Write a function to find the maximum value of: 𝑓(𝑥) = 𝑥−5 + sqrt(4−𝑥^2) by applying gradient descent: 𝑥𝑡+1=𝑥𝑡−𝜂𝑓′(𝑥𝑡)

Question

The return value of x in f_gradient_descent(0, 0.01, 1000) is not -2.17 (max). Would you please have a look? Thank you so much!

import math

def f(x):
    return x-5 + math.sqrt(4-x**2)
 
def f_derivative(x):
    return 1 + 1/(2*math.sqrt(4-x**2))*(-2*x)

def f_gradient_descent(x0, eta, n_step):
"""
    Parameter:
        x0: Start point
        eta: Learning rate
        n_step: algorithm will stop after `n_step` cycle
"""
    x0 = 1.23
    eta = 0.001
    for _ in range(n_step):
        x0 = x0 - eta * f_derivative(x0)
        if abs(f_derivative(x0)) < 0.00001:
            break
    return f(x0)

assert f_gradient_descent(0, 0.01, 1000) - (-2.17) < 1e-4

Asked By: Chris

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Answer 1

There are quite a few mistakes in the code.

To arrive at the peak you have to ascend, not descend. So you have to increase x0 if the derivative is positive.
The values for x0 and eta that are passed to the function are ignored because they are overwritten inside. eta = 0.001 might be too small to arrive at the maximum within 1000 steps.
In order to evaluate the result, you have to use the absolute value of the difference between the actual and the expected result.
The actual maximum is -2.1716…, not -2.17, which is more than 1e-4 away.

Answered By: mkrieger1

Write a function to find the maximum value of: 𝑓(𝑥) = 𝑥−5 + sqrt(4−𝑥^2) by applying gradient descent: 𝑥𝑡+1=𝑥𝑡−𝜂𝑓′(𝑥𝑡)

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