To implement your code here spots

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 raise NotImplementedError() return wgrad, bgrad

def test_grad1(): X = np.random.rand(25,5) # generate n random vectors with d dimensions w = np.random.rand(5) # define a random weight vector b = np.random.rand(1) # define a bias y = (np.random.rand(25)>0.5)*2-1 # set labels all-(+1) wgrad, bgrad = gradient(X, y, w, b) # compute the gradient using your function return wgrad.shape == w.shape and np.isscalar(bgrad) def test_grad2(): X = np.random.rand(25,5) # generate n random vectors with d dimensions w = np.random.rand(5) # define a random weight vector b = np.random.rand(1) # define a bias y = (np.random.rand(25)>0.5)*2-1 # set labels all-(+1) wgrad, bgrad = gradient(X, y, w, b) # compute the gradient using your function wgrad2, bgrad2 = gradient_grader(X, y, w, b) # compute the gradient using ground truth return np.linalg.norm(wgrad - wgrad2)<1e-06 and np.linalg.norm(bgrad - bgrad2) < 1e-06 # test if they match def test_grad3(): X = np.random.rand(25,5) # generate n random vectors with d dimensions y = (np.random.rand(25)>0.5)*2-1 # set labels all-(+1) w = np.random.rand(5) # define a random weight vector b = np.random.rand(1) w_s = np.random.rand(5)*1e-05 # define tiny random step b_s = np.random.rand(1)*1e-05 # define tiny random step ll1 = log_loss(X,y,w+w_s, b+b_s) # compute log-likelihood after taking a step ll = log_loss(X,y,w,b) # use Taylor's expansion to approximate new loss with gradient wgrad, bgrad =gradient(X,y,w,b) # compute gradient ll2=ll+ wgrad @ w_s + bgrad * b_s # take linear step with Taylor's approximation return np.linalg.norm(ll1-ll2)<1e-05 # test if they match def test_grad4(): w1, b1, losses1 = logistic_regression_grader(features, labels, 1000, 1e-03, gradient) w2, b2, losses2 = logistic_regression_grader(features, labels, 1000, 1e-03) return(np.abs(losses1[-1]-losses2[-1])<0.1) runtest(test_grad1, 'test_grad1') runtest(test_grad2, 'test_grad2') runtest(test_grad3, 'test_grad3') runtest(test_grad4, 'test_grad4')

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 raise NotImplementedError() 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 :-)

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]