Not necessarily, may have state-actions with identical optimal values Bolei Zhou IERG5350 Reinforcement Learning September 15, 2020 40 / 63 It gets a reward of 10 for leaving the bottom-middle square and a punishment of 100 for leaving the top-left square. A nite planning horizon arises naturally in many decision problems. 10. Value iteration converges. EE266: In nite Horizon Markov Decision Problems In nite horizon Markov decision problems In nite horizon dynamic programming Example 1 At convergence, we have found the optimal value function V* for the discounted infinite horizon Optimal cost-to go functions J is the unique solution to the Bellman equation TJ= Jand iterates Start with value function U 0 for each state Let π 1 be greedy policy based on U 0. This is a stationary MDP with an infinite horizon. Solve infinite-horizon discounted MDPs in finite time. Evaluate π 1 and let U 1 be the resulting value function. The task is to develop a plan that minimizes the expected cost (or maximize expected reward) over some number of stages. Value Iteration Convergence Theorem. We will also introduce the The agent can only be in one of the six locations. For finite horizon settings, sample complexity only characterizes the performance (i.e., V π (s 1) − V ∗ 1 (s 1)) of a policy π at the starting state of episodes s 1. Optimal policy for a MDP in an infinite horizon problem (agent acts forever) is 1 Deterministic 2 Stationary (does not depend on time step) 3 Unique? CSE571/Fall 2013/ASU: MDP--Value of a policy; finding optimal policies for finite horizon MDPs; start of Infinite horizon MDP In general, a policy may depend on the entire history of the MDP, but it is well-known that stationary Markov policies are optimal for infinite-horizon MDPs with stationary transition probabilities and rewards (Puterman, 1994, §6.2). We will see examples of both cases. We develop several new algorithms for learning Markov Decision Processes in an infinite-horizon average-reward setting with linear function approximation. In nite Horizon and Inde nite Horizon MDPs Lecture 2 / #15 In nite Horizon Discounted MDPs: Main Results Cost-to go functions J is the unique solution to the equation T J= Jand iterates of the relation J k+1 = T J k converge to J at a geometric rate. Each policy is an improvement until optimal policy is MDP Planning Problem: Input: an MDP (S,A,R,T) Output: a policy that achieves an “optimal value” This depends on how we define the value of a policy There are several choices and the solution algorithms depend on the choice We will consider two common choices Finite-Horizon Value Infinite Horizon Discounted Value Finite horizon decision problems This chapter will treat stochastic decision problems de ned over a nite period. On the contrary, for infinite horizon … Sometimes the planning period is exogeneously pre-determined. • Infinite Horizon, Discounted Reward Maximization MDP • • Most often studied in machine learning, economics, operations research communities • Goal-directed, Finite Horizon, Prob. It gets the reward/punishment in a particular cell when it leaves the cell. One important issue is caused by the difference in sample complexity for finite and infinite horizon MDP. Therefore, there are no associated termination actions. \$ Note: the infinite horizon optimal policy is stationary, i.e., the optimal action at a state s is the same action at all times. Let π t+1 be greedy policy for U t Let U t+1 be value of π t+1. 3 Infinite-Horizon Problems In stochastic control theory and artificial intelligence research, most problems considered to date do not specify a goal set. (Efficient to store!) Horizon … This is a stationary MDP with an infinite horizon … This is a stationary MDP an. 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