def graph_PI():
n_states = 10
n_actions = 2
det_pis = utils.get_deterministic_policies(n_states, n_actions)
print('n pis: {}'.format(len(det_pis)))
mdp = utils.build_random_sparse_mdp(n_states, n_actions, 0.5)
A = graph.mdp_topology(det_pis)
G = nx.from_numpy_array(A)
pos = nx.spring_layout(G, iterations=200)
basis = graph.construct_mdp_basis(det_pis, mdp)
init_pi = utils.softmax(np.random.standard_normal((n_states, n_actions)))
init_v = utils.value_functional(mdp.P, mdp.r, init_pi, mdp.discount).squeeze()
a = graph.sparse_coeffs(basis, init_v, lr=0.1)
pis = utils.solve(search_spaces.policy_iteration(mdp), init_pi)
print("\n{} policies to vis".format(len(pis)))
for i, pi in enumerate(pis[:-1]):
print('Iteration: {}'.format(i))
v = utils.value_functional(mdp.P, mdp.r, pi, mdp.discount).squeeze()
a = graph.sparse_coeffs(basis, v, lr=0.1, a_init=a)
plt.figure(figsize=(16,16))
nx.draw(G, pos, node_color=a, node_size=150)
# plt.show()
plt.savefig('figs/pi_graphs/{}.png'.format(i))
plt.close()
def graph_PG():
# ffmpeg -framerate 10 -start_number 0 -i %d.png -c:v libx264 -r 30 -pix_fmt yuv420p out.mp4
n_states = 6
n_actions = 4
det_pis = utils.get_deterministic_policies(n_states, n_actions)
print('n pis: {}'.format(len(det_pis)))
mdp = utils.build_random_mdp(n_states, n_actions, 0.9)
A = graph.mdp_topology(det_pis)
G = nx.from_numpy_array(A)
pos = nx.spring_layout(G, iterations=200)
basis = graph.construct_mdp_basis(det_pis, mdp)
init_logits = np.random.standard_normal((n_states, n_actions))
init_v = utils.value_functional(mdp.P, mdp.r, utils.softmax(init_logits), mdp.discount).squeeze()
a = graph.sparse_coeffs(basis, init_v, lr=0.1)
print('\nSolving PG')
pis = utils.solve(search_spaces.policy_gradient_iteration_logits(mdp, 0.1), init_logits)
print("\n{} policies to vis".format(len(pis)))
n = len(pis)
# pis = pis[::n//100]
pis = pis[0:20]
for i, pi in enumerate(pis[:-1]):
print('Iteration: {}'.format(i))
v = utils.value_functional(mdp.P, mdp.r, pi, mdp.discount).squeeze()
a = graph.sparse_coeffs(basis, v, lr=0.1, a_init=a)
plt.figure()
nx.draw(G, pos, node_color=a)
# plt.show()
plt.savefig('figs/pg_graphs/{}.png'.format(i))
plt.close()
def generate_model_cs():
"""
Compare using all deterministic policies versus fewer mixed policies.
Starts to get interesting in higher dims?
"""
n_states = 32
n_actions = 2
lr = 0.01
k = 64
mdp = utils.build_random_mdp(n_states, n_actions, 0.5)
init = rnd.standard_normal((mdp.S * mdp.S * mdp.A + mdp.S * mdp.A))
pi_star = utils.solve(policy_iteration(mdp),
utils.softmax(rnd.standard_normal(
(mdp.S, mdp.A))))[-1]
print('pi_star\n', pi_star)
# adversarial pis
# apis = utils.get_deterministic_policies(mdp.S, mdp.A)
apis = np.stack([utils.random_det_policy(mdp.S, mdp.A) for _ in range(k)])
update_fn = model_iteration(mdp, lr, apis)
params = utils.solve(update_fn, init)
p_logits, r = parse_model_params(mdp.S, mdp.A, params[-1])
error = np.mean(
(utils.value_functional(mdp.P, mdp.r, pi_star, mdp.discount) -
utils.value_functional(utils.softmax(p_logits), r, pi_star,
mdp.discount))**2)
print('\n', error)
new_mdp = utils.MDP(mdp.S, mdp.A, utils.softmax(p_logits), r, mdp.discount,
mdp.d0)
pi_star = utils.solve(policy_iteration(new_mdp),
utils.softmax(rnd.standard_normal(
(mdp.S, mdp.A))))[-1]
print(pi_star)
apis = np.stack([utils.random_policy(mdp.S, mdp.A) for _ in range(k)])
update_fn = model_iteration(mdp, lr, apis)
params = utils.solve(update_fn, init)
p_logits, r = parse_model_params(mdp.S, mdp.A, params[-1])
error = np.mean(
(utils.value_functional(mdp.P, mdp.r, pi_star, mdp.discount) -
utils.value_functional(utils.softmax(p_logits), r, pi_star,
mdp.discount))**2)
print('\n', error)
new_mdp = utils.MDP(mdp.S, mdp.A, utils.softmax(p_logits), r, mdp.discount,
mdp.d0)
pi_star = utils.solve(policy_iteration(new_mdp),
utils.softmax(rnd.standard_normal(
(mdp.S, mdp.A))))[-1]
print(pi_star)
def value_graph():
# vs = [np.sum(utils.value_functional(mdp.P, mdp.r, pi, mdp.discount).squeeze()**2) for pi in det_pis]
# plt.figure(figsize=(16,16))
# nx.draw(G, pos, node_color=vs, node_size=150)
# plt.savefig('figs/pi_graphs/val.png')
# plt.close()
n_states = 10
n_actions = 2
det_pis = utils.get_deterministic_policies(n_states, n_actions)
n = len(det_pis)
print('n pis: {}'.format(n))
# how does discount effect these!?
mdp = utils.build_random_mdp(n_states, n_actions, 0.5)
values = [utils.value_functional(mdp.P, mdp.r, pi, mdp.discount).squeeze() for pi in det_pis]
vs = np.stack(values).reshape((n, n_states))
W = 1/(np.abs(np.sum(vs[None, :, :] - vs[:, None, :], axis=-1)) + 1e-8)
A = graph.mdp_topology(det_pis)
adj = A*W
G = nx.from_numpy_array(adj)
pos = nx.spring_layout(G, iterations=200)
plt.figure(figsize=(16,16))
nx.draw(G, pos, node_color=[np.sum(v) for v in values], node_size=150)
plt.savefig('figs/value_graphs/value_graph-{}-{}.png'.format(n_states, n_actions))
plt.close()
def generate_iteration_figures(mdp, pis, iteration_fn, name):
"""
How many steps to converge to the optima from different starting points.
"""
n = 3
lrs = np.linspace(0.0001, 0.1, n**2) # 0.5 - 0.00195...
plt.figure(figsize=(16, 16))
value = vmap(
lambda pi: utils.value_functional(mdp.P, mdp.r, pi, mdp.discount))
Vs = value(np.stack(pis))[:, :, 0]
# pool = multiprocessing.Pool(n**2)
# # couldnt serialise the mdp collection. so just unwrap them here.
# lens_n_pi_stars = pool.map(iteration_fn, [(mdp.P, mdp.r, mdp.discount, mdp.d0, pis, lr) for lr in lrs])
# for i, lr, results in zip(range(n**2), lrs, lens_n_pi_stars):
# len, pi_star = results
for i, lr in enumerate(lrs):
print('\n{}: {}\n'.format(i, lr))
lens, pi_stars = iteration_fn(
(mdp.P, mdp.r, mdp.discount, mdp.d0, pis, lr))
plt.subplot(n, n, i + 1)
plt.title('Learning rate: {}'.format(lr))
fig = plt.scatter(Vs[:, 0], Vs[:, 1], c=lens, s=5)
fig.axes.get_xaxis().set_visible(False)
fig.axes.get_yaxis().set_visible(False)
plt.tight_layout()
plt.savefig('figs/iteration-lrs/0-{}.png'.format(name))
def model_iteration(mdp, lr, pis):
V_true = vmap(lambda pi: utils.value_functional(mdp.P, mdp.r, pi, mdp.discount))
V_guess = vmap(lambda P, r, pi: utils.value_functional(P, r, pi, mdp.discount), in_axes=(None, None, 0))
def loss_fn(params):
p_logits, r = parse_model_params(mdp.S, mdp.A, params)
return np.sum((V_true(pis) - V_guess(utils.softmax(p_logits), r, pis))**2)
dLdp = grad(loss_fn)
@jit
def update_fn(params):
return params - lr*utils.clip_by_norm(dLdp(params), 100)
return update_fn
def Q(init, M, f):
# solve
V_init = utils.value_functional(M.P, M.r, init, M.discount)
Q_init = utils.bellman_operator(M.P, M.r, V_init, M.discount)
Q_star = utils.solve(ss.q_learning(M, 0.01), Q_init)[-1]
# lift
return np.dot(f.T, np.max(Q_star, axis=1, keepdims=True))
def onoffpolicy_abstraction(mdp, pis):
tol = 0.01
init = np.random.random((mdp.S, mdp.A))
init = init / np.sum(init, axis=1, keepdims=True)
# ### all policy abstraction
# # n x |S| x |A|
# Qs = np.stack([utils.bellman_operator(mdp.P, mdp.r, utils.value_functional(mdp.P, mdp.r, pi, mdp.discount), mdp.discount) for pi in pis], axis=0)
# similar_states = np.sum(np.sum(np.abs(Qs[:, :, None, :] - Qs[:, None, :, :]), axis=3), axis=0) # |S| x |S|
# all_idx, all_abstracted_mdp, all_f = abs.build_state_abstraction(similar_states, mdp)
### optimal policy abstraction
pi_star = utils.solve(ss.policy_iteration(mdp), np.log(init))[-1]
Q_star = utils.bellman_operator(
mdp.P, mdp.r,
utils.value_functional(mdp.P, mdp.r, pi_star, mdp.discount),
mdp.discount)
# similar_states = np.sum(np.abs(Q_star[:, None, :] - Q_star[None, :, :]), axis=-1) # |S| x |S|. preserves optimal policy's value (for all actions)
# similar_states = np.abs(np.max(Q_star[:, None, :],axis=-1) - np.max(Q_star[None, :, :],axis=-1)) # |S| x |S|. preserves optimal action's value
#
V = utils.value_functional(mdp.P, mdp.r, init, mdp.discount)
similar_states = np.abs(V[None, :, :] - V[:, None, :])[:, :, 0]
optimal_idx, optimal_abstracted_mdp, optimal_f = abs.build_state_abstraction(
similar_states, mdp, tol)
mdps = [mdp, optimal_abstracted_mdp]
names = ['ground', 'optimal_abstracted_mdp']
solvers = [abs.Q, abs.SARSA, abs.VI]
lifts = [np.eye(mdp.S), optimal_f]
idxs = [range(mdp.S), optimal_idx]
# if all_f.shape[0] == optimal_f.shape[0]:
# raise ValueError('Abstractions are the same so we probs wont see any difference...')
print('\nAbstraction:', optimal_f.shape)
truth = abs.PI(init, mdp, np.eye(mdp.S))
results = []
for n, M, idx, f in zip(names, mdps, idxs, lifts):
for solve in solvers:
err = np.max(np.abs(truth - solve(init[idx, :], M, f)))
results.append((n, solve.__name__, err))
return results
def generate_vi(mdp, c, lr=0.1):
init_pi = utils.random_policy(mdp.S,mdp.A)
init_v = utils.value_functional(mdp.P, mdp.r, init_pi, mdp.discount)
vs = np.stack(utils.solve(ss.value_iteration(mdp, lr), init_v))[:,:,0]
n = vs.shape[0]
plt.scatter(vs[0, 0], vs[0, 1], c=c, s=30, label='{}'.format(n))
plt.scatter(vs[1:-1, 0], vs[1:-1, 1], c=range(n-2), cmap='viridis', s=10)
plt.scatter(vs[-1, 0], vs[-1, 1], c='m', marker='x')
def VI(init, M, f):
# solve
V_init = utils.value_functional(M.P, M.r, init, M.discount)
V_star = utils.solve(ss.value_iteration(M, 0.01), V_init)[-1]
# lift
return np.dot(f.T, V_star)
def generate_pg(mdp, c, lr=0.01):
init_pi = utils.random_policy(mdp.S,mdp.A)
init_logit = np.log(init_pi)
logits = utils.solve(ss.policy_gradient_iteration_logits(mdp, lr), init_logit)
vs = np.stack([utils.value_functional(mdp.P, mdp.r, utils.softmax(logit), mdp.discount) for logit in logits])[:,:,0]
n = vs.shape[0]
plt.scatter(vs[0, 0], vs[0, 1], c=c, s=30, label='{}'.format(n))
plt.scatter(vs[1:-1, 0], vs[1:-1, 1], c=range(n-2), cmap='viridis', s=10)
plt.scatter(vs[-1, 0], vs[-1, 1], c='m', marker='x')
def value_iteration(mdp, pis, lr):
trajs = []
for pi in pis:
init_V = utils.value_functional(mdp.P, mdp.r, pi, mdp.discount)
traj = utils.solve(ss.value_iteration(mdp, lr), init_V)
v_star = traj[-1]
trajs.append(traj)
return trajs
def compare_mdp_lmdp():
n_states, n_actions = 2, 2
mdp = utils.build_random_mdp(n_states, n_actions, 0.9)
pis = utils.gen_grid_policies(7)
vs = utils.polytope(mdp.P, mdp.r, mdp.discount, pis)
plt.figure(figsize=(16, 16))
plt.scatter(vs[:, 0], vs[:, 1], s=10, alpha=0.75)
# solve via LMDPs
p, q = lmdps.mdp_encoder(mdp.P, mdp.r)
u, v = lmdps.lmdp_solver(p, q, mdp.discount)
pi_u_star = lmdps.lmdp_decoder(u, mdp.P)
pi_p = lmdps.lmdp_decoder(p, mdp.P)
# solve MDP
init = np.random.standard_normal((n_states, n_actions))
pi_star = utils.solve(search_spaces.policy_iteration(mdp), init)[-1]
# pi_star = onehot(np.argmax(qs, axis=1), n_actions)
# evaluate both policies.
v_star = utils.value_functional(mdp.P, mdp.r, pi_star, mdp.discount)
v_u_star = utils.value_functional(mdp.P, mdp.r, pi_u_star, mdp.discount)
v_p = utils.value_functional(mdp.P, mdp.r, pi_p, mdp.discount)
plt.scatter(v_star[0, 0],
v_star[1, 0],
c='m',
alpha=0.5,
marker='x',
label='mdp')
plt.scatter(v_u_star[0, 0],
v_u_star[1, 0],
c='g',
alpha=0.5,
marker='x',
label='lmdp')
plt.scatter(v_p[0, 0], v_p[1, 0], c='k', marker='x', alpha=0.5, label='p')
plt.legend()
plt.show()
def generate_pi(mdp, c):
init_pi = utils.random_policy(mdp.S,mdp.A)
pis = utils.solve(ss.policy_iteration(mdp), init_pi)
vs = np.stack([utils.value_functional(mdp.P, mdp.r, pi, mdp.discount) for pi in pis])[:,:,0]
n = vs.shape[0]
plt.scatter(vs[0, 0], vs[0, 1], c=c, s=30, label='{}'.format(n-2))
plt.scatter(vs[1:-1, 0], vs[1:-1, 1], c=range(n-2), cmap='viridis', s=10)
plt.scatter(vs[-1, 0], vs[-1, 1], c='m', marker='x')
for i in range(len(vs)-2):
dv = 0.1*(vs[i+1, :] - vs[i, :])
plt.arrow(vs[i, 0], vs[i, 1], dv[0], dv[1], color=c, alpha=0.5, width=0.005)
def value_iteration(mdp, pis):
lens, pi_stars = [], []
for pi in pis:
init_V = utils.value_functional(mdp.P, mdp.r, pi, mdp.discount)
pi_traj = utils.solve(ss.value_iteration(mdp, 0.01), init_V)
pi_star = pi_traj[-1]
pi_stars.append(pi_star)
lens.append(len(pi_traj))
return lens, pi_stars
def generate_model_iteration():
n_states, n_actions = 2, 2
mdp = utils.build_random_mdp(n_states, n_actions, 0.5)
pis = utils.gen_grid_policies(7)
init = rnd.standard_normal(
(mdp.S * mdp.S * mdp.A + mdp.S * mdp.A)
) # needs its own init. alternatively could find init that matches value of other inits?!?
vs = utils.polytope(mdp.P, mdp.r, mdp.discount, pis)
plt.figure(figsize=(16, 16))
plt.scatter(vs[:, 0], vs[:, 1], c='b', s=10, alpha=0.75)
lr = 0.01
pi_star = utils.solve(policy_iteration(mdp),
utils.softmax(rnd.standard_normal(
(mdp.S, mdp.A))))[-1]
# adversarial pis
apis = utils.get_deterministic_policies(mdp.S, mdp.A)
apis = np.stack(apis)
update_fn = model_iteration(mdp, lr, apis)
params = utils.solve(update_fn, init)
params = [parse_model_params(mdp.S, mdp.A, p) for p in params]
vs = np.vstack([
utils.value_functional(utils.softmax(p_logits), r, pi_star,
mdp.discount).T for p_logits, r in params
])
n = vs.shape[0]
plt.scatter(vs[0, 0], vs[0, 1], c='g', label='PG')
plt.scatter(vs[1:-1, 0], vs[1:-1, 1], c=range(n - 2), cmap='spring', s=10)
plt.scatter(vs[-1, 0], vs[-1, 1], c='g', marker='x')
p_logits, r = params[-1]
vs = utils.polytope(utils.softmax(p_logits), r, mdp.discount, pis)
plt.scatter(vs[:, 0], vs[:, 1], c='r', s=10, alpha=0.75)
plt.title('Model iteration')
plt.xlabel('Value of state 1')
plt.ylabel('Value of state 2')
# plt.show()
plt.savefig('figs/model_iteration_1.png')
learned_mdp = utils.MDP(mdp.S, mdp.A, utils.softmax(p_logits), r,
mdp.discount, mdp.d0)
pi_star_approx = utils.solve(
policy_iteration(learned_mdp),
utils.softmax(rnd.standard_normal((mdp.S, mdp.A))))[-1]
print(pi_star_approx, '\n', pi_star)
def compare_acc():
n_states, n_actions = 2, 2
lmdp = []
lmdp_rnd = []
for _ in range(10):
mdp = utils.build_random_mdp(n_states, n_actions, 0.5)
# solve via LMDPs
p, q = lmdps.mdp_encoder(mdp.P, mdp.r)
u, v = lmdps.lmdp_solver(p, q, mdp.discount)
pi_u_star = lmdps.lmdp_decoder(u, mdp.P)
# solve MDP
init = np.random.standard_normal((n_states, n_actions))
pi_star = utils.solve(search_spaces.policy_iteration(mdp), init)[-1]
# solve via LMDPs
# with p set to the random dynamics
p, q = lmdps.mdp_encoder(mdp.P, mdp.r)
p = np.einsum('ijk,jk->ij', mdp.P,
np.ones((n_states, n_actions)) / n_actions)
# q = np.max(mdp.r, axis=1, keepdims=True)
u, v = lmdps.lmdp_solver(p, q, mdp.discount)
pi_u_star_random = lmdps.lmdp_decoder(u, mdp.P)
# evaluate both policies.
v_star = utils.value_functional(mdp.P, mdp.r, pi_star, mdp.discount)
v_u_star = utils.value_functional(mdp.P, mdp.r, pi_u_star,
mdp.discount)
v_u_star_random = utils.value_functional(mdp.P, mdp.r,
pi_u_star_random,
mdp.discount)
lmdp.append(np.isclose(v_star, v_u_star, 1e-3).all())
lmdp_rnd.append(np.isclose(v_star, v_u_star_random, 1e-3).all())
print([np.sum(lmdp), np.sum(lmdp_rnd)])
plt.bar(range(2), [np.sum(lmdp), np.sum(lmdp_rnd)])
plt.show()
def test_sparse_estimation():
n_states = 5
n_actions = 2
mdp = utils.build_random_mdp(n_states, n_actions, 0.9)
det_pis = utils.get_deterministic_policies(mdp.S, mdp.A)
basis = graph.construct_mdp_basis(det_pis, mdp)
v = utils.value_functional(mdp.P, mdp.r, det_pis[2],
mdp.discount).squeeze()
a = graph.sparse_coeffs(basis, v)
print(a)
def thompson(mdp, lr):
"""
Can we do TD with values from different MDPs?
"""
V_true = vmap(
lambda pi: utils.value_functional(mdp.P, mdp.r, pi, mdp.discount))
V_guess = vmap(lambda params, pi: utils.value_functional(
*mdp_sampler(params), pi, mdp.discount),
in_axes=(None, None, 0))
dLdp = grad(lambda params: mse(V_true(pis), V_guess(params, pis)))
@jit
def update_fn(params, Q):
m = symmetric_sampler(params)
Q_ = utils.bellman_optimality_operator(m.P, m.r, Q, m.discount)
params_tp1 -= lr * dLdp(
params
) # done based on observations... could use model iteration!?
Q_tp1 = Q + lr * (Q_ - Q)
return Q_tp1, params_tp1
return update_fn
def mom_value_iteration(mdp, pis):
lens, pi_stars = [], []
for pi in pis:
init_V = utils.value_functional(mdp.P, mdp.r, pi, mdp.discount)
pi_traj = utils.solve(
ss.momentum_bundler(ss.value_iteration(mdp, 0.01), 0.9),
(init_V, np.zeros_like(init_V)))
pi_star, _ = pi_traj[-1]
pi_stars.append(pi_star)
lens.append(len(pi_traj))
return lens, pi_stars
def mom_param_value_iteration(mdp, pis):
lens, pi_stars = [], []
core_init = ss.random_parameterised_matrix(2, 2, 32, 4)
for pi in pis:
init_V = utils.value_functional(mdp.P, mdp.r, pi, mdp.discount)
core_init = ss.approximate(init_V, core_init)
params = utils.solve(
ss.momentum_bundler(
ss.parameterised_value_iteration(mdp, 0.01 / len(core_init)),
0.8), (core_init, [np.zeros_like(c) for c in core_init]))
pi_star, _ = params[-1]
pi_stars.append(pi_star)
lens.append(len(params))
return lens, pi_stars
def test_everything():
n_states = 5
n_actions = 2
det_pis = utils.get_deterministic_policies(n_states, n_actions)
mdp = utils.build_random_mdp(n_states, n_actions, 0.9)
A = graph.mdp_topology(det_pis)
basis = graph.construct_mdp_basis(det_pis, mdp)
# v = np.random.random((n_states, ))
v = utils.value_functional(mdp.P, mdp.r, det_pis[2],
mdp.discount).squeeze()
a = graph.sparse_coeffs(basis, v)
G = nx.from_numpy_array(A)
pos = nx.spring_layout(G, iterations=200)
nx.draw(G, pos, node_color=a)
plt.show()
def param_value_iteration(mdp, pis):
# hypothesis. we are going to see some weirdness in the mom partitions.
# oscillations will depend on shape of the polytope?!?
lens, pi_stars = [], []
core_init = ss.random_parameterised_matrix(2, 2, 32, 4)
for pi in pis:
init_V = utils.value_functional(mdp.P, mdp.r, pi, mdp.discount)
core_init = ss.approximate(init_V, core_init)
params = utils.solve(
ss.parameterised_value_iteration(mdp, 0.01 / len(core_init)),
core_init)
pi_star = params[-1]
pi_stars.append(pi_star)
lens.append(len(params))
return lens, pi_stars
def policy_gradient_iteration_logits(mdp, lr):
# this doesnt seem to behave nicely in larger state spaces!?
# d/dlogits V = E_{\pi}[V] = E[V . d/dlogit log \pi]
# dlogpi_dlogit = jacrev(lambda logits: np.log(utils.softmax(logits)+1e-8))
dHdlogit = grad(lambda logits: utils.entropy(utils.softmax(logits)))
dVdlogit = grad(lambda logits: np.sum(utils.value_functional(mdp.P, mdp.r, utils.softmax(logits), mdp.discount)))
@jit
def update_fn(logits):
# NOTE this is actually soft A2C.
# V = utils.value_functional(mdp.P, mdp.r, utils.softmax(logits), mdp.discount)
# Q = utils.bellman_operator(mdp.P, mdp.r, V, mdp.discount)
# A = Q-V
# g = np.einsum('ijkl,ij->kl', dlogpi_dlogit(logits), A)
g = dVdlogit(logits)
return logits + lr * utils.clip_by_norm(g, 500) + 1e-8*dHdlogit(logits)
return update_fn
def value_graph_laplacian():
n_states = 8
n_actions = 2
det_pis = utils.get_deterministic_policies(n_states, n_actions)
n = len(det_pis)
print('n pis: {}'.format(n))
mdp = utils.build_random_mdp(n_states, n_actions, 0.5)
values = [utils.value_functional(mdp.P, mdp.r, pi, mdp.discount).squeeze() for pi in det_pis]
Vs = np.stack(values).reshape((n, n_states))
A = graph.mdp_topology(det_pis)
W = 1/(np.abs(np.sum(Vs[None, :, :] - Vs[:, None, :], axis=-1)) + 1e-8)
adj = A*W
G = nx.from_numpy_array(adj)
pos = nx.spring_layout(G, iterations=200)
plt.figure(figsize=(16,16))
nx.draw(G, pos, node_color=[np.sum(v) for v in values], node_size=150)
plt.savefig('figs/value_graphs/value_graph-{}-{}.png'.format(n_states, n_actions))
plt.close()
# how can you calulate expected eignenvalues!?
# observation. the underlying complexity of the value topology is linear!?!?
# how hard is it to estimate the main eigen vec from noisy observations!?
# that would tell us the complexity!?!?
for i, alpha in enumerate(np.linspace(0, 1, 10)):
us = []
for _ in range(50):
vs = Vs + alpha*np.random.standard_normal(Vs.shape)
W = 1/(np.abs(np.sum(vs[None, :, :] - vs[:, None, :], axis=-1)) + 1e-8)
adj = A*W
u, v = graph_laplacian_spectra(adj)
us.append(u)
us = np.stack(us, axis=0)
mean = np.mean(us, axis=0)
var = np.var(us, axis=0)
plt.bar(range(len(mean)), mean, yerr=np.sqrt(var))
plt.savefig('figs/value_graphs/{}-lap.png'.format(i))
plt.close()
def plot():
n_states = 2
n_actions = 2
mdp = utils.build_random_mdp(n_states, n_actions, 0.5)
value = vmap(
lambda pi: utils.value_functional(mdp.P, mdp.r, pi, mdp.discount))
pis = np.stack(utils.gen_grid_policies(101), axis=0)
vs = value(pis)
plt.scatter(vs[:, 0], vs[:, 1], s=10)
pis = np.stack(utils.get_deterministic_policies(2, 2), axis=0)
vs = value(pis)
plt.scatter(vs[:, 0], vs[:, 1], s=10, c='r')
plt.xlabel('The value of state 1')
plt.ylabel('The value of state 2')
plt.title('The value polytope')
plt.show()
def generate_iteration_figures(mdps, pis, iteration_fn, name):
"""
How many steps to converge to the optima from different starting points.
"""
n = np.sqrt(len(mdps))
plt.figure(figsize=(16, 16))
for i, mdp in enumerate(mdps):
print(i)
Vs = np.hstack([
utils.value_functional(mdp.P, mdp.r, pi, mdp.discount)
for pi in pis
])
lens, pi_stars = iteration_fn(mdp, pis)
plt.subplot(n, n, i + 1)
fig = plt.scatter(Vs[0, :], Vs[1, :], c=lens, s=5)
fig.axes.get_xaxis().set_visible(False)
fig.axes.get_yaxis().set_visible(False)
plt.tight_layout()
plt.savefig('figs/iterations/{}.png'.format(name))
def value_graph_laplacians():
n_states = 8
n_actions = 2
det_pis = utils.get_deterministic_policies(n_states, n_actions)
N = len(det_pis)
print('n pis: {}'.format(N))
for i in range(1):
mdp = utils.build_random_mdp(n_states, n_actions, 0.5)
values = [utils.value_functional(mdp.P, mdp.r, pi, mdp.discount).squeeze() for pi in det_pis]
Vs = np.stack(values).reshape((N, n_states))
A = graph.mdp_topology(det_pis)
W = np.exp(-np.linalg.norm(Vs[None, :, :] - Vs[:, None, :], ord=np.inf, axis=-1)+1e-8)
# mVs = np.mean(Vs, axis=0) # n_states
# W = np.dot((Vs - mVs) , (Vs - mVs).T)
adj = W * A
G = nx.from_numpy_array(adj)
pos = nx.spectral_layout(G) #, iterations=500)
plt.figure(figsize=(16,16))
nx.draw(G, pos, node_color=[np.sum(v) for v in values], node_size=150)
plt.savefig('figs/value_graphs/{}-value_graph-{}-{}.png'.format(i, n_states, n_actions))
plt.close()
u, v = graph_laplacian_spectra(adj)
plt.figure(figsize=(8,8))
plt.bar(range(len(u)), u)
plt.savefig('figs/value_graphs/{}-lap.png'.format(i))
plt.close()
plt.figure(figsize=(16,16))
n = 5
for j in range(n*n):
plt.subplot(n,n,j+1)
nx.draw(G, pos, node_color=u[10*j] * v[10*j], node_size=150)
plt.savefig('figs/value_graphs/{}-spectra.png'.format(i, n_states, n_actions))
plt.close()
def find_symmetric_mdp(n_states, n_actions, discount, lr=1e-2):
"""
Approximately find a mdp with ??? symmetry
"""
model_init = rnd.standard_normal(n_states * n_states * n_actions +
n_states * n_actions)
pis = utils.get_deterministic_policies(n_states, n_actions)
# pis = [utils.random_policy(n_states, n_actions) for _ in range(100)]
pis = np.stack(pis)
# print(pis.shape)
V = vmap(lambda P, r, pi: utils.value_functional(P, r, pi, discount),
in_axes=(None, None, 0))
def loss_fn(model_params):
# policy symmetry
P, r = ss.parse_model_params(n_states, n_actions, model_params)
return np.sum(
np.square(
V(utils.softmax(P), r, pis) -
V(utils.softmax(P), r, np.flip(pis, 1))))
# def loss_fn(model_params):
# # value symmetry
# P, r = ss.parse_model_params(n_states, n_actions, model_params)
# vals = V(utils.softmax(P), r, pis)
# n = n_states//2
# return np.sum(np.square(vals[:, :n] - vals[:, n:]))
dldp = grad(loss_fn)
update_fn = lambda model: model - lr * dldp(model)
init = (model_init, np.zeros_like(model_init))
model_params, momentum_var = utils.solve(
ss.momentum_bundler(update_fn, 0.9), init)[-1]
P, r = ss.parse_model_params(n_states, n_actions, model_params)
d0 = rnd.random((n_states, 1))
return utils.MDP(n_states, n_actions, P, r, discount, d0)
def variance(P, r, pi, discount):
# var = int_s' int_a p(s'|s, a) pi(a|s) (r(s,a) + \gamma V(s') + V(s))
V = utils.value_functional(P, r, pi, discount)[:, 0]
d = (r[None, :, :] + discount * V[:, None, None] - V[None, :, None])**2
expected_d = np.einsum('ijk,ijk->j', P * pi[None, :, :], d) # aka variance
return np.sum(expected_d)
