代码未收敛香草梯度下降

Question

I have a specific analytical gradient I am using to calculate my cost f(x,y), and gradients dx and dy. It runs, but I can't tell if my gradient descent is broken. Should I plot my partial derivatives x and y?

import math

gamma = 0.00001 # learning rate
iterations = 10000 #steps
theta = np.array([0,5]) #starting value
thetas = []
costs = []

# calculate cost of any point
def cost(theta):
    x = theta[0]
    y = theta[1]
    return 100*x*math.exp(-0.5*x*x+0.5*x-0.5*y*y-y+math.pi)

def gradient(theta):
    x = theta[0]
    y = theta[1]
    dx = 100*math.exp(-0.5*x*x+0.5*x-0.0035*y*y-y+math.pi)*(1+x*(-x + 0.5))
    dy = 100*x*math.exp(-0.5*x*x+0.5*x-0.05*y*y-y+math.pi)*(-y-1)
    gradients = np.array([dx,dy])
    return gradients

#for 2 features
for step in range(iterations):
    theta = theta - gamma*gradient(theta)
    value = cost(theta)
    thetas.append(theta)
    costs.append(value)

thetas = np.array(thetas)
X = thetas[:,0]
Y = thetas[:,1]
Z = np.array(costs)

iterations = [num for num in range(iterations)]

plt.plot(Z)
plt.xlabel("num. iteration")
plt.ylabel("cost")

Answer 1

I strongly recommend you check whether or not your analytic gradient is working correcly by first evaluating it against a numerical gradient. Ie make sure that your f'(x) = (f(x+h) - f(x)) / h for some small h.

After that, make sure your updates are actually in the right direction by picking a point where you know x or y should decrease and then checking the sign of your gradient function output.

Of course make sure your goal is actually minimization vs maximization.