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There is the notion of the return G t, which is the total discounted rewards from time step t. This is what we care about, the goal is to maximize this return ,. It informs the agent of how much it should care about rewards now to rewards in the future. You might be confused, why put a discounting factor?.
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It turns out to be mathematically convenient to discount rewards, here we guarantee that the algorithm will converge, and avoid infinite returns in loopy Markov processes. It informs the agent of how much reward to expect if it takes a particular action in a particular state. The state-value function of an MRP is the expected return starting from state s ,.
Bellman Equation. The agent tries to get the most expected sum of rewards from every state it lands in. In order to achieve that we must try to get the optimal value function, i. Bellman equation will help us to do so.
We unroll the return G t,. That gives us the Bellman equation for MRPs,.
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So, for each state in the state space, the Bellman equation gives us the value of that state,. The value of the state S is the reward we get upon leaving that state, plus a discounted average over next possible successor states, where the value of each possible successor state is multiplied by the probability that we land in it.
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For our example state space, the value of Class 3 is the reward -2 that we get upon leaving that state added to the discounted average over next possible successor states. This Bellman equation is a linear equation, i. This direct solution is only possible for small MRPs. For larger MRPs, there are many iterative methods, e. Markov Decision Process.
This is what we want to solve. R is the reward function,. A policy fully defines the behavior of an agent,.
MDP policies depend on the current state, not the history, i. Policies can be stochastic to allow us to do exploration in the state space. The same goes for the reward function,. We average over all possible rewards associated with different possible actions from state S.
We already have the value function for MRPs, but there was no decisions. Now we got a policy, i. It tells us how good is it to take a particular action from a particular state,.