected. For example, a recognizer for the temporally extended action of picking up a cup would not specify which hand is to be used, or what the motion should be at all different positions of the cup. The recognizer would recognize a whole variety of directions of motion and poses as part of picking the cup. The advantage of this strategy is not that one might prefer a multitude of different behaviors, but that the behavior may be based on a variety of different strategies, all of which are relevant, and we would like to learn from any of them. In general, a recognizer is a function that recognizes or accepts a space of different ways of behaving and thus, can learn from a wider range of data. Recognizers have two advantages over direct speciﬁcation of a target policy: 1) they are a natural and easy way to specify a target policy for which importance sampling will be well conditioned, and 2) they do not require the behavior policy to be known. The latter is important because in many cases we may have little knowledge of the behavior policy, or a stationary behavior policy may not even exist. We show that for the case of state aggregation, even if the behavior policy is unknown, convergence to a good model is achieved. 1 Non-sequential example The beneﬁts of using recognizers in off-policy learning can be most easily seen in a nonsequential context with a single continuous action. Suppose you are given a sequence of sample actions ai ∈ [0, 1], selected i.i.d. according to probability density b : [0, 1] → ℜ+ (the behavior density). For example, suppose the behavior density is of the oscillatory form shown as a red line in Figure 1. For each each action, ai , we observe a corresponding outcome, zi ∈ ℜ, a random variable whose distribution depends only on ai . Thus the behavior density induces an outcome density. The on-policy problem is to estimate the mean mb of the outcome density. This problem can be solved simply by averaging the sample outcomes: mb = (1/n) ∑n zi . The off-policy problem is to use this same data to learn what ˆ i=1 the mean would be if actions were selected in some way other than b, for example, if the actions were restricted to a designated range, such as between 0.7 and 0.9. There are two natural ways to pose this off-policy problem. The most straightforward way is to be equally interested in all actions within the designated region. One professes to be interested in actions selected according to a target density π : [0, 1] → ℜ+ , which in the example would be 5.0 between 0.7 and 0.9, and zero elsewhere, as in the dashed line in 12 Probability density functions 1.5 Target policy with recognizer 1 Target policy w/o recognizer without recognizer .5 Behavior policy 0 0 Action 0.7 Empirical variances (average of 200 sample variances) 0.9 1 0 10 with recognizer 100 200 300 400 500 Number of sample actions Figure 1: The left panel shows the behavior policy and the target policies for the formulations of the problem with and without recognizers. The right panel shows empirical estimates of the variances for the two formulations as a function of the number sample actions. The lowest line is for the formulation using empirically-estimated recognition probabilities. Figure 1 (left). The importance- sampling estimate of the mean outcome is 1 n π(ai ) mπ = ∑ ˆ zi . n i=1 b(ai ) (1) This approach is problematic if there are parts of the region of interest where the behavior density is zero or very nearly so, such as near 0.72 and 0.85 in the example. Here the importance sampling ratios are exceedingly large and the estimate is poorly conditioned (large variance). The upper curve in Figure 1 (right) shows the empirical variance of this estimate as a function of the number of samples. The spikes and uncertain decline of the empirical variance indicate that the distribution is very skewed and that the estimates are very poorly conditioned. The second way to pose the problem uses recognizers. One professes to be interested in actions to the extent that they are both selected by b and within the designated region. This leads to the target policy shown in blue in the left panel of Figure 1 (it is taller because it still must sum to 1). For this problem, the variance of (1) is much smaller, as shown in the lower two lines of Figure 1 (right). To make this way of posing the problem clear, we introduce the notion of a recognizer function c : A → ℜ+ . The action space in the example is A = [0, 1] and the recognizer is c(a) = 1 for a between 0.7 and 0.9 and is zero elsewhere. The target policy is deﬁned in general by c(a)b(a) c(a)b(a) = . (2) π(a) = µ ∑x c(x)b(x) where µ = ∑x c(x)b(x) is a constant, equal to the probability of recognizing an action from the behavior policy. Given π, mπ from (1) can be rewritten in terms of the recognizer as ˆ n π(ai ) 1 n c(ai )b(ai ) 1 1 n c(ai ) 1 mπ = ∑ zi ˆ = ∑ zi = ∑ zi (3) n i=1 b(ai ) n i=1 µ b(ai ) n i=1 µ Note that the target density does not appear at all in the last expression and that the behavior distribution appears only in µ, which is independent of the sample action. If this constant is known, then this estimator can be computed with no knowledge of π or b. The constant µ can easily be estimated as the fraction of recognized actions in the sample. The lowest line in Figure 1 (right) shows the variance of the estimator using this fraction in place of the recognition probability. Its variance is low, no worse than that of the exact algorithm, and apparently slightly lower. Because this algorithm does not use the behavior density, it can be applied when the behavior density is unknown or does not even exist. For example, suppose actions were selected in some deterministic, systematic way that in the long run produced an empirical distribution like b. This would be problematic for the other algorithms but would require no modiﬁcation of the recognition-fraction algorithm. 2 Recognizers improve conditioning of off-policy learning The main use of recognizers is in formulating a target density π about which we can successfully learn predictions, based on the current behavior being followed. Here we formalize this intuition. Theorem 1 Let A = {a1 , . . . ak } ⊆ A be a subset of all the possible actions. Consider a ﬁxed behavior policy b and let πA be the class of policies that only choose actions from A, i.e., if π(a) > 0 then a ∈ A. Then the policy induced by b and the binary recognizer cA is the policy with minimum-variance one-step importance sampling corrections, among those in πA : π(ai ) 2 π as given by (2) = arg min Eb (4) π∈πA b(ai ) Proof: Denote π(ai ) = πi , b(ai ) = bi . Then the expected variance of the one-step importance sampling corrections is: Eb πi bi πi bi 2 2 − Eb = ∑ bi i πi bi 2 −1 = ∑ i π2 i − 1, bi where the summation (here and everywhere below) is such that the action ai ∈ A. We want to ﬁnd πi that minimizes this expression, subject to the constraint that ∑i πi = 1. This is a constrained optimization problem. To solve it, we write down the corresponding Lagrangian: π2 L(πi , β) = ∑ i − 1 + β(∑ πi − 1) i i bi We take the partial derivatives wrt πi and β and set them to 0: βbi ∂L 2 = πi + β = 0 ⇒ πi = − ∂πi bi 2 (5) ∂L = πi − 1 = 0 ∂β ∑ i (6) By taking (5) and plugging into (6), we get the following expression for β: − β 2 bi = 1 ⇒ β = − 2∑ ∑i bi i By substituting β into (5) we obtain: πi = bi ∑i b i This is exactly the policy induced by the recognizer deﬁned by c(ai ) = 1 iff ai ∈ A. We also note that it is advantageous, from the point of view of minimizing the variance of the updates, to have recognizers that accept a broad range of actions: Theorem 2 Consider two binary recognizers c1 and c2 , such that µ1 > µ2 . Then the importance sampling corrections for c1 have lower variance than the importance sampling corrections for c2 . Proof: From the previous theorem, we have the variance of a recognizer cA : Var = ∑ i π2 bi i −1 = ∑ bi ∑ j∈A b j i 2 1 1 1 −1 = −1 = −1 bi µ ∑ j∈A b j 3 Formal framework for sequential problems We turn now to the full case of learning about sequential decision processes with function approximation. We use the standard framework in which an agent interacts with a stochastic environment. At each time step t, the agent receives a state st and chooses an action at . We assume for the moment that actions are selected according to a ﬁxed behavior policy, b : S × A → [0, 1] where b(s, a) is the probability of selecting action a in state s. The behavior policy is used to generate a sequence of experience (observations, actions and rewards). The goal is to learn, from this data, predictions about different ways of behaving. In this paper we focus on learning predictions about expected returns, but other predictions can be tackled as well (for instance, predictions of transition models for options (Sutton, Precup & Singh, 1999), or predictions speciﬁed by a TD-network (Sutton & Tanner, 2005; Sutton, Rafols & Koop, 2006)). We assume that the state space is large or continuous, and function approximation must be used to compute any values of interest. In particular, we assume a space of feature vectors Φ and a mapping φ : S → Φ. We denote by φs the feature vector associated with s. An option is deﬁned as a triple o = I, π, β where I ⊆ S is the set of states in which the option can be initiated, π is the internal policy of the option and β : S → [0, 1] is a stochastic termination condition. In the option work (Sutton, Precup & Singh, 1999), each of these elements has to be explicitly speciﬁed and ﬁxed in order for an option to be well deﬁned. Here, we will instead deﬁne options implicitly, using the notion of a recognizer. A recognizer is deﬁned as a function c : S × A → [0, 1], where c(s, a) indicates to what extent the recognizer allows action a in state s. An important special case, which we treat in this paper, is that of binary recognizers. In this case, c is an indicator function, specifying a subset of actions that are allowed, or recognized, given a particular state. Note that recognizers do not specify policies; instead, they merely give restrictions on the policies that are allowed or recognized. A recognizer c together with a behavior policy b generates a target policy π, where: b(s, a)c(s, a) b(s, a)c(s, a) π(s, a) = (7) = µ(s) ∑x b(s, x)c(s, x) The denominator of this fraction, µ(s) = ∑x b(s, x)c(s, x), is the recognition probability at s, i.e., the probability that an action will be accepted at s when behavior is generated according to b. The policy π is only deﬁned at states for which µ(s) > 0. The numerator gives the probability that action a is produced by the behavior and recognized in s. Note that if the recognizer accepts all state-action pairs, i.e. c(s, a) = 1, ∀s, a, then π is the same as b. Since a recognizer and a behavior policy can specify together a target policy, we can use recognizers as a way to specify policies for options, using (7). An option can only be initiated at a state for which at least one action is recognized, so µ(s) > 0, ∀s ∈ I. Similarly, the termination condition of such an option, β, is deﬁned as β(s) = 1 if µ(s) = 0. In other words, the option must terminate if no actions are recognized at a given state. At all other states, β can be deﬁned between 0 and 1 as desired. We will focus on computing the reward model of an option o, which represents the expected total return. The expected values of different features at the end of the option can be estimated similarly. The quantity that we want to compute is Eo {R(s)} = E{r1 + r2 + . . . + rT |s0 = s, π, β} where s ∈ I, experience is generated according to the policy of the option, π, and T denotes the random variable representing the time step at which the option terminates according to β. We assume that linear function approximation is used to represent these values, i.e. Eo {R(s)} ≈ θT φs where θ is a vector of parameters. 4 Off-policy learning algorithm In this section we present an adaptation of the off-policy learning algorithm of Precup, Sutton & Dasgupta (2001) to the case of learning about options. Suppose that an option’s policy π was used to generate behavior. In this case, learning the reward model of the option is a special case of temporal-difference learning of value functions. The forward ¯ (n) view of this algorithm is as follows. Let Rt denote the truncated n-step return starting at ¯ (0) time step t and let yt denote the 0-step truncated return, Rt . By the deﬁnition of the n-step truncated return, we have: ¯ (n) ¯ (n−1) Rt = rt+1 + (1 − βt+1 )Rt+1 . This is similar to the case of value functions, but it accounts for the possibility of terminating the option at time step t + 1. The λ-return is deﬁned in the usual way: ∞ ¯ (n) ¯ Rtλ = (1 − λ) ∑ λn−1 Rt . n=1 The parameters of the linear function approximator are updated on every time step proportionally to: ¯ ¯ ∆θt = Rtλ − yt ∇θ yt (1 − β1 ) · · · (1 − βt ). In our case, however, trajectories are generated according to the behavior policy b. The main idea of the algorithm is to use importance sampling corrections in order to account for the difference in the state distribution of the two policies. Let ρt = (n) Rt , π(st ,at ) b(st ,at ) be the importance sampling ratio at time step t. The truncated n-step return, satisﬁes: (n) (n−1) Rt = ρt [rt+1 + (1 − βt+1 )Rt+1 ]. The update to the parameter vector is proportional to: ∆θt = Rtλ − yt ∇θ yt ρ0 (1 − β1 ) · · · ρt−1 (1 − βt ). The following result shows that the expected updates of the on-policy and off-policy algorithms are the same. Theorem 3 For every time step t ≥ 0 and any initial state s, ¯ Eb [∆θt |s] = Eπ [∆θt |s]. (n) (n) ¯ Proof: First we will show by induction that Eb {Rt |s} = Eπ {Rt |s}, ∀n (which implies ¯ that Eb {Rtλ |s} = Eπ (Rtλ |s}). For n = 0, the statement is trivial. Assuming that it is true for n − 1, we have (n) Eb Rt |s = a ∑b(s, a)∑Pss ρ(s, a) a = s ∑∑ a Pss b(s, a) a s = a ∑π(s, a)∑Pss a (n−1) a rss + (1 − β(s ))Eb Rt+1 |s π(s, a) a ¯ (n−1) r + (1 − β(s ))Eπ Rt+1 |s b(s, a) ss a ¯ (n−1) rss + (1 − β(s ))Eπ Rt+1 |s ¯ (n) = Eπ Rt |s . s Now we are ready to prove the theorem’s main statement. Deﬁning Ωt to be the set of all trajectory components up to state st , we have: Eb {∆θt |s} = ∑ ω∈Ωt Pb (ω|s)Eb (Rtλ − yt )∇θ yt |ω t−1 ∏ ρi (1 − βi+1 ) i=0 πi (1 − βi+1 ) i=0 bi t−1 = t−1 ∑ ∏ bi Psaiisi+1 ω∈Ωt Eb Rtλ |st − yt ∇θ yt ∏ i=0 t−1 = ∑ ∏ πi Psaiisi+1 ω∈Ωt = ∑ ω∈Ωt ¯ Eπ Rtλ |st − yt ∇θ yt (1 − β1 )...(1 − βt ) i=0 ¯ ¯ Pπ (ω|s)Eπ (Rtλ − yt )∇θ yt |ω (1 − β1 )...(1 − βt ) = Eπ ∆θt |s . Note that we are able to use st and ω interchangeably because of the Markov property. ¯ Since we have shown that Eb [∆θt |s] = Eπ [∆θt |s] for any state s, it follows that the expected updates will also be equal for any distribution of the initial state s. When learning the model of options with data generated from the behavior policy b, the starting state distribution with respect to which the learning is performed, I0 is determined by the stationary distribution of the behavior policy, as well as the initiation set of the option I. We note also that the importance sampling corrections only have to be performed for the trajectory since the initiation of the updates for the option. No corrections are required for the experience prior to this point. This should generate updates that have signiﬁcantly lower variance than in the case of learning values of policies (Precup, Sutton & Dasgupta, 2001). Because of the termination condition of the option, β, ∆θ can quickly decay to zero. To avoid this problem, we can use a restart function g : S → [0, 1], such that g(st ) speciﬁes the extent to which the updating episode is considered to start at time t. Adding restarts generates a new forward update: t ∆θt = (Rtλ − yt )∇θ yt ∑ gi ρi ...ρt−1 (1 − βi+1 )...(1 − βt ), (8) i=0 where Rtλ is the same as above. With an adaptation of the proof in Precup, Sutton & Dasgupta (2001), we can show that we get the same expected value of updates by applying this algorithm from the original starting distribution as we would by applying the algorithm without restarts from a starting distribution deﬁned by I0 and g. We can turn this forward algorithm into an incremental, backward view algorithm in the following way: • Initialize k0 = g0 , e0 = k0 ∇θ y0 • At every time step t: δt = θt+1 = kt+1 = et+1 = ρt (rt+1 + (1 − βt+1 )yt+1 ) − yt θt + αδt et ρt kt (1 − βt+1 ) + gt+1 λρt (1 − βt+1 )et + kt+1 ∇θ yt+1 Using a similar technique to that of Precup, Sutton & Dasgupta (2001) and Sutton & Barto (1998), we can prove that the forward and backward algorithm are equivalent (omitted due to lack of space). This algorithm is guaranteed to converge if the variance of the updates is ﬁnite (Precup, Sutton & Dasgupta, 2001). In the case of options, the termination condition β can be used to ensure that this is the case. 5 Learning when the behavior policy is unknown In this section, we consider the case in which the behavior policy is unknown. This case is generally problematic for importance sampling algorithms, but the use of recognizers will allow us to deﬁne importance sampling corrections, as well as a convergent algorithm. Recall that when using a recognizer, the target policy of the option is deﬁned as: c(s, a)b(s, a) π(s, a) = µ(s) and the recognition probability becomes: π(s, a) c(s, a) = b(s, a) µ(s) Of course, µ(s) depends on b. If b is unknown, instead of µ(s), we will use a maximum likelihood estimate µ : S → [0, 1]. The structure used to compute µ will have to be compatible ˆ ˆ with the feature space used to represent the reward model. We will make this more precise below. Likewise, the recognizer c(s, a) will have to be deﬁned in terms of the features used to represent the model. We will then deﬁne the importance sampling corrections as: c(s, a) ˆ ρ(s, a) = µ(s) ˆ ρ(s, a) = We consider the case in which the function approximator used to model the option is actually a state aggregator. In this case, we will deﬁne recognizers which behave consistently in each partition, i.e., c(s, a) = c(p, a), ∀s ∈ p. This means that an action is either recognized or not recognized in all states of the partition. The recognition probability µ will have one ˆ entry for every partition p of the state space. Its value will be: N(p, c = 1) µ(p) = ˆ N(p) where N(p) is the number of times partition p was visited, and N(p, c = 1) is the number of times the action taken in p was recognized. In the limit, w.p.1, µ converges to ˆ ∑s d b (s|p) ∑a c(p, a)b(s, a) where d b (s|p) is the probability of visiting state s from partiˆ ˆ tion p under the stationary distribution of b. At this limit, π(s, a) = ρ(s, a)b(s, a) will be a ˆ well-deﬁned policy (i.e., ∑a π(s, a) = 1). Using Theorem 3, off-policy updates using imˆ portance sampling corrections ρ will have the same expected value as on-policy updates ˆ ˆ using π. Note though that the learning algorithm never uses π; the only quantities needed ˆ are ρ, which are learned incrementally from data. For the case of general linear function approximation, we conjecture that a similar idea can be used, where the recognition probability is learned using logistic regression. The development of this part is left for future work. Acknowledgements The authors gratefully acknowledge the ideas and encouragement they have received in this work from Eddie Rafols, Mark Ring, Lihong Li and other members of the rlai.net group. We thank Csaba Szepesvari and the reviewers of the paper for constructive comments. This research was supported in part by iCore, NSERC, Alberta Ingenuity, and CFI. References Baird, L. C. (1995). Residual algorithms: Reinforcement learning with function approximation. In Proceedings of ICML. Precup, D., Sutton, R. S. and Dasgupta, S. (2001). Off-policy temporal-difference learning with function approximation. In Proceedings of ICML. Sutton, R.S., Precup D. and Singh, S (1999). Between MDPs and semi-MDPs: A framework for temporal abstraction in reinforcement learning. Artiﬁcial Intelligence, vol . 112, pp. 181–211. Sutton,, R.S. and Tanner, B. (2005). Temporal-difference networks. In Proceedings of NIPS-17. Sutton R.S., Raffols E. and Koop, A. (2006). Temporal abstraction in temporal-difference networks”. In Proceedings of NIPS-18. Tadic, V. (2001). On the convergence of temporal-difference learning with linear function approximation. In Machine learning vol. 42, pp. 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.

4 0.2715089 153 nips-2005-Policy-Gradient Methods for Planning

Author: Douglas Aberdeen

Abstract: Probabilistic temporal planning attempts to ﬁnd good policies for acting in domains with concurrent durative tasks, multiple uncertain outcomes, and limited resources. These domains are typically modelled as Markov decision problems and solved using dynamic programming methods. This paper demonstrates the application of reinforcement learning — in the form of a policy-gradient method — to these domains. Our emphasis is large domains that are infeasible for dynamic programming. Our approach is to construct simple policies, or agents, for each planning task. The result is a general probabilistic temporal planner, named the Factored Policy-Gradient Planner (FPG-Planner), which can handle hundreds of tasks, optimising for probability of success, duration, and resource use. 1

5 0.24938722 186 nips-2005-TD(0) Leads to Better Policies than Approximate Value Iteration

Author: Benjamin V. Roy

Abstract: We consider approximate value iteration with a parameterized approximator in which the state space is partitioned and the optimal cost-to-go function over each partition is approximated by a constant. We establish performance loss bounds for policies derived from approximations associated with ﬁxed points. These bounds identify beneﬁts to having projection weights equal to the invariant distribution of the resulting policy. Such projection weighting leads to the same ﬁxed points as TD(0). Our analysis also leads to the ﬁrst performance loss bound for approximate value iteration with an average cost objective. 1 Preliminaries Consider a discrete-time communicating Markov decision process (MDP) with a ﬁnite state space S = {1, . . . , |S|}. At each state x ∈ S, there is a ﬁnite set Ux of admissible actions. If the current state is x and an action u ∈ Ux is selected, a cost of gu (x) is incurred, and the system transitions to a state y ∈ S with probability pxy (u). For any x ∈ S and u ∈ Ux , y∈S pxy (u) = 1. Costs are discounted at a rate of α ∈ (0, 1) per period. Each instance of such an MDP is deﬁned by a quintuple (S, U, g, p, α). A (stationary deterministic) policy is a mapping µ that assigns an action u ∈ Ux to each state x ∈ S. If actions are selected based on a policy µ, the state follows a Markov process with transition matrix Pµ , where each (x, y)th entry is equal to pxy (µ(x)). The restriction to communicating MDPs ensures that it is possible to reach any state from any other state. Each policy µ is associated with a cost-to-go function Jµ ∈ |S| , deﬁned by Jµ = ∞ t t −1 gµ , where, with some abuse of notation, gµ (x) = gµ(x) (x) t=0 α Pµ gµ = (I − αPµ ) for each x ∈ S. A policy µ is said to be greedy with respect to a function J if µ(x) ∈ argmin(gu (x) + α y∈S pxy (u)J(y)) for all x ∈ S. u∈Ux The optimal cost-to-go function J ∗ ∈ |S| is deﬁned by J ∗ (x) = minµ Jµ (x), for all x ∈ S. A policy µ∗ is said to be optimal if Jµ∗ = J ∗ . It is well-known that an optimal policy exists. Further, a policy µ∗ is optimal if and only if it is greedy with respect to J ∗ . Hence, given the optimal cost-to-go function, optimal actions can computed be minimizing the right-hand side of the above inclusion. Value iteration generates a sequence J converging to J ∗ according to J +1 = T J , where T is the dynamic programming operator, deﬁned by (T J)(x) = minu∈Ux (gu (x) + α y∈S pxy (u)J(y)), for all x ∈ S and J ∈ |S| . This sequence converges to J ∗ for any initialization of J0 . 2 Approximate Value Iteration The state spaces of relevant MDPs are typically so large that computation and storage of a cost-to-go function is infeasible. One approach to dealing with this obstacle involves partitioning the state space S into a manageable number K of disjoint subsets S1 , . . . , SK and approximating the optimal cost-to-go function with a function that is constant over each partition. This can be thought of as a form of state aggregation – all states within a given partition are assumed to share a common optimal cost-to-go. To represent an approximation, we deﬁne a matrix Φ ∈ |S|×K such that each kth column is an indicator function for the kth partition Sk . Hence, for any r ∈ K , k, and x ∈ Sk , (Φr)(x) = rk . In this paper, we study variations of value iteration, each of which computes a vector r so that Φr approximates J ∗ . The use of such a policy µr which is greedy with respect to Φr is justiﬁed by the following result (see [10] for a proof): ˜ Theorem 1 If µ is a greedy policy with respect to a function J ∈ Jµ − J ∗ ≤ ∞ 2α ˜ J∗ − J 1−α |S| then ∞. One common way of approximating a function J ∈ |S| with a function of the form Φr involves projection with respect to a weighted Euclidean norm · π . The weighted Euclidean 1/2 |S| 2 norm: J 2,π = . Here, π ∈ + is a vector of weights that assign x∈S π(x)J (x) relative emphasis among states. The projection Ππ J is the function Φr that attains the minimum of J −Φr 2,π ; if there are multiple functions Φr that attain the minimum, they must form an afﬁne space, and the projection is taken to be the one with minimal norm Φr 2,π . Note that in our context, where each kth column of Φ represents an indicator function for the kth partition, for any π, J, and x ∈ Sk , (Ππ J)(x) = y∈Sk π(y)J(y)/ y∈Sk π(y). Approximate value iteration begins with a function Φr(0) and generates a sequence according to Φr( +1) = Ππ T Φr( ) . It is well-known that the dynamic programming operator T is a contraction mapping with respect to the maximum norm. Further, Ππ is maximum-norm nonexpansive [16, 7, 8]. (This is not true for general Φ, but is true in our context in which columns of Φ are indicator functions for partitions.) It follows that the composition Ππ T is a contraction mapping. By the contraction mapping theorem, Ππ T has a unique ﬁxed point Φ˜, which is the limit of the sequence Φr( ) . Further, the following result holds: r Theorem 2 For any MDP, partition, and weights π with support intersecting every partition, if Φ˜ = Ππ T Φ˜ then r r Φ˜ − J ∗ r ∞ ≤ 2 min J ∗ − Φr 1 − α r∈ K and (1 − α) Jµr − J ∗ ˜ ∞ ≤ ∞, 4α min J ∗ − Φr 1 − α r∈ K ∞. The ﬁrst inequality of the theorem is an approximation error bound, established in [16, 7, 8] for broader classes of approximators that include state aggregation as a special case. The second is a performance loss bound, derived by simply combining the approximation error bound and Theorem 1. Note that Jµr (x) ≥ J ∗ (x) for all x, so the left-hand side of the performance loss bound ˜ is the maximal increase in cost-to-go, normalized by 1 − α. This normalization is natural, since a cost-to-go function is a linear combination of expected future costs, with coefﬁcients 1, α, α2 , . . ., which sum to 1/(1 − α). Our motivation of the normalizing constant begs the question of whether, for ﬁxed MDP parameters (S, U, g, p) and ﬁxed Φ, minr J ∗ − Φr ∞ also grows with 1/(1 − α). It turns out that minr J ∗ − Φr ∞ = O(1). To see why, note that for any µ, Jµ = (I − αPµ )−1 gµ = 1 λ µ + hµ , 1−α where λµ (x) is the expected average cost if the process starts in state x and is controlled by policy µ, τ −1 1 t λµ = lim Pµ gµ , τ →∞ τ t=0 and hµ is the discounted differential cost function hµ = (I − αPµ )−1 (gµ − λµ ). Both λµ and hµ converge to ﬁnite vectors as α approaches 1 [3]. For an optimal policy µ∗ , limα↑1 λµ∗ (x) does not depend on x (in our context of a communicating MDP). Since constant functions lie in the range of Φ, lim min J ∗ − Φr α↑1 r∈ K ∞ ≤ lim hµ∗ α↑1 ∞ < ∞. The performance loss bound still exhibits an undesirable dependence on α through the coefﬁcient 4α/(1 − α). In most relevant contexts, α is close to 1; a representative value might be 0.99. Consequently, 4α/(1 − α) can be very large. Unfortunately, the bound is sharp, as expressed by the following theorem. We will denote by 1 the vector with every component equal to 1. Theorem 3 For any δ > 0, α ∈ (0, 1), and ∆ ≥ 0, there exists MDP parameters (S, U, g, p) and a partition such that minr∈ K J ∗ − Φr ∞ = ∆ and, if Φ˜ = Ππ T Φ˜ r r with π = 1, 4α min J ∗ − Φr ∞ − δ. (1 − α) Jµr − J ∗ ∞ ≥ ˜ 1 − α r∈ K This theorem is established through an example in [22]. The choice of uniform weights (π = 1) is meant to point out that even for such a simple, perhaps natural, choice of weights, the performance loss bound is sharp. Based on Theorems 2 and 3, one might expect that there exists MDP parameters (S, U, g, p) and a partition such that, with π = 1, (1 − α) Jµr − J ∗ ˜ ∞ =Θ 1 min J ∗ − Φr 1 − α r∈ K ∞ . In other words, that the performance loss is both lower and upper bounded by 1/(1 − α) times the smallest possible approximation error. It turns out that this is not true, at least if we restrict to a ﬁnite state space. However, as the following theorem establishes, the coefﬁcient multiplying minr∈ K J ∗ − Φr ∞ can grow arbitrarily large as α increases, keeping all else ﬁxed. Theorem 4 For any L and ∆ ≥ 0, there exists MDP parameters (S, U, g, p) and a partition such that limα↑1 minr∈ K J ∗ − Φr ∞ = ∆ and, if Φ˜ = Ππ T Φ˜ with π = 1, r r lim inf (1 − α) (Jµr (x) − J ∗ (x)) ≥ L lim min J ∗ − Φr ∞ , ˜ α↑1 α↑1 r∈ K for all x ∈ S. This Theorem is also established through an example [22]. For any µ and x, lim ((1 − α)Jµ (x) − λµ (x)) = lim(1 − α)hµ (x) = 0. α↑1 α↑1 Combined with Theorem 4, this yields the following corollary. Corollary 1 For any L and ∆ ≥ 0, there exists MDP parameters (S, U, g, p) and a partition such that limα↑1 minr∈ K J ∗ − Φr ∞ = ∆ and, if Φ˜ = Ππ T Φ˜ with π = 1, r r ∗ lim inf (λµr (x) − λµ∗ (x)) ≥ L lim min J − Φr ∞ , ˜ α↑1 α↑1 r∈ K for all x ∈ S. 3 Using the Invariant Distribution In the previous section, we considered an approximation Φ˜ that solves Ππ T Φ˜ = Φ˜ for r r r some arbitrary pre-selected weights π. We now turn to consider use of an invariant state distribution πr of Pµr as the weight vector.1 This leads to a circular deﬁnition: the weights ˜ ˜ are used in deﬁning r and now we are deﬁning the weights in terms of r. What we are ˜ ˜ really after here is a vector r that satisﬁes Ππr T Φ˜ = Φ˜. The following theorem captures ˜ r r ˜ the associated beneﬁts. (Due to space limitations, we omit the proof, which is provided in the full length version of this paper [22].) Theorem 5 For any MDP and partition, if Φ˜ = Ππr T Φ˜ and πr has support intersecting r r ˜ ˜ T every partition, (1 − α)πr (Jµr − J ∗ ) ≤ 2α minr∈ K J ∗ − Φr ∞ . ˜ ˜ When α is close to 1, which is typical, the right-hand side of our new performance loss bound is far less than that of Theorem 2. The primary improvement is in the omission of a factor of 1 − α from the denominator. But for the bounds to be compared in a meaningful way, we must also relate the left-hand-side expressions. A relation can be based on the fact that for all µ, limα↑1 (1 − α)Jµ − λµ ∞ = 0, as explained in Section 2. In particular, based on this, we have lim(1 − α) Jµ − J ∗ ∞ = |λµ − λ∗ | = λµ − λ∗ = lim π T (Jµ − J ∗ ), α↑1 α↑1 for all policies µ and probability distributions π. Hence, the left-hand-side expressions from the two performance bounds become directly comparable as α approaches 1. Another interesting comparison can be made by contrasting Corollary 1 against the following immediate consequence of Theorem 5. Corollary 2 For all MDP parameters (S, U, g, p) and partitions, if Φ˜ = Ππr T Φ˜ and r r ˜ lim inf α↑1 x∈Sk πr (x) > 0 for all k, ˜ lim sup λµr − λµ∗ ∞ ≤ 2 lim min J ∗ − Φr ∞ . ˜ α↑1 α↑1 r∈ K The comparison suggests that solving Φ˜ = Ππr T Φ˜ is strongly preferable to solving r r ˜ Φ˜ = Ππ T Φ˜ with π = 1. r r 1 By an invariant state distribution of a transition matrix P , we mean any probability distribution π such that π T P = π T . In the event that Pµr has multiple invariant distributions, πr denotes an ˜ ˜ arbitrary choice. 4 Exploration If a vector r solves Φ˜ = Ππr T Φ˜ and the support of πr intersects every partition, Theorem ˜ r r ˜ ˜ 5 promises a desirable bound. However, there are two signiﬁcant shortcomings to this solution concept, which we will address in this section. First, in some cases, the equation Ππr T Φ˜ = Φ˜ does not have a solution. It is easy to produce examples of this; though r r ˜ no example has been documented for the particular class of approximators we are using here, [2] offers an example involving a different linearly parameterized approximator that captures the spirit of what can happen. Second, it would be nice to relax the requirement that the support of πr intersect every partition. ˜ To address these shortcomings, we introduce stochastic policies. A stochastic policy µ maps state-action pairs to probabilities. For each x ∈ S and u ∈ Ux , µ(x, u) is the probability of taking action u when in state x. Hence, µ(x, u) ≥ 0 for all x ∈ S and u ∈ Ux , and u∈Ux µ(x, u) = 1 for all x ∈ S. Given a scalar > 0 and a function J, the -greedy Boltzmann exploration policy with respect to J is deﬁned by µ(x, u) = e−(Tu J)(x)(|Ux |−1)/ e . −(Tu J)(x)(|Ux |−1)/ e u∈Ux e For any > 0 and r, let µr denote the -greedy Boltzmann exploration policy with respect to Φr. Further, we deﬁne a modiﬁed dynamic programming operator that incorporates Boltzmann exploration: (T J)(x) = u∈Ux e−(Tu J)(x)(|Ux |−1)/ e (Tu J)(x) . −(Tu J)(x)(|Ux |−1)/ e u∈Ux e As approaches 0, -greedy Boltzmann exploration policies become greedy and the modiﬁed dynamic programming operators become the dynamic programming operator. More precisely, for all r, x, and J, lim ↓0 µr (x, µr (x)) = 1 and lim ↓1 T J = T J. These are immediate consequences of the following result (see [4] for a proof). Lemma 1 For any n, v ∈ mini vi . n , mini vi + ≥ i e−vi (n−1)/ e vi / i e−vi (n−1)/ e ≥ Because we are only concerned with communicating MDPs, there is a unique invariant state distribution associated with each -greedy Boltzmann exploration policy µr and the support of this distribution is S. Let πr denote this distribution. We consider a vector r that ˜ solves Φ˜ = Ππr T Φ˜. For any > 0, there exists a solution to this equation (this is an r r ˜ immediate extension of Theorem 5.1 from [4]). We have the following performance loss bound, which parallels Theorem 5 but with an equation for which a solution is guaranteed to exist and without any requirement on the resulting invariant distribution. (Again, we omit the proof, which is available in [22].) Theorem 6 For any MDP, partition, and > 0, if Φ˜ = Ππr T Φ˜ then (1 − r r ˜ T ∗ ∗ α)(πr ) (Jµr − J ) ≤ 2α minr∈ K J − Φr ∞ + . ˜ ˜ 5 Computation: TD(0) Though computation is not a focus of this paper, we offer a brief discussion here. First, we describe a simple algorithm from [16], which draws on ideas from temporal-difference learning [11, 12] and Q-learning [23, 24] to solve Φ˜ = Ππ T Φ˜. It requires an abilr r ity to sample a sequence of states x(0) , x(1) , x(2) , . . ., each independent and identically distributed according to π. Also required is a way to efﬁciently compute (T Φr)(x) = minu∈Ux (gu (x) + α y∈S pxy (u)(Φr)(y)), for any given x and r. This is typically possible when the action set Ux and the support of px· (u) (i.e., the set of states that can follow x if action u is selected) are not too large. The algorithm generates a sequence of vectors r( ) according to r( +1) = r( ) + γ φ(x( ) ) (T Φr( ) )(x( ) ) − (Φr( ) )(x( ) ) , where γ is a step size and φ(x) denotes the column vector made up of components from the xth row of Φ. In [16], using results from [15, 9], it is shown that under appropriate assumptions on the step size sequence, r( ) converges to a vector r that solves Φ˜ = Ππ T Φ˜. ˜ r r The equation Φ˜ = Ππ T Φ˜ may have no solution. Further, the requirement that states r r are sampled independently from the invariant distribution may be impractical. However, a natural extension of the above algorithm leads to an easily implementable version of TD(0) that aims at solving Φ˜ = Ππr T Φ˜. The algorithm requires simulation of a trajectory r r ˜ x0 , x1 , x2 , . . . of the MDP, with each action ut ∈ Uxt generated by the -greedy Boltzmann exploration policy with respect to Φr(t) . The sequence of vectors r(t) is generated according to r(t+1) = r(t) + γt φ(xt ) (T Φr(t) )(xt ) − (Φr(t) )(xt ) . Under suitable conditions on the step size sequence, if this algorithm converges, the limit satisﬁes Φ˜ = Ππr T Φ˜. Whether such an algorithm converges and whether there are r r ˜ other algorithms that can effectively solve Φ˜ = Ππr T Φ˜ for broad classes of relevant r r ˜ problems remain open issues. 6 Extensions and Open Issues Our results demonstrate that weighting a Euclidean norm projection by the invariant distribution of a greedy (or approximately greedy) policy can lead to a dramatic performance gain. It is intriguing that temporal-difference learning implicitly carries out such a projection, and consequently, any limit of convergence obeys the stronger performance loss bound. This is not the ﬁrst time that the invariant distribution has been shown to play a critical role in approximate value iteration and temporal-difference learning. In prior work involving approximation of a cost-to-go function for a ﬁxed policy (no control) and a general linearly parameterized approximator (arbitrary matrix Φ), it was shown that weighting by the invariant distribution is key to ensuring convergence and an approximation error bound [17, 18]. Earlier empirical work anticipated this [13, 14]. The temporal-difference learning algorithm presented in Section 5 is a version of TD(0), This is a special case of TD(λ), which is parameterized by λ ∈ [0, 1]. It is not known whether the results of this paper can be extended to the general case of λ ∈ [0, 1]. Prior research has suggested that larger values of λ lead to superior results. In particular, an example of [1] and the approximation error bounds of [17, 18], both of which are restricted to the case of a ﬁxed policy, suggest that approximation error is ampliﬁed by a factor of 1/(1 − α) as λ is changed from 1 to 0. The results of Sections 3 and 4 suggest that this factor vanishes if one considers a controlled process and performance loss rather than approximation error. Whether the results of this paper can be extended to accommodate approximate value iteration with general linearly parameterized approximators remains an open issue. In this broader context, error and performance loss bounds of the kind offered by Theorem 2 are unavailable, even when the invariant distribution is used to weight the projection. Such error and performance bounds are available, on the other hand, for the solution to a certain linear program [5, 6]. Whether a factor of 1/(1 − α) can similarly be eliminated from these bounds is an open issue. Our results can be extended to accommodate an average cost objective, assuming that the MDP is communicating. With Boltzmann exploration, the equation of interest becomes Φ˜ = Ππr (T Φ˜ − λ1). r r ˜ ˜ ˜ The variables include an estimate λ ∈ of the minimal average cost λ∗ ∈ and an approximation Φ˜ of the optimal differential cost function h∗ . The discount factor α is set r to 1 in computing an -greedy Boltzmann exploration policy as well as T . There is an average-cost version of temporal-difference learning for which any limit of convergence ˜ ˜ (λ, r) satisﬁes this equation [19, 20, 21]. Generalization of Theorem 2 does not lead to a useful result because the right-hand side of the bound becomes inﬁnite as α approaches 1. On the other hand, generalization of Theorem 6 yields the ﬁrst performance loss bound for approximate value iteration with an average-cost objective: Theorem 7 For any communicating MDP with an average-cost objective, partition, and r ˜ > 0, if Φ˜ = Ππr (T Φ˜ − λ1) then r ˜ λµr − λ∗ ≤ 2 min h∗ − Φr ˜ r∈ K ∞ + . Here, λµr ∈ denotes the average cost under policy µr , which is well-deﬁned because the ˜ ˜ process is irreducible under an -greedy Boltzmann exploration policy. This theorem can be proved by taking limits on the left and right-hand sides of the bound of Theorem 6. It is easy to see that the limit of the left-hand side is λµr − λ∗ . The limit of minr∈ K J ∗ − Φr ∞ ˜ on the right-hand side is minr∈ K h∗ − Φr ∞ . (This follows from the analysis of [3].) Acknowledgments This material is based upon work supported by the National Science Foundation under Grant ECS-9985229 and by the Ofﬁce of Naval Research under Grant MURI N00014-001-0637. The author’s understanding of the topic beneﬁted from collaborations with Dimitri Bertsekas, Daniela de Farias, and John Tsitsiklis. A full length version of this paper has been submitted to Mathematics of Operations Research and has beneﬁted from a number of useful comments and suggestions made by reviewers. References [1] D. P. Bertsekas. A counterexample to temporal-difference learning. Neural Computation, 7:270–279, 1994. [2] D. P. Bertsekas and J. N. Tsitsiklis. Neuro-Dynamic Programming. Athena Scientiﬁc, Belmont, MA, 1996. [3] D. Blackwell. Discrete dynamic programming. Annals of Mathematical Statistics, 33:719–726, 1962. [4] D. P. de Farias and B. Van Roy. On the existence of ﬁxed points for approximate value iteration and temporal-difference learning. Journal of Optimization Theory and Applications, 105(3), 2000. [5] D. P. de Farias and B. Van Roy. Approximate dynamic programming via linear programming. In Advances in Neural Information Processing Systems 14. MIT Press, 2002. [6] D. P. de Farias and B. Van Roy. The linear programming approach to approximate dynamic programming. Operations Research, 51(6):850–865, 2003. [7] G. J. Gordon. Stable function approximation in dynamic programming. Technical Report CMU-CS-95-103, Carnegie Mellon University, 1995. [8] G. J. Gordon. Stable function approximation in dynamic programming. In Machine Learning: Proceedings of the Twelfth International Conference (ICML), San Francisco, CA, 1995. [9] T. Jaakkola, M. I. Jordan, and S. P. Singh. On the Convergence of Stochastic Iterative Dynamic Programming Algorithms. Neural Computation, 6:1185–1201, 1994. [10] S. P. Singh and R. C. Yee. An upper-bound on the loss from approximate optimalvalue functions. Machine Learning, 1994. [11] R. S. Sutton. Temporal Credit Assignment in Reinforcement Learning. PhD thesis, University of Massachusetts, Amherst, Amherst, MA, 1984. [12] R. S. Sutton. Learning to predict by the methods of temporal differences. Machine Learning, 3:9–44, 1988. [13] R. S. Sutton. On the virtues of linear learning and trajectory distributions. In Proceedings of the Workshop on Value Function Approximation, Machine Learning Conference, 1995. [14] R. S. Sutton. Generalization in reinforcement learning: Successful examples using sparse coarse coding. In Advances in Neural Information Processing Systems 8, Cambridge, MA, 1996. MIT Press. [15] J. N. Tsitsiklis. Asynchronous stochastic approximation and Q-learning. Machine Learning, 16:185–202, 1994. [16] J. N. Tsitsiklis and B. Van Roy. Feature–based methods for large scale dynamic programming. Machine Learning, 22:59–94, 1996. [17] J. N. Tsitsiklis and B. Van Roy. An analysis of temporal–difference learning with function approximation. IEEE Transactions on Automatic Control, 42(5):674–690, 1997. [18] J. N. Tsitsiklis and B. Van Roy. Analysis of temporal-difference learning with function approximation. In Advances in Neural Information Processing Systems 9, Cambridge, MA, 1997. MIT Press. [19] J. N. Tsitsiklis and B. Van Roy. Average cost temporal-difference learning. In Proceedings of the IEEE Conference on Decision and Control, 1997. [20] J. N. Tsitsiklis and B. Van Roy. Average cost temporal-difference learning. Automatica, 35(11):1799–1808, 1999. [21] J. N. Tsitsiklis and B. Van Roy. On average versus discounted reward temporaldifference learning. Machine Learning, 49(2-3):179–191, 2002. [22] B. Van Roy. Performance loss bounds for approximate value iteration with state aggregation. Under review with Mathematics of Operations Research, available at www.stanford.edu/ bvr/psﬁles/aggregation.pdf, 2005. [23] C. J. C. H. Watkins. Learning From Delayed Rewards. PhD thesis, Cambridge University, Cambridge, UK, 1989. [24] C. J. C. H. Watkins and P. Dayan. Q-learning. Machine Learning, 8:279–292, 1992.

6 0.24038965 87 nips-2005-Goal-Based Imitation as Probabilistic Inference over Graphical Models

7 0.14394183 91 nips-2005-How fast to work: Response vigor, motivation and tonic dopamine

8 0.13751455 53 nips-2005-Cyclic Equilibria in Markov Games

9 0.13625659 199 nips-2005-Value Function Approximation with Diffusion Wavelets and Laplacian Eigenfunctions

10 0.13599236 72 nips-2005-Fast Online Policy Gradient Learning with SMD Gain Vector Adaptation

11 0.10518681 119 nips-2005-Learning to Control an Octopus Arm with Gaussian Process Temporal Difference Methods

12 0.098311543 151 nips-2005-Pattern Recognition from One Example by Chopping

13 0.093432769 187 nips-2005-Temporal Abstraction in Temporal-difference Networks

14 0.084155068 36 nips-2005-Bayesian models of human action understanding

15 0.083066002 200 nips-2005-Variable KD-Tree Algorithms for Spatial Pattern Search and Discovery

16 0.071378268 113 nips-2005-Learning Multiple Related Tasks using Latent Independent Component Analysis

17 0.069668829 26 nips-2005-An exploration-exploitation model based on norepinepherine and dopamine activity

18 0.069073968 195 nips-2005-Transfer learning for text classification

19 0.067213856 12 nips-2005-A PAC-Bayes approach to the Set Covering Machine

20 0.06592577 154 nips-2005-Preconditioner Approximations for Probabilistic Graphical Models

similar papers computed by lsi model

lsi for this paper:

topicId topicWeight

[(0, 0.236), (1, 0.018), (2, 0.492), (3, -0.169), (4, -0.065), (5, 0.067), (6, 0.045), (7, 0.033), (8, 0.046), (9, -0.083), (10, -0.008), (11, -0.091), (12, 0.065), (13, -0.008), (14, 0.015), (15, 0.016), (16, 0.028), (17, -0.166), (18, 0.03), (19, -0.124), (20, 0.042), (21, 0.094), (22, -0.121), (23, -0.036), (24, 0.023), (25, 0.109), (26, -0.083), (27, -0.138), (28, -0.086), (29, 0.025), (30, 0.035), (31, 0.064), (32, 0.032), (33, 0.016), (34, 0.106), (35, -0.089), (36, -0.011), (37, -0.05), (38, 0.027), (39, -0.071), (40, -0.019), (41, 0.064), (42, -0.059), (43, 0.002), (44, 0.071), (45, 0.013), (46, -0.012), (47, 0.041), (48, 0.085), (49, 0.052)]

similar papers list:

simIndex simValue paperId paperTitle

same-paper 1 0.96703237 78 nips-2005-From Weighted Classification to Policy Search

Author: Doron Blatt, Alfred O. Hero

2 0.83193284 145 nips-2005-On Local Rewards and Scaling Distributed Reinforcement Learning

Author: Drew Bagnell, Andrew Y. Ng

3 0.77596849 186 nips-2005-TD(0) Leads to Better Policies than Approximate Value Iteration

Author: Benjamin V. Roy

4 0.7672531 153 nips-2005-Policy-Gradient Methods for Planning

Author: Douglas Aberdeen

5 0.76357508 144 nips-2005-Off-policy Learning with Options and Recognizers

Author: Doina Precup, Cosmin Paduraru, Anna Koop, Richard S. Sutton, Satinder P. Singh

6 0.68520856 119 nips-2005-Learning to Control an Octopus Arm with Gaussian Process Temporal Difference Methods

7 0.63374782 72 nips-2005-Fast Online Policy Gradient Learning with SMD Gain Vector Adaptation

8 0.62732375 87 nips-2005-Goal-Based Imitation as Probabilistic Inference over Graphical Models

9 0.46638212 91 nips-2005-How fast to work: Response vigor, motivation and tonic dopamine

10 0.39038551 53 nips-2005-Cyclic Equilibria in Markov Games

11 0.38554177 199 nips-2005-Value Function Approximation with Diffusion Wavelets and Laplacian Eigenfunctions

12 0.32066229 151 nips-2005-Pattern Recognition from One Example by Chopping

13 0.31767941 187 nips-2005-Temporal Abstraction in Temporal-difference Networks

14 0.30996519 130 nips-2005-Modeling Neuronal Interactivity using Dynamic Bayesian Networks

15 0.30845964 12 nips-2005-A PAC-Bayes approach to the Set Covering Machine

16 0.29661521 200 nips-2005-Variable KD-Tree Algorithms for Spatial Pattern Search and Discovery

17 0.29627788 195 nips-2005-Transfer learning for text classification

18 0.28703204 113 nips-2005-Learning Multiple Related Tasks using Latent Independent Component Analysis

19 0.28410178 154 nips-2005-Preconditioner Approximations for Probabilistic Graphical Models

20 0.27047014 175 nips-2005-Sequence and Tree Kernels with Statistical Feature Mining

similar papers computed by lda model

lda for this paper:

topicId topicWeight

[(1, 0.077), (3, 0.057), (9, 0.012), (10, 0.053), (27, 0.046), (31, 0.201), (34, 0.121), (50, 0.016), (55, 0.038), (69, 0.049), (73, 0.021), (77, 0.02), (86, 0.016), (88, 0.105), (91, 0.061)]

similar papers list:

simIndex simValue paperId paperTitle

1 0.94367504 65 nips-2005-Estimating the wrong Markov random field: Benefits in the computation-limited setting

Author: Martin J. Wainwright

Abstract: Consider the problem of joint parameter estimation and prediction in a Markov random ﬁeld: i.e., the model parameters are estimated on the basis of an initial set of data, and then the ﬁtted model is used to perform prediction (e.g., smoothing, denoising, interpolation) on a new noisy observation. Working in the computation-limited setting, we analyze a joint method in which the same convex variational relaxation is used to construct an M-estimator for ﬁtting parameters, and to perform approximate marginalization for the prediction step. The key result of this paper is that in the computation-limited setting, using an inconsistent parameter estimator (i.e., an estimator that returns the “wrong” model even in the inﬁnite data limit) is provably beneﬁcial, since the resulting errors can partially compensate for errors made by using an approximate prediction technique. En route to this result, we analyze the asymptotic properties of M-estimators based on convex variational relaxations, and establish a Lipschitz stability property that holds for a broad class of variational methods. We show that joint estimation/prediction based on the reweighted sum-product algorithm substantially outperforms a commonly used heuristic based on ordinary sum-product. 1 Keywords: Markov random ﬁelds; variational method; message-passing algorithms; sum-product; belief propagation; parameter estimation; learning. 1

2 0.93471587 145 nips-2005-On Local Rewards and Scaling Distributed Reinforcement Learning

Author: Drew Bagnell, Andrew Y. Ng

same-paper 3 0.93388963 78 nips-2005-From Weighted Classification to Policy Search

Author: Doron Blatt, Alfred O. Hero

4 0.92283773 204 nips-2005-Walk-Sum Interpretation and Analysis of Gaussian Belief Propagation

Author: Dmitry Malioutov, Alan S. Willsky, Jason K. Johnson

Abstract: This paper presents a new framework based on walks in a graph for analysis and inference in Gaussian graphical models. The key idea is to decompose correlations between variables as a sum over all walks between those variables in the graph. The weight of each walk is given by a product of edgewise partial correlations. We provide a walk-sum interpretation of Gaussian belief propagation in trees and of the approximate method of loopy belief propagation in graphs with cycles. This perspective leads to a better understanding of Gaussian belief propagation and of its convergence in loopy graphs. 1

5 0.92150199 108 nips-2005-Layered Dynamic Textures

Author: Antoni B. Chan, Nuno Vasconcelos

Abstract: A dynamic texture is a video model that treats a video as a sample from a spatio-temporal stochastic process, speciﬁcally a linear dynamical system. One problem associated with the dynamic texture is that it cannot model video where there are multiple regions of distinct motion. In this work, we introduce the layered dynamic texture model, which addresses this problem. We also introduce a variant of the model, and present the EM algorithm for learning each of the models. Finally, we demonstrate the efﬁcacy of the proposed model for the tasks of segmentation and synthesis of video.

6 0.88906556 164 nips-2005-Representing Part-Whole Relationships in Recurrent Neural Networks

7 0.84932297 154 nips-2005-Preconditioner Approximations for Probabilistic Graphical Models

8 0.83703482 153 nips-2005-Policy-Gradient Methods for Planning

9 0.83479673 46 nips-2005-Consensus Propagation

10 0.82585412 142 nips-2005-Oblivious Equilibrium: A Mean Field Approximation for Large-Scale Dynamic Games

11 0.81642771 96 nips-2005-Inference with Minimal Communication: a Decision-Theoretic Variational Approach

12 0.81051099 144 nips-2005-Off-policy Learning with Options and Recognizers

13 0.80267704 90 nips-2005-Hot Coupling: A Particle Approach to Inference and Normalization on Pairwise Undirected Graphs

14 0.79754066 111 nips-2005-Learning Influence among Interacting Markov Chains

15 0.79582989 184 nips-2005-Structured Prediction via the Extragradient Method

16 0.79387057 124 nips-2005-Measuring Shared Information and Coordinated Activity in Neuronal Networks

17 0.79128939 43 nips-2005-Comparing the Effects of Different Weight Distributions on Finding Sparse Representations

18 0.78927088 72 nips-2005-Fast Online Policy Gradient Learning with SMD Gain Vector Adaptation

19 0.78371882 187 nips-2005-Temporal Abstraction in Temporal-difference Networks

20 0.77867895 139 nips-2005-Non-iterative Estimation with Perturbed Gaussian Markov Processes