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Fitting a single classifier does not take long, fitting hundreds takes a while. To find the best hyperparameters you need to fit a lot of classifiers. What to do?
This post explores the inner workings of an algorithm you can use to reduce the number of hyperparameter sets you need to try before finding the best set. The algorithm goes under the name of bayesian optimisation. If you are looking for a production ready implementation check out: MOE, metric optimisation engine developed by Yelp.
Gaussian processe regression is a useful tool in general and is used heavily here. Check out my post on Gaussian processes with george for a short introduction.
This post starts with an example where we know the true form of the scoring function. Followed by pitting random grid search against Bayesian optimisation to find the best hyper-parameter for a real classifier.
As usual first some setup and importing:
%matplotlib inline
import random
import numpy as np
np.random.seed(9)
from scipy.stats import randint as sp_randint
import matplotlib.pyplot as plt
import seaborn as sns
sns.set_style(‘whitegrid‘)
sns.set_context("talk")
Bayesian optimisation uses gaussian processes to fit a regression model to the previously evaluated points in hyper-parameter space. This model is then used to suggest the next (best) point in hyper-parameter space to evaluate the model at.
To choose the best point we need to define a criterion, in this case we use "expected improvement". As we only know the score to with a certain precision we do not want to simply choose the point with the best score. Instead we pick the point which promises the largest expected improvement. This allows us to incorporate the uncertainty about our estimation of the scoring function into the procedure. It leads to a mixture of exploitation and exploration of the parameter space.
Below we setup a toy scoring function (−xsinx), sample a two points from it, and fit our gaussian process model to it.
import george
from george.kernels import ExpSquaredKernel
score_func = lambda x: -x*np.sin(x)
x = np.arange(0, 10, 0.1)
# Generate some fake, noisy data. These represent
# the points in hyper-parameter space for which
# we already trained our classifier and evaluated its score
xp = 10 * np.sort(np.random.rand(2))
yerr = 0.2 * np.ones_like(xp)
yp = score_func(xp) + yerr * np.random.randn(len(xp))
# Set up a Gaussian process
kernel = ExpSquaredKernel(1)
gp = george.GP(kernel)
gp.compute(xp, yerr)
mu, cov = gp.predict(yp, x)
std = np.sqrt(np.diag(cov))
def basic_plot():
fig, ax = plt.subplots()
ax.plot(x, mu, label="GP median")
ax.fill_between(x, mu-std, mu+std, alpha=0.5)
ax.plot(x, score_func(x), ‘--‘, label=" True score function (unknown)")
# explicit zorder to draw points and errorbars on top of everything
ax.errorbar(xp, yp, yerr=yerr, fmt=‘ok‘, zorder=3, label="samples")
ax.set_ylim(-9,6)
ax.set_ylabel("score")
ax.set_xlabel(‘hyper-parameter X‘)
ax.legend(loc=‘best‘)
return fig,ax
basic_plot()
The dashed green line represents the true value of the scoring function as a function of our hypothetical hyper-parameter X. The black dots (and their errorbars) represent points at which we evaluated our classifier and calculated the score. In blue our regression model trying to predict the value of the score function. The shaded area represents the uncertainty on the median (solid blue line) value of the estimated score function value.
Next let‘s calculate the expected improvement at every value of the hyper-parameter X. We also build a multistart optimisation routine (next_sample) which uses the expected improvement to suggest which point to sample next.
from scipy.optimize import minimize
from scipy import stats
def expected_improvement(points, gp, samples, bigger_better=False):
# are we trying to maximise a score or minimise an error?
if bigger_better:
best_sample = samples[np.argmax(samples)]
mu, cov = gp.predict(samples, points)
sigma = np.sqrt(cov.diagonal())
Z = (mu-best_sample)/sigma
ei = ((mu-best_sample) * stats.norm.cdf(Z) + sigma*stats.norm.pdf(Z))
# want to use this as objective function in a minimiser so multiply by -1
return -ei
else:
best_sample = samples[np.argmin(samples)]
mu, cov = gp.predict(samples, points)
sigma = np.sqrt(cov.diagonal())
Z = (best_sample-mu)/sigma
ei = ((best_sample-mu) * stats.norm.cdf(Z) + sigma*stats.norm.pdf(Z))
# want to use this as objective function in a minimiser so multiply by -1
return -ei
def next_sample(gp, samples, bounds=(0,10), bigger_better=False):
"""Find point with largest expected improvement"""
best_x = None
best_ei = 0
# EI is zero at most values -> often get trapped
# in a local maximum -> multistarting to increase
# our chances to find the global maximum
for rand_x in np.random.uniform(bounds[0], bounds[1], size=30):
res = minimize(expected_improvement, rand_x,
bounds=[bounds],
method=‘L-BFGS-B‘,
args=(gp, samples, bigger_better))
if res.fun <