def demo():
X = np.asfortranarray(np.random.randn(1000, 100))
srft = SubsampledRandomizedFourrierTransform(600)
Y = srft.fit_transform(X)
sigma_X = np.linalg.norm(X, 2)
sigma_Y = np.linalg.norm(Y, 2)
print(np.abs(sigma_X - sigma_Y) / sigma_Y)
def demo():
X = np.asfortranarray(np.random.randn(1000, 100))
srft = SubsampledRandomizedFourrierTransform(600)
Y = srft.fit_transform(X)
sigma_X = np.linalg.norm(X, 2)
sigma_Y = np.linalg.norm(Y, 2)
print(np.abs(sigma_X - sigma_Y) / sigma_Y)
def demo_1d():
x = np.random.randn(1000)
srft = SubsampledRandomizedFourrierTransform(500)
tic = time()
srft.fit(x, prefit=True)
print 'Prefit:', time() - tic
tic = time()
y = srft.transform_1d(x)
print 'Transform:', time() - tic
def demo_1d():
x = np.random.randn(1000)
srft = SubsampledRandomizedFourrierTransform(500)
tic = time()
srft.fit(x, prefit=True)
print 'Prefit:', time() - tic
tic = time()
y = srft.transform_1d(x)
print 'Transform:', time() - tic
def __init__(self, wrt, fprime, eta, lamb, delta, k, n_classes=None, f=None, args=None):
"""Create a RadaGrad object.
Parameters
----------
wrt : array_like
Array that represents the solution. Will be operated upon in
place. ``fprime`` should accept this array as a first argument.
fprime : callable
Callable that given a solution vector as first parameter and *args
and **kwargs drawn from the iterations ``args`` returns a
search direction, such as a gradient.
step_rate : scalar or array_like, optional [default: 1]
Value to multiply steps with before they are applied to the
parameter vector.
args : iterable
Iterator over arguments which ``fprime`` will be called with.
"""
super(Radagrad, self).__init__(wrt, args=args)
self.n_classes = n_classes
self.f = f
if self.n_classes is not None:
self.k_perc = k
self.k_full = k * n_classes + n_classes
# self.k_full = k * n_classes
# self.k_perc = k + 1
# self.k_full = k * n_classes + n_classes
else:
self.k_perc = k
self.k_full = k
self.fprime = fprime
self.g_avg = zero_like(wrt)
self.P_g_avg = np.zeros(self.k_full)
# self.Gt = np.zeros((self.k_full, self.k_full))
self.Gt = np.eye(self.k_full, self.k_full)
self.eta = eta
self.lamb = lamb
self.delta = delta
self.I_delta = np.diag(np.ones(self.Gt.shape[0]) * delta)
# self.P_Lamb_inv_P = np.diag(np.ones(k) * wrt.shape[0] / (k * 2 * lamb))
# self.P_Lamb_inv_P = np.diag(np.ones(k) / (2 * lamb))
# self.lamb_inv = 1 / (2 * lamb)
# self.eye_Gt = np.eye(*self.Gt.shape)
self.srft = SubsampledRandomizedFourrierTransform(self.k_perc)
if self.n_classes is None:
self.srft.fit(wrt)
else:
self.n_param = wrt.shape[0]
self.n_features = (self.n_param-self.n_classes)/self.n_classes
self.srft.fit(wrt[:self.n_features])
def demo_I():
k, d = 2, 11
# X = np.eye(d)
nrm_X, nrm_Y2 = 0, 0
for i in range(100):
X = np.eye(d)
srft = SubsampledRandomizedFourrierTransform(k)
srft.fit(X)
Y1 = srft.transform(np.asfortranarray(X.copy()))
Y2 = srft.transform(np.asfortranarray(Y1.copy().T)).T
# print X
# print Y2
# print srft.srht_const ** 2
# print np.linalg.norm(X[0, :]), np.linalg.norm(Y2[0, :])
# print np.linalg.norm(X[:, 0]), np.linalg.norm(Y2[:, 0])
nrm_X += np.linalg.norm(X)
nrm_Y2 += np.linalg.norm(Y2)
print nrm_X / 100, nrm_Y2 / 100
def demo_I():
k, d = 2, 11
# X = np.eye(d)
nrm_X, nrm_Y2 = 0, 0
for i in range(100):
X = np.eye(d)
srft = SubsampledRandomizedFourrierTransform(k)
srft.fit(X)
Y1 = srft.transform(np.asfortranarray(X.copy()))
Y2 = srft.transform(np.asfortranarray(Y1.copy().T)).T
# print X
# print Y2
# print srft.srht_const ** 2
# print np.linalg.norm(X[0, :]), np.linalg.norm(Y2[0, :])
# print np.linalg.norm(X[:, 0]), np.linalg.norm(Y2[:, 0])
nrm_X += np.linalg.norm(X)
nrm_Y2 += np.linalg.norm(Y2)
print nrm_X / 100, nrm_Y2 / 100
class Radagrad(Minimizer):
"""RadaGrad optimizer.
RadaGrad [krummenacher_mcwilliams2014]_ is a method that ...
Let :math:`f'(\\theta_t)` be the derivative of the loss with respect to the
parameters at time step :math:`t`. In its
basic form, given a step rate :math:`\\alpha`, a decay term
:math:`\\gamma` and an offset :math:`\\epsilon` we perform the following
updates:
.. math::
g_t &=& (1 - \\gamma)~f'(\\theta_t)^2 + \\gamma g_{t-1}
where :math:`g_0 = 0`. Let :math:`s_0 = 0` for updating the parameters:
.. math::
\\Delta \\theta_t &=& \\alpha {\sqrt{s_{t-1} + \\epsilon} \over \sqrt{g_t + \\epsilon}}~f'(\\theta_t), \\\\
\\theta_{t+1} &=& \\theta_t + \\Delta \\theta_t.
Subsequently we adapt the moving average of the steps:
.. math::
s_t &=& (1 - \\gamma)~\\Delta\\theta_t^2 + \\gamma s_{t-1}.
To extend this with Nesterov's accelerated gradient, we need a momentum
coefficient :math:`\\beta` and incorporate it by using slightly different
formulas:
.. math::
\\theta_{t + {1 \over 2}} &=& \\theta_t + \\beta \\Delta \\theta_{t-1}, \\\\
g_t &=& (1 - \\gamma)~f'(\\theta_{t + {1 \over 2}})^2 + \\gamma g_{t-1}, \\\\
\\Delta \\theta_t &=& \\alpha {\sqrt{s_{t-1} + \\epsilon} \over \sqrt{g_t + \\epsilon}}~f'(\\theta_{t + {1 \over 2}}).
In its original formulation, the case :math:`\\alpha = 1, \\beta = 0` was
considered only.
.. [krummenacher_mcwilliams2014] Krummenacher, G. and McWilliams, B.
"RadaGrad: Random Projections for Adaptive Stochastic Optimization","
preprint http://people.inf.ethz.ch/kgabriel/publications/krummenacher_mcwilliams14radagrad.pdf (2014).
"""
state_fields = 'n_iter g_avg P_g_avg Gt eta lamb delta'.split()
def __init__(self, wrt, fprime, eta, lamb, delta, k, n_classes=None, f=None, args=None):
"""Create a RadaGrad object.
Parameters
----------
wrt : array_like
Array that represents the solution. Will be operated upon in
place. ``fprime`` should accept this array as a first argument.
fprime : callable
Callable that given a solution vector as first parameter and *args
and **kwargs drawn from the iterations ``args`` returns a
search direction, such as a gradient.
step_rate : scalar or array_like, optional [default: 1]
Value to multiply steps with before they are applied to the
parameter vector.
args : iterable
Iterator over arguments which ``fprime`` will be called with.
"""
super(Radagrad, self).__init__(wrt, args=args)
self.n_classes = n_classes
self.f = f
if self.n_classes is not None:
self.k_perc = k
self.k_full = k * n_classes + n_classes
# self.k_full = k * n_classes
# self.k_perc = k + 1
# self.k_full = k * n_classes + n_classes
else:
self.k_perc = k
self.k_full = k
self.fprime = fprime
self.g_avg = zero_like(wrt)
self.P_g_avg = np.zeros(self.k_full)
# self.Gt = np.zeros((self.k_full, self.k_full))
self.Gt = np.eye(self.k_full, self.k_full)
self.eta = eta
self.lamb = lamb
self.delta = delta
self.I_delta = np.diag(np.ones(self.Gt.shape[0]) * delta)
# self.P_Lamb_inv_P = np.diag(np.ones(k) * wrt.shape[0] / (k * 2 * lamb))
# self.P_Lamb_inv_P = np.diag(np.ones(k) / (2 * lamb))
# self.lamb_inv = 1 / (2 * lamb)
# self.eye_Gt = np.eye(*self.Gt.shape)
self.srft = SubsampledRandomizedFourrierTransform(self.k_perc)
if self.n_classes is None:
self.srft.fit(wrt)
else:
self.n_param = wrt.shape[0]
self.n_features = (self.n_param-self.n_classes)/self.n_classes
self.srft.fit(wrt[:self.n_features])
# self.srft.fit(wrt[:self.n_features + 1])
def _iterate(self):
for args, kwargs in self.args:
gradient = self.fprime(self.wrt, *args, **kwargs)
print self.f(self.wrt, *args, **kwargs)
if self.n_classes is None:
P_gt = self.srft.transform_1d(gradient)
else:
P_gt = np.r_[np.array([self.srft.transform_1d(gradient[self.n_features * i:self.n_features * (i + 1)]) for i in range(self.n_classes)]).flatten(), gradient[self.n_param - self.n_classes:]]
# P_gt = np.array([self.srft.transform_1d(gradient[self.n_features * i:self.n_features * (i + 1)]) for i in range(self.n_classes)]).flatten()
# P_gt = np.r_[np.array([self.srft.transform_1d(gradient[np.r_[np.arange(self.n_features * i, self.n_features * (i + 1)), self.n_param - self.n_classes + i].astype(int)]) for i in range(self.n_classes)]).flatten()]
# P_gt = gradient
self.Gt += np.outer(P_gt, P_gt)
# St = self._my_sqrtm(np.diag(np.diag(self.Gt)))
St = self._my_sqrtm(self.I_delta + self.Gt)
Ht_inv = np.linalg.inv(St)
if self.n_classes is None:
uppro = self.srft.inverse_transform_1d(np.dot(Ht_inv, P_gt))
else:
vec = np.dot(Ht_inv, P_gt)
uppro = np.r_[np.array([self.srft.inverse_transform_1d(vec[self.k_perc * i:self.k_perc * (i + 1)]) for i in range(self.n_classes)]).flatten(), vec[self.k_full - self.n_classes:]]
# uppro_i = np.r_[np.array([self.srft.inverse_transform_1d(vec[self.k_perc * i:self.k_perc * (i + 1)]) for i in range(self.n_classes)]).flatten()]
# uppro = np.r_[np.array([uppro_i[(self.n_features + 1) * i:(self.n_features + 1) * (i + 1) - 1] for i in range(self.n_classes)]).flatten(), np.array([uppro_i[(self.n_features + 1) * i + self.n_features] for i in range(self.n_classes)]).flatten()]
# uppro = np.dot(Ht_inv, P_gt)
self.wrt -= self.eta * uppro
self.n_iter += 1
yield {
'n_iter': self.n_iter,
'gradient': gradient,
'args': args,
'kwargs': kwargs,
}
def _iterate_rda(self):
for args, kwargs in self.args:
t = self.n_iter + 1
gradient = self.fprime(self.wrt, *args, **kwargs)
self.g_avg = ((t - 1) / t) * self.g_avg + (1 / t) * gradient
P_gt = self.srft.transform_1d(gradient)
# P_gt = gradient
self.P_g_avg = ((t - 1) / t) * self.P_g_avg + (1 / t) * P_gt
self.Gt += np.outer(P_gt, P_gt)
St = self._my_sqrtm(self.Gt)
Ht_inv = np.linalg.inv(self.I_delta + St)
Ht_reg_inv = scipy_pinv(Ht_inv + 1 / (t * self.eta) * self.P_Lamb_inv_P)
uppro = self.srft.inverse_transform_1d(np.dot(Ht_reg_inv, self.P_g_avg))
# uppro = np.dot(Ht_reg_inv, self.P_g_avg)
self.wrt = -self.lamb_inv * (self.g_avg - 1 / (t * self.eta) * self.lamb_inv * uppro)
self.n_iter += 1
yield {
'n_iter': self.n_iter,
'gradient': gradient,
'args': args,
'kwargs': kwargs,
}
def _my_sqrtm(self, X):
# tol = 1e-7
return np.real(sqrtm(X)) # + self.eye_Gt * tol)
def _my_sqrtm_polar(self, X):
tol = 1e-10
L = np.linalg.cholesky(X + np.diag(np.ones(X.shape[0]) * tol))
U, P = polar(L, side="right")
return P