Need help with the following: see log_loss and gradient at the end for reference

Part Five: Weight Update of Gradient Ascent [Graded]

Write code below to implement the weight update of gradient descent on the log_loss function. Hint: use the gradient and log_loss functions from above. Please use a constant learning rate throughout (i.e. do not decrease the learning rate).

def logistic_regression(X, y, max_iter, alpha):

 n, d = X.shape

 w = np.zeros(d)

 b = 0.0

 losses = np.zeros(max_iter) 

 for step in range(max_iter):

 # YOUR CODE HERE

 return w, b, losses

weight, b, losses = logistic_regression(features, labels, 1000, 1e-04)

plot(losses)

xlabel('iterations')

ylabel('log_loss')

# your loss should go down :-)

#code must pass the follwingTesting

def test_logistic_regression1():

 XUnit = np.array([[-1,1],[-1,0],[0,-1],[-1,2],[1,-2],[1,-1],[1,0],[0,1],[1,-2],[-1,2]])

 YUnit = np.hstack((np.ones(5), -np.ones(5)))

 w1, b1, _ = logistic_regression(XUnit, YUnit, 30000, 5e-5)

 w2, b2, _ = logistic_regression_grader(XUnit, YUnit, 30000, 5e-5)

 return (np.linalg.norm(w1 - w2) < 1e-5) and (np.linalg.norm(b1 - b2) < 1e-5)

def test_logistic_regression2():

 X = np.vstack((np.random.randn(50, 5), np.random.randn(50, 5) + 2))

 Y = np.hstack((np.ones(50), -np.ones(50)))

 max_iter = 300

 alpha = 1e-5

 w1, b1, _ = logistic_regression(X, Y, max_iter, alpha)

 w2, b2, _ = logistic_regression_grader(X, Y, max_iter, alpha)

 return (np.linalg.norm(w1 - w2) < 1e-5) and (np.linalg.norm(b1 - b2) < 1e-5)

def test_logistic_regression3(): # check if losses match predictions

 X = np.vstack((np.random.randn(50, 5), np.random.randn(50, 5) + 2))

 Y = np.hstack((np.ones(50), -np.ones(50)))

 max_iter = 30

 alpha = 1e-5

 w1, b1, losses1 = logistic_regression(X, Y, max_iter, alpha)

 return np.abs(log_loss(X,Y,w1,b1)-losses1[-1])<1e-09

def test_logistic_regression4(): # check if loss decreases

 X = np.vstack((np.random.randn(50, 5), np.random.randn(50, 5) + 2))

 Y = np.hstack((np.ones(50), -np.ones(50)))

 max_iter = 30

 alpha = 1e-5

 w1, b1, losses1 = logistic_regression(X, Y, max_iter, alpha)

 return losses[-1] 
 
runtest(test_logistic_regression1, 'test_logistic_regression1')
 
runtest(test_logistic_regression2, 'test_logistic_regression2')
 
runtest(test_logistic_regression3, 'test_logistic_regression3')
 
runtest(test_logistic_regression4, 'test_logistic_regression4')
 
#Reference Log loss and gradient, already solved below
 
def log_loss(X, y, w, b=0): # Input: # X: nxd matrix # y: n-dimensional vector with labels (+1 or -1) # w: d-dimensional vector # Output: # a scalar assert np.sum(np.abs(y))==len(y) # check if all labels in y are either +1 or -1 predicted = y_pred(X, w, b) nll = -(1/2)*np.sum((y+1)*np.log(predicted)-(y-1)*np.log(1-predicted)) return nll
 
Part Four: Compute Gradient [Graded]
 
Now, verify that the gradient of the log-loss with respect to the weight vector is:
 
 (,,,)==1((+)).NLL(X,y,w,b)w=i=1nyi(yi(wxi+b))xi.
 
 
 
 (,,,)==1((+)).NLL(X,y,w,b)b=i=1nyi(yi(wxi+b)).
 
Implement the function gradient which returns the first derivative with respect to w, b for a given X, y, w, b.
 
Hint: remember that you derived earlier that ()=()(1())(z)=(z)(1(z)).
 
def gradient(X, y, w, b):
 
 # Input:
 
 # X: nxd matrix
 
 # y: n-dimensional vector with labels (+1 or -1)
 
 # w: d-dimensional vector
 
 # b: a scalar bias term
 
 # Output:
 
 # wgrad: d-dimensional vector with gradient
 
 # bgrad: a scalar with gradient
 
 
 
 n, d = X.shape
 
 wgrad = np.zeros(d)
 
 bgrad = 0.0
 
 # YOUR CODE HERE
 
 m = X.shape[1]
 
 A = sigmoid(np.multiply((-1)*y,(np.dot(X,w) + b))) 
 
 A = A.reshape(n,1) 
 
 y = y.reshape(n,1) 
 
 wgrad = (1/m)*np.sum(np.dot(np.dot(X.T,A),(-1)*y.T), axis=1)
 
 bgrad = (1/m)*np.sum((np.multiply(-y,A)))
 
 return wgrad, bgrad