def init_bfgs(self, X, y):
"""
Initializes the BFGS algorithm
Parameters
----------
X : N x D matrix composed of numerical features, where each feature is 1 x D
y : N X 1 matrix, where y is a bit vector corresponding to whether each of the
N features is in class 1 or 0
Returns
-------
W : initial weights
B : 2 x D x D "matrix", where the first is the initial pseudo-Hessian matrix and
the second is empty
G : Gradient matrix containing the initial gradient vector and an empty slot for the next
gradient vector
"""
W = np.zeros(shape=(2, X.shape[1])) # initializing weights
B = np.zeros(shape=(2, X.shape[1], X.shape[1]))
B[0] = np.diag(np.ones(shape=X.shape[1])
) # Initializing pseudo-hessian to identity matrix
pi = expit(safedot(X, W[0]))
G = np.empty(shape=(2, X.shape[1]))
G[0] = safedot(X.T, (pi - y))
return W, B, G
def l2_distance(self, X, x, gen_dist, kernel, **kwargs):
"""
Computes Euclidean distance between a single feature observation from testing
data and each observation from training data
Parameters
----------
X : N x D matrix consisting of N observations of data
x : D x 1 vector consisting of a single observation
gen_dist : pre computed squared norm of each row vector in X
kernel : function that will apply a pseudo projection of the features into a space
of different dimensions
Returns
-------
N x 1 vector of distances between x and each observation in X
"""
if kernel is None:
t_1 = gen_dist
t_2 = safedot(x, x)
t_3 = 2 * safedot(X, x)
return t_1 + t_2 - t_3
else:
distances = np.zeros(X.shape[0])
for i in range(X.shape[0]):
t_2 = kernel(x, x, **kwargs)
t_3 = kernel(X[i], x, **kwargs)
distances[i] = gen_dist[i] + t_2 - (2 * t_3)
return distances
def gaussian(self, x, x_, **kwargs):
"""
Computes the Gaussian(RBF) kernel of the two given vectors
Parameters
----------
x : D x 1 feature vector
x_ : D x 1 feature vector
sigma : parameter controlling the bandwith of the kernel
Returns
-------
kernelized inner product of the two given vectors
"""
if kwargs:
try:
sigma = kwargs['sigma']
except KeyError:
raise ValueError(
'Must use proper parameters of Gaussian(RBF) kernel')
else:
sigma = 30
gamma = 1 / (2 * np.square(sigma))
sq_norm = safedot(x, x) + safedot(x_, x_) - (2 * safedot(x, x_))
n = gamma * sq_norm
return np.exp(-n)
def conjugate_gradient(self, A, b, epsilon, x=None):
"""
Solves linear equation Ax = b using conjugate gradients. This Algorithm
can be found on pg. 111 of "Numerical Analysis"
Parameters
----------
A : In the context of newton methods, A is the Hessian matrix
b : In the context of newton methods, b is the gradient vector
epsilon : Since covergence to 0 would be a waste of computational resources, epsilon is used
to determine if the residual is "close-enough" to 0
x : can be given if there is a desired starting
Returns
-------
x : approximate solution to the linear equation of the form Ax = b
"""
if x is None:
x = np.ones(b.shape[0]) # Initialize random x
r_0 = safedot(A, x) - b # Calculating residual
p = -r_0 # original direction
while la.norm(r_0) > epsilon:
alpha = safedot(r_0, r_0) / safedot(safedot(p, A),
p) # Creating 1-D minimizer
x = x + (alpha * p) # Updating x
r_1 = r_0 + (alpha * safedot(A, p)) # Updating residual
beta = safedot(r_1, r_1) / safedot(
r_0,
r_0) # matrix to ensure conjugacy between new direct p and A
p = -r_1 + safedot(beta, p) # updating direction
r_0 = r_1
return x
def get_bernoulli_likelihood(self, X):
"""
Calculates the likelihood of each observation in X being in a
class given that the features in X are either 0 or 1
Parameters
----------
X : N x D matrix of 1 x D feature vectors
"""
log_p = np.log(self.p_matrix)
log_p_not = np.log(1 - self.p_matrix)
a = safedot(1 - X, log_p_not.T)
b = safedot(X, log_p.T)
self.predictions = np.argmax(np.log(self.priors) + a + b, axis = 1)
def fit(self, X, y, max_iters=10000, save_weights=False, epsilon=1e-5):
"""
Finds optimal weights by adjusting them only when incorrect observations are made
Parameters
----------
X : N x D matrix composed of numerical features, where each feature is 1 x D
y : N X 1 matrix, where y is a bit vector corresponding to whether each of the
N features is in class 1 or 0
max_iters : maximum number of iterations before the algorithm will terminate
save_weights : whether or not the weights obtain from each iteration should be saved
epsilon : small number to be used to test for approximate convergence when determining
the direction to be descended
"""
if not np.allclose(np.unique(y), np.array([-1, 1])):
y[y == 0] = -1
w = -np.ones(shape=X.shape[1]) + np.finfo('float').resolution
W = np.zeros(shape=(max_iters, w.shape[0]))
s = np.zeros(w.shape[0])
for i in range(max_iters):
idx = i % (X.shape[0] - 1)
x = X[idx]
y_hat = np.sign(safedot(x, w))
# Only updates when an incorrect predict is made
if not np.allclose(y_hat, y[idx]):
w += y[idx] * x
W[i] = w
if i > 50 and la.norm(W[i] - W[i - 50]) < epsilon:
break
self.w = w
self.W = W
def polynomial(self, x, x_, **kwargs):
"""
Computes the polynomial kernel of the two given vectors
Parameters
----------
x : D x 1 feature vector
x_ : D x 1 feature vector
c : C >= 0 is a free parameter that controls the influence of higher order
terms versus lower order terms in polynomial
d : scaling degree
Returns
-------
Kernelized inner product of the two given vectors
"""
if kwargs:
try:
c = kwargs['c']
d = kwargs['d']
except KeyError:
raise ValueError(
'Must use proper arguments for polynomial kernel')
else:
c = 1
d = 2
n = safedot(x, x_) + c
return np.power(n, d)
def irls(self, X, y, iterations=50, l2_reg=0, save_weights=False):
"""
Solving the likelihood function using Iteratively Reweighted Least Squares method
Parameters
----------
X : N x D matrix composed of numerical features, where each feature is 1 x D
y : N X 1 matrix, where y is a bit vector corresponding to whether each of the
N features is in class 1 or 0
max_iters : maximum number of iterations before the algorithm will terminate
l2_reg : amount of l2 regularization to be applied
save_weights : whether or not the weights obtain from each iteration should be saved
"""
w = np.zeros(X.shape[1])
prediction = np.ones(X.shape[1])
if save_weights:
W = np.empty(shape=((iterations), w.shape[0]))
i = 0
while i < iterations:
n = safedot(X, w)
prediction = expit(
n) # predicting new y values based on current weights
s = prediction * (1 - prediction)
s[s == 0] = np.finfo(
'float'
).resolution # Ensuring that matrix S will be invertible
S = np.diag(s)
z = n + self.score_irls(y, prediction, S) # response variable
if np.allclose(z, n):
if (save_weights):
W = W[:i]
break
w_0 = la.inv(la.multi_dot((X.T, S, X)))
w_1 = la.multi_dot((X.T, S, z))
w_n = safedot(w_0, w_1)
w = w_n + (l2_reg * w)
if save_weights:
W[i] = w
i += 1
self.w = w
if save_weights:
self.W = W
def update_B(self, B, W, G):
"""
Performs the update of the pseudo-Hessian Matrix
Parameters
----------
B : 2 x D x D "martix" containing pseudo-Hessian matrix and an empty slot for the next one
W : 2 x D matrix containing the two weights calculated from the past two iterations of
the BFGS algorithm
G : 2 x D matrix containing the two gradients calculated from the past two iterations
of the BFGS algorithm
Returns
-------
B : 2 x D x D matrix containing past two pseudo hessian matrices
"""
dW = W[1] - W[0]
dG = G[1] - G[0]
t_1a = safedot(dG, dG)
t_1b = safedot(dG, dW)
if not t_1b:
t_1 = 0
else:
t_1 = t_1a / t_1b
t_2a = safedot(safedot(B[0], dW), safedot(B[0], dW))
t_2b = safedot(safedot(dW, B[0]), dW)
B[1] = B[0] + t_1 + (t_2a / t_2b)
return B[1]
def classify(self, X):
"""
Uses weights calculating from fitting to form classification of new features
Parameters
----------
X : M x D matrix composed of numerical features, where each feature is 1 x D
"""
preds = np.sign(safedot(X, self.w))
preds[preds < 1] = 0
self.predictions = preds
def bfgs(self, X, y, iterations=20, save_weights=False, t=0, k=.12):
"""
Implementation of the Broyden–Fletcher–Goldfarb–Shanno algorithm
Parameters
----------
X : N x D matrix composed of numerical features, where each feature is 1 x D
y : N X 1 matrix, where y is a bit vector corresponding to whether each of the
N features is in class 1 or 0
max_iters : maximum number of iterations before the algorithm will terminate
save_weights : whether or not the weights obtain from each iteration should be saved
k : must be in range (0,1) and larger values will decrease step sizes more
t : larger values will decrease the step size taken
Note
----
Significantly faster convergence with normalized X
"""
W, B, G = self.init_bfgs(X, y)
weights = np.empty(shape=(iterations, W.shape[1]))
for i in range(iterations):
n = safedot(X, W[0])
G[1] = (safedot(X.T, expit(n) - y)) / G.shape[1]
d = safedot(la.pinv(B[0]), G[1])
a = self.step_size(iter_num=(i + 1) * 10, t=t, k=k)
W[1] = W[0] - (a * d)
B[0] = self.update_B(B, W, G)
G[0] = G[1]
W[0] = W[1]
if save_weights:
weights[i] = W[0]
self.w = W[0]
if save_weights:
self.W = weights
def binary_newton_cg(self,
X,
y,
max_iters=10,
l2_reg=0,
save_weights=False,
epsilon=1e-4):
"""
Solves the optimization of the likelihood function using newton's
method with conjugate gradients
Parameters
----------
X : N x D matrix composed of numerical features, where each feature is 1 x D
y : N X 1 matrix, where y is a bit vector corresponding to whether each of the
N features is in class 1 or 0
max_iters : maximum number of iterations before the algorithm will terminate
l2_reg : determines the amount of regularization that will be applied at each iteration
save_weights : whether or not the weights obtain from each iteration should be saved
epsilon : small number to be used to test for approximate convergence when determining
the direction to be descended
"""
w = np.zeros(X.shape[1])
if save_weights:
W = np.zeros(shape=(max_iters, X.shape[1]))
for i in range(max_iters):
mu = expit(safedot(X, w)) # Calculating predicted probabilities
g = safedot(X.T, (mu - y)) # gradient vector
H = safedot(safedot(X.T, np.diag(mu)), X) # hessian matrix
n = self.conjugate_gradient(H, g, epsilon) #
w = (w - n) + (l2_reg * w
) # updating weights with l2 regularization
if save_weights:
W[i] = w
if save_weights:
self.W = W # weights of each iteration
self.w = w # final weights
def create_variables(self, X):
"""
Defining variables to make softmax function more convienient in the future
Parameters
----------
X : N x D matrix of 1 x D feature vectors
"""
try:
inv_sigma = la.inv(self.pooled_sigma)
except:
inv_sigma = la.pinv(self.pooled_sigma)
self.gamma = np.zeros(shape=self.priors.shape[0])
self.beta = np.zeros(shape=(self.gamma.shape[0], X.shape[1]))
for i in range(self.priors.shape[0]):
a = safedot(self.means[i], inv_sigma)
b = safedot(a, self.means[i]) / (-2)
self.gamma[i] = b + np.log(self.priors[i])
self.beta[i] = safedot(inv_sigma, self.means[i])
def get_multinomial_likelihood(self, X):
"""
Calculates the likelihood of each observation in X being in a
class given that the features in X are of a multinomial distribution
Parameters
----------
X : N x D matrix of 1 x D feature vectors
"""
ll = safedot(X, self.l_p_matrix.T)
l_prior = np.log(self.priors)
self.predictions = np.argmax(ll + l_prior, axis = 1)
def classify(self, X):
"""
Performs classification of observations depending on selected model
Parameters
----------
X : N x D matrix of 1 x D feature vectors
"""
if (self.model == 'Linear'):
n = safedot(X, self.beta.T) + self.gamma
self.predictions = np.argmax(self.softmax(n), axis=1)
elif (self.model == 'Quadratic' or self.model == 'Regularized'):
t_1 = (-0.5) * self.determinants
t_2 = (-0.5) * self.compute_mahalanobis(X)
self.predictions = np.argmax(t_1 + t_2, axis=1)
def score_irls(self, y, pi, S):
"""
Gives the working response after each update it irls algorithm
Parameters
----------
y : Actual label for each feature vector
pi : Prediction from [0,1], where closer proximity to 0 or 1 means it is more likely for
that feature vector to be either a 0 or 1 respectively
S : Diagonal matrix calculating from the decomposition of the Hessian
Returns
-------
error : response of predictions compared to actual values
"""
error = safedot(la.pinv(S), (y - pi))
return error
def convert_labels_binary(self, X, w, threshold=0.5):
"""
Converts values to 0 or 1 based on thresholding of expit function
Parameters
----------
X : N x D matrix composed of numerical features
w : Weights determined by chosen solver
of what class each observation belongs to
threshold : Some value in the range (0, 1) that decides the decision boundary
of what class each observation belongs to
Returns
-------
y: bit vector containing 0 or 1 depending on which class each belongs to
"""
n_pred = expit(safedot(X, w))
y = self.threshold(n_pred, threshold)
return y
def sigmoid(self, x, x_, **kwargs):
"""
Computes the sigmoid kernel of the two given vectors
Parameters
----------
x : D x 1 feature vector
x_ : D x 1 feature vector
alpha : scaling parameter
c = : shifting parameter
"""
if kwargs:
try:
alpha = kwargs['alpha']
c = kwargs['c']
except KeyError:
raise ValueError(
'Must use proper arguments for sigmoid kernel function.')
else:
alpha = 1e-10
c = 0.5
n = alpha * safedot(x, x_) + c
return np.tanh(n)
