Policies.SparseUCB module

The SparseUCB 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 SparseUCB algorithm requires to known exactly the value of s.

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

Warning

This algorithm only works for sparse Gaussian (or sub-Gaussian) stochastic bandits.

class Policies.SparseUCB.Phase

Bases: enum.Enum

Different states during the SparseUCB 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.SparseUCB'

Policies.SparseUCB.ALPHA = 4

Default parameter for \(\alpha\) for the UCB indexes.

class Policies.SparseUCB.SparseUCB(nbArms, sparsity=None, alpha=4, lower=0.0, amplitude=1.0)

Bases: Policies.UCBalpha.UCBalpha

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

• By default, assume sparsity = nbArms.

__init__(nbArms, sparsity=None, alpha=4, 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.

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 \(\mathcal{J}(t)\):

\[\mathcal{J}(t) = \left\{ k \in [1,...,K]\;, \frac{X_k(t)}{N_k(t)} \geq \sqrt{\frac{\alpha \log(N_k(t))}{N_k(t)}} \right\}.\]

update_k()

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

\[\mathcal{K}(t) = \left\{ k \in [1,...,K]\;, \frac{X_k(t)}{N_k(t)} \geq \sqrt{\frac{\alpha \log(t)}{N_k(t)}} \right\}.\]

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 \(\mathcal{J}(t) \setminus \mathcal{K}(t)\).

• Otherwise,

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

__module__ = 'Policies.SparseUCB'