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goal state and placing it at a new random initial state As noted earlier, if we begin with all Q values initialized to zero, then after the first full episode only one entry in the agent's Q table will have been changed: the entry corresponding to the final transition into the goal state Note that if the agent happens to follow the same sequence of actions from the same random initial state in its second full episode, then a second table entry would be made nonzero, and so on If we run repeated identical episodes in this fashion, the frontier of nonzero Q values will creep backward from the goal state at the rate of one new state-action transition per episode Now consider training on these same state-action transitions, but in reverse chronological order for each episode That is, we apply the same update rule from Equation (137) for each transition considered, but perform these updates in reverse order In this case, after the first full episode the agent will have updated its Q estimate for every transition along the path it took to the goal This training process will clearly converge in fewer iterations, although it requires that the agent use more memory to store the entire episode before beginning the training for that episode A second strategy for improving the rate of convergence is to store past state-action transitions, along with the immediate reward that was received, and retrain on them periodically Although at first it might seem a waste of effort to a, retrain on the same transition, recall that the updated ~ ( s ) value is determined by the values ~ ( s 'a,) of the successor state s' = 6(s, a) Therefore, if subsequent ' training changes one of the ~ ( s a ,) values, then retraining on the transition ( s ,a ) may result in an altered value for ~ ( s , In general, the degree to which we wish a) to replay old transitions versus obtain new ones from the environment depends on the relative costs of these two operations in the specific problem domain For example, in a robot domain with navigation actions that might take several seconds to perform, the delay in collecting a new state-action transition from the external world might be several orders of magnitude more costly than internally replaying a previously observed transition This difference can be very significant given that Q learning can often require thousands of training iterations to converge Note throughout the above discussion we have kept our assumption that the agent does not know the state-transition function 6(s,a ) used by the environment to create the successor state s' = S(s, a ) , or the function r(s, a ) used to generate rewards If it does know these two functions, then many more efficient methods are possible For example, if performing external actions is expensive the agent may simply ignore the environment and instead simulate it internally, efficiently generating simulated actions and assigning the appropriate simulated rewards Sutton (1991) describes the DYNA architecture that performs a number of simulated actions after each step executed in the external world Moore and Atkeson (1993) describe an approach called prioritized sweeping that selects promising states to update next, focusing on predecessor states when the current state is found to have a large update Peng and Williams (1994) describe a similar approach A large number of efficient algorithms from the field of dynamic programming can be applied when the functions 6 and r are known Kaelbling et al (1996) survey a number of these.

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