def optimize(self, fitness_function=None):
start_optimization = time.time()
if (fitness_function is None) and (self.fitness_function != None):
fitness_function = self.fitness_function
# initialize
m = MuCommaLambda._get_m(self)
start_evaluation = time.time()
y = fitness_function(m)
n_evaluations, time_evaluations = 1, time.time() - start_evaluation
best_so_far_x, best_so_far_y = np.copy(m), np.copy(y)
if self.save_fitness_data: fitness_data = [y]
if self.save_best_so_far_x: history_x = np.hstack((n_evaluations, best_so_far_x))
else: history_x = None
# iterate
n_evolution_paths = self.n_evolution_paths
termination = "max_evaluations"
x_z1, x_z2 = np.sqrt(1 - self.c_cov), np.sqrt(self.c_cov) # Line 9 of Algorithm 1
q_1, q_2 = 1 - self.c_c, np.sqrt(self.c_c * (2 - self.c_c)) # Line 15 of Algorithm 1
Z_1, Z_2 = 1 - self.c_s, np.sqrt(self.c_s * (2 - self.c_s)) # Line 22 of Algorithm 1
is_restart, n_restart = True, 0
while n_evaluations < self.max_evaluations:
if is_restart:
if n_restart > 0:
m = self.rng.uniform(self.lower_boundary,
self.upper_boundary, (self.ndim_problem,))
start_evaluation = time.time()
y = fitness_function(m)
time_evaluations += (time.time() - start_evaluation)
n_evaluations += 1
if self.save_fitness_data: fitness_data.append(np.copy(y))
if best_so_far_y > y: best_so_far_x, best_so_far_y = np.copy(m), np.copy(y)
if self.save_best_so_far_x and not(n_evaluations % self.freq_best_so_far_x):
history_x = np.vstack((history_x, np.hstack((n_evaluations, best_so_far_x))))
self.n_individuals = self.n_individuals * 2
self.n_parents = int(np.floor(self.n_individuals / 2))
self.d_sigma = self.d_sigma * 2
# set weights for parents
w = np.log(np.arange(1, self.n_parents + 1))
w = (np.log(self.n_parents + 1) - w) / (
self.n_parents * np.log(self.n_parents + 1) - np.sum(w))
mu_eff = 1 / np.sum(np.power(w, 2))
W = np.tile(w[:, np.newaxis], (1, self.ndim_problem))
# initialize m search directions
# note that in the original paper Q is a n*m matrix (v.s. a m*n matrix here)
Q = np.empty((n_evolution_paths, self.ndim_problem))
for i in range(n_evolution_paths):
Q[i, :] = 1e-10 * self.rng.standard_normal((self.ndim_problem,))
sigma = self.step_size
RR = np.arange(1, self.n_individuals * 2 + 1)
Z = 0
U_mean = (self.n_individuals ** 2) / 2
U_var = np.sqrt((self.n_individuals ** 2) * (2 * self.n_individuals + 1) / 12)
d_sigma, p_star = self.d_sigma, self.p_star
X = np.empty((self.n_individuals, self.ndim_problem)) # population
Y = np.tile(y, (self.n_individuals,)) # fitness of population
is_restart = False
Y_bak = np.copy(Y)
for i in range(self.n_individuals):
z1 = self.rng.standard_normal((self.ndim_problem,))
z2 = self.rng.standard_normal((n_evolution_paths,))
X[i, :] = m + sigma * (x_z1 * z1 + x_z2 * np.dot(z2, Q))
start_evaluation = time.time()
y = fitness_function(X[i, :])
time_evaluations += (time.time() - start_evaluation)
n_evaluations += 1
Y[i] = y
if self.save_fitness_data: fitness_data.append(np.copy(y))
# update best-so-far x and y
if best_so_far_y > y: best_so_far_x, best_so_far_y = np.copy(X[i, :]), np.copy(y)
if self.save_best_so_far_x and not(n_evaluations % self.freq_best_so_far_x):
history_x = np.vstack((history_x, np.hstack((n_evaluations, best_so_far_x))))
# check three termination criteria
is_break, termination = MuCommaLambda._check_terminations(
self, n_evaluations, time.time() - start_optimization, best_so_far_y)
if is_break: break
# check three termination criteria
runtime = time.time() - start_optimization
is_break, termination = MuCommaLambda._check_terminations(
self, n_evaluations, runtime, best_so_far_y)
if is_break: break
if sigma <= self.threshold_step_size:
is_restart, n_restart = True, n_restart + 1
# update distribution mean
index = np.argsort(Y)
X, Y = X[index, :], Y[index]
m_bak = m
m = np.sum(X[:self.n_parents, :] * W, 0)
# update search directions
z = np.sqrt(mu_eff) * (m - m_bak) / sigma
for i in range(n_evolution_paths):
Q[i, :] = q_1 * Q[i, :] + q_2 * z
t = (np.sum(z * Q[i, :])) / (np.sum(Q[i, :] * Q[i, :]))
z = 1 / np.sqrt(1 + t ** 2) * (z - t * Q[i, :])
# update step-size
F = np.hstack((Y, Y_bak))
R1 = np.sum(RR[np.argsort(F) >= self.n_individuals])
U = R1 - self.n_individuals * (self.n_individuals + 1) / 2
Z = Z_1 * Z + Z_2 * (U - U_mean) / U_var
sigma = sigma * np.exp((st.norm.cdf(Z) / (1 - p_star) - 1) / d_sigma)
fitness_data, time_compression = MuCommaLambda._save_data(self, history_x, fitness_data)
results = {"best_so_far_x": best_so_far_x,
"best_so_far_y": best_so_far_y,
"n_evaluations": n_evaluations,
"runtime": runtime,
"fitness_data": fitness_data,
"termination": termination,
"time_evaluations": time_evaluations,
"time_compression": time_compression,
"m": m,
"step_size": sigma,
"n_individuals": self.n_individuals,
"n_parents": self.n_parents,
"d_sigma": self.d_sigma,
"n_restart": n_restart}
return results
def optimize(self, fitness_function=None):
start_optimization = time.time()
if (fitness_function is None) and (self.fitness_function != None):
fitness_function = self.fitness_function
# initialize
parent = MuCommaLambda._get_m(self)
start_evaluation = time.time()
y = fitness_function(parent)
n_evaluations, time_evaluations = 1, time.time() - start_evaluation
best_so_far_x, best_so_far_y = np.copy(parent), np.copy(y)
if self.save_fitness_data: fitness_data = [y]
if self.save_best_so_far_x:
history_x = np.hstack((n_evaluations, best_so_far_x))
else:
history_x = None
# initialize variables involved for all individuals
beta = np.sqrt(1 / self.ndim_problem) # adaptation vs precision
beta_scal = self.beta_scal # in the range (0, 1)
# beta_scal: small values facilitate a precise but time-consuming adaptaton
# iterate
termination = "max_evaluations"
is_restart, n_restart = True, 0
while n_evaluations < self.max_evaluations:
if is_restart:
if n_restart > 0:
parent = self.rng.uniform(self.lower_boundary,
self.upper_boundary,
(self.ndim_problem, ))
start_evaluation = time.time()
y = fitness_function(parent)
time_evaluations += (time.time() - start_evaluation)
n_evaluations += 1
if best_so_far_y > y:
best_so_far_x, best_so_far_y = np.copy(
parent), np.copy(y)
if self.save_best_so_far_x and not (
n_evaluations % self.freq_best_so_far_x):
history_x = np.vstack(
(history_x,
np.hstack((n_evaluations, best_so_far_x))))
if self.save_fitness_data: fitness_data.append(np.copy(y))
self.n_individuals *= 2
if self.n_individuals >= 1000: self.n_individuals = 1000
Y = np.tile(y, (self.n_individuals, )) # fitness of population
X = np.empty(
(self.n_individuals, self.ndim_problem)) # population
delta = np.ones((self.ndim_problem, )) # individual step-sizes
xi = np.empty((self.n_individuals, )) # global step-size
Z = np.empty(
(self.n_individuals, self.ndim_problem)) # Gaussian noises
is_restart = False
# 1. creation of lambda offspring
for k in range(self.n_individuals):
if self.rng.uniform(0, 1, 1) > 0.5: xi[k] = 1.4
else: xi[k] = 1 / 1.4
Z[k, :] = self.rng.standard_normal((self.ndim_problem, ))
X[k, :] = parent + xi[k] * delta * Z[k, :]
start_evaluation = time.time()
Y[k] = fitness_function(X[k, :])
time_evaluations += time.time() - start_evaluation
n_evaluations += 1
if self.save_fitness_data: fitness_data.append(Y[k])
# update best-so-far x and y
if best_so_far_y > Y[k]:
best_so_far_x, best_so_far_y = np.copy(X[k, :]), np.copy(
Y[k])
if self.save_best_so_far_x and not (n_evaluations %
self.freq_best_so_far_x):
history_x = np.vstack(
(history_x, np.hstack((n_evaluations, best_so_far_x))))
# check three termination criteria
is_break, termination = MuCommaLambda._check_terminations(
self, n_evaluations,
time.time() - start_optimization, best_so_far_y)
if is_break: break
# 2. selection / adaptation
sel = np.argmin(Y)
parent = np.copy(X[sel, :])
delta *= (np.power(xi[sel], beta) * np.power(
np.exp(np.abs(Z[sel, :]) - np.sqrt(2 / np.pi)), beta_scal))
# check termination criteria
runtime = time.time() - start_optimization
is_break, termination = MuCommaLambda._check_terminations(
self, n_evaluations, runtime, best_so_far_y)
if is_break: break
if np.min(delta) <= self.threshold_step_size:
is_restart, n_restart = True, n_restart + 1
if np.max(delta) >= 3 * np.min(self.upper_boundary -
self.lower_boundary):
is_restart, n_restart = True, n_restart + 1
fitness_data, time_compression = MuCommaLambda._save_data(
self, history_x, fitness_data)
results = {
"best_so_far_x": best_so_far_x,
"best_so_far_y": best_so_far_y,
"n_evaluations": n_evaluations,
"runtime": runtime,
"fitness_data": fitness_data,
"termination": termination,
"time_evaluations": time_evaluations,
"time_compression": time_compression,
"delta": delta,
"n_restart": n_restart
}
return results
def optimize(self, fitness_function=None):
start_optimization = time.time()
if (fitness_function is None) and (self.fitness_function != None):
fitness_function = self.fitness_function
# initialize distribution mean
m = MuCommaLambda._get_m(self) # distribution mean
start_evaluation = time.time()
y = fitness_function(m)
time_evaluations, n_evaluations = time.time() - start_evaluation, 1
best_so_far_x, best_so_far_y = np.copy(m), np.copy(y)
if self.save_fitness_data: fitness_data = [y]
if self.save_best_so_far_x:
history_x = np.hstack((n_evaluations, best_so_far_x))
else:
history_x = None
# iterate / evolve
termination = "max_evaluations"
# Here introduce 3 new constant symbols to simplify code
# constant symbols for Line 5 of Algorithm 1
m_1, m_2 = np.sqrt(1 - self.c_cov), np.sqrt(self.c_cov)
# constant symbol for Line 12 of Algorithm 1
p_1 = 1 - self.c
is_restart, n_restart = True, 0
while n_evaluations < self.max_evaluations:
if is_restart:
is_restart = False
if n_restart > 0:
m = self.rng.uniform(self.lower_boundary,
self.upper_boundary,
(self.ndim_problem, ))
start_evaluation = time.time()
y = fitness_function(m)
time_evaluations += (time.time() - start_evaluation)
n_evaluations += 1
if best_so_far_y > y:
best_so_far_x, best_so_far_y = np.copy(m), np.copy(y)
if self.save_fitness_data: fitness_data.append(np.copy(y))
if self.save_best_so_far_x and not (
n_evaluations % self.freq_best_so_far_x):
history_x = np.vstack(
(history_x,
np.hstack((n_evaluations, best_so_far_x))))
self.n_individuals *= 2
self.n_parents = int(np.floor(self.n_individuals / 2))
self.d_sigma *= 2
# set weights for parents selection
w, w_1 = np.log(np.arange(1, self.n_parents +
1)), np.log(self.n_parents + 1)
w = (w_1 - w) / (self.n_parents * w_1 - np.sum(w))
# normalization factor for principal search direction adaptation
mu_eff = 1 / np.sum(np.power(w, 2))
sigma = self.step_size # mutation strength (global step-size)
p = np.zeros((self.ndim_problem,
)) # principal search direction (evolution path)
s = 0 # cumulative rank rate
p_2 = np.sqrt(self.c * (2 - self.c) * mu_eff)
RR = np.arange(1, self.n_parents * 2 +
1) # ranks for R_t, R_(t+1)
X = np.empty(
(self.n_individuals, self.ndim_problem)) # population
Y = np.tile(y, (self.n_individuals, )) # fitness of population
Y_bak = np.copy(Y)
for i in range(self.n_individuals): # one generation
z = self.rng.standard_normal((self.ndim_problem, ))
r = self.rng.standard_normal()
# sample (Line 5 of Algorithm 1)
X[i, :] = m + sigma * (m_1 * z + m_2 * r * p)
start_evaluation = time.time()
y = fitness_function(X[i, :])
time_evaluations += (time.time() - start_evaluation)
n_evaluations += 1
Y[i] = y
# update best-so-far x and y
if best_so_far_y > y:
best_so_far_x, best_so_far_y = np.copy(X[i, :]), np.copy(y)
if self.save_fitness_data: fitness_data.append(np.copy(y))
if self.save_best_so_far_x and not (n_evaluations %
self.freq_best_so_far_x):
history_x = np.vstack(
(history_x, np.hstack((n_evaluations, best_so_far_x))))
# check three termination criteria
runtime = time.time() - start_optimization
is_break, termination = MuCommaLambda._check_terminations(
self, n_evaluations, runtime, best_so_far_y)
if is_break: break
# check four termination criteria
is_break, termination = MuCommaLambda._check_terminations(
self, n_evaluations, runtime, best_so_far_y, sigma)
if termination == "threshold_step_size (lower)":
is_restart, n_restart = True, n_restart + 1
elif is_break:
break
# update distribution mean
index = np.argsort(Y)
X, Y = X[index, :], Y[index] # Line 10 of Algorithm 1
m_bak = m
# Line 11 of Algorithm 1
m = np.zeros((self.ndim_problem, ))
for j in range(self.n_parents):
m += w[j] * X[j, :]
# update principal search direction (Line 12 of Algorithm 1)
p = p_1 * p + p_2 * ((m - m_bak) / sigma)
# adapt mutation strength (rank-based success rule, RSR)
F = np.hstack((Y_bak[:self.n_parents], Y[:self.n_parents]))
R = np.argsort(F)
# Line 13 of Algorithm 1
R_t, R_t1 = RR[R < self.n_parents], RR[R >= self.n_parents]
q = np.sum(w * (R_t - R_t1)) / self.n_parents
s = (1 - self.c_s) * s + self.c_s * (q - self.q_star)
sigma *= np.exp(s / self.d_sigma)
fitness_data, time_compression = MuCommaLambda._save_data(
self, history_x, fitness_data)
results = {
"best_so_far_x": best_so_far_x,
"best_so_far_y": best_so_far_y,
"n_evaluations": n_evaluations,
"runtime": runtime,
"fitness_data": fitness_data,
"termination": termination,
"time_evaluations": time_evaluations,
"time_compression": time_compression,
"m": m,
"p": p,
"s": s,
"step_size": sigma,
"n_individuals": self.n_individuals,
"n_parents": self.n_parents,
"d_sigma": self.d_sigma,
"n_restart": n_restart
}
return results
def optimize(self, fitness_function=None):
start_optimization = time.time()
if (fitness_function is None) and (self.fitness_function != None):
fitness_function = self.fitness_function
# initialize distribution mean
x = MuCommaLambda._get_m(self) # distribution mean
start_evaluation = time.time()
y = fitness_function(x)
time_evaluations, n_evaluations = time.time() - start_evaluation, 1
best_so_far_x, best_so_far_y = np.copy(x), np.copy(y)
if self.save_fitness_data: fitness_data = [y]
if self.save_best_so_far_x:
history_x = np.hstack((n_evaluations, best_so_far_x))
else:
history_x = None
# iterate / evolve
n_iterations = 1 # n_iter in train.py
n_failure_cholesky = 0 # number of failure of cholesky decomposition
G = [] # all gradients obtained during optmization
m, v = np.zeros((self.ndim_problem, )), np.zeros(
(self.ndim_problem, )) # for Adam optimizer
while n_evaluations < self.max_evaluations:
if n_iterations < self.iota: # do full sampling before iota iterations
UUT = np.zeros([self.ndim_problem,
self.ndim_problem]) # covariance matrix
n_t, alpha = self.n_t, self.alpha # n_samples in es.py
else: # use PCA decomposition to obtain subspace
pca = PCA()
pca_fit = pca.fit(G) # SVD
# take top n_t directions of maximum variance to construct covariance matrix
n_t = max(
np.argmax(
np.cumsum(pca_fit.explained_variance_ratio_) >
self.epsilon) + 1, self.min_n_t)
U, U_ort = pca_fit.components_[:n_t], pca_fit.components_[n_t:]
UUT, UUT_ort = np.matmul(U.T, U), np.matmul(U_ort.T, U_ort)
if n_iterations == self.iota: n_t = self.n_t
# sample from hybrid Gaussian distribution
cov = ((alpha / self.ndim_problem) * np.eye(self.ndim_problem) +
((1 - alpha) / n_t) * UUT) * self.sigma
A = np.zeros((
n_t,
self.ndim_problem)) # search directions (perturbation vectors)
try:
search_directions = cholesky(cov,
check_finite=False,
overwrite_a=True) # l in es.py
for i in range(n_t):
try:
A[i] = search_directions.dot(
self.rng.standard_normal((self.ndim_problem, )))
except LinAlgError:
A[i] = self.rng.standard_normal((self.ndim_problem, ))
except LinAlgError:
n_failure_cholesky += 1
for i in range(n_t):
A[i] = self.rng.standard_normal((self.ndim_problem, ))
# renormalize
A /= np.linalg.norm(A, axis=-1)[:, np.newaxis]
# estimate gradient via antithetic sampling
antithetic_fitness = np.zeros((n_t, 2)) # all_rollouts in es.py
for i in range(n_t):
up_x = x + A[i, :]
start_evaluation = time.time()
up_y = fitness_function(up_x)
time_evaluations += (time.time() - start_evaluation)
n_evaluations += 1
# update best-so-far x and y
if best_so_far_y > up_y:
best_so_far_x, best_so_far_y = np.copy(up_x), np.copy(up_y)
if self.save_fitness_data: fitness_data.append(np.copy(up_y))
if self.save_best_so_far_x and not (n_evaluations %
self.freq_best_so_far_x):
history_x = np.vstack(
(history_x, np.hstack((n_evaluations, best_so_far_x))))
if n_evaluations >= self.max_evaluations: break
down_x = x - A[i, :]
start_evaluation = time.time()
down_y = fitness_function(down_x)
time_evaluations += (time.time() - start_evaluation)
n_evaluations += 1
# update best-so-far x and y
if best_so_far_y > down_y:
best_so_far_x, best_so_far_y = np.copy(down_x), np.copy(
down_y)
if self.save_fitness_data: fitness_data.append(np.copy(down_y))
if self.save_best_so_far_x and not (n_evaluations %
self.freq_best_so_far_x):
history_x = np.vstack(
(history_x, np.hstack((n_evaluations, best_so_far_x))))
if n_evaluations >= self.max_evaluations: break
antithetic_fitness[i, :] = np.array([up_y, down_y])
if n_evaluations >= self.max_evaluations: break
antithetic_fitness = (antithetic_fitness -
np.mean(antithetic_fitness)) / (
np.std(antithetic_fitness) + 1e-8)
fitness_diff = antithetic_fitness[:,
0] - antithetic_fitness[:,
1] # m in es.py
gradient = np.zeros((self.ndim_problem, )) # g in es.py
for i in range(n_t):
gradient += (A[i, :] * fitness_diff[i])
gradient /= (2 * self.sigma)
# adaptive exploration mechanism
if n_iterations >= self.iota:
alpha = np.linalg.norm(np.dot(
gradient, UUT_ort)) / np.linalg.norm(np.dot(gradient, UUT))
# add current gradient to G
if n_iterations == 1: G = np.copy(gradient)
else: G = np.vstack([self.gamma * G, gradient])
gradient /= (np.linalg.norm(gradient) / self.ndim_problem + 1e-8)
# update gradient via Adam optimizer
m = 0.9 * m + (1 - 0.9) * gradient
mt = m / (1 - 0.9**n_iterations)
v = 0.999 * v + (1 - 0.999) * (gradient**2)
vt = v / (1 - 0.999**n_iterations)
x -= (self.eta * mt / (np.sqrt(vt) + 1e-8))
n_iterations += 1
fitness_data, time_compression = MuCommaLambda._save_data(
self, history_x, fitness_data)
results = {
"best_so_far_x": best_so_far_x,
"best_so_far_y": best_so_far_y,
"n_evaluations": n_evaluations,
"runtime": time.time() - start_optimization,
"fitness_data": fitness_data,
"termination": "max_evaluations",
"time_evaluations": time_evaluations,
"time_compression": time_compression,
"n_iterations": n_iterations,
"x": x,
"gradient": gradient,
"n_t": n_t,
"alpha": alpha,
"n_failure_cholesky": n_failure_cholesky,
"shape_G": G.shape
}
return results
def optimize(self, fitness_function=None):
start_optimization = time.time()
if (fitness_function is None) and (self.fitness_function != None):
fitness_function = self.fitness_function
# initialize distribution mean
m = MuCommaLambda._get_m(self) # distribution mean == _cur_mean
cur_std = self.init_std # distribution std
start_evaluation = time.time()
y = fitness_function(m)
time_evaluations, n_evaluations = time.time() - start_evaluation, 1
best_so_far_x, best_so_far_y = np.copy(m), np.copy(y)
if self.save_fitness_data: fitness_data = [y]
if self.save_best_so_far_x: history_x = np.hstack((n_evaluations, best_so_far_x))
else: history_x = None
# iterate / evolve
termination = "max_evaluations"
n_epoch = 0
# self.ndim_problem == _n_params
X = np.empty((self.n_individuals, self.ndim_problem)) # population
Y = np.tile(y, (self.n_individuals,)) # fitness of population
while n_evaluations < self.max_evaluations:
for i in range(self.n_individuals): # one generation / epoch
# sample
extra_var_mult = max(1.0 - n_epoch / self.extra_decay_time, 0)
sample_std = np.sqrt(np.square(cur_std) + np.square(self.extra_std) * extra_var_mult)
z = self.rng.standard_normal((self.ndim_problem,))
X[i, :] = m + sample_std * z
start_evaluation = time.time()
y = fitness_function(X[i, :])
time_evaluations += (time.time() - start_evaluation)
n_evaluations += 1
Y[i] = y
# update best-so-far x and y
if best_so_far_y > y: best_so_far_x, best_so_far_y = np.copy(X[i, :]), np.copy(y)
if self.save_fitness_data: fitness_data.append(np.copy(y))
if self.save_best_so_far_x and not(n_evaluations % self.freq_best_so_far_x):
history_x = np.vstack((history_x, np.hstack((n_evaluations, best_so_far_x))))
# check three termination criteria
runtime = time.time() - start_optimization
is_break, termination = MuCommaLambda._check_terminations(
self, n_evaluations, runtime, best_so_far_y)
if is_break: break
# check three termination criteria
is_break, termination = MuCommaLambda._check_terminations(
self, n_evaluations, runtime, best_so_far_y)
if is_break: break
# update distribution mean + std via Maximum Likelihood Estimation (MLE)
index = np.argsort(Y)
X = X[index, :]
m = np.mean(X[:self.n_best, :], axis=0)
cur_std = np.std(X[:self.n_best, :], axis=0) # Here it is a vector rather than a scalar
n_epoch += 1
fitness_data, time_compression = MuCommaLambda._save_data(self, history_x, fitness_data)
results = {"best_so_far_x": best_so_far_x,
"best_so_far_y": best_so_far_y,
"n_evaluations": n_evaluations,
"runtime": runtime,
"fitness_data": fitness_data,
"termination": termination,
"time_evaluations": time_evaluations,
"time_compression": time_compression,
"m": m,
"cur_std": cur_std,
"n_epoch": n_epoch}
return results
def optimize(self, fitness_function=None):
start_optimization = time.time()
if (fitness_function is None) and (self.fitness_function != None):
fitness_function = self.fitness_function
# initialize distribution mean
# Here not confuse distribution mean (m) with number of multiple evolution paths (m)
m = MuCommaLambda._get_m(self) # distribution mean
start_evaluation = time.time()
y = fitness_function(m)
time_evaluations, n_evaluations = time.time() - start_evaluation, 1
best_so_far_x, best_so_far_y = np.copy(m), np.copy(y)
if self.save_fitness_data: fitness_data = [y]
else: fitness_data = None
if self.save_best_so_far_x:
history_x = np.hstack((n_evaluations, best_so_far_x))
else:
history_x = None
# set weights for parents selection
w, w_1 = np.log(np.arange(1, self.n_parents +
1)), np.log(self.n_parents + 1)
w = (w_1 - w) / (self.n_parents * w_1 - np.sum(w))
# normalization factor for search direction adaptation
mu_eff = 1 / np.sum(np.power(w, 2))
# iterate / evolve
termination = "max_evaluations"
sigma = self.step_size # mutation strength (global step-size)
n_evolution_paths = self.n_evolution_paths # m
# multiple evolution paths
MEP = np.zeros((n_evolution_paths, self.ndim_problem)) # P
t_hat = np.zeros(
(n_evolution_paths, )) # direction set for Algorithm 2
p = np.zeros((self.ndim_problem,
)) # principal search direction (evolution path)
s = 0 # cumulative rank rate
t = 0 # generation index
# Here introduce 6 new constant symbols to simplify code
# constant symbols for Line 4 of Algorithm 3
a, b = np.sqrt(1 - self.c_cov), np.sqrt(self.c_cov)
a_m = a**n_evolution_paths
# constant symbols for Line 11 of Algorithm 3
p_1, p_2 = 1 - self.c, np.sqrt(self.c * (2 - self.c) * mu_eff)
# constant symbols for Line 13 of Algorithm 3
RR = np.arange(1, self.n_parents * 2 + 1) # ranks for R_t, R_(t+1)
X = np.empty((self.n_individuals, self.ndim_problem)) # population
Y = np.tile(y, (self.n_individuals, )) # fitness of population
while n_evaluations < self.max_evaluations:
Y_bak = np.copy(Y)
for i in range(self.n_individuals): # one generation
z = self.rng.standard_normal((self.ndim_problem, ))
# sample (Line 4 of Algorithm 3)
sum_p = np.zeros((self.ndim_problem, ))
for j in range(1, n_evolution_paths + 1):
r = self.rng.standard_normal()
sum_p += (a**(n_evolution_paths - j)) * r * MEP[j - 1, :]
X[i, :] = m + sigma * (a_m * z + b * sum_p)
start_evaluation = time.time()
y = fitness_function(X[i, :])
time_evaluations += (time.time() - start_evaluation)
n_evaluations += 1
Y[i] = y
# update best-so-far x and y
if best_so_far_y > y:
best_so_far_x, best_so_far_y = np.copy(X[i, :]), np.copy(y)
if self.save_fitness_data: fitness_data.append(np.copy(y))
if self.save_best_so_far_x and not (n_evaluations %
self.freq_best_so_far_x):
history_x = np.vstack(
(history_x, np.hstack((n_evaluations, best_so_far_x))))
# check three termination criteria
runtime = time.time() - start_optimization
is_break, termination = MuCommaLambda._check_terminations(
self, n_evaluations, runtime, best_so_far_y)
if is_break: break
# check four termination criteria
is_break, termination = MuCommaLambda._check_terminations(
self, n_evaluations, runtime, best_so_far_y, sigma)
if is_break: break
# update distribution mean
index = np.argsort(Y)
X, Y = X[index, :], Y[index] # Line 9 of Algorithm 3
m_bak = m
# Line 10 of Algorithm 3
m = np.zeros((self.ndim_problem, ))
for j in range(self.n_parents):
m += w[j] * X[j, :]
# update principal search direction (Line 11 of Algorithm 3)
p = p_1 * p + p_2 * ((m - m_bak) / sigma)
# update multiple evolution paths (Algorithm 2 or Line 12 of Algorithm 3)
T_min = np.min(np.diff(t_hat))
if (T_min > self.T) or (t < n_evolution_paths):
for i in range(n_evolution_paths - 1):
MEP[i, :], t_hat[i] = MEP[i + 1, :], t_hat[i + 1]
else:
i_apostrophe = np.argmin(np.diff(t_hat))
for i in range(i_apostrophe, n_evolution_paths - 1):
MEP[i, :], t_hat[i] = MEP[i + 1, :], t_hat[i + 1]
MEP[n_evolution_paths - 1, :] = p
t_hat[n_evolution_paths - 1] = t
t += 1 # Line 14 of Algorithm 3
# adapt mutation strength (rank-based success rule, RSR) -> Line 13 of Algorithm 3
F = np.hstack((Y_bak[:self.n_parents], Y[:self.n_parents]))
R = np.argsort(F)
R_t, R_t1 = RR[R < self.n_parents], RR[R >= self.n_parents]
q = np.sum(w * (R_t - R_t1)) / self.n_parents
s = (1 - self.c_s) * s + self.c_s * (q - self.q_star)
sigma *= np.exp(s / self.d_sigma)
fitness_data, time_compression = MuCommaLambda._save_data(
self, history_x, fitness_data)
results = {
"best_so_far_x": best_so_far_x,
"best_so_far_y": best_so_far_y,
"n_evaluations": n_evaluations,
"runtime": runtime,
"fitness_data": fitness_data,
"termination": termination,
"time_evaluations": time_evaluations,
"time_compression": time_compression,
"m": m,
"p": p,
"s": s,
"step_size": sigma,
"t_hat": t_hat
}
return results
def optimize(self, fitness_function=None):
start_optimization = time.time()
if (fitness_function is None) and (self.fitness_function != None):
fitness_function = self.fitness_function
# initialize distribution mean
m = MuCommaLambda._get_m(self) # y in Fig. 3
start_evaluation = time.time()
y = fitness_function(m)
n_evaluations, time_evaluations = 1, time.time() - start_evaluation
best_so_far_x, best_so_far_y = np.copy(m), np.copy(y)
Y = np.tile(y, (self.n_individuals,)) # fitness of population
if self.save_fitness_data: fitness_data = [y]
if self.save_best_so_far_x: history_x = np.hstack((n_evaluations, best_so_far_x))
else: history_x = None
# set weights for parents
w = np.log(np.arange(1, self.n_parents + 1))
w = (np.log((self.n_individuals + 1) / 2) - w) / (
self.n_parents * np.log((self.n_individuals + 1) / 2) - np.sum(w))
mu_eff = 1 / np.sum(np.power(w, 2))
# initialize transformation matrix
s = np.zeros((self.ndim_problem,))
M = np.diag(np.ones((self.ndim_problem,)))
c_s = (mu_eff + 2) / (mu_eff + self.ndim_problem + 5)
s_1, s_2 = 1 - c_s, np.sqrt(mu_eff * c_s * (2 - c_s))
c_1 = self.alpha_cov / (np.power(self.ndim_problem + 1.3, 2) + mu_eff)
c_w = min(1 - c_1, self.alpha_cov * (mu_eff + 1 / mu_eff - 2) / (
np.power(self.ndim_problem + 2, 2) + self.alpha_cov * mu_eff / 2))
d_sigma = 1 + c_s + 2 * max(0, np.sqrt((mu_eff - 1) / (self.ndim_problem + 1)) - 1)
# iterate
sigma, step_size_data = self.step_size, [self.step_size if self.save_step_size_data else None]
Z = np.empty((self.n_individuals, self.ndim_problem)) # Guassian noise for mutation
D = np.empty((self.n_individuals, self.ndim_problem)) # search directions
X = np.empty((self.n_individuals, self.ndim_problem)) # population
I = np.diag(np.ones((self.ndim_problem,)))
while n_evaluations < self.max_evaluations:
for i in range(self.n_individuals): # l in Fig. 3
Z[i, :] = self.rng.standard_normal((self.ndim_problem,)) # z_l in Fig. 3
D[i, :] = np.transpose(np.dot(M, Z[i, :][:, np.newaxis])) # d_l in Fig. 3
X[i, :] = m + sigma * D[i, :]
start_evaluation = time.time()
y = fitness_function(X[i, :]) # f_l in Fig. 3
time_evaluations += (time.time() - start_evaluation)
n_evaluations += 1
Y[i] = y
if self.save_fitness_data: fitness_data.append(np.copy(y))
# update best-so-far x and y
if best_so_far_y > y: best_so_far_x, best_so_far_y = np.copy(X[i, :]), np.copy(y)
if self.save_best_so_far_x and not(n_evaluations % self.freq_best_so_far_x):
history_x = np.vstack((history_x, np.hstack((n_evaluations, best_so_far_x))))
# check three termination criteria
is_break, termination = MuCommaLambda._check_terminations(
self, n_evaluations, time.time() - start_optimization, best_so_far_y)
if is_break: break
# check four termination criteria
runtime = time.time() - start_optimization
is_break, termination = MuCommaLambda._check_terminations(
self, n_evaluations, runtime, best_so_far_y)
if is_break: break
if sigma <= self.threshold_step_size:
termination = "threshold_step_size (lower)"
break
index = np.argsort(Y)
Z, D, Y = Z[index, :], D[index, :], Y[index]
d_w = np.zeros((self.ndim_problem,)) # for (M9) in Fig. 3
z_w = np.zeros((self.ndim_problem,)) # for (M10) in Fig. 3
zzt_w = np.zeros((self.ndim_problem, self.ndim_problem)) # for (M11) in Fig. 3
for j in range(self.n_parents):
d_w += (w[j] * D[j, :])
z_w += (w[j] * Z[j, :])
zzt_w += (w[j] * np.dot(Z[j, :][:, np.newaxis], Z[j, :][np.newaxis, :]))
# update distribution mean
m += (sigma * d_w)
# update transformation matrix
s = s_1 * s + s_2 * z_w
M1 = (c_1 / 2) * (np.dot(s[:, np.newaxis], s[np.newaxis, :]) - I)
M2 = (c_w / 2) * (zzt_w - I)
M = np.dot(M, I + M1 + M2)
# update step-size
sigma *= np.exp(c_s / d_sigma * (np.linalg.norm(s) / self.expectation_chi - 1))
if self.save_step_size_data: step_size_data.append(sigma)
fitness_data, time_compression = MuCommaLambda._save_data(self, history_x, fitness_data)
results = {"best_so_far_x": best_so_far_x,
"best_so_far_y": best_so_far_y,
"n_evaluations": n_evaluations,
"runtime": runtime,
"fitness_data": fitness_data,
"termination": termination,
"time_evaluations": time_evaluations,
"time_compression": time_compression,
"m": m,
"step_size": sigma,
"step_size_data": step_size_data}
return results