CVX Log2 Approximation

cvx_log2.py

import matplotlib.pyplot as plt
import numpy as np
import cvxpy as cp

# broke the degree parameter for the time being.
def get_cvx(fn, deg, lo, hi, rel=True, m=500):
    # create an array of x-values
    x = np.linspace(lo, hi, m)

    # compute the true y-values
    y = fn(x)
    y_recip = 1 / y

    # create a vector of coefficients for our polynomial
    A = cp.Variable()
    B = cp.Variable()

    # returns an array: [x**deg, x**(deg - 1), ... x]
    def X(x):
        return [x**i for i in range(deg, 0, -1)]
    
    # create the objective
    obj = 0
    
    # this is wrong
    #for i in range(m):
    #    expr = A @ X(x[i] - 1.0) - y[i]
    #    if rel: expr /= y[i]
    #    obj += cp.max(cp.abs(expr))
    
    # not super tidy, but I think this works correctly
    if rel:
        obj = cp.max( cp.abs( cp.multiply((A * (x - 1)**2) + (B * (x - 1)) - y, y_recip) ) )
    else:
        obj = cp.max( cp.abs( ((A * (x - 1)**2) + (B * (x - 1)) - y) ) ) 

    problem = cp.Problem(cp.Minimize(obj), [])
    problem.solve()

    return [A.value, B.value]


def model(x, a, b):
    return a * (x - 1.0)**2 + b * (x - 1.0)

u = get_cvx(np.log2, 2, 0.75, 1.5, rel=True)
v = get_cvx(np.log2, 2, 0.75, 1.5, rel=False)

x = np.arange(0.75, 1.5, 0.01)
x = x[np.abs(x - 1.0) > 0.001]
y = np.log2(x)

errf_cvx_mine_rel = (model(x, u[0], u[1]) - y) / y
errf_cvx_mine_abs = (model(x, v[0], v[1]) - y) / y
errf_cvx_ebay = (model(x, -0.629673, 1.466967) - y) / y

plt.plot(x, errf_cvx_mine_rel, label="CVX Mine, Rel.")
plt.plot(x, errf_cvx_mine_abs, label="CVX Mine, Abs.")
plt.plot(x, errf_cvx_ebay, label="CVX Ebay")

plt.xlabel("x")
plt.ylabel("rel. err")

plt.legend(loc="upper left")
plt.show()