"""Markov Decision Processes (Chapter 17) First we define an MDP, and the special case of a GridMDP, in which states are laid out in a 2-dimensional grid. We also represent a policy as a dictionary of {state:action} pairs, and a Utility function as a dictionary of {state:number} pairs. We then define the value_iteration and policy_iteration algorithms.""" from utils import * class MDP: """A Markov Decision Process, defined by an initial state, transition model, and reward function. We also keep track of a gamma value, for use by algorithms. The transition model is represented somewhat differently from the text. Instead of T(s, a, s') being probability number for each state/action/state triplet, we instead have T(s, a) return a list of (p, s') pairs. We also keep track of the possible states, terminal states, and actions for each state. [page 615]""" def __init__(self, init, actlist, terminals, gamma=.9): update(self, init=init, actlist=actlist, terminals=terminals, gamma=gamma, states=set(), reward={}) def R(self, state): "Return a numeric reward for this state." return self.reward[state] def T(state, action): """Transition model. From a state and an action, return a list of (result-state, probability) pairs.""" abstract def actions(self, state): """Set of actions that can be performed in this state. By default, a fixed list of actions, except for terminal states. Override this method if you need to specialize by state.""" if state in self.terminals: return [None] else: return self.actlist class GridMDP(MDP): """A two-dimensional grid MDP, as in [Figure 17.1]. All you have to do is specify the grid as a list of lists of rewards; use None for an obstacle (unreachable state). Also, you should specify the terminal states. An action is an (x, y) unit vector; e.g. (1, 0) means move east.""" def __init__(self, grid, terminals, init=(0, 0), gamma=.9): grid.reverse() ## because we want row 0 on bottom, not on top MDP.__init__(self, init, actlist=orientations, terminals=terminals, gamma=gamma) update(self, grid=grid, rows=len(grid), cols=len(grid[0])) for x in range(self.cols): for y in range(self.rows): self.reward[x, y] = grid[y][x] if grid[y][x] is not None: self.states.add((x, y)) def T(self, state, action): if action == None: return [(0.0, state)] else: return [(0.8, self.go(state, action)), (0.1, self.go(state, turn_right(action))), (0.1, self.go(state, turn_left(action)))] def go(self, state, direction): "Return the state that results from going in this direction." state1 = vector_add(state, direction) return if_(state1 in self.states, state1, state) def to_grid(self, mapping): """Convert a mapping from (x, y) to v into a [[..., v, ...]] grid.""" return list(reversed([[mapping.get((x,y), None) for x in range(self.cols)] for y in range(self.rows)])) def to_arrows(self, policy): chars = {(1, 0):'>', (0, 1):'^', (-1, 0):'<', (0, -1):'v', None: '.'} return self.to_grid(dict([(s, chars[a]) for (s, a) in policy.items()])) #______________________________________________________________________________ Fig[17,1] = GridMDP([[-0.04, -0.04, -0.04, +1], [-0.04, None, -0.04, -1], [-0.04, -0.04, -0.04, -0.04]], terminals=[(3, 2), (3, 1)]) #______________________________________________________________________________ def value_iteration(mdp, epsilon=0.001): "Solving an MDP by value iteration. [Fig. 17.4]" U1 = dict([(s, 0) for s in mdp.states]) R, T, gamma = mdp.R, mdp.T, mdp.gamma while True: U = U1.copy() delta = 0 for s in mdp.states: U1[s] = R(s) + gamma * max([sum([p * U[s1] for (p, s1) in T(s, a)]) for a in mdp.actions(s)]) delta = max(delta, abs(U1[s] - U[s])) if delta < epsilon * (1 - gamma) / gamma: return U def best_policy(mdp, U): """Given an MDP and a utility function U, determine the best policy, as a mapping from state to action. (Equation 17.4)""" pi = {} for s in mdp.states: pi[s] = argmax(mdp.actions(s), lambda a:expected_utility(a, s, U, mdp)) return pi def expected_utility(a, s, U, mdp): "The expected utility of doing a in state s, according to the MDP and U." return sum([p * U[s1] for (p, s1) in mdp.T(s, a)]) #______________________________________________________________________________ def policy_iteration(mdp): "Solve an MDP by policy iteration [Fig. 17.7]" U = dict([(s, 0) for s in mdp.states]) pi = dict([(s, random.choice(mdp.actions(s))) for s in mdp.states]) while True: U = policy_evaluation(pi, U, mdp) unchanged = True for s in mdp.states: a = argmax(mdp.actions(s), lambda a: expected_utility(a,s,U,mdp)) if a != pi[s]: pi[s] = a unchanged = False if unchanged: return pi def policy_evaluation(pi, U, mdp, k=20): """Return an updated utility mapping U from each state in the MDP to its utility, using an approximation (modified policy iteration).""" R, T, gamma = mdp.R, mdp.T, mdp.gamma for i in range(k): for s in mdp.states: U[s] = R(s) + gamma * sum([p * U[s] for (p, s1) in T(s, pi[s])]) return U