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| #%% gradient descent | |
| # *********************************************************** | |
| def run_gradient_descent(func, initial_point, gradients, verbose=False): | |
| """ | |
| Simple implementation of gradient descent for a convex function of 2 arguments. | |
| Arguments: | |
| func: the function of 2 variables to minimize | |
| initial_point: a 2D tuple representing the initial point | |
| gradients: a set of 2 functions, representing the partial derivatives of func | |
| with respect to each input parameter | |
| verbose: to print intermediate results in the console | |
| Returns: | |
| results: a dictionary containing the best solution found | |
| along with the intermediate points evaluated. | |
| """ | |
| k = 0 # iteration number | |
| x_k = initial_point[0] | |
| y_k = initial_point[1] | |
| epsilon = -0.01 # stopping criterion; small negative value for minimization problems | |
| alpha = 0.01 # step size | |
| results = { 'Ks': [], 'Xs': [], 'Ys': [], 'Fs': [] } | |
| improvement = epsilon | |
| # repeat until convergence | |
| while improvement <= epsilon: | |
| k += 1 | |
| # evaluate function | |
| f_k = func(x_k, y_k) | |
| if k>1: | |
| improvement = f_k - results['Fs'][-1] | |
| # store statistics | |
| results['Ks'].append(k) | |
| results['Xs'].append(x_k) | |
| results['Ys'].append(y_k) | |
| results['Fs'].append(f_k) | |
| if verbose: | |
| print('iteration k={};\t f={};\t point (x={}, y={})'.format(k,f_k,x_k,y_k)) | |
| # follow gradient and move to next point | |
| x_k -= alpha * gradients[0](x_k, y_k) | |
| y_k -= alpha * gradients[1](x_k, y_k) | |
| # function has stopped improving | |
| results['optimal_solution'] = (results['Xs'][-1], results['Ys'][-1]) | |
| results['optimal_cost'] = results['Fs'][-1] | |
| if verbose: | |
| print('gradient descent finished' | |
| '; result=', results['optimal_solution'], | |
| '; cost=', results['optimal_cost']) | |
| return results | |
| # example | |
| from mpl_toolkits.mplot3d import Axes3D | |
| import matplotlib.pyplot as plt | |
| import numpy as np | |
| import pandas as pd | |
| def f(x, y): | |
| return x**2 + y**2 - np.log(x-2) + 10 - np.log(y-2) | |
| def grad_x(x, y): | |
| return 2*x - 1/(x-2) | |
| def grad_y(x, y): | |
| return 2*y - 1/(y-2) | |
| solution = run_gradient_descent(f, (3.5, 3.5), (grad_x, grad_y), True) | |
| # plot | |
| x = np.arange(2.001, 4, 1/100.) | |
| y = np.arange(2.001, 4, 1/100.) | |
| X, Y = np.meshgrid(x, y) | |
| zs = np.array(f(np.ravel(X), np.ravel(Y))) | |
| Z = zs.reshape(X.shape) | |
| fig = plt.figure(figsize=(14, 5), dpi=100) | |
| ax = fig.add_subplot(121, projection="3d") | |
| ax.plot(solution['Xs'], solution['Ys'], [x for x in solution['Fs']], marker='o', color='#f89406', zorder=2) | |
| ax.plot_surface(X, Y, Z, cmap=plt.cm.viridis, alpha=0.5, zorder=1) | |
| plt.title(r'$f(x,y)=x^2+y^2+10-log(x-2)-log(y-2)$') | |
| ax = fig.add_subplot(122) | |
| plt.title('value of f(x,y) at each iteration') | |
| ax.scatter(solution['Ks'], solution['Fs'], color='#f89406') | |
| ax.set_xlabel('iteration k') | |
| plt.show() |
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