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implementations.py
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implementations.py
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import numpy as np
# Gradient based methods for linear systems
def compute_mse(y, tx, w):
"""
Compute the Mean Square Error as defined in class.
Takes as input the targeted y, the sample matrix X and the feature fector w.
"""
e = y - tx@w
mse = e.T.dot(e) /(2*len(e))
return mse
def least_squares(y, tx):
"""
Compute an esimated solution of the problem y = tx @ w, and the associated error. This method is equivalent
to the minimization problem of finding w such that |y-tx@w||^2 is minimal. Note that this methods provides the global optimum.
The error is the mean square error of the targeted y and the solution produced by the least square function.
Takes as input the targeted y, and the sample matrix X.
"""
w = np.linalg.solve (tx.T.dot(tx),tx.T.dot(y))
mse = compute_mse(y, tx, w)
return w, mse
def least_squares_gradient(y, tx, w):
"""
Compute the gradient of the mean square error with respect to w, and the current error vector e.
Takes as input the targeted y, the sample matrix w and the feature vector w.
This function is used when solving gradient based method, such that least_squares_GD() and least_squares_SGD().
"""
e = y - tx.dot(w)
grad = -tx.T.dot(e) / len(e)
return grad, e
def least_squares_GD(y, tx, initial_w=None, max_iters=50, gamma=0.1):
"""
Compute an estimated solution of the problem y = tx @ w and the associated error using Gradient Descent.
This method is equivalent to the minimization problem of finding w such that |y-tx@w||^2 is minimal. Note that
this method may output a local minimum, while least_squares() provides the global minimum.
Takes as input:
* the targeted y
* the sample matrix w
* the initial guess for w, by default set as a vector of zeros
* the number of iterations for Gradient Descent
* the learning rate gamma
"""
# Define parameters to store w and loss
if np.all(initial_w == None): initial_w = np.zeros(tx.shape[1])
ws = [initial_w] # Initial guess w0 generated randomly
losses = []
w = ws[0]
for n_iter in range(max_iters):
# compute loss, gradient
grad, err = least_squares_gradient(y, tx, w)
loss = compute_mse(y,tx,w)
# gradient w by descent update
w = w - gamma * grad
# store w and loss
ws.append(w)
losses.append(loss)
#if (n_iter % int(max_iters/5)) == 0:
#print("Gradient Descent({bi}/{ti}): loss={l}".format(bi=n_iter, ti=max_iters,l=loss))
return w,loss
def batch_iter(y, tx, batch_size, num_batches=1, shuffle=True):
"""
Generate a minibatch iterator for a dataset.
Takes as input two iterables (here the output desired values 'y' and the input data 'tx')
Outputs an iterator which gives mini-batches of `batch_size` matching elements from `y` and `tx`.
Data can be randomly shuffled to avoid ordering in the original data messing with the randomness of the minibatches.
Example of use :
for minibatch_y, minibatch_tx in batch_iter(y, tx, 32):
<DO-SOMETHING>
"""
data_size = len(y)
if shuffle:
shuffle_indices = np.random.permutation(np.arange(data_size))
shuffled_y = y[shuffle_indices]
shuffled_tx = tx[shuffle_indices]
else:
shuffled_y = y
shuffled_tx = tx
for batch_num in range(num_batches):
start_index = batch_num * batch_size
end_index = min((batch_num + 1) * batch_size, data_size)
if start_index != end_index:
yield shuffled_y[start_index:end_index], shuffled_tx[start_index:end_index]
def least_squares_SGD(y, tx, initial_w=None, batch_size=1, max_iters=50, gamma=0.00005):
"""
Compute an estimated solution of the problem y = tx @ w and the associated error using Stochastic Gradient Descent.
Takes as input:
* the targeted y
* the sample matrix w
* the initial guess for w, by default set as a vector of zeros
* the batch_size, which is the number of samples on which the new gradient is computed. If set to 1 it corresponds
to Stochastic Gradient Descent, to the full number of samples it is identifical to least_squares_GD().
* the number of iterations for Gradient Descent
* the learning rate gamma
"""
# Define parameters to store w and loss
if np.all(initial_w == None): initial_w = np.zeros(tx.shape[1])
losses = []
w = initial_w
for n_iter in range(max_iters):
for y_batch, tx_batch in batch_iter(y, tx, batch_size=batch_size, num_batches=1):
# compute a stochastic gradient and loss
grad, _ = least_squares_gradient(y_batch, tx_batch, np.array(w))
# update w through the stochastic gradient update
w = w - gamma * grad
# calculate loss
loss = compute_mse(y, tx, w)
# store w and loss
losses.append(loss)
#if n_iter % int(max_iters/5) == 0:
#print("SGD({bi}/{ti}): loss={l}".format(bi=n_iter, ti=max_iters - 1, l=loss))
return w,loss
# RIDGE REGRESSION
def ridge_regression(y, tx, lambda_):
"""
Compute an esimated solution of the problem y = tx @ w , and the associated error. Note that this method
is a variant of least_square() but with an added regularization term lambda_.
This method is equivalent to the minimization problem of finding w such that |y-tx@w||^2 + lambda_*||w||^2 is minimal.
The error is the mean square error of the targeted y and the solution produced by the least square function.
Takes as input the targeted y, the sample matrix X and the regulariation term lambda_.
"""
x_t = tx.T
lambd = lambda_ * 2 * len(y)
w = np.linalg.solve (np.dot(x_t, tx) + lambd * np.eye(tx.shape[1]), np.dot(x_t,y))
loss = compute_mse(y, tx, w)
return w,loss
#LOGISTIC REGRESSION
def sigmoid(t):
"""
Apply the sigmoid function on t.
"""
return np.exp(t)/(1+np.exp(t))
def calculate_loss(y, tx, w):
"""
Compute the negative log likelihood as defined in class.
Takes as input the targeted y, the sample matrix X and the feature fector w.
"""
return np.sum(np.log(1+np.exp(tx@w))-y*(tx@w))
def calculate_gradient(y, tx, w):
"""
Compute the gradient of the negative log likelihood with respect to w, and the current error vector e.
Takes as input the targeted y, the sample matrix w and the feature vector w.
This function is used when solving gradient based method, such that logistic_regression() and reg_logistic_regression().
"""
return tx.T@(sigmoid(tx@w)-y)
def learning_by_gradient_descent(y, tx, w, gamma):
"""
Compute one step of gradient descent for logistic regression.
Takes as input the targeted y, the sample matrix w, the feature w and the learning rate gamma.
Return the feature vector w and the error defined as the negative log likelihood.
"""
loss = calculate_loss(y,tx,w)
grad = calculate_gradient(y,tx,w)
w = w-gamma*grad
return w, loss
def logistic_regression(y, tx, initial_w=None, max_iters=100, gamma=0.009, batch_size=1):
"""
Compute an estimated solution of the problem y = sigmoid(tx @ w) and the associated error using Gradient Descent.
This method is equivalent to the minimization problem of finding w such that the negative log likelihood is minimal. Note that
this method may output a local minimum.
Takes as input:
* the targeted y
* the sample matrix w
* the initial guess for w, by default set as a vector of zeros
* the number of iterations for Stochastic Gradient Descent
* the learning rate gamma
* the batch_size, which is the number of samples on which the new gradient is computed. If set to 1 it corresponds
to Stochastic Gradient Descent, to the full number of samples it is Gradient Descent.
"""
# init parameters
if np.all(initial_w == None): initial_w = np.zeros(tx.shape[1])
threshold = 1e-8
losses = []
y = (1 + y) / 2
# build tx
w = initial_w
# start the logistic regression
for i in range(max_iters):
# get loss and update w.
for y_batch, tx_batch in batch_iter(y, tx, batch_size=batch_size, num_batches=1):
w, _ = learning_by_gradient_descent(y_batch, tx_batch, w, gamma)
# converge criterion
losses.append(calculate_loss(y,tx,w))
if len(losses) > 1 and np.abs(losses[-1] - losses[-2]) < threshold:
break
#if i % int(max_iters/5) == 0:
#print(losses[-1],i,'/{tot}'.format(tot=max_iters))
return w,losses[-1]
# Regularized LOGISTIC REGRESSION
def learning_by_penalized_gradient_descent(y, tx, w, gamma, lambda_):
"""
Compute one step of gradient descent for regularized logistic regression.
Takes as input the targeted y, the sample matrix w, the feature w and the learning rate gamma.
Return the feature vector w and the error defined as the negative log likelihood.
"""
loss = calculate_loss(y, tx, w) + lambda_ * np.squeeze(w.T.dot(w))
grad = calculate_gradient(y, tx, w) + 2 * lambda_ * w
w = w-gamma*grad
return w, loss
def reg_logistic_regression(y, tx, lambda_, initial_w=None, max_iters=100, gamma=0.009, batch_size=1):
"""
Compute an estimated solution of the problem y = sigmoid(tx @ w) and the associated error using Gradient Descent.
Note that this method is a variant of logistic_regression() but with an added regularization term lambda_.
This method is equivalent to the minimization problem of finding w such that the negative log likelihood is minimal. Note that
this method may output a local minimum.
Takes as input:
* the targeted y
* the sample matrix w
* the initial guess for w, by default set as a vector of zeros
* the number of iterations for Stochastic Gradient Descent
* the learning rate gamma
* the batch_size, which is the number of samples on which the new gradient is computed. If set to 1 it corresponds
to Stochastic Gradient Descent, to the full number of samples it is Gradient Descent.
"""
# init parameters
if np.all(initial_w == None): initial_w = np.zeros(tx.shape[1])
threshold = 1e-8
losses = []
y = (1 + y) / 2
# build tx
w = initial_w
# start the logistic regression
for iter in range(max_iters):
# get loss and update w.
for y_batch, tx_batch in batch_iter(y, tx, batch_size=batch_size, num_batches=1):
w, loss = learning_by_penalized_gradient_descent(y_batch, tx_batch, w, gamma, lambda_)
# converge criterion
loss = calculate_loss(y, tx, w) + lambda_ * np.squeeze(w.T.dot(w))
losses.append(loss)
if len(losses) > 1 and np.abs(losses[-1] - losses[-2]) < threshold:
break
#if iter % int(max_iters/5) == 0:
#print(losses[-1],iter,'/{tot}'.format(tot=max_iters))
return w,losses[-1]