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In this section we formulate the problem of learning sequential control strategies more precisely Note there are many ways to do so For example, we might assume the agent's actions are deterministic or that they are nondeterministic We might assume that the agent can predict the next state that will result from each action, or that it cannot We might assume that the agent is trained by an expert who shows it examples of optimal action sequences, or that it must train itself by performing actions of its own choice Here we define one quite general formulation of the problem, based on Markov decision processes This formulation of the problem follows the problem illustrated in Figure 131 In a Markov decision process (MDP) the agent can perceive a set S of distinct states of its environment and has a set A of actions that it can perform At each discrete time step t , the agent senses the current state st, chooses a current action a,, and performs it The environment responds by giving the agent a reward r, = r (st,a,) and by producing the succeeding state s,+l = 6(s,, a,) Here the functions 6 and r are part of the environment and are not necessarily known to the agent In an MDP, the functions 6(st,a,) and r(s,,a,) depend only on the current state and action, and not on earlier states or actions In this chapter we consider only the case in which S and A are finite In general, 6 and r may be nondeterministic functions, but we begin by considering only the deterministic case The task of the agent is to learn a policy, n : S + A, for selecting its next action a, based on the current observed state st; that is, n(s,) = a, How shall we specify precisely which policy n we would like the agent to learn One obvious approach is to require the policy that produces the greatest possible cumulative reward for the robot over time To state this requirement more precisely, we define the cumulative value Vn(s,)achieved by following an arbitrary policy n from an arbitrary initial state st as follows:.

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.

(continued)

where the sequence of rewards rt+i is generated by beginning at state s, and by repeatedly using the policy n to select actions as described above (ie, a, = n(st), etc) a,+l = n ( ~ , + ~ ) , Here 0 5 y < 1 is a constant that determines the relative value of delayed versus immediate rewards In particular, rewards received i time steps into the future are discounted exponentially by a factor of y Note if we set y = 0, only the immediate reward is considered As we set y closer to 1, future rewards are given greater emphasis relative to the immediate reward The quantity VX(s)defined by Equation (131) is often called the discounted cumulative reward achieved by policy n from initial state s It is reasonable to discount future rewards relative to immediate rewards because, in many cases, we prefer to obtain the reward sooner rather than later However, other definitions of total reward have also been explored For example, jinite horizon reward, rt+i, considers the undiscounted sum of rewards over a finite number h of steps Another possibility is average reward, limb,, rt+i, which considers the average reward per time step over the entire lifetime of the agent In this chapter we restrict ourselves to considering discounted reward as defined by Equation (131) Mahadevan (1996) provides a discussion of reinforcement learning when the criterion to be optimized is average reward We are now in a position to state precisely the agent's learning task We require that the agent learn a policy n that maximizes V"(s) for all states s We will call such a policy an optimal policy and denote it by T*