# PoliciesMultiPlayers.rhoLearnExp3 module¶

rhoLearnExp3: implementation of a variant of the multi-player policy from [Distributed Algorithms for Learning…, Anandkumar et al., 2010](http://ieeexplore.ieee.org/document/5462144/), using the Exp3 learning algorithm instead of a random exploration for choosing the rank.

• Each child player is selfish, and plays according to an index policy (any index policy, e.g., UCB, Thompson, KL-UCB, BayesUCB etc),

• But instead of aiming at the best (the 1-st best) arm, player i aims at the rank_i-th best arm,

• At first, every player has a random rank_i from 1 to M, and when a collision occurs, rank_i is given by a second learning algorithm, playing on arms = ranks from [1, .., M], where M is the number of player.

• If rankSelection = Uniform, this is like rhoRand, but if it is a smarter policy (like Exp3 here), it might be better! Warning: no theoretical guarantees exist!

• Reference: [Proof-of-Concept System for Opportunistic Spectrum Access in Multi-user Decentralized Networks, S.J.Darak, C.Moy, J.Palicot, EAI 2016](https://doi.org/10.4108/eai.5-9-2016.151647), algorithm 2. (for BayesUCB only)

Note

This is not fully decentralized: as each child player needs to know the (fixed) number of players.

For the Exp3 algorithm:

PoliciesMultiPlayers.rhoLearnExp3.binary_feedback(sensing, collision)[source]

Count 1 iff the sensing authorized to communicate and no collision was observed.

$\begin{split}\mathrm{reward}(\text{user}\;j, \text{time}\;t) &:= r_{j,t} = F_{m,t} \times (1 - c_{m,t}), \\ \text{where}\;\; F_{m,t} &\; \text{is the sensing feedback (1 iff channel is free)}, \\ \text{and} \;\; c_{m,t} &\; \text{is the collision feedback (1 iff user j experienced a collision)}.\end{split}$
PoliciesMultiPlayers.rhoLearnExp3.ternary_feedback(sensing, collision)[source]

Count 1 iff the sensing authorized to communicate and no collision was observed, 0 if no communication, and -1 iff communication but a collision was observed.

$\begin{split}\mathrm{reward}(\text{user}\;j, \text{time}\;t) &:= F_{m,t} \times (2 r_{m,t} - 1), \\ \text{where}\;\; r_{j,t} &:= F_{m,t} \times (1 - c_{m,t}), \\ \text{and} \;\; F_{m,t} &\; \text{is the sensing feedback (1 iff channel is free)}, \\ \text{and} \;\; c_{m,t} &\; \text{is the collision feedback (1 iff user j experienced a collision)}.\end{split}$
PoliciesMultiPlayers.rhoLearnExp3.generic_ternary_feedback(sensing, collision, bonus=1, malus=-1)[source]

Count ‘bonus’ iff the sensing authorized to communicate and no collision was observed, ‘malus’ iff communication but a collision was observed, and 0 if no communication.

PoliciesMultiPlayers.rhoLearnExp3.make_generic_ternary_feedback(bonus=1, malus=-1)[source]
PoliciesMultiPlayers.rhoLearnExp3.generic_continuous_feedback(sensing, collision, bonus=1, malus=-1)[source]

Count ‘bonus’ iff the sensing authorized to communicate and no collision was observed, ‘malus’ iff communication but a collision was observed, but possibly does not count 0 if no communication.

$\begin{split}\mathrm{reward}(\text{user}\;j, \text{time}\;t) &:= \mathrm{malus} + (\mathrm{bonus} - \mathrm{malus}) \times \frac{r'_{j,t} + 1}{2}, \\ \text{where}\;\; r'_{j,t} &:= F_{m,t} \times (2 r_{m,t} - 1), \\ \text{where}\;\; r_{j,t} &:= F_{m,t} \times (1 - c_{m,t}), \\ \text{and} \;\; F_{m,t} &\; \text{is the sensing feedback (1 iff channel is free)}, \\ \text{and} \;\; c_{m,t} &\; \text{is the collision feedback (1 iff user j experienced a collision)}.\end{split}$
PoliciesMultiPlayers.rhoLearnExp3.make_generic_continuous_feedback(bonus=1, malus=-1)[source]
PoliciesMultiPlayers.rhoLearnExp3.reward_from_decoupled_feedback(sensing, collision)

Decide the default function to use. FIXME try all of them!

PoliciesMultiPlayers.rhoLearnExp3.CHANGE_RANK_EACH_STEP = False

Should oneRhoLearnExp3 players select a (possibly new) rank at each step ? The algorithm P2 from https://doi.org/10.4108/eai.5-9-2016.151647 suggests to do so. But I found it works better without this trick.

class PoliciesMultiPlayers.rhoLearnExp3.oneRhoLearnExp3(maxRank, rankSelectionAlgo, change_rank_each_step, feedback_function, *args, **kwargs)[source]

Class that acts as a child policy, but in fact it pass all its method calls to the mother class, who passes it to its i-th player.

• Except for the handleCollision method: a (possibly new) rank is sampled after observing a collision, from the rankSelection algorithm.

• When no collision is observed on a arm, a small reward is given to the rank used for this play, in order to learn the best ranks with rankSelection.

• And the player does not aim at the best arm, but at the rank-th best arm, based on her index policy.

__init__(maxRank, rankSelectionAlgo, change_rank_each_step, feedback_function, *args, **kwargs)[source]

Initialize self. See help(type(self)) for accurate signature.

maxRank = None

Max rank, usually nbPlayers but can be different

rank = None

Current rank, starting to 1

change_rank_each_step = None

Change rank at each step?

feedback_function = None

Feedback function: (sensing, collision) -> reward

__str__()[source]

Return str(self).

startGame()[source]

Initialize both rank and arm selection algorithms.

getReward(arm, reward)[source]

Give a “good” reward to the rank selection algorithm (no collision), give reward to the arm selection algorithm, and if self.change_rank_each_step, select a (possibly new) rank.

handleCollision(arm, reward)[source]

Give a “bad” reward to the rank selection algorithm, and select a (possibly new) rank.

__module__ = 'PoliciesMultiPlayers.rhoLearnExp3'
class PoliciesMultiPlayers.rhoLearnExp3.rhoLearnExp3(nbPlayers, nbArms, playerAlgo, rankSelectionAlgo=<class 'Policies.Exp3.Exp3Decreasing'>, maxRank=None, change_rank_each_step=False, feedback_function=<function binary_feedback>, lower=0.0, amplitude=1.0, *args, **kwargs)[source]

rhoLearnExp3: implementation of the multi-player policy from [Distributed Algorithms for Learning…, Anandkumar et al., 2010](http://ieeexplore.ieee.org/document/5462144/), using a learning algorithm instead of a random exploration for choosing the rank.

__init__(nbPlayers, nbArms, playerAlgo, rankSelectionAlgo=<class 'Policies.Exp3.Exp3Decreasing'>, maxRank=None, change_rank_each_step=False, feedback_function=<function binary_feedback>, lower=0.0, amplitude=1.0, *args, **kwargs)[source]
• nbPlayers: number of players to create (in self._players).

• playerAlgo: class to use for every players.

• nbArms: number of arms, given as first argument to playerAlgo.

• rankSelectionAlgo: algorithm to use for selecting the ranks.

• maxRank: maximum rank allowed by the rhoRand child (default to nbPlayers, but for instance if there is 2 × rhoRand[UCB] + 2 × rhoRand[klUCB], maxRank should be 4 not 2).

• *args, **kwargs: arguments, named arguments, given to playerAlgo.

Example:

>>> from Policies import *
>>> import random; random.seed(0); import numpy as np; np.random.seed(0)
>>> nbArms = 17
>>> nbPlayers = 6
>>> s = rhoLearnExp3(nbPlayers, nbArms, UCB)
>>> [ child.choice() for child in s.children ]
[0, 1, 9, 0, 10, 3]
>>> [ child.choice() for child in s.children ]
[11, 2, 0, 0, 4, 5]

• To get a list of usable players, use s.children.

• Warning: s._players is for internal use ONLY!

maxRank = None

Max rank, usually nbPlayers but can be different

nbPlayers = None

Number of players

children = None

List of children, fake algorithms

rankSelectionAlgo = None

Policy to use to chose the ranks

nbArms = None

Number of arms

change_rank_each_step = None

Change rank at every steps?

__module__ = 'PoliciesMultiPlayers.rhoLearnExp3'
__str__()[source]

Return str(self).