Ex. 2

# define function def f(x1, x2): ''' :param x1: value for x1. :param x2: value for x2. :return: rosenbrock value for the given x1, x2. ''' return (1 - x1)**2 + 100 * (x2 - x1**2)**2

def y_per_steps(list_x1, list_x2, opt_func): ''' Function to plot the values of y for each step. :param list_x1: list of all x1 values in the path. :param list_x2: list of all x2 values in the path. :param opt_func: the function being analysed. ''' steps = list(range(len(list_x1))) y_values = opt_func(np.array(list_x1), np.array(list_x2)) plt.plot(steps, y_values) plt.xlabel('Number of updates') plt.ylabel('f(x1, x2)')

%matplotlib inline import numpy as np import matplotlib.pyplot as plt from matplotlib.colors import LogNorm def plot_function(list_x1, list_x2, opt_func): ''' Function to plot the optimization path of a function. :param list_x1: list of all x1 values in the path. :param list_x2: list of all x2 values in the path. :param opt_func: the function being analysed. ''' xmin, xmax, xstep = -2., 2, .1 ymin, ymax, ystep = -2., 2, .1 x1, x2 = np.meshgrid(np.arange(xmin, xmax + xstep, xstep), np.arange(ymin, ymax + ystep, ystep)) y = opt_func(x1, x2) minima = np.array([1., 1.]).reshape(-1, 1) list_x1 = np.array(list_x1).reshape(-1,1) list_x2 = np.array(list_x2).reshape(-1,1) fig, ax = plt.subplots(figsize=(15, 9)) ax.contour(x1, x2, y, levels=np.logspace(0, 5, 10), norm=LogNorm(), cmap=plt.cm.jet) ax.quiver(list_x1[:-1], list_x2[:-1], list_x1[1:]-list_x1[:-1], list_x2[1:]-list_x2[:-1], scale_units='xy', angles='xy', scale=1, color='k') ax.plot(*minima, 'r*', markersize=18) ax.set_xlabel('$x_1$') ax.set_ylabel('$x_2$') ax.set_xlim((xmin, xmax)) ax.set_ylim((ymin, ymax))

def derivative_function(x1,x2): grad_x1 = -2 * (1 - x1) - 400 * x1 * (x2 - (x1)**2) grad_x2 = 200 * (x2 - (x1)**2) return grad_x1, grad_x2

def gradient_descent(initial_x1, initial_x2, convergence_test, lr, lr_red=False): ''' Function to compute the gradient descent. :param initial_x1: initial value for x1. :param initial_x2: initial value for x2. :param convergence_test: tolerance value. :param lr: learning rate. :param lr_red: flag to inform if learning rate will change among steps. :return: func returns the final values for x1,x2 and a list of all x1 and x2 values in the optimization path. ''' x1 = initial_x1 x2 = initial_x2 x1_list = [x1] x2_list = [x2] diff_x = 1e+10 step = 0 # print(f'Step: {step} - f(x1, x2): {f(x1, x2)}') while diff_x > convergence_test and step <= 50000: try: grad_x1, grad_x2 = derivative_function(x1,x2) x1_prev = x1 x2_prev = x2 x1 = x1_prev - (grad_x1 * lr) x2 = x2_prev - (grad_x2 * lr) f_x_old = f(x1_prev, x2_prev) f_x_new = f(x1, x2) diff_x = abs(f_x_new - f_x_old) step += 1 # print(f'Step: {step} - f(x1, x2): {f(x1, x2)}') if lr_red: lr = lr * 0.999 x1_list.append(x1) x2_list.append(x2) except OverflowError: print(f'Número de atualizações : {step}') print(f'Valor final do par (x1, x2) : ({x1}, {x2})') return x1, x2, x1_list, x2_list print(f'Número de atualizações : {step}') print(f'Valor final do par (x1, x2) : ({x1}, {x2})') print(f'Valor final da função f(x1, x2) : ({f(x1, x2)})') return x1, x2, x1_list, x2_list

x1, x2, list_x1, list_x2 = gradient_descent(0, 0, 1e-5, 1e-3)

y_per_steps(list_x1, list_x2, f)

plot_function(list_x1, list_x2, f)

x1, x2, list_x1, list_x2 = gradient_descent(0, 0, 1e-5, 1e-4)

y_per_steps(list_x1, list_x2, f)

plot_function(list_x1, list_x2, f)

x1, x2, list_x1, list_x2 = gradient_descent(0, 0, 1e-5, 1e-2)

y_per_steps(list_x1, list_x2, f)

plot_function(list_x1, list_x2, f)

x1, x2, list_x1, list_x2 = gradient_descent(0, 0, 1e-5, 5e-3, lr_red=True)

y_per_steps(list_x1, list_x2, f)

plot_function(list_x1, list_x2, f)

import tensorflow as tf

def reset(initial_value=0.0): ''' Function to reset the variables for a initial value. :param initial_value: value to be set as initial in reset method. :return: the x1, x2 value as tf variable. ''' x1 = tf.Variable(initial_value) x2 = tf.Variable(initial_value) return x1, x2

def gradient_descent_tf(convergence_test, lr): ''' Function to compute the gradient descent using tensorflow. :param convergence_test: tolerance value :param lr: learning rate :return: func returns the list of all x1 and x2 values in the optimization path ''' lr = tf.Variable(lr) x1, x2 = reset() opt = tf.keras.optimizers.SGD(learning_rate=lr) list_x1 = [x1.numpy()] list_x2 = [x2.numpy()] diff_x = 1e+10 step = 0 f_x_new = 1e+10 while diff_x > convergence_test and step <= 50000: f_x_old = f(x1, x2) with tf.GradientTape(persistent=True) as tape: tape.watch(x1) tape.watch(x2) y = f(x1, x2) grads = tape.gradient(y, [x1, x2]) processed_grads = [g for g in grads] x1 = x1 - processed_grads[0] * lr x2 = x2 - processed_grads[1] * lr list_x1.append(x1.numpy()) list_x2.append(x2.numpy()) f_x_new = f(x1, x2) diff_x = abs(f_x_new - f_x_old) step += 1 print(f'Número de atualizações : {step}') print(f'Valor final do par (x1, x2) : ({x1.numpy()}, {x2.numpy()})') return list_x1, list_x2

list_x1, list_x2 = gradient_descent_tf(convergence_test=1e-5, lr=1e-3)

y_per_steps(list_x1, list_x2, f)

plot_function(list_x1, list_x2, f)