Natural Evolution Strategies (NES) toy example that optimizes a quadratic function

nes.py

"""
A bare bones examples of optimizing a black-box function (f) using
Natural Evolution Strategies (NES), where the parameter distribution is a 
gaussian of fixed standard deviation.
"""

import numpy as np
np.random.seed(0)

# the function we want to optimize
def f(w):
  # here we would normally:
  # ... 1) create a neural network with weights w
  # ... 2) run the neural network on the environment for some time
  # ... 3) sum up and return the total reward

  # but for the purposes of an example, lets try to minimize
  # the L2 distance to a specific solution vector. So the highest reward
  # we can achieve is 0, when the vector w is exactly equal to solution
  reward = -np.sum(np.square(solution - w))
  return reward

# hyperparameters
npop = 50 # population size
sigma = 0.1 # noise standard deviation
alpha = 0.001 # learning rate

# start the optimization
solution = np.array([0.5, 0.1, -0.3])
w = np.random.randn(3) # our initial guess is random
for i in range(300):

  # print current fitness of the most likely parameter setting
  if i % 20 == 0:
    print('iter %d. w: %s, solution: %s, reward: %f' % 
          (i, str(w), str(solution), f(w)))

  # initialize memory for a population of w's, and their rewards
  N = np.random.randn(npop, 3) # samples from a normal distribution N(0,1)
  R = np.zeros(npop)
  for j in range(npop):
    w_try = w + sigma*N[j] # jitter w using gaussian of sigma 0.1
    R[j] = f(w_try) # evaluate the jittered version

  # standardize the rewards to have a gaussian distribution
  A = (R - np.mean(R)) / np.std(R)
  # perform the parameter update. The matrix multiply below
  # is just an efficient way to sum up all the rows of the noise matrix N,
  # where each row N[j] is weighted by A[j]
  w = w + alpha/(npop*sigma) * np.dot(N.T, A)

# when run, prints:
# iter 0. w: [ 1.76405235  0.40015721  0.97873798], solution: [ 0.5  0.1 -0.3], reward: -3.323094
# iter 20. w: [ 1.63796944  0.36987244  0.84497941], solution: [ 0.5  0.1 -0.3], reward: -2.678783
# iter 40. w: [ 1.50042904  0.33577052  0.70329169], solution: [ 0.5  0.1 -0.3], reward: -2.063040
# iter 60. w: [ 1.36438269  0.29247833  0.56990397], solution: [ 0.5  0.1 -0.3], reward: -1.540938
# iter 80. w: [ 1.2257328   0.25622233  0.43607161], solution: [ 0.5  0.1 -0.3], reward: -1.092895
# iter 100. w: [ 1.08819889  0.22827364  0.30415088], solution: [ 0.5  0.1 -0.3], reward: -0.727430
# iter 120. w: [ 0.95675286  0.19282042  0.16682465], solution: [ 0.5  0.1 -0.3], reward: -0.435164
# iter 140. w: [ 0.82214521  0.16161165  0.03600742], solution: [ 0.5  0.1 -0.3], reward: -0.220475
# iter 160. w: [ 0.70282088  0.12935569 -0.09779598], solution: [ 0.5  0.1 -0.3], reward: -0.082885
# iter 180. w: [ 0.58380424  0.11579811 -0.21083135], solution: [ 0.5  0.1 -0.3], reward: -0.015224
# iter 200. w: [ 0.52089064  0.09897718 -0.2761225 ], solution: [ 0.5  0.1 -0.3], reward: -0.001008
# iter 220. w: [ 0.50861791  0.10220363 -0.29023563], solution: [ 0.5  0.1 -0.3], reward: -0.000174
# iter 240. w: [ 0.50428202  0.10834192 -0.29828744], solution: [ 0.5  0.1 -0.3], reward: -0.000091
# iter 260. w: [ 0.50147991  0.1044559  -0.30255291], solution: [ 0.5  0.1 -0.3], reward: -0.000029
# iter 280. w: [ 0.50208135  0.0986722  -0.29841024], solution: [ 0.5  0.1 -0.3], reward: -0.000009

I didnt run this code but it seems there is a minor error

def f(w):
  # here we would normally:
  # ... 1) create a neural network with weights w
  # ... 2) run the neural network on the environment for some time
  # ... 3) sum up and return the total reward

  # but for the purposes of an example, lets try to minimize
  # the L2 distance to a specific solution vector. So the highest reward
  # we can achieve is 0, when the vector w is exactly equal to solution
  reward = -np.sum(np.square(solution - w))
  return reward

there is no solution variable passing in this function

@potholiday the solution is a global variable, there is no need to pass it to the function.

@xombelle thanks you are right and still now I thought variable inside a function is treated as local. Now I have to rewrite all of my previous codes.

I just wondering, if I know solution, why I need to calculate w? Just assign like this w = solution, DONE?

@refactor I think it's just a example to introduce Natural Evolution Strategies. This program uses solution in the f function, but f is always calculated without knowing solution in real problem.

@Thinginitself Got it. So I guess the 'solution vector' is desired outputs from an ideal neural network, solution is NOT weights. the example f function use it to measure the distance from imperfect neural network, just for demo.

Thanks! This is really useful as it's a simple and intuitive way of doing RL. One small issue is that A = (R - np.mean(R)) / np.std(R) will result in a division by 0 when all R's are equal which can be somewhat common for simpler problems.

Why do we standardize the reward? The algorithm in the paper does not standardize the reward and looks right. I understand that by standardizing we move in the opposite direction to those weights that yielded rewards less than the mean reward. But this is not included in the paper where the derivation of the gradient is transparent with rigor and no standardization?

Section 3.2 in the paper shows how not standardizing is actually equivalent to standardizing

@siddpr It might just be a convenience here to make the code work faster. I tried with and without standardizing, and the standardized version runs faster (though both work). Intuitively, this is because we get more controlled rewards about zero which help to quickly drive weights to appropriate directions. If a weight is negative, but has to be positive, and our rewards are all negative, the best we can do is to multiply the change by zero to keep it unchanged. Beyond that, standardizing is just generally useful in many cases.

tried to understand NES paper, but not fully got it, as i understand it may be good for stable solution.
just played with little change to update solution each iteration, seems its very sensitive to too big moves of solution (animal is running away too quickly), but with small move is able still to catch solution

with too quickly moving solution, the normalization didnt helped, and the target was quickly lost
with A=R, was still able to catch somehow moving target

--# lets move the solution, as imitation of moving target
moving_target = np.random.randint(0,3); # values from 0 till 2
solution_jitter = 1+moving_target * alpha/2.5
solution = solution*solution_jitter

iter 0. w: [ 1.76405235 0.40015721 0.97873798], solution: [10 3 -2], reward: -83.462896
iter 260. w: [ 4.07337088 1.17450879 0.15430971], solution: [ 11.12672453 3.33801736 -2.22534491], reward: -60.093323
iter 1000. w: [ 10.78848973 3.22619184 -1.88405165], solution: [ 15.08420061 4.52526018 -3.01684012], reward: -21.423920
iter 2980. w: [ 29.06941533 8.73460539 -5.81538955], solution: [ 32.50510382 9.75153115 -6.50102076], reward: -13.308184

Why is sigma multiplied for the perturbations and divided for the update?

w_try = w + sigma*N[j] # jitter w using gaussian of sigma 0.1
w = w + alpha/(npop*sigma) * np.dot(N.T, A)

I appreciate for your effort. I have a question.
In my thought(as you said), Evolution Strategy (ES) could be useful in case we do not know exact gradient. Therefore, we alternatively compute gradients from randomly re-generated samples. I think such idea is analogous to the random-mutation behavior in genetic algorithm. Then, Is there any way of implementing crossover operation which is a basic behavior in the genetic algorithm?

Thanks.

I was looking for a general module for this and I couldn't find it so I developed one based on the mentioned algorithm. the code is available here if anyone's looking for it too. And I got some pretty interesting results here. I hope this helps.