# Copyright (c) Facebook, Inc. and its affiliates. All Rights Reserved.
#
# This source code is licensed under the MIT license found in the
# LICENSE file in the root directory of this source tree.
from typing import Any, List, Callable
from math import exp, sqrt, tanh
import numpy as np
from nevergrad.parametrization import discretization
from nevergrad.common.decorators import Registry
registry: Registry[Callable[[np.ndarray], float]] = Registry()
def onemax(x: List[int]) -> float:
"""onemax(x) is the most classical case of discrete functions, adapted to minimization.
It is originally designed for lists of bits. It just counts the number of 1,
and returns len(x) - number of ones..
It also works in the continuous case but in that cases discretizes the
input domain by ]0.5,1.5] --> 1 and 0 everywhere else.
"""
return len(x) - sum(1 if int(round(w)) == 1 else 0 for w in x)
def leadingones(x: List[int]) -> float:
"""leadingones is the second most classical discrete function, adapted for minimization.
Returns len(x) - number of initial 1. I.e.
leadingones([0 1 1 1]) = 4,
leadingones([1 1 1 1]) = 0,
leadingones([1 0 0 0]) = 1.
"""
for i, x_ in enumerate(list(x)):
if int(round(x_)) != 1:
return len(x) - i
return 0
def jump(x: List[int]) -> float: # TODO: docstring?
"""There exists variants of jump functions; we are in minimization.
The principle of a jump function is that local descent does not succeed.
Jumps are necessary.
"""
n = len(x)
m = n // 4
o = n - onemax(x)
if o == n or o <= n - m:
return n - m - o
return o # Deceptive part.
def _styblinksitang(x: np.ndarray, noise: float) -> float:
"""Classical function for testing noisy optimization."""
x2 = x ** 2
val = x2.dot(x2) + np.sum(5 * x - 16 * x2)
# return a positive value for maximization
return float(39.16599 * len(x) + 0.5 * val + noise * np.random.normal(size=val.shape))
class DelayedSphere:
def __call__(self, x: np.ndarray) -> float:
return float(np.sum(x ** 2))
def compute_pseudotime(self, input_parameter: Any, value: float) -> float: # pylint: disable=unused-argument
x = input_parameter[0][0]
return float(abs(1.0 / x[0]) / 1000.0) if x[0] != 0.0 else 0.0
registry.register(DelayedSphere())
@registry.register
def sphere(x: np.ndarray) -> float:
"""The most classical continuous optimization testbed.
If you do not solve that one then you have a bug."""
assert x.ndim == 1
return float(x.dot(x))
@registry.register
def sphere1(x: np.ndarray) -> float:
"""Translated sphere function."""
return sphere(x - 1.0)
@registry.register
def sphere2(x: np.ndarray) -> float:
"""A bit more translated sphere function."""
return sphere(x - 2.0)
@registry.register
def sphere4(x: np.ndarray) -> float:
"""Even more translated sphere function."""
return sphere(x - 4.0)
@registry.register
def maxdeceptive(x: np.ndarray) -> float:
dec = 3 * x ** 2 - (2 / (3 ** (x - 2) ** 2 + 0.1))
return float(np.max(dec))
@registry.register
def sumdeceptive(x: np.ndarray) -> float:
dec = 3 * x ** 2 - (2 / (3 ** (x - 2) ** 2 + 0.1))
return float(np.sum(dec))
@registry.register
def altcigar(x: np.ndarray) -> float:
"""Similar to cigar, but variables in inverse order.
E.g. for pointing out algorithms not invariant to the order of variables."""
return float(x[-1]) ** 2 + 1000000.0 * sphere(x[:-1])
@registry.register
def discus(x: np.ndarray) -> float:
"""Only one variable is very penalized."""
return sphere(x[1:]) + 1000000.0 * float(x[0]) ** 2
@registry.register
def cigar(x: np.ndarray) -> float:
"""Classical example of ill conditioned function.
The other classical example is ellipsoid.
"""
return float(x[0]) ** 2 + 1000000.0 * sphere(x[1:])
@registry.register
def bentcigar(x: np.ndarray) -> float:
"""Classical example of ill conditioned function, but bent."""
y = np.asarray([x[i] ** (1 + .5 * np.sqrt(x[i]) * (i - 1) / (len(x) - 1)) if x[i] > 0. else x[i] for i in range(len(x))])
return float(y[0]) ** 2 + 1000000.0 * sphere(y[1:])
@registry.register
def multipeak(x: np.ndarray) -> float:
"""Inspired by M. Gallagher's Gaussian peaks function."""
v = 10000.
for a in range(101):
x_ = np.asarray([np.cos(a + np.sqrt(i)) for i in range(len(x))])
v = min(v, a / 101. + np.exp(sphere(x - x_)))
return v
@registry.register
def altellipsoid(y: np.ndarray) -> float:
"""Similar to Ellipsoid, but variables in inverse order.
E.g. for pointing out algorithms not invariant to the order of variables."""
return ellipsoid(y[::-1])
def step(s: float) -> float:
return float(np.exp(int(np.log(s))))
@registry.register
def stepellipsoid(x: np.ndarray) -> float:
"""Classical example of ill conditioned function.
But we add a 'step', i.e. we set the gradient to zero everywhere.
Compared to some existing testbeds, we decided to have infinitely many steps.
"""
dim = x.size
weights = 10 ** np.linspace(0, 6, dim)
return float(step(weights.dot(x ** 2)))
@registry.register
def ellipsoid(x: np.ndarray) -> float:
"""Classical example of ill conditioned function.
The other classical example is cigar.
"""
dim = x.size
weights = 10 ** np.linspace(0, 6, dim)
return float(weights.dot(x ** 2))
@registry.register
def rastrigin(x: np.ndarray) -> float:
"""Classical multimodal function."""
cosi = float(np.sum(np.cos(2 * np.pi * x)))
return float(10 * (len(x) - cosi) + sphere(x))
@registry.register
def bucherastrigin(x: np.ndarray) -> float:
"""Classical multimodal function. No box-constraint penalization here."""
s = np.asarray([x[i] * (10 if x[i] > 0. and i % 2 else 1) * (10**((i - 1) / (2 * (len(x) - 1)))) for i in range(len(x))])
cosi = float(np.sum(np.cos(2 * np.pi * s)))
return float(10 * (len(x) - cosi) + sphere(s))
@registry.register
def doublelinearslope(x: np.ndarray) -> float:
"""We decided to use two linear slopes rather than having a constraint artificially added for
not having the optimum at infinity."""
return float(np.abs(np.sum(x)))
@registry.register
def stepdoublelinearslope(x: np.ndarray) -> float:
return step(np.abs(np.sum(x)))
@registry.register
def hm(x: np.ndarray) -> float:
"""New multimodal function (proposed for Nevergrad)."""
return float((x ** 2).dot(1.1 + np.cos(1.0 / x)))
@registry.register
def rosenbrock(x: np.ndarray) -> float:
x_m_1 = x[:-1] - 1
x_diff = x[:-1] ** 2 - x[1:]
return float(100 * x_diff.dot(x_diff) + x_m_1.dot(x_m_1))
@registry.register
def ackley(x: np.ndarray) -> float:
dim = x.size
sum_cos = np.sum(np.cos(2 * np.pi * x))
return -20.0 * exp(-0.2 * sqrt(sphere(x) / dim)) - exp(sum_cos / dim) + 20 + exp(1)
@registry.register
def schwefel_1_2(x: np.ndarray) -> float:
cx = np.cumsum(x)
return sphere(cx)
@registry.register
def griewank(x: np.ndarray) -> float:
"""Multimodal function, often used in Bayesian optimization."""
part1 = sphere(x)
part2 = np.prod(np.cos(x / np.sqrt(1 + np.arange(len(x)))))
return 1 + (float(part1) / 4000.0) - float(part2)
@registry.register
def deceptiveillcond(x: np.ndarray) -> float:
"""An extreme ill conditioned functions. Most algorithms fail on this.
The condition number increases to infinity as we get closer to the optimum."""
assert len(x) >= 2
return float(
max(np.abs(np.arctan(x[1] / x[0])), np.sqrt(x[0] ** 2.0 + x[1] ** 2.0), 1.0 if x[0] > 0 else 0.0) if x[0] != 0.0 else float("inf")
)
@registry.register
def deceptivepath(x: np.ndarray) -> float:
"""A function which needs following a long path. Most algorithms fail on this.
The path becomes thiner as we get closer to the optimum."""
assert len(x) >= 2
distance = np.sqrt(x[0] ** 2 + x[1] ** 2)
if distance == 0.0:
return 0.0
angle = np.arctan(x[0] / x[1]) if x[1] != 0.0 else np.pi / 2.0
invdistance = (1.0 / distance) if distance > 0.0 else 0.0
if np.abs(np.cos(invdistance) - angle) > 0.1:
return 1.0
return float(distance)
@registry.register
def deceptivemultimodal(x: np.ndarray) -> float:
"""Infinitely many local optima, as we get closer to the optimum."""
assert len(x) >= 2
distance = np.sqrt(x[0] ** 2 + x[1] ** 2)
if distance == 0.0:
return 0.0
angle = np.arctan(x[0] / x[1]) if x[1] != 0.0 else np.pi / 2.0
invdistance = int(1.0 / distance) if distance > 0.0 else 0.0
if np.abs(np.cos(invdistance) - angle) > 0.1:
return 1.0
return float(distance)
@registry.register
def lunacek(x: np.ndarray) -> float:
"""Multimodal function.
Based on https://www.cs.unm.edu/~neal.holts/dga/benchmarkFunction/lunacek.html."""
problemDimensions = len(x)
s = 1.0 - (1.0 / (2.0 * np.sqrt(problemDimensions + 20.0) - 8.2))
mu1 = 2.5
mu2 = -np.sqrt(abs((mu1 ** 2 - 1.0) / s))
firstSum = 0.0
secondSum = 0.0
thirdSum = 0.0
for i in range(problemDimensions):
firstSum += (x[i] - mu1) ** 2
secondSum += (x[i] - mu2) ** 2
thirdSum += 1.0 - np.cos(2 * np.pi * (x[i] - mu1))
return min(firstSum, 1.0 * problemDimensions + secondSum) + 10 * thirdSum
# following functions using discretization should not be used with translation/rotation
@registry.register_with_info(no_transform=True)
def hardonemax(y: np.ndarray) -> float:
"""Onemax, with a discretization in 2 by threshold 0 (>0 or <0)."""
return onemax(discretization.threshold_discretization(y))
@registry.register_with_info(no_transform=True)
def hardjump(y: np.ndarray) -> float:
"""Hardjump, with a discretization in 2 by threshold 0 (>0 or <0)."""
return jump(discretization.threshold_discretization(y))
@registry.register_with_info(no_transform=True)
def hardleadingones(y: np.ndarray) -> float:
"""Leading ones, with a discretization in 2 by threshold 0 (>0 or <0)."""
return leadingones(discretization.threshold_discretization(y))
@registry.register_with_info(no_transform=True)
def hardonemax5(y: np.ndarray) -> float:
"""Hardonemax, with a discretization by 5 with 4 thresholds (quantiles of Gaussian)."""
return onemax(discretization.threshold_discretization(y, 5))
@registry.register_with_info(no_transform=True)
def hardjump5(y: np.ndarray) -> float:
"""Jump, with a discretization by 5 with 4 thresholds (quantiles of Gaussian)."""
return jump(discretization.threshold_discretization(y, 5))
@registry.register_with_info(no_transform=True)
def hardleadingones5(y: np.ndarray) -> float:
"""Leadingones, with a discretization by 5 with 4 thresholds (quantiles of Gaussian)."""
return leadingones(discretization.threshold_discretization(y, 5))
@registry.register_with_info(no_transform=True)
def onemax5(y: np.ndarray) -> float:
"""Softmax discretization of onemax with 5 possibles values.
This multiplies the dimension by 5."""
return onemax(discretization.Encoder(y.reshape(-1, 5), np.random).encode().tolist())
@registry.register_with_info(no_transform=True)
def jump5(y: np.ndarray) -> float:
"""Softmax discretization of jump with 5 possibles values.
This multiplies the dimension by 5."""
return jump(discretization.Encoder(y.reshape(-1, 5), np.random).encode().tolist())
@registry.register_with_info(no_transform=True)
def leadingones5(y: np.ndarray) -> float:
"""Softmax discretization of leadingones with 5 possibles values.
This multiplies the dimension by 5."""
return leadingones(discretization.Encoder(y.reshape(-1, 5), np.random).encode().tolist())
@registry.register_with_info(no_transform=True)
def genzcornerpeak(y: np.ndarray) -> float:
"""One of the Genz functions, originally used in integration,
tested in optim because why not."""
value = float(1 + np.mean(np.tanh(y)))
if value == 0:
return float("inf")
return value ** (-len(y) - 1)
@registry.register_with_info(no_transform=True)
def minusgenzcornerpeak(y: np.ndarray) -> float:
"""One of the Genz functions, originally used in integration,
tested in optim because why not."""
return -genzcornerpeak(y)
@registry.register
def genzgaussianpeakintegral(x: np.ndarray) -> float:
"""One of the Genz functions, originally used in integration,
tested in optim because why not."""
return exp(-sphere(x) / 4.0)
@registry.register
def minusgenzgaussianpeakintegral(x: np.ndarray) -> float:
"""One of the Genz functions, originally used in integration,
tested in optim because why not."""
return -genzgaussianpeakintegral(x)
@registry.register
def slope(x: np.ndarray) -> float:
return sum(x)
@registry.register
def linear(x: np.ndarray) -> float:
return tanh(x[0])
@registry.register
def st0(x: np.ndarray) -> float:
"""Styblinksitang function with 0 noise."""
return _styblinksitang(x, 0)
@registry.register
def st1(x: np.ndarray) -> float:
"""Styblinksitang function with noise 1."""
return _styblinksitang(x, 1)
@registry.register
def st10(x: np.ndarray) -> float:
"""Styblinksitang function with noise 10."""
return _styblinksitang(x, 10)
@registry.register
def st100(x: np.ndarray) -> float:
"""Styblinksitang function with noise 100."""
return _styblinksitang(x, 100)
