In this small notebook, I implement various Kullback-Leibler divergence functions, in Python, using different approaches: naive Python, and using Numba and Cython.
I also implement KL-UCB indexes, in the three approaches, and finally I present some basic benchmarks to compare the time and memory efficiency of the different approaches, for each function.
Requirements:
%load_ext watermark
%watermark -v -m -a "Lilian Besson (Naereen)" -p numpy,numba -g
import numpy as np
I will copy and paste parts of this file from my SMPyBandits library.
eps = 1e-15 #: Threshold value: everything in [0, 1] is truncated to [eps, 1 - eps]
I will include docstrings and examples only for the naive implementation.
def klBern(x, y):
r""" Kullback-Leibler divergence for Bernoulli distributions. https://en.wikipedia.org/wiki/Bernoulli_distribution#Kullback.E2.80.93Leibler_divergence
.. math:: \mathrm{KL}(\mathcal{B}(x), \mathcal{B}(y)) = x \log(\frac{x}{y}) + (1-x) \log(\frac{1-x}{1-y})."""
x = min(max(x, eps), 1 - eps)
y = min(max(y, eps), 1 - eps)
return x * np.log(x / y) + (1 - x) * np.log((1 - x) / (1 - y))
klBern(0.5, 0.5)
klBern(0.1, 0.9)
klBern(0.9, 0.1)
klBern(0.4, 0.5)
klBern(0.01, 0.99)
klBern(0, 1)
def klBin(x, y, n):
r""" Kullback-Leibler divergence for Binomial distributions. https://math.stackexchange.com/questions/320399/kullback-leibner-divergence-of-binomial-distributions
- It is simply the n times :func:`klBern` on x and y.
.. math:: \mathrm{KL}(\mathrm{Bin}(x, n), \mathrm{Bin}(y, n)) = n \times \left(x \log(\frac{x}{y}) + (1-x) \log(\frac{1-x}{1-y}) \right).
.. warning:: The two distributions must have the same parameter n, and x, y are p, q in (0, 1).
"""
x = min(max(x, eps), 1 - eps)
y = min(max(y, eps), 1 - eps)
return n * (x * np.log(x / y) + (1 - x) * np.log((1 - x) / (1 - y)))
klBin(0.5, 0.5, 10)
klBin(0.1, 0.9, 10)
klBin(0.9, 0.1, 10)
klBin(0.4, 0.5, 10)
klBin(0.01, 0.99, 10)
klBin(0, 1, 10)
def klPoisson(x, y):
r""" Kullback-Leibler divergence for Poison distributions. https://en.wikipedia.org/wiki/Poisson_distribution#Kullback.E2.80.93Leibler_divergence
.. math:: \mathrm{KL}(\mathrm{Poisson}(x), \mathrm{Poisson}(y)) = y - x + x \times \log(\frac{x}{y}).
"""
x = max(x, eps)
y = max(y, eps)
return y - x + x * np.log(x / y)
klPoisson(3, 3)
klPoisson(2, 1)
klPoisson(1, 2)
klPoisson(3, 6)
klPoisson(6, 8)
klPoisson(1, 0)
klPoisson(0, 0)
def klExp(x, y):
r""" Kullback-Leibler divergence for exponential distributions. https://en.wikipedia.org/wiki/Exponential_distribution#Kullback.E2.80.93Leibler_divergence
.. math::
\mathrm{KL}(\mathrm{Exp}(x), \mathrm{Exp}(y)) = \begin{cases}
\frac{x}{y} - 1 - \log(\frac{x}{y}) & \text{if} x > 0, y > 0\\
+\infty & \text{otherwise}
\end{cases}
"""
if x <= 0 or y <= 0:
return float('+inf')
else:
x = max(x, eps)
y = max(y, eps)
return x / y - 1 - np.log(x / y)
klExp(3, 3)
klExp(3, 6)
klExp(1, 2)
klExp(2, 1)
klExp(4, 2)
klExp(6, 8)
klExp(-3, 2)
klExp(3, -2)
klExp(-3, -2)
def klGamma(x, y, a=1):
r""" Kullback-Leibler divergence for gamma distributions. https://en.wikipedia.org/wiki/Gamma_distribution#Kullback.E2.80.93Leibler_divergence
- It is simply the a times :func:`klExp` on x and y.
.. math::
\mathrm{KL}(\Gamma(x, a), \Gamma(y, a)) = \begin{cases}
a \times \left( \frac{x}{y} - 1 - \log(\frac{x}{y}) \right) & \text{if} x > 0, y > 0\\
+\infty & \text{otherwise}
\end{cases}
.. warning:: The two distributions must have the same parameter a.
"""
if x <= 0 or y <= 0:
return float('+inf')
else:
x = max(x, eps)
y = max(y, eps)
return a * (x / y - 1 - np.log(x / y))
klGamma(3, 3)
klGamma(3, 6)
klGamma(1, 2)
klGamma(2, 1)
klGamma(4, 2)
klGamma(6, 8)
klGamma(-3, 2)
klGamma(3, -2)
klGamma(-3, -2)
def klNegBin(x, y, r=1):
r""" Kullback-Leibler divergence for negative binomial distributions. https://en.wikipedia.org/wiki/Negative_binomial_distribution
.. math:: \mathrm{KL}(\mathrm{NegBin}(x, r), \mathrm{NegBin}(y, r)) = r \times \log((r + x) / (r + y)) - x \times \log(y \times (r + x) / (x \times (r + y))).
.. warning:: The two distributions must have the same parameter r.
"""
x = max(x, eps)
y = max(y, eps)
return r * np.log((r + x) / (r + y)) - x * np.log(y * (r + x) / (x * (r + y)))
klNegBin(0.5, 0.5)
klNegBin(0.1, 0.9)
klNegBin(0.9, 0.1)
klNegBin(0.4, 0.5)
klNegBin(0.01, 0.99)
klBern(0, 1)
klNegBin(0.5, 0.5, r=2)
klNegBin(0.1, 0.9, r=2)
klNegBin(0.1, 0.9, r=4)
klNegBin(0.9, 0.1, r=2)
klNegBin(0.4, 0.5, r=2)
klNegBin(0.01, 0.99, r=2)
def klGauss(x, y, sig2x=0.25, sig2y=None):
r""" Kullback-Leibler divergence for Gaussian distributions of means ``x`` and ``y`` and variances ``sig2x`` and ``sig2y``, :math:`\nu_1 = \mathcal{N}(x, \sigma_x^2)` and :math:`\nu_2 = \mathcal{N}(y, \sigma_x^2)`:
.. math:: \mathrm{KL}(\nu_1, \nu_2) = \frac{(x - y)^2}{2 \sigma_y^2} + \frac{1}{2}\left( \frac{\sigma_x^2}{\sigma_y^2} - 1 \log\left(\frac{\sigma_x^2}{\sigma_y^2}\right) \right).
See https://en.wikipedia.org/wiki/Normal_distribution#Other_properties
- By default, sig2y is assumed to be sig2x (same variance).
"""
if sig2y is None or - eps < (sig2y - sig2x) < eps:
return (x - y) ** 2 / (2. * sig2x)
else:
return (x - y) ** 2 / (2. * sig2y) + 0.5 * ((sig2x/sig2y)**2 - 1 - np.log(sig2x/sig2y))
klGauss(3, 3)
klGauss(3, 6)
klGauss(1, 2)
klGauss(2, 1)
klGauss(4, 2)
klGauss(6, 8)
klGauss(-3, 2)
klGauss(3, -2)
klGauss(-3, -2)
klGauss(3, 2)
klGauss(3, 3, sig2x=10)
klGauss(3, 6, sig2x=10)
klGauss(1, 2, sig2x=10)
klGauss(2, 1, sig2x=10)
klGauss(4, 2, sig2x=10)
klGauss(6, 8, sig2x=10)
klGauss(0, 0, sig2x=0.25, sig2y=0.5)
klGauss(0, 0, sig2x=0.25, sig2y=1.0)
klGauss(0, 0, sig2x=0.5, sig2y=0.25)
klGauss(0, 1, sig2x=0.25, sig2y=0.5)
klGauss(0, 1, sig2x=0.25, sig2y=1.0)
klGauss(0, 1, sig2x=0.5, sig2y=0.25)
klGauss(1, 0, sig2x=0.25, sig2y=0.5)
klGauss(1, 0, sig2x=0.25, sig2y=1.0)
klGauss(1, 0, sig2x=0.5, sig2y=0.25)
def klucb(x, d, kl, upperbound, lowerbound=float('-inf'), precision=1e-6, max_iterations=50):
""" The generic KL-UCB index computation.
- x: value of the cum reward,
- d: upper bound on the divergence,
- kl: the KL divergence to be used (:func:`klBern`, :func:`klGauss`, etc),
- upperbound, lowerbound=float('-inf'): the known bound of the values x,
- precision=1e-6: the threshold from where to stop the research,
- max_iterations: max number of iterations of the loop (safer to bound it to reduce time complexity).
.. note:: It uses a **bisection search**, and one call to ``kl`` for each step of the bisection search.
"""
value = max(x, lowerbound)
u = upperbound
_count_iteration = 0
while _count_iteration < max_iterations and u - value > precision:
_count_iteration += 1
m = (value + u) / 2.
if kl(x, m) > d:
u = m
else:
value = m
return (value + u) / 2.
For example, for klucbBern, the two steps are to first compute an upperbound (as precise as possible) and the compute the kl-UCB index:
x, d = 0.9, 0.2
upperbound = 1
klucb(x, d, klBern, upperbound, lowerbound=0, precision=1e-3, max_iterations=10)
klucb(x, d, klBern, upperbound, lowerbound=0, precision=1e-6, max_iterations=10)
klucb(x, d, klBern, upperbound, lowerbound=0, precision=1e-3, max_iterations=50)
klucb(x, d, klBern, upperbound, lowerbound=0, precision=1e-6, max_iterations=100)
def klucbGauss(x, d, sig2x=0.25, precision=0.):
""" KL-UCB index computation for Gaussian distributions.
- Note that it does not require any search.
.. warning:: it works only if the good variance constant is given.
"""
return x + np.sqrt(2 * sig2x * d)
klucbGauss(0.1, 0.2)
klucbGauss(0.5, 0.2)
klucbGauss(0.9, 0.2)
klucbGauss(0.1, 0.4)
klucbGauss(0.1, 0.9)
klucbGauss(0.5, 0.4)
klucbGauss(0.5, 0.9)
klucbGauss(0.9, 0.4)
klucbGauss(0.9, 0.9)
def klucbBern(x, d, precision=1e-6):
""" KL-UCB index computation for Bernoulli distributions, using :func:`klucb`."""
upperbound = min(1., klucbGauss(x, d, sig2x=0.25)) # variance 1/4 for [0,1] bounded distributions
# upperbound = min(1., klucbPoisson(x, d)) # also safe, and better ?
return klucb(x, d, klBern, upperbound, precision)
klucbBern(0.1, 0.2)
klucbBern(0.5, 0.2)
klucbBern(0.9, 0.2)
klucbBern(0.1, 0.4)
klucbBern(0.1, 0.9)
klucbBern(0.5, 0.4)
klucbBern(0.5, 0.9)
klucbBern(0.9, 0.4)
klucbBern(0.9, 0.9)
def klucbPoisson(x, d, precision=1e-6):
""" KL-UCB index computation for Poisson distributions, using :func:`klucb`."""
upperbound = x + d + np.sqrt(d * d + 2 * x * d) # looks safe, to check: left (Gaussian) tail of Poisson dev
return klucb(x, d, klPoisson, upperbound, precision)
klucbPoisson(0.1, 0.2)
klucbPoisson(0.5, 0.2)
klucbPoisson(0.9, 0.2)
klucbPoisson(0.1, 0.4)
klucbPoisson(0.1, 0.9)
klucbPoisson(0.5, 0.4)
klucbPoisson(0.5, 0.9)
klucbPoisson(0.9, 0.4)
klucbPoisson(0.9, 0.9)
def klucbExp(x, d, precision=1e-6):
""" KL-UCB index computation for exponential distributions, using :func:`klucb`."""
if d < 0.77: # XXX where does this value come from?
upperbound = x / (1 + 2. / 3 * d - np.sqrt(4. / 9 * d * d + 2 * d))
# safe, klexp(x,y) >= e^2/(2*(1-2e/3)) if x=y(1-e)
else:
upperbound = x * np.exp(d + 1)
if d > 1.61: # XXX where does this value come from?
lowerbound = x * np.exp(d)
else:
lowerbound = x / (1 + d - np.sqrt(d * d + 2 * d))
return klucb(x, d, klGamma, upperbound, lowerbound, precision)
klucbExp(0.1, 0.2)
klucbExp(0.5, 0.2)
klucbExp(0.9, 0.2)
klucbExp(0.1, 0.4)
klucbExp(0.1, 0.9)
klucbExp(0.5, 0.4)
klucbExp(0.5, 0.9)
klucbExp(0.9, 0.4)
klucbExp(0.9, 0.9)
We could do the same for more distributions, but that's enough.
from numba import jit
As much as possible, one should call @jit(nopython=True) to be sure that numba does not fall back silently to naive Python code. With nopython=True, any call to the generated function will fail if the compilation could not succeed.
@jit(nopython=True)
def klBern_numba(x, y):
x = min(max(x, eps), 1 - eps)
y = min(max(y, eps), 1 - eps)
return x * np.log(x / y) + (1 - x) * np.log((1 - x) / (1 - y))
@jit(nopython=True)
def klBin_numba(x, y, n):
x = min(max(x, eps), 1 - eps)
y = min(max(y, eps), 1 - eps)
return n * (x * np.log(x / y) + (1 - x) * np.log((1 - x) / (1 - y)))
@jit(nopython=True)
def klPoisson_numba(x, y):
x = max(x, eps)
y = max(y, eps)
return y - x + x * np.log(x / y)
@jit(nopython=True)
def klExp_numba(x, y):
if x <= 0 or y <= 0:
return inf
else:
x = max(x, eps)
y = max(y, eps)
return x / y - 1 - np.log(x / y)
@jit(nopython=True)
def klGamma_numba(x, y, a=1):
if x <= 0 or y <= 0:
return inf
else:
x = max(x, eps)
y = max(y, eps)
return a * (x / y - 1 - np.log(x / y))
@jit(nopython=True)
def klNegBin_numba(x, y, r=1):
x = max(x, eps)
y = max(y, eps)
return r * np.log((r + x) / (r + y)) - x * np.log(y * (r + x) / (x * (r + y)))
@jit(nopython=True)
def klGauss_numba(x, y, sig2x=0.25, sig2y=0.25):
if - eps < (sig2y - sig2x) and (sig2y - sig2x) < eps:
return (x - y) ** 2 / (2. * sig2x)
else:
return (x - y) ** 2 / (2. * sig2y) + 0.5 * ((sig2x/sig2y)**2 - 1 - np.log(sig2x/sig2y))
@jit
def klucb_numba(x, d, kl, upperbound,
lowerbound=float('-inf'), precision=1e-6, max_iterations=50):
value = max(x, lowerbound)
u = upperbound
_count_iteration = 0
while _count_iteration < max_iterations and u - value > precision:
_count_iteration += 1
m = (value + u) / 2.
if kl(x, m) > d:
u = m
else:
value = m
return (value + u) / 2.
For example, for klucbBern, the two steps are to first compute an upperbound (as precise as possible) and the compute the kl-UCB index:
x, d = 0.9, 0.2
upperbound = 1
klucb_numba(x, d, klBern_numba, upperbound, lowerbound=0, precision=1e-3, max_iterations=10)
klucb_numba(x, d, klBern_numba, upperbound, lowerbound=0, precision=1e-6, max_iterations=10)
klucb_numba(x, d, klBern_numba, upperbound, lowerbound=0, precision=1e-3, max_iterations=50)
klucb_numba(x, d, klBern_numba, upperbound, lowerbound=0, precision=1e-6, max_iterations=100)
@jit(nopython=True)
def klucbGauss_numba(x, d, sig2x=0.25, precision=0.):
return x + np.sqrt(2 * sig2x * d)
Here, the nopython=True fails as numba has a hard time typing linked function calls.
@jit
def klucbBern_numba(x, d, precision=1e-6):
upperbound = min(1., klucbGauss_numba(x, d, sig2x=0.25)) # variance 1/4 for [0,1] bounded distributions
# upperbound = min(1., klucbPoisson(x, d)) # also safe, and better ?
return klucb_numba(x, d, klBern_numba, upperbound, precision)
@jit
def klucbPoisson_numba(x, d, precision=1e-6):
upperbound = x + d + np.sqrt(d * d + 2 * x * d) # looks safe, to check: left (Gaussian) tail of Poisson dev
return klucb_numba(x, d, klPoisson_numba, upperbound, precision)
@jit
def klucbExp_numba(x, d, precision=1e-6):
if d < 0.77: # XXX where does this value come from?
upperbound = x / (1 + 2. / 3 * d - np.sqrt(4. / 9 * d * d + 2 * d))
# safe, klexp(x,y) >= e^2/(2*(1-2e/3)) if x=y(1-e)
else:
upperbound = x * np.exp(d + 1)
if d > 1.61: # XXX where does this value come from?
lowerbound = x * np.exp(d)
else:
lowerbound = x / (1 + d - np.sqrt(d * d + 2 * d))
return klucb_numba(x, d, klGamma_numba, upperbound, lowerbound, precision)
%load_ext cython
A cell can now be written in Cython. For instance, we can define a simple example function in Python, and then write a Cython version, simply by declaring variables and tagging their types, like this:
def some_loop(n: int) -> int:
s = 0
for i in range(0, n, 2):
s += i
return s
%%cython
def some_loop_cython(int n) -> int:
cdef int s = 0
cdef int i = 0
for i in range(0, n, 2):
s += i
return s
%timeit np.random.randint(1000)
%timeit some_loop(np.random.randint(1000))
%timeit some_loop_cython(np.random.randint(1000))
Here we observe a large speed-up. But how large? $6$ times or $50$ times?
It's really important to include the time taken by the Pseudo-Random Number Generator:
14.6 / 2.21
(14.6 - 1.95) / (2.21 - 1.95)
%%cython
from libc.math cimport log
eps = 1e-15 #: Threshold value: everything in [0, 1] is truncated to [eps, 1 - eps]
def klBern_cython(float x, float y) -> float:
x = min(max(x, eps), 1 - eps)
y = min(max(y, eps), 1 - eps)
return x * log(x / y) + (1 - x) * log((1 - x) / (1 - y))
%%cython
from libc.math cimport log
eps = 1e-15 #: Threshold value: everything in [0, 1] is truncated to [eps, 1 - eps]
def klBin_cython(float x, float y, int n) -> float:
x = min(max(x, eps), 1 - eps)
y = min(max(y, eps), 1 - eps)
return n * (x * log(x / y) + (1 - x) * log((1 - x) / (1 - y)))
%%cython
from libc.math cimport log
eps = 1e-15 #: Threshold value: everything in [0, 1] is truncated to [eps, 1 - eps]
def klPoisson_cython(float x, float y) -> float:
x = max(x, eps)
y = max(y, eps)
return y - x + x * log(x / y)
%%cython
from libc.math cimport log
eps = 1e-15 #: Threshold value: everything in [0, 1] is truncated to [eps, 1 - eps]
def klExp_cython(float x, float y) -> float:
if x <= 0 or y <= 0:
return float('+inf')
else:
x = max(x, eps)
y = max(y, eps)
return x / y - 1 - log(x / y)
%%cython
from libc.math cimport log
eps = 1e-15 #: Threshold value: everything in [0, 1] is truncated to [eps, 1 - eps]
def klGamma_cython(float x, float y, float a=1) -> float:
if x <= 0 or y <= 0:
return float('+inf')
else:
x = max(x, eps)
y = max(y, eps)
return a * (x / y - 1 - log(x / y))
%%cython
from libc.math cimport log
eps = 1e-15 #: Threshold value: everything in [0, 1] is truncated to [eps, 1 - eps]
def klNegBin_cython(float x, float y, float r=1) -> float:
x = max(x, eps)
y = max(y, eps)
return r * log((r + x) / (r + y)) - x * log(y * (r + x) / (x * (r + y)))
%%cython
from libc.math cimport log
eps = 1e-15 #: Threshold value: everything in [0, 1] is truncated to [eps, 1 - eps]
def klGauss_cython(float x, float y, float sig2x=0.25, float sig2y=0.25) -> float:
if - eps < (sig2y - sig2x) < eps:
return (x - y) ** 2 / (2. * sig2x)
else:
return (x - y) ** 2 / (2. * sig2y) + 0.5 * ((sig2x/sig2y)**2 - 1 - log(sig2x/sig2y))
For these, they need previously defined functions, which have to be rewritten from inside the cython cell to be accessible from Cython.
To minimize repetitions, I use only one cell to define all functions.
%%cython
from libc.math cimport sqrt, log, exp
eps = 1e-15 #: Threshold value: everything in [0, 1] is truncated to [eps, 1 - eps]
def klucbGauss_cython(float x, float d, float sig2x=0.25, float precision=0.) -> float:
return x + sqrt(2 * sig2x * d)
cdef float klucbGauss_cython_x(float x, float d, float sig2x=0.25, float precision=0.):
return x + sqrt(2 * sig2x * d)
def klucb_cython(float x, float d, kl, float upperbound,
float lowerbound=float('-inf'),
float precision=1e-6, int max_iterations=50) -> float:
cdef float value = max(x, lowerbound)
cdef float u = upperbound
cdef int _count_iteration = 0
cdef float m = 0
while _count_iteration < max_iterations and u - value > precision:
_count_iteration += 1
m = (value + u) / 2.
if kl(x, m) > d:
u = m
else:
value = m
return (value + u) / 2.
cdef float klBern_cython_x(float x, float y):
x = min(max(x, eps), 1 - eps)
y = min(max(y, eps), 1 - eps)
return x * log(x / y) + (1 - x) * log((1 - x) / (1 - y))
def klucbBern_cython(float x, float d, float precision=1e-6) -> float:
cdef float upperbound = min(1., klucbGauss_cython_x(x, d, sig2x=0.25)) # variance 1/4 for [0,1] bounded distributions
# upperbound = min(1., klucbPoisson(x, d)) # also safe, and better ?
return klucb_cython(x, d, klBern_cython_x, upperbound, precision)
cdef float klPoisson_cython_x(float x, float y):
x = max(x, eps)
y = max(y, eps)
return y - x + x * log(x / y)
def klucbPoisson_cython(float x, float d, float precision=1e-6) -> float:
cdef float upperbound = x + d + sqrt(d * d + 2 * x * d) # looks safe, to check: left (Gaussian) tail of Poisson dev
return klucb_cython(x, d, klPoisson_cython_x, upperbound, precision)
cdef float klGamma_cython_x(float x, float y):
if x <= 0 or y <= 0:
return float('+inf')
else:
x = max(x, eps)
y = max(y, eps)
return x / y - 1 - log(x / y)
def klucbExp_cython(float x, float d, float precision=1e-6) -> float:
cdef float upperbound = 1
cdef float lowerbound = 0
if d < 0.77: # XXX where does this value come from?
upperbound = x / (1 + 2. / 3 * d - sqrt(4. / 9 * d * d + 2 * d))
# safe, klexp(x,y) >= e^2/(2*(1-2e/3)) if x=y(1-e)
else:
upperbound = x * exp(d + 1)
if d > 1.61: # XXX where does this value come from?
lowerbound = x * exp(d)
else:
lowerbound = x / (1 + d - sqrt(d * d + 2 * d))
return klucb_cython(x, d, klGamma_cython_x, upperbound, lowerbound, precision)
For example, for klucbBern_cython, the two steps are to first compute an upperbound (as precise as possible) and the compute the kl-UCB index:
x, d = 0.9, 0.2
upperbound = 1
klucb_cython(x, d, klBern_cython, upperbound, lowerbound=0, precision=1e-3, max_iterations=10)
klucb_cython(x, d, klBern_cython, upperbound, lowerbound=0, precision=1e-6, max_iterations=10)
klucb_cython(x, d, klBern_cython, upperbound, lowerbound=0, precision=1e-3, max_iterations=50)
klucb_cython(x, d, klBern_cython, upperbound, lowerbound=0, precision=1e-6, max_iterations=100)
It is more tedious, and won't be included here, but Python can easily be extended using C. It is the best way to obtain close-to-optimal performance for some parts of your code, and I will let you read the introduction to the official documentation if you are curious.
For my SMPyBandits, I reused some code from the py/maBandits project, and the authors implemented some of the previously defined KL-divergences and KL-UCB indexes in pure Python as well as in C optimized. I copied the compiled library in the current directory, and it can be imported:
%%bash
ls -larth *kullback*
[ -f kullback.py ] && mv -vf kullback.py kullback.py.old
!ls -larth kullback*.so
import kullback
help(kullback.klBern)
[ s for s in dir(kullback) if not s.startswith('_') ]
klBern_c = kullback.klBern
klBin_c = kullback.klBin
klExp_c = kullback.klExp
klGamma_c = kullback.klGamma
klGauss_c = kullback.klGauss
klPoisson_c = kullback.klPoisson
klucbBern_c = kullback.klucbBern
klucbExp_c = kullback.klucbExp
klucbGamma_c = kullback.klucbGamma
klucbGauss_c = kullback.klucbGauss
klucbPoisson_c = kullback.klucbPoisson
If you want to reproduce this notebook, download the kullback_py3.c and follow the build instructions.
For each of the functions defined in three approaches above, I will do some numerical tests to compare their speed − and memory − efficiency. Simple.
The benchmark will be to test the computation time on random entries. It includes a constant time: creating random values! So I also compare the time to simply generate the values.
r = np.random.random
rn = lambda: np.random.randint(1000)
%timeit (r(), r())
%timeit (r(), r(), rn())
$\implies$ we will remove this $700$ ns or $2.5$ µs overhead when computing speed-up ratio between naive Python and numb or Cython versions.
But we also need to test that the three versions of each function gives the same results (up-to approximation errors less than $10^{-6}$ (at least)).
def test_fs(fs, inputs, tolerance=1e-5, nb_tests=100):
for _ in range(nb_tests):
args = inputs()
ref_f = fs[0] # Python version
output = ref_f(*args)
for other_f in fs[1:]:
other_output = other_f(*args)
if abs(output) > 1:
rel_diff = (output - other_output) / output
else:
rel_diff = (output - other_output)
assert abs(rel_diff) <= tolerance, "Error: function {} gave {} and function {} gave {} on inputs {}, and the two outputs are too different.".format(ref_f, output, other_f, other_output, args)
WARNING in the following, I use a very manual approach: I copied the time of each '%timeit' example, to compare speed-up ratios. So when I rerun the cells, the times might vary (a little bit), and I cannot keep an up-to-date versions of the computations of each ratio, so bear with me the (tiny) incoherences.
test_fs([klBern, klBern_numba, klBern_cython, klBern_c], lambda: (r(), r()))
%timeit klBern(r(), r())
%timeit klBern_numba(r(), r())
%timeit klBern_cython(r(), r())
%timeit klBern_c(r(), r())
This is a speed-up ratio of about $12$ times faster for Numba and Cython, and $25$ times faster for the C version.
(6280 - 576) / (1000 - 576) # for Python vs numba
(6280 - 576) / (882 - 576) # for Python vs Cython
(6280 - 576) / (811 - 576) # for Python vs C
test_fs([klBin, klBin_numba, klBin_cython, klBin_c], lambda: (r(), r(), rn()))
Too much numerical difference? Let's try again with a larger tolerance:
test_fs([klBin, klBin_numba, klBin_cython, klBin_c], lambda: (r(), r(), rn()), tolerance=1e-3)
%timeit klBin(r(), r(), rn())
%timeit klBin_numba(r(), r(), rn())
%timeit klBin_cython(r(), r(), rn())
%timeit klBin_c(r(), r(), rn())
This is a speed-up ratio of about $5$ times faster for both Numba and Cython. Not so great, but still something!
(7005 - 2350) / (3070 - 2350) # for Python vs numba
(7005 - 2350) / (3331 - 2350) # for Python vs Cython
(7005 - 2350) / (2980 - 2350) # for Python vs C
test_fs([klPoisson, klPoisson_numba, klPoisson_cython, klPoisson_c], lambda: (r(), r()))
%timeit klPoisson(r(), r())
%timeit klPoisson_numba(r(), r())
%timeit klPoisson_cython(r(), r())
%timeit klPoisson_c(r(), r())
This is a speed-up ratio of about $7.5$ times faster for Numba, and about $7$ times for Cython and C.
(2350 - 576) / (935 - 576) # for Python vs numba
(2350 - 576) / (859 - 576) # for Python vs Cython
(2350 - 576) / (811 - 576) # for Python vs C
test_fs([klExp, klExp_numba, klExp_cython, klExp_c], lambda: (r(), r()))
%timeit klExp(r(), r())
%timeit klExp_numba(r(), r())
%timeit klExp_cython(r(), r())
%timeit klExp_c(r(), r())
This is a speed-up ratio of about $3$ times faster for Numba and $6$ times faster for Cython. Cython starts to win the race!
(2210 - 576) / (1070 - 576) # for Python vs numba
(2210 - 576) / (842 - 576) # for Python vs Cython
(2210 - 576) / (869 - 576) # for Python vs C
klGamma_c = lambda x, y: kullback.klGamma(x, y, 1)
test_fs([klGamma, klGamma_numba, klGamma_cython, klGamma_c], lambda: (r(), r()))
%timeit klGamma(r(), r())
%timeit klGamma_numba(r(), r())
%timeit klGamma_cython(r(), r())
%timeit klGamma_c(r(), r())
This is a speed-up ratio of about $6$ times faster for Numba, and $6$ or $5$ times faster for Cython and C.
Note that the C version probably looses a little bit due to this lambda , y: ... trick.
(2700 - 576) / (1070 - 576) # for Python vs numba
(2700 - 576) / (889 - 576) # for Python vs Cython
(2700 - 576) / (997 - 576) # for Python vs C
test_fs([klNegBin, klNegBin_numba, klNegBin_cython], lambda: (r(), r()))
%timeit klNegBin(r(), r())
%timeit klNegBin_numba(r(), r())
%timeit klNegBin_cython(r(), r())
This is a speed-up ratio of about $5$ times faster for Numba and $10$ times faster for Cython.
(3890 - 576) / (1160 - 576) # for Python vs numba
(3890 - 576) / (901 - 576) # for Python vs Cython
klGauss_c = lambda x, y: kullback.klGauss(x, y, 0.25)
test_fs([klGauss, klGauss_numba, klGauss_cython, klGauss_c], lambda: (r(), r()))
%timeit klGauss(r(), r())
%timeit klGauss_numba(r(), r())
%timeit klGauss_cython(r(), r())
%timeit klGauss_c(r(), r())
This is a speed-up ratio of about $45$ times faster for Cython, but Numba completely here!
Why? No idea!
C is also slower than the Python version here, due to this lambda x,y: ... trick.
(852 - 576) / (1070 - 576) # for Python vs numba
(852 - 576) / (745 - 576) # for Python vs Cython
(852 - 576) / (911 - 576) # for Python vs C
klucbGauss_c = lambda x, y: kullback.klucbGauss(x, y, 0.25)
test_fs([klucbGauss, klucbGauss_numba, klucbGauss_cython, klucbGauss_c], lambda: (r(), r()))
%timeit klucbGauss(r(), r())
%timeit klucbGauss_numba(r(), r())
%timeit klucbGauss_cython(r(), r())
%timeit klucbGauss_c(r(), r())
This is a speed-up ratio of about $14$ times faster for Cython and $4$ times for C, and one more failure case for Numba.
The C version looses again due to the lambda x,y trick.
(1960 - 576) / (31300 - 576) # for Python vs numba
(1960 - 576) / (676 - 576) # for Python vs Cython
(1960 - 576) / (939 - 576) # for Python vs C
klucbBern_c = lambda x, y: kullback.klucbBern(x, y, 1e-6)
test_fs([klucbBern, klucbBern_numba, klucbBern_cython, klucbBern_c], lambda: (r(), r()))
%timeit klucbBern(r(), r())
%timeit klucbBern_numba(r(), r())
%timeit klucbBern_cython(r(), r())
%timeit klucbBern_c(r(), r())
This is a speed-up ratio of about $15$ times faster for Cython, and one more failure case for Numba. The speed-up for the C version is CRAZY here, and it shows that optimizing loops is the most important!
(91900 - 576) / (170000 - 576) # for Python vs numba
(91900 - 576) / (6930 - 576) # for Python vs Cython
(91900 - 576) / abs(314 - 576) # for Python vs C
klucbPoisson_c = lambda x, y: kullback.klucbPoisson(x, y, 1e-6)
test_fs([klucbPoisson, klucbPoisson_numba, klucbPoisson_cython, klucbPoisson_c], lambda: (r(), r()))
%timeit klucbPoisson(r(), r())
%timeit klucbPoisson_numba(r(), r())
%timeit klucbPoisson_cython(r(), r())
%timeit klucbPoisson_c(r(), r())
This is a speed-up ratio of about $15$ times faster for Cython, and one more failure case for Numba. It's again a strong victory for the C version, with a speed up of about $45$ !
(72600 - 576) / (167000 - 576) # for Python vs numba
(72600 - 576) / (5330 - 576) # for Python vs Cython
(72600 - 576) / (2180 - 576) # for Python vs Cython
klucbExp_c = lambda x, y: kullback.klucbExp(x, y, 1e-6)
test_fs([klucbExp, klucbExp_numba, klucbExp_cython, klucbExp_c], lambda: (r(), r()))
%timeit klucbExp(r(), r())
%timeit klucbExp_numba(r(), r())
%timeit klucbExp_cython(r(), r())
%timeit klucbExp_c(r(), r())
This is a speed-up ratio of about $17$ times faster for Cython, and one more failure case for Numba. Strong victory for the C version, a speed-up of $50$ is impressive!
(78700 - 576) / (156000 - 576) # for Python vs numba
(78700 - 576) / (4410 - 576) # for Python vs Cython
(78700 - 576) / (2040 - 576) # for Python vs Cython
%%bash
[ -f kullback.py.old ] && mv -vf kullback.py.old kullback.py
nopython code, and on some examples the nopython code can be slower than naive Python (like, crazily slower). No idea why, and the point was precisely not to try too much optimizing this use of Numba.The take away messages are the following:
numba.jit to speed them up,My advice for using Cython are the following:
%%cython magic is very easy!.pyx file, and use pyximport from your Python code to automatically compile and import it. It works perfectly fine, believe me!It's not too hard, but it's certainly not as easy as Cython.
My approach from now will to not even consider writing C code, as Cython offers a very good speedup when carefully used. And it's much easier to just write import pyximport before importing your Cython-accelerated module, in comparison to writing a setupy.py and requiring a compilation step for a C-optimized module!
That's it for today, folks! See this page for other notebooks I wrote recently.