## A generalization error for Q-Learning (2005)

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Venue: | JOURNAL OF MACHINE LEARNING RESEARCH |

Citations: | 15 - 5 self |

### BibTeX

@ARTICLE{Murphy05ageneralization,

author = {Susan A. Murphy},

title = {A generalization error for Q-Learning},

journal = {JOURNAL OF MACHINE LEARNING RESEARCH},

year = {2005},

volume = {6},

pages = {1073--1097}

}

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### Abstract

Planning problems that involve learning a policy from a single training set of finite horizon trajectories arise in both social science and medical fields. We consider Q-learning with function approximation for this setting and derive an upper bound on the generalization error. This upper bound is in terms of quantities minimized by a Q-learning algorithm, the complexity of the approximation space and an approximation term due to the mismatch between Q-learning and the goal of learning a policy that maximizes the value function.

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Citation Context ...ting is Q-learning c○2005 Susan A. Murphy.MURPHY (Watkins, 1989) since the actions in the training set are chosen according to a (non-optimal) exploration policy; Q-learning is an off-policy method (=-=Sutton and Barto, 1998-=-). When the observables are vectors of continuous variables or are otherwise of high dimension, Q-learning must be combined with function approximation. The contributions of this paper are as follows.... |

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Citation Context ...e in value functions or more specifically the average generalization error. Here the generalization error for batch Q-learning is defined analogous to the generalization error in supervised learning (=-=Schapire et al., 1998-=-); it is the average diffe! rence in value when using the optimal policy as compared to using the greedy policy (from Q-learning) in generating a separate test set. The performance guarantees are anal... |

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Citation Context ...policy as compared to using the greedy policy (from Q-learning) in generating a separate test set. The performance guarantees are analogous to performance guarantees available in supervised learning (=-=Anthony and Bartlett, 1999-=-). The upper bounds on the average generalization error permit an additional contribution. These upper bounds illuminate the mismatch between Q-learning with function approximation and the goal of fin... |

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Citation Context ...At), ft+1; this conditional distribution does not depend on the policy.) In Section 4 we express the difference in value functions for policy ˜π and policy π in terms of the advantages (as defined in =-=Baird, 1993-=-). The time t advantage is µπ,t(ot,at) = Qπ,t(ot,at) −Vπ,t(ot,at−1). The advantage can be interpreted as the gain in performance obtained by following action at at time t and thereafter policy π as co... |

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Citation Context ...tion error: maxo[V ∗ (o) −Vπ(o)] (Bertsekas and Tsitsiklis, 1996). However here we consider an average generalization error as in Kakade (2003) (see also Fiechter, 1997; Kearns, Mansour and Ng, 2000; =-=Peshkin and Shelton, 2002-=-); that is R o [V ∗ (o) −Vπ(o)]dF(o) for a specified distribution F on the observation space. The choice of F with density f = f0 ( f0 is the density of O0 in likelihoods (1) and (2)) is particularly ... |

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Citation Context ... number of ongoing large clinical trials for chronic disorders in which, each time an individual relapses, the individual is re-randomized to one of several further treatments (Schneider et al.,2001; =-=Fava et al., 2003-=-; Thall et al., 2000). These are finite horizon problems with T generally quite small, T = 2 − 4, with known exploration policy. Scientists want to estimate the best “strategies,” i.e. policies, for m... |

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Citation Context ... π is evaluated in terms of maximum generalization error: maxo[V ∗ (o) −Vπ(o)] (Bertsekas and Tsitsiklis, 1996). However here we consider an average generalization error as in Kakade (2003) (see also =-=Fiechter, 1997-=-; Kearns, Mansour and Ng, 2000; Peshkin and Shelton, 2002); that is R o [V ∗ (o) −Vπ(o)]dF(o) for a specified distribution F on the observation space. The choice of F with density f = f0 ( f0 is the d... |

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Citation Context ...t, Ot+1) for rt a reward function and for each 0 ≤ t ≤ T (if the Markov assumption holds then replace Ot with Ot and At with At). We assume that the rewards are bounded, taking values in the interval =-=[0, 1]-=-. We assume the trajectories are sampled at random according to a fixed distribution denoted by P . Thus the trajectories are generated by one fixed distribution. This distribution is composed of the ... |

1 |
Less is more? STI in acute and chronic HIV-1 infection. Nature Medicine 7:881–884
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Citation Context ... training set of trajectories are not unusual and can be expected to increase due to the widespread use of policies in the social and behavioral/medical sciences (see, for example, Rush et al., 2003; =-=Altfeld and Walker, 2001-=-; Brooner, and Kidorf, 2002); at this time these policies are formulated using expert opinion, clinical experience and/or theoretical models. However there is growing interest in formulating these pol... |

1 | TMAP Research Group. Texas medication algorithm project, phase 3 (TMAP-3): Rationale and study design - Rush, Crismon, et al. |

1 | et al. National Institute of Mental Health clinical antipsychotic trials of intervention effectiveness (CATIE - Schneider, Tariot - 2001 |

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