def feature_expectations2(self, model, initial, agent):
'''
compute
mu = Phi*inv(I - gamma*P)
where
P(i,j) = P(s'=i|s=j,pi) = E_{a}[ P(s'=i|s=j,a) ] = sum_{a} P(a|s) P(s'=i|s=j,a)
Phi(:,j) = phi(s=j) = E_{a}[ phi(s=j,a) ] = sum_{a} P(a|s) phi(s,a)
mu(:,j) = mu(pi)(s=j) = E[sum_t gamma^t phi(s_t,a_t) | s_0=s]
assumes agent chooses policy deterministically.
'''
# Index states
S = {}
for (i, s) in enumerate(model.S()):
S[s] = i
#Initialize matrices
ff = model.reward_function
k = ff.dim
n_S = len(S)
Phi = np.zeros((k, n_S))
P = np.zeros((n_S, n_S))
# build Phi
for s in S:
Ta = agent.actions(s)
for a in Ta:
j = S[s]
Phi[:, j] += ff.features(s, a) * Ta[a]
# Build P
for s in S:
Ta = agent.actions(s)
for a in Ta:
Ts = model.T(s, a)
for s_p in Ts:
i = S[s_p]
j = S[s]
P[i, j] += Ts[s_p] * Ta[a]
# Calculate mu
mu = np.dot(Phi, np.linalg.pinv(np.eye(n_S) - model.gamma * P))
# Calculate E_{s0}[ phi(s) ]
result = np.zeros(k)
for s in initial:
j = S[s]
result += initial[s] * mu[:, j]
return result
def feature_expectations2(self, model, initial, agent):
'''
compute
mu = Phi*inv(I - gamma*P)
where
P(i,j) = P(s'=i|s=j,pi) = E_{a}[ P(s'=i|s=j,a) ] = sum_{a} P(a|s) P(s'=i|s=j,a)
Phi(:,j) = phi(s=j) = E_{a}[ phi(s=j,a) ] = sum_{a} P(a|s) phi(s,a)
mu(:,j) = mu(pi)(s=j) = E[sum_t gamma^t phi(s_t,a_t) | s_0=s]
assumes agent chooses policy deterministically.
'''
# Index states
S = {}
for (i,s) in enumerate( model.S() ):
S[s] = i
#Initialize matrices
ff = model.reward_function
k = ff.dim
n_S = len(S)
Phi = np.zeros( (k,n_S) )
P = np.zeros( (n_S, n_S) )
# build Phi
for s in S:
Ta = agent.actions(s)
for a in Ta:
j = S[s]
Phi[:,j] += ff.features(s,a)*Ta[a]
# Build P
for s in S:
Ta = agent.actions(s)
for a in Ta:
Ts = model.T(s,a)
for s_p in Ts:
i = S[s_p]
j = S[s]
P[i,j] += Ts[s_p]*Ta[a]
# Calculate mu
mu = np.dot( Phi, np.linalg.pinv(np.eye(n_S) - model.gamma*P) )
# Calculate E_{s0}[ phi(s) ]
result = np.zeros( k )
for s in initial:
j = S[s]
result += initial[s]*mu[:,j]
return result
def lstdq_exact(self, samples, model, agent, feature_f):
'''
Return w such that
Phi*w = R + gamma*P*Phi*w
by solving
w = inv(A)*b
where
A = Phi'*Delta*(Phi-gamma*P*Phi)
b = Delta*Phi*R
Phi[i,:] = Phi( (s,a)=i )
P[i,j] = P( (s',a')=j | (s,a) = i )
R[i] = R( (s,a)=i )
Delta[i,i] = P( (s,a)=i ) in samples
'''
# Make index
k = feature_f.dim
SA = {}
for (i, sa) in enumerate(itertools.product(model.S(), model.A())):
SA[sa] = i
n_SA = len(SA)
# initialize matrices
Phi = np.zeros((n_SA, k))
Delta = np.zeros((n_SA, n_SA))
P = np.zeros((n_SA, n_SA))
R = np.zeros(n_SA)
# Feature Matrix
for sa in SA:
i = SA[sa]
Phi[i, :] = feature_f.features(*sa)
# Transition Matrix
for sa in SA:
Ts = model.T(*sa)
for s_p in Ts:
Ta = agent.actions(s_p)
for a_p in Ta:
sa_p = (s_p, a_p)
j = SA[sa_p]
i = SA[sa]
P[i, j] = Ta[a_p] * Ts[s_p]
# Weighting Matrix
delta = util.classes.NumMap()
for (s, a, r, s_p) in samples:
delta[(s, a)] += 1
delta = delta.normalize()
for sa in SA:
i = SA[sa]
Delta[i, i] = delta[sa]
# Reward Vector
for sa in SA:
i = SA[sa]
R[i] = model.R(*sa)
A = np.dot(np.dot(Phi.T, Delta), Phi - model.gamma * np.dot(Phi, P))
b = np.dot(np.dot(Phi.T, Delta), R)
return np.dot(np.linalg.pinv(A), b)
def lstdq_exact(self, samples, model, agent, feature_f):
'''
Return w such that
Phi*w = R + gamma*P*Phi*w
by solving
w = inv(A)*b
where
A = Phi'*Delta*(Phi-gamma*P*Phi)
b = Delta*Phi*R
Phi[i,:] = Phi( (s,a)=i )
P[i,j] = P( (s',a')=j | (s,a) = i )
R[i] = R( (s,a)=i )
Delta[i,i] = P( (s,a)=i ) in samples
'''
# Make index
k = feature_f.dim
SA = {}
for (i,sa) in enumerate( itertools.product( model.S(), model.A() ) ):
SA[sa] = i
n_SA = len(SA)
# initialize matrices
Phi = np.zeros( (n_SA, k) )
Delta = np.zeros( (n_SA, n_SA) )
P = np.zeros( (n_SA, n_SA) )
R = np.zeros( n_SA )
# Feature Matrix
for sa in SA:
i = SA[sa]
Phi[ i,: ] = feature_f.features(*sa)
# Transition Matrix
for sa in SA:
Ts = model.T(*sa)
for s_p in Ts:
Ta = agent.actions(s_p)
for a_p in Ta:
sa_p = (s_p,a_p)
j = SA[ sa_p ]
i = SA[ sa ]
P[i,j] = Ta[a_p]*Ts[s_p]
# Weighting Matrix
delta = util.classes.NumMap()
for (s,a,r,s_p) in samples:
delta[ (s,a) ] += 1
delta = delta.normalize()
for sa in SA:
i = SA[sa]
Delta[i,i] = delta[sa]
# Reward Vector
for sa in SA:
i = SA[sa]
R[i] = model.R(*sa)
A = np.dot( np.dot( Phi.T, Delta), Phi-model.gamma*np.dot(Phi,P) )
b = np.dot( np.dot( Phi.T, Delta), R )
return np.dot( np.linalg.pinv(A), b)
def iter(self, model, agent, V):
VV = util.classes.NumMap()
for s in model.S():
pi = agent.actions(s)
vv = 0
for (a,t_pi) in pi.items():
v = model.R(s,a)
T = model.T(s,a)
v += model.gamma*sum( [t*V[s_prime] for (s_prime,t) in T.items()] )
vv += t_pi*v
VV[s] = vv
return VV
def iter(self, model, agent, V):
VV = util.classes.NumMap()
for s in model.S():
pi = agent.actions(s)
vv = 0
for (a, t_pi) in pi.items():
v = model.R(s, a)
T = model.T(s, a)
v += model.gamma * sum(
[t * V[s_prime] for (s_prime, t) in T.items()])
vv += t_pi * v
VV[s] = vv
return VV
def feature_expectations(self, model, initial, agent):
'''
Calculate mu(s,a) = E[sum_t gamma^ phi(s_t,a_t) | s_0=s, a_0=a] via repeated applications of
mu(s,a) <--- phi(s,a) + gamma*sum_s' P(s'|s,a) P(a'|s') mu(s',a')
Then returns sum_s0 sum_a P(s0) P(a|s0) mu(s0,a).
Assumes sup_{ (s,a) } ||phi(s,a)||_{\infty} <= 1.0
'''
ff = model.reward_function
i = 0
# Initialize feature expectations
mu = {}
for s in model.S():
for a in model.A(s):
mu[(s, a)] = numpy.zeros(ff.dim)
# Until error is less than 1% (assuming ||phi(s,a)||_{inf} <= 1 for all (s,a) )
# mu(s,a) = phi(s,a) + gamma*sum_{s'} P(s'|s,a) *sum_{a'} P(a'|s') mu(s',a')
while model.gamma**i >= 0.01:
i += 1
mu2 = {}
for s in model.S():
for a in model.A(s):
v = ff.features(s, a)
for (s_prime, t_s) in model.T(s, a).items():
for (a_prime, t_a) in agent.actions(s_prime).items():
v += model.gamma * t_s * t_a * mu[(s_prime,
a_prime)]
mu2[(s, a)] = v
mu = mu2
result = numpy.zeros(ff.dim)
# result = sum_{s} sum_{a} P(s)*P(a|s)*mu(s,a)
for s in initial:
pi = agent.actions(s)
for a in pi:
result += initial[s] * pi[a] * mu[(s, a)]
return result
def feature_expectations(self, model, initial, agent):
'''
Calculate mu(s,a) = E[sum_t gamma^ phi(s_t,a_t) | s_0=s, a_0=a] via repeated applications of
mu(s,a) <--- phi(s,a) + gamma*sum_s' P(s'|s,a) P(a'|s') mu(s',a')
Then returns sum_s0 sum_a P(s0) P(a|s0) mu(s0,a).
Assumes sup_{ (s,a) } ||phi(s,a)||_{\infty} <= 1.0
'''
ff = model.reward_function
i = 0
# Initialize feature expectations
mu = {}
for s in model.S():
for a in model.A(s):
mu[ (s,a) ] = numpy.zeros( ff.dim )
# Until error is less than 1% (assuming ||phi(s,a)||_{inf} <= 1 for all (s,a) )
# mu(s,a) = phi(s,a) + gamma*sum_{s'} P(s'|s,a) *sum_{a'} P(a'|s') mu(s',a')
while model.gamma**i >= 0.01:
i += 1
mu2 = {}
for s in model.S():
for a in model.A(s):
v = ff.features(s,a)
for (s_prime,t_s) in model.T(s,a).items():
for (a_prime,t_a) in agent.actions(s_prime).items():
v += model.gamma*t_s*t_a*mu[ (s_prime, a_prime) ]
mu2[ (s,a) ] = v
mu = mu2
result = numpy.zeros( ff.dim )
# result = sum_{s} sum_{a} P(s)*P(a|s)*mu(s,a)
for s in initial:
pi = agent.actions(s)
for a in pi:
result += initial[s]*pi[a]*mu[ (s,a) ]
return result
def iter(self, model, agent, V):
VV = util.classes.NumMap()
delta = 0
for s in model.S():
pi = agent.actions(s)
vv = 0
for (a, t_pi) in pi.items():
v = model.R(s, a)
T = model.T(s, a)
v += model.gamma * sum(
[t * V[s_prime] for (s_prime, t) in T.items()])
vv += t_pi * v
VV[s] = vv
if (s in V):
delta = max(delta, abs(V[s] - VV[s]))
else:
delta = max(delta, abs(VV[s]))
return (VV, delta)
def lstd_exact(self, samples, model, feature_f, gamma, agent):
'''
Use least squares to estimate V^{\pi} ~~ Phi*w with
w = inv(A)*b
where
A = Phi'*Delta*(Phi-gamma*P*Phi)
b = Phi'*R
Phi[i,:] = phi(s=i)
Delta[i,i] = P(s=i) according to stationary distr of samples
P[i,j] = P(s'=j|s=i) = sum_a P(a|s) P(s'|s,a)
R = E[R(s)] = sum_a P(a|s) R(s,a)
'''
k = feature_f.dim
# Make index
S = {}
for (i, s) in enumerate(model.S()):
S[s] = i
n_S = len(S)
# Initialize Matrices
Phi = np.zeros((n_S, k))
Delta = np.zeros((n_S, n_S))
P = np.zeros((n_S, n_S))
R = np.zeros(n_S)
# Phi
for s in S:
i = S[s]
Phi[i, :] = feature_f.features(s)
# P
for s in S:
Ta = agent.actions(s)
for a in model.A(s):
Ts = model.T(s, a)
for s_p in Ts:
i = S[s]
j = S[s_p]
P[i, j] += Ta[a] * Ts[s_p]
# R
for s in S:
Ta = agent.actions(s)
for a in Ta:
i = S[s]
R[i] += model.R(s, a) * Ta[a]
# Delta
delta = util.classes.NumMap()
for (s, a, r, s_p) in samples:
delta[s] += 1
delta = delta.normalize()
for s in delta:
i = S[s]
Delta[i, i] = delta[s]
# Actual computation
A = np.dot(np.dot(Phi.T, Delta), Phi - gamma * np.dot(P, Phi))
b = np.dot(Phi.T, np.dot(Delta, R))
w = np.dot(np.linalg.pinv(A), b)
V = util.classes.NumMap()
for s in model.S():
V[s] = np.dot(feature_f.features(s), w)
return V
def evaluate_policy(self, model, agent):
'''
Use linear algebra to solve
Q = R + gamma*P*PI*Q
where R is (m*n x 1)
P is (m*n x n)
PI is (n x m*n) with n copies of P(a=i|s=j) along each row
Q is (m*n x 1)
m = number of actions
n = number of states
'''
# State + Actions
S = list(model.S())
A = list(model.A())
SA = []
for s in S:
for a in A:
SA.append((s, a))
S_dict = {}
for (i, s) in enumerate(S):
S_dict[s] = i
A_dict = {}
for (j, a) in enumerate(A):
A_dict[a] = j
SA_dict = {}
for (i, s) in enumerate(S):
for (j, a) in enumerate(A):
SA_dict[(s, a)] = i * len(A) + j
gamma = model.gamma
(n, m) = (len(S), len(A))
R = np.zeros(m * n)
P = np.zeros([m * n, n])
PI = np.zeros([n, m * n])
# Fill R
for ((s, a), i) in SA_dict.items():
R[i] = model.R(s, a)
# Fill P
for ((s, a), i) in SA_dict.items():
T = model.T(s, a)
for (s2, p) in T.items():
j = S_dict[s2]
P[i, j] = p
# Fill PI
pis = {}
for s in S:
pi = agent.actions(s)
for a in A:
pis[(s, a)] = pi[a]
for (i, s) in enumerate(S):
for (j, (s_p, a)) in enumerate(SA):
if s_p != s:
continue
PI[i, j] = pis[(s, a)]
# Solve Q = R + gamma*P*PI*Q
Q = numpy.linalg.solve(np.eye(n * m) - gamma * np.dot(P, PI), R)
# Build V = max_{A} Q(s,a)
V = util.classes.NumMap()
for (i, s) in enumerate(S):
acts = util.classes.NumMap()
for a in A:
acts[a] = Q[SA_dict[(s, a)]]
V[s] = acts.max()
return V
def lstd_exact(self, samples, model, feature_f, gamma, agent):
'''
Use least squares to estimate V^{\pi} ~~ Phi*w with
w = inv(A)*b
where
A = Phi'*Delta*(Phi-gamma*P*Phi)
b = Phi'*R
Phi[i,:] = phi(s=i)
Delta[i,i] = P(s=i) according to stationary distr of samples
P[i,j] = P(s'=j|s=i) = sum_a P(a|s) P(s'|s,a)
R = E[R(s)] = sum_a P(a|s) R(s,a)
'''
k = feature_f.dim
# Make index
S = {}
for (i,s) in enumerate(model.S()):
S[s] = i
n_S = len(S)
# Initialize Matrices
Phi = np.zeros( (n_S,k) )
Delta = np.zeros( (n_S,n_S) )
P = np.zeros( (n_S, n_S) )
R = np.zeros( n_S )
# Phi
for s in S:
i = S[s]
Phi[i,:] = feature_f.features(s)
# P
for s in S:
Ta = agent.actions(s)
for a in model.A(s):
Ts = model.T(s,a)
for s_p in Ts:
i = S[s]
j = S[s_p]
P[i,j] += Ta[a]*Ts[s_p]
# R
for s in S:
Ta = agent.actions(s)
for a in Ta:
i = S[s]
R[i] += model.R(s,a)*Ta[a]
# Delta
delta = util.classes.NumMap()
for (s,a,r,s_p) in samples:
delta[s] += 1
delta = delta.normalize()
for s in delta:
i = S[s]
Delta[i,i] = delta[s]
# Actual computation
A = np.dot( np.dot( Phi.T,Delta ), Phi - gamma*np.dot(P,Phi) )
b = np.dot( Phi.T, np.dot(Delta, R) )
w = np.dot( np.linalg.pinv(A),b )
V = util.classes.NumMap()
for s in model.S():
V[s] = np.dot( feature_f.features(s), w)
return V
def evaluate_policy(self, model, agent):
'''
Use linear algebra to solve
Q = R + gamma*P*PI*Q
where R is (m*n x 1)
P is (m*n x n)
PI is (n x m*n) with n copies of P(a=i|s=j) along each row
Q is (m*n x 1)
m = number of actions
n = number of states
'''
# State + Actions
S = list( model.S() )
A = list( model.A() )
SA = []
for s in S:
for a in A:
SA.append((s,a))
S_dict = {}
for (i,s) in enumerate(S):
S_dict[s] = i
A_dict = {}
for (j,a) in enumerate(A):
A_dict[a] = j
SA_dict = {}
for (i,s) in enumerate(S):
for (j,a) in enumerate(A):
SA_dict[(s,a)] = i*len(A)+j
gamma = model.gamma
(n,m) = ( len(S), len(A) )
R = np.zeros( m*n )
P = np.zeros( [m*n,n])
PI= np.zeros( [n,m*n])
# Fill R
for ((s,a),i) in SA_dict.items():
R[i] = model.R(s,a)
# Fill P
for ((s,a),i) in SA_dict.items():
T = model.T(s,a)
for (s2,p) in T.items():
j = S_dict[s2]
P[i,j] = p
# Fill PI
pis = {}
for s in S:
pi = agent.actions(s)
for a in A:
pis[(s,a)] = pi[a]
for (i,s) in enumerate(S):
for (j,(s_p,a)) in enumerate(SA):
if s_p != s:
continue
PI[i,j] = pis[(s,a)]
# Solve Q = R + gamma*P*PI*Q
Q = numpy.linalg.solve(np.eye(n*m)-gamma*np.dot(P,PI), R)
# Build V = max_{A} Q(s,a)
V = util.classes.NumMap()
for (i,s) in enumerate(S):
acts = util.classes.NumMap()
for a in A:
acts[a] = Q[ SA_dict[(s,a)] ]
V[s] = acts.max()
return V