Policies.SparseklUCB module

The SparseklUCB policy, designed to tackle sparse stochastic bandit problems:

• This means that only a small subset of size s of the K arms has non-zero means.

• The SparseklUCB algorithm requires to known exactly the value of s.

• This SparseklUCB is my version. It uses the KL-UCB index for both the decision in the UCB phase and the construction of the sets \(\mathcal{J}(t)\) and \(\mathcal{K}(t)\).

• The usual UCB indexes can be used for the sets by setting the flag use_ucb_for_sets to true.

• Reference: [[“Sparse Stochastic Bandits”, by J. Kwon, V. Perchet & C. Vernade, COLT 2017](https://arxiv.org/abs/1706.01383)] who introduced SparseUCB.

Warning

This algorithm only works for sparse Gaussian (or sub-Gaussian) stochastic bandits, of known variance.

class Policies.SparseklUCB.Phase

Bases: enum.Enum

Different states during the SparseklUCB algorithm.

• RoundRobin means all are sampled once.

• ForceLog uniformly explores arms that are in the set \(\mathcal{J}(t) \setminus \mathcal{K}(t)\).

• UCB is the phase that the algorithm should converge to, when a normal UCB selection is done only on the “good” arms, i.e., \(\mathcal{K}(t)\).

ForceLog = 2

RoundRobin = 1

UCB = 3

__module__ = 'Policies.SparseklUCB'

Policies.SparseklUCB.c = 1.0

default value, as it was in pymaBandits v1.0

Policies.SparseklUCB.USE_UCB_FOR_SETS = False

Default value for the flag controlling whether the usual UCB indexes are used for the sets \(\mathcal{J}(t)\) and \(\mathcal{K}(t)\). Default it to use the KL-UCB indexes, which should be more efficient.

class Policies.SparseklUCB.SparseklUCB(nbArms, sparsity=None, tolerance=0.0001, klucb=CPUDispatcher(<function klucbBern>), c=1.0, use_ucb_for_sets=False, lower=0.0, amplitude=1.0)

Bases: Policies.klUCB.klUCB

The SparseklUCB policy, designed to tackle sparse stochastic bandit problems.

• By default, assume sparsity = nbArms.

__init__(nbArms, sparsity=None, tolerance=0.0001, klucb=CPUDispatcher(<function klucbBern>), c=1.0, use_ucb_for_sets=False, lower=0.0, amplitude=1.0)

New generic index policy.

• nbArms: the number of arms,

• lower, amplitude: lower value and known amplitude of the rewards.

sparsity = None

Known value of the sparsity of the current problem.

use_ucb_for_sets = None

Whether the usual UCB indexes are used for the sets \(\mathcal{J}(t)\) and \(\mathcal{K}(t)\).

phase = None

Current phase of the algorithm.

force_to_see = None

Binary array for the set \(\mathcal{J}(t)\).

goods = None

Binary array for the set \(\mathcal{K}(t)\).

offset = None

Next arm to sample, for the Round-Robin phase

__str__()

-> str

startGame()

Initialize the policy for a new game.

update_j()

Recompute the set \(\mathcal{J}(t)\):

\[\begin{split}\hat{\mu}_k(t) &= \frac{X_k(t)}{N_k(t)}, \\ U^{\mathcal{J}}_k(t) &= \sup\limits_{q \in [a, b]} \left\{ q : \mathrm{kl}(\hat{\mu}_k(t), q) \leq \frac{c \log(N_k(t))}{N_k(t)} \right\},\\ \mathcal{J}(t) &= \left\{ k \in [1,...,K]\;, \hat{\mu}_k(t) \geq U^{\mathcal{J}}_k(t) - \hat{\mu}_k(t) \right\}.\end{split}\]

• If use_ucb_for_sets is True, the same formula from Policies.SparseUCB is used.

update_k()

Recompute the set \(\mathcal{K}(t)\):

\[\begin{split}\hat{\mu}_k(t) &= \frac{X_k(t)}{N_k(t)}, \\ U^{\mathcal{K}}_k(t) &= \sup\limits_{q \in [a, b]} \left\{ q : \mathrm{kl}(\hat{\mu}_k(t), q) \leq \frac{c \log(t)}{N_k(t)} \right\},\\ \mathcal{J}(t) &= \left\{ k \in [1,...,K]\;, \hat{\mu}_k(t) \geq U^{\mathcal{K}}_k(t) - \hat{\mu}_k(t) \right\}.\end{split}\]

• If use_ucb_for_sets is True, the same formula from Policies.SparseUCB is used.

__module__ = 'Policies.SparseklUCB'

choice()

Choose the next arm to play:

• If still in a Round-Robin phase, play the next arm,

• Otherwise, recompute the set \(\mathcal{J}(t)\),

• If it is too small, if \(\mathcal{J}(t) < s\):

  • Start a new Round-Robin phase from arm 0.

• Otherwise, recompute the second set \(\mathcal{K}(t)\),

• If it is too small, if \(\mathcal{K}(t) < s\):

  • Play a Force-Log step by choosing an arm uniformly at random from the set \(\mathcal{J}(t) \setminus K(t)\).

• Otherwise,

  • Play a UCB step by choosing an arm with highest KL-UCB index from the set \(\mathcal{K}(t)\).