# -*- coding: utf-8 -*-
""" The generic KL-UCB policy for one-parameter exponential distributions.
- By default, it assumes Bernoulli arms.
- Reference: [Garivier & Cappé - COLT, 2011](https://arxiv.org/pdf/1102.2490.pdf).
"""
from __future__ import division, print_function # Python 2 compatibility
__author__ = "Olivier Cappé, Aurélien Garivier, Lilian Besson"
__version__ = "0.6"
from math import log
import numpy as np
np.seterr(divide='ignore') # XXX dangerous in general, controlled here!
try:
from .kullback import klucbBern
from .IndexPolicy import IndexPolicy
except (ImportError, SystemError):
from kullback import klucbBern
from IndexPolicy import IndexPolicy
#: Default value for the constant c used in the computation of KL-UCB index.
c = 1. #: default value, as it was in pymaBandits v1.0
# c = 1. #: as suggested in the Theorem 1 in https://arxiv.org/pdf/1102.2490.pdf
#: Default value for the tolerance for computing numerical approximations of the kl-UCB indexes.
TOLERANCE = 1e-4
# --- Class
[docs]class klUCB(IndexPolicy):
""" The generic KL-UCB policy for one-parameter exponential distributions.
- By default, it assumes Bernoulli arms.
- Reference: [Garivier & Cappé - COLT, 2011](https://arxiv.org/pdf/1102.2490.pdf).
"""
[docs] def __init__(self, nbArms, tolerance=TOLERANCE, klucb=klucbBern, c=c, lower=0., amplitude=1.):
super(klUCB, self).__init__(nbArms, lower=lower, amplitude=amplitude)
self.c = c #: Parameter c
self.klucb = klucb #: kl function to use
self.klucb_vect = np.vectorize(klucb) #: kl function to use, in a vectorized way using :func:`numpy.vectorize`.
self.klucb_vect.__name__ = klucb.__name__
self.tolerance = tolerance #: Numerical tolerance
[docs] def __str__(self):
name = self.klucb.__name__[5:]
if name == "Bern": name = ""
complement = "{}{}".format(name, "" if self.c == 1 else r"$c={:.3g}$".format(self.c))
if complement != "": complement = "({})".format(complement)
return r"kl-UCB{}".format(complement)
[docs] def computeIndex(self, arm):
r""" Compute the current index, at time t and after :math:`N_k(t)` pulls of arm k:
.. math::
\hat{\mu}_k(t) &= \frac{X_k(t)}{N_k(t)}, \\
U_k(t) &= \sup\limits_{q \in [a, b]} \left\{ q : \mathrm{kl}(\hat{\mu}_k(t), q) \leq \frac{c \log(t)}{N_k(t)} \right\},\\
I_k(t) &= U_k(t).
If rewards are in :math:`[a, b]` (default to :math:`[0, 1]`) and :math:`\mathrm{kl}(x, y)` is the Kullback-Leibler divergence between two distributions of means x and y (see :mod:`Arms.kullback`),
and c is the parameter (default to 1).
"""
if self.pulls[arm] < 1:
return float('+inf')
else:
# XXX We could adapt tolerance to the value of self.t
return self.klucb(self.rewards[arm] / self.pulls[arm], self.c * log(self.t) / self.pulls[arm], self.tolerance)
[docs] def computeAllIndex(self):
""" Compute the current indexes for all arms, in a vectorized manner."""
indexes = self.klucb_vect(self.rewards / self.pulls, self.c * np.log(self.t) / self.pulls, self.tolerance)
indexes[self.pulls < 1] = float('+inf')
self.index[:] = indexes