from __future__ import print_function from __future__ import division from __future__ import unicode_literals from argparse import ArgumentParser from time import time import numpy as np import matplotlib as mpl mpl.use('TkAgg') import matplotlib.pyplot as plt P = 0.1 ALPHA = 0.90 EPSILON = 1e-8 # Convergence criterium # Global state MAZE = None # Map of the environment STATE_MASK = None # Fields of maze belonging to state space S_TO_IJ = None # Mapping of state vector to coordinates IJ_TO_S = None # Mapping of coordinates to state vector U_OF_X = None # The allowed action space matrix representation PW_OF_X_U = None # The probability distribution of disturbance G1_X = None # The cost function vector representation (depends only on state) G2_X = None # The second cost function vector representation F_X_U_W = None # The state function SN = None # Number of states A2 = np.array([ [-1, 0], [1, 0], [0, -1], [0, 1], [0, 0] ]) ACTIONS = { 'UP': (-1, 0), 'DOWN': (1, 0), 'LEFT': (0, -1), 'RIGHT': (0, 1), 'IDLE': (0, 0) } def _ij_to_s(ij): return np.argwhere(np.all(ij == S_TO_IJ, axis=1)).flatten()[0] # TODO: for all x and u in one go def h_function(x, u, j, g): """Return E_pi_w[g(x, pi(x), w) + alpha*J(f(x, pi(x), w))].""" pw = pw_of_x_u(x, u) expectation = sum( pw[w] * (g(x, u, w) + ALPHA*j[_ij_to_s(f(x, u, w))]) for w in pw ) return expectation def h_matrix(j, g): result = (PW_OF_X_U * (g[F_X_U_W] + ALPHA*j[F_X_U_W])).sum(axis=2) result[~U_OF_X] = np.inf # discard invalid policies return result def f(x, u, w): return _move(_move(x, ACTIONS[u]), ACTIONS[w]) def cost_treasure(x, u, w): xt = f(x, u, w) options = { 'T': 50, 'G': -1, } return options.get(MAZE[xt], 0) def cost_energy(x, u, w): xt = f(x, u, w) options = { 'T': 50, 'G': 0 } return options.get(MAZE[xt], 1) def _move(start, move): return start[0] + move[0], start[1] + move[1] def _valid_target(target): return ( 0 <= target[0] < MAZE.shape[0] and 0 <= target[1] < MAZE.shape[1] and MAZE[tuple(target)] != '1' ) def _init_global(maze_file): global MAZE, STATE_MASK, SN, S_TO_IJ, IJ_TO_S global U_OF_X, PW_OF_X_U, F_X_U_W, G1_X, G2_X # Basic maze structure initialization MAZE = np.genfromtxt( maze_file, dtype=str, ) STATE_MASK = (MAZE != '1') S_TO_IJ = np.indices(MAZE.shape).transpose(1, 2, 0)[STATE_MASK] SN = len(S_TO_IJ) IJ_TO_S = np.zeros(MAZE.shape, dtype=np.int32) IJ_TO_S[STATE_MASK] = np.arange(SN) # One step cost functions initialization maze_cost = np.zeros(MAZE.shape) maze_cost[MAZE == '1'] = np.nan maze_cost[(MAZE == '0') | (MAZE == 'S')] = 0 maze_cost[MAZE == 'T'] = 50 maze_cost[MAZE == 'G'] = -1 G1_X = maze_cost.copy()[STATE_MASK] maze_cost[maze_cost < 1] += 1 # assert np.nan < whatever == True G2_X = maze_cost.copy()[STATE_MASK] # Actual environment modelling U_OF_X = np.zeros((SN, len(A2)), dtype=np.bool) PW_OF_X_U = np.zeros((SN, len(A2), len(A2))) F_X_U_W = np.zeros(PW_OF_X_U.shape, dtype=np.int32) for ix, x in enumerate(S_TO_IJ): for iu, u in enumerate(A2): if _valid_target(x + u): U_OF_X[ix, iu] = True if iu in (0, 1): possible_iw = [2, 3] elif iu in (2, 3): possible_iw = [0, 1] for iw in possible_iw: if _valid_target(x + u + A2[iw]): PW_OF_X_U[ix, iu, iw] = P F_X_U_W[ix, iu, iw] = IJ_TO_S[tuple(x + u + A2[iw])] # IDLE w is always possible PW_OF_X_U[ix, iu, -1] = 1 - PW_OF_X_U[ix, iu].sum() F_X_U_W[ix, iu, -1] = IJ_TO_S[tuple(x + u)] def u_of_x(x): """Return a list of allowed actions for the given state x.""" return [u for u in ACTIONS if _valid_target(_move(x, ACTIONS[u]))] def pw_of_x_u(x, u): """Calculate probabilities of disturbances given state and action. Parameters ---------- x : tuple of ints The state coordinate (it is up to user to ensure this is a valid state). u : str The name of the action (again, up to the user to ensure validity). Returns ------- dict A mapping of valid disturbances to their probabilities. """ if u in ('LEFT', 'RIGHT'): possible_w = ('UP', 'IDLE', 'DOWN') elif u in ('UP', 'DOWN'): possible_w = ('LEFT', 'IDLE', 'RIGHT') else: # I assume that the IDLE action is deterministic possible_w = ('IDLE',) allowed_w = [ w for w in possible_w if _valid_target(f(x, u, w)) ] probs = {w: P for w in allowed_w if w != 'IDLE'} probs['IDLE'] = 1 - sum(probs.values()) return probs def plot_j_policy_on_maze(j, policy): heatmap = np.ones(MAZE.shape) * np.nan # Ugly heatmap[STATE_MASK] = j # Even uglier cmap = mpl.cm.get_cmap('coolwarm') cmap.set_bad(color='black') plt.imshow(heatmap, cmap=cmap) plt.colorbar() plt.quiver(S_TO_IJ[:,1], S_TO_IJ[:,0], A2[policy, 1], -A2[policy, 0]) plt.gca().get_xaxis().set_visible(False) plt.gca().get_yaxis().set_visible(False) def plot_cost_history(hist): error = [((h - hist[-1])**2).sum()**0.5 for h in hist[:-1]] plt.xlabel('Number of iterations') plt.ylabel('Cost function error') plt.plot(error) def _policy_improvement(j, g): h_mat = h_matrix(j, g) return np.argmin(h_mat, axis=1), h_mat.min(axis=1) def _evaluate_policy(policy, g): pw_pi = PW_OF_X_U[np.arange(SN), policy] # p(w) given policy for all x targs = F_X_U_W[np.arange(SN), policy] # all f(x, u(x)) G = (pw_pi * g[targs]).sum(axis=1) M = np.zeros((SN, SN)) # Markov matrix for given determ policy x_from = [x_ff for x_f, nz in zip(np.arange(SN), np.count_nonzero(pw_pi, axis=1)) for x_ff in [x_f] * nz] M[x_from, targs[pw_pi > 0]] = pw_pi[pw_pi > 0] # M[np.arange(SN), F_X_U_W[PW_OF_X_U > 0]] = PW_OF_X_U[PW_OF_X_U > 0] # for x, u in zip(S_TO_IJ, policy): # pw = pw_of_x_u(x, u) # G.append(sum(pw[w] * g(x, u, w) for w in pw)) # targets = [(_ij_to_s(f(x, u, w)), pw[w]) for w in pw] # iox = _ij_to_s(x) # for t, pww in targets: # M[iox, t] = pww # G = np.array(G) return np.linalg.solve(np.eye(SN) - ALPHA*M, G) def value_iteration(g, return_history=False): j = np.zeros(SN) history = [j] while True: # print(j) policy, j_new = _policy_improvement(j, g) j_old = j j = j_new if return_history: history.append(j) if np.abs(j - j_old).max() < EPSILON: break if not return_history: return j, policy else: return history def policy_iteration(g, return_history=False): j = None policy = np.full(SN, len(A2) - 1) history = [] while True: j_old = j j = _evaluate_policy(policy, g) history.append(j) if j_old is not None and max(abs(j - j_old)) < EPSILON: break policy, _ = _policy_improvement(j, g) if not return_history: return j, policy else: return history if __name__ == '__main__': # Argument Parsing ap = ArgumentParser() ap.add_argument('maze_file', help='Path to maze file') args = ap.parse_args() # start = time() # Initialization start = time() _init_global(args.maze_file) # J / policy for both algorithms for both cost functions for 3 alphas costs = {'g1': G1_X, 'g2': G2_X} optimizers = {'Value Iteration': value_iteration, 'Policy Iteration': policy_iteration} for a in [0.9, 0.5, 0.01]: plt.figure() plt.suptitle('DISCOUNT = ' + str(a)) i = 1 for opt in ['Value Iteration', 'Policy Iteration']: for g in ['g1', 'g2']: name = ' / '.join([opt, g]) ALPHA = a j, policy = optimizers[opt](costs[g]) print(name, j) plt.subplot(2, 2, i) plt.gca().set_title(name) plot_j_policy_on_maze(j, policy) i += 1 # Error graphs for opt in ['Value Iteration', 'Policy Iteration']: plt.figure() plt.suptitle(opt) i = 1 for g in ['g1', 'g2']: for a in [0.9, 0.8, 0.7]: name = 'Cost: {}, discount: {}'.format(g, a) ALPHA = a history = optimizers[opt](costs[g], return_history=True) plt.subplot(2, 3, i) plt.gca().set_title(name) plot_cost_history(history) i += 1 print('I ran in {} seconds'.format(time() - start)) plt.show()