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Choosing between several options in uncertain environments
1. METAGAMING:
Bandits with simple regret and small
budget
Chen-Wei Chou, Ping-Chiang Chou,
Chang-Shing Lee, David Lupien St-Pierre,
Olivier Teytaud, Mei-Hui Wang, Li-Wen Wu
and Shi-Jim Yen
2. Outline:
- what is a bandit problem ?
- what is a strategic bandit problem ?
- is a strategic bandit different from a bandit ?
- algorithms
- results
3. What is a bandit problem ?
A finite number of time steps
A (finite) number of options,
each of them equipped with a (unknown) proba distribution
At each time step:
- you choose one option
- you get a reward, distributed according to its proba distribution
At the end:
- you choose one option (you can not change anymore...)
- your reward is the expected reward associated to this option
4. What is a bandit problem ?
A finite number of time steps
A (finite) number of options,
each of them equipped with a (unknown) proba distribution
At each time step:
- you choose one option
- you get a reward, distributed according to its proba distribution
At the end:
- you choose one option (you can not change anymore...)
- your reward is the expected reward associated to this option
Here we collect
information
5. What is a bandit problem ?
A finite number of time steps
A (finite) number of options,
each of them equipped with a (unknown) proba distribution
At each time step:
- you choose one option
- you get a reward, distributed according to its proba distribution
At the end:
- you choose one option (you can not change anymore...)
- your reward is the expected reward associated to this option
Here we use
information for
the final choice
6. What is a bandit problem ?
A finite number of time steps
A (finite) number of options,
each of them equipped with a (unknown) proba distribution
At each time step:
- you choose one option
- you get a reward, distributed according to its proba distribution
At the end:
- you choose one option (you can not change anymore...)
- your reward is the expected reward associated to this option
Here, we
explore
7. What is a bandit problem ?
A finite number of time steps
A (finite) number of options,
each of them equipped with a (unknown) proba distribution
At each time step:
- you choose one option
- you get a reward, distributed according to its proba distribution
At the end:
- you choose one option (you can not change anymore...)
- your reward is the expected reward associated to this option
Here, we take no risk
8. What is a bandit problem ?
A finite number of time steps
A (finite) number of options,
each of them equipped with a (unknown) proba distribution
At each time step (exploration):
- you choose one option
- you get a reward, distributed according to its proba distribution
At the end (recommendation):
- you choose one option (you can not change anymore...)
- your reward is the expected reward associated to this option
9. Which kind of bandit ?
- in the bandit literature, options are
also termed “arms”
- here the criterion is the expected reward
of the option chosen at the end
(sometimes it is the sum
of the rewards during exploration)
- we presented here stochastic bandits
(a probability distribution
per option) ==> next slide is different
10. And adversarial bandit ?
A finite number of time steps
A (finite) number of options for player 1,
and a finite number of options for player 2.
An unknown probability distribution for each pair of options
At each time step:
- you choose one option for P1 and one option for P2
- you get a reward, distributed according to the
corresponding proba distribution
At the end:
- you choose one **probabilistic** option for P1
(you can not change anymore...)
- your reward is the expected reward associated to this option,
for the worst choice by P2
11. What is meta-gaming ?
What is “strategic choice” ?
Strategic choices:
- decisions once and for all, at a high level
- ≠ from tactical level
Meta-gaming: choice at a strategic level, in games:
- choosings cards, in card games
- choosing handicap positioning, in Go
==> once and for all, at the beginning of the game
12. Example of stochastic bandit
(i.e. 1P strategic choice)
Game of Go handicap bandit problem, at each time step:
- you choose one handicap positioning
- then you simulate one game from this position
==> only one player has a strategic choice
==> stochastic bandit
13. Example of adversarial bandit
(i.e. 2P strategic choice)
Urban Rivals bandit problem, at each time step:
- you choose
- one set of cards for you (P1)
- one set of cards for P2
- then you simulate one Urban Rivals game from this position
PLAYER 1:
PLAYER 2:
==> two players have a strategic choice
==> adversarial bandit
14. Is a strategic bandit problem
different from
a classical bandit problem ?
No difference in nature
Just a much
smaller budget
15. Algorithms
Reminder:
- two algorithms needed:
- one for choosing during N exploration steps
- one for choosing during 1 recommendation step
- two settings
- one-player case
- two-player case
16. Algorithms for exploration
Uniform: test all options uniformly
Bernstein races:
- uniformly among non discarded options,
- discard options with statistical tests
Successive reject:
- uniformly among non discarded options,
- discard periodically the worst option
UCB: choose option with best average result + bonus
for options weakly sampled,
Adaptive-UCB-E: a variant of UCB aimed at removing
hyper-parameters
EXP3: empirically best option + random perturbation
17. Algorithms for recommendation
Empirically Best Arm: choose empirically best option
Most Played Arm: choose most simulated option
Successive reject:: the only non discarded option
UCB: choose option with best average result + bonus
for options weakly sampled.
LCB: choose option with best average result + malus for
options weakly sampled.
Empirical distribution of play: an option has its
frequency (during exploration) as probability (for
recommendation)
TEXP3: idem, but discard low probability options
23. Do you know killall-Go ?
Black has stones in advance (e.g. 8 in 13x13).
If white makes life, white wins.
If black kills everything, black wins.
Black choose stones
positioning
(strategic decisions).
24. Left: human is Black and chooses E3 C4.
Right: computer is Black and chooses D3 D5.
White won both.
Human said that the computer choice D3 D5 is good.
25. Killall Go, H8 (left) H9 (right)
Left: Human Pro Player (5P) as black has 8 handicap stones.
White (computer) makes life and wins.
Right: Human Pro Player (5P) as black has 9 handicap stones
and kills everything and wins.
26. CONCLUSIONS
1 player case:
UCB for exploration,
LCB or MPA for recommendation
2 player case:
TEXP3 performs best.
Killall-Go
Win against pro with H2 in 7x7 Killall-Go as white.
Loss against pro with H2 in 7x7 Killall-Go as black.
13x13: Computer won as white with H8, lost with H9.
13x13: Computer lost as black with H8 and with H9.
Further work:
Structured bandit: some options are close to each other.
Batoo: Go with strategic choice for both players; nice test case.
Industry: choosing investments for power grid simulations – in progress.