Multi-Armed Bandits. Credit: David Silver. Google DeepMind. Presenter: Tianlu Wang

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1 Multi-Armed Bandits Credit: David Silver Google DeepMind Presenter: Tianlu Wang Credit: David Silver (DeepMind) Multi-Armed Bandits Presenter: Tianlu Wang 1 / 27

2 Outline 1 Introduction Exploration vs. Exploitation Dilemma How to do Exploration 2 Multi-Armed Bandits The Multi-Armed Bandit Regret Greedy and ɛ-greedy algorithms Lower Bound Upper Confidence Bound Credit: David Silver (DeepMind) Multi-Armed Bandits Presenter: Tianlu Wang 2 / 27

3 Outline 1 Introduction Exploration vs. Exploitation Dilemma How to do Exploration 2 Multi-Armed Bandits The Multi-Armed Bandit Regret Greedy and ɛ-greedy algorithms Lower Bound Upper Confidence Bound Credit: David Silver (DeepMind) Multi-Armed Bandits Presenter: Tianlu Wang 3 / 27

4 Exploration vs. Exploitation Dilemma Online decision-making involves a fundamental choice: Exploitation Make the best decision given current information Exploration Gather more information The best long-term strategy may involve short-term sacrifices Gather enough information to make the best overall decisions Examples: Online Banner Advertisements: Exploitation Show the most successful advert; Exploration Show a different advert Game Playing: Exploitation Play the move you believe is best; Exploration Play an experimental move Credit: David Silver (DeepMind) Multi-Armed Bandits Presenter: Tianlu Wang 4 / 27

5 Outline 1 Introduction Exploration vs. Exploitation Dilemma How to do Exploration 2 Multi-Armed Bandits The Multi-Armed Bandit Regret Greedy and ɛ-greedy algorithms Lower Bound Upper Confidence Bound Credit: David Silver (DeepMind) Multi-Armed Bandits Presenter: Tianlu Wang 5 / 27

6 How to do Exploration Random exploration: ɛ-greedy: Pull a random chosen arm a fraction ɛ of the time and the other 1 ɛ time, pull the arm which estimated to be the most profitable.(devote a fraction ɛ of resources to testing) Credit: David Silver (DeepMind) Multi-Armed Bandits Presenter: Tianlu Wang 6 / 27

$How to do Exploration Random exploration: ɛ-greedy: Pull a random chosen arm a fraction ɛ of the time and the other 1 ɛ time, pull the arm which$

7 How to do Exploration Random exploration: ɛ-greedy: Pull a random chosen arm a fraction ɛ of the time and the other 1 ɛ time, pull the arm which estimated to be the most profitable.(devote a fraction ɛ of resources to testing) Credit: David Silver (DeepMind) Multi-Armed Bandits Presenter: Tianlu Wang 6 / 27

8 How to do Exploration Optimism in the face of uncertainty: Estimate uncertainty on value Prefer to explore states/actions with highest uncertainty Credit: David Silver (DeepMind) Multi-Armed Bandits Presenter: Tianlu Wang 7 / 27

9 How to do Exploration Information state space(most correct but computationally difficult): Consider agent s information as part of its state Look ahead to see how information helps reward Credit: David Silver (DeepMind) Multi-Armed Bandits Presenter: Tianlu Wang 8 / 27

10 Outline 1 Introduction Exploration vs. Exploitation Dilemma How to do Exploration 2 Multi-Armed Bandits The Multi-Armed Bandit Regret Greedy and ɛ-greedy algorithms Lower Bound Upper Confidence Bound Credit: David Silver (DeepMind) Multi-Armed Bandits Presenter: Tianlu Wang 9 / 27

11 The Multi-Armed Bandit A multi-armed bandit is a tuple < A, R > A is a known set of m actions(or arms ) R a (r) = P[r a] is an unknown probability distribution over rewards At each step t the agent selects an action a t A The environment generates a reward r t R at The goal is to maximise cumulative reward Σ t τ=1 r τ Credit: David Silver (DeepMind) Multi-Armed Bandits Presenter: Tianlu Wang 10 / 27

12 Outline 1 Introduction Exploration vs. Exploitation Dilemma How to do Exploration 2 Multi-Armed Bandits The Multi-Armed Bandit Regret Greedy and ɛ-greedy algorithms Lower Bound Upper Confidence Bound Credit: David Silver (DeepMind) Multi-Armed Bandits Presenter: Tianlu Wang 11 / 27

13 Regret The action-value is the mean reward for action a: The optimal value V is Q(a) = E[r a] V = Q(a ) = max a A Q(a) The regret is the opportunity loss for one step: I t = E[V Q(a t )] The total regret is the total opportunity loss L t = E[Σ t τ=1v Q(a τ )] Maximise cumulative reward minimise total regret Credit: David Silver (DeepMind) Multi-Armed Bandits Presenter: Tianlu Wang 12 / 27

14 Counting Regret The count N t (a) is expected number of selections for action a The gap a is the difference in value between action a and optimal action a, = V Q(a) Regret is a function of gaps and the counts: L t = E[Σ t τ=1v Q(a τ )] = Σ a A E[N t (a)](v Q(a)) = Σ a A E[N t (a)] a (1) A good algorithm ensures small counts for large gaps Problem: gaps are not known. Credit: David Silver (DeepMind) Multi-Armed Bandits Presenter: Tianlu Wang 13 / 27

15 Linear or Sublinear regret If an algorithm forever explores it will have linear total regret If an algorithm never explores it will have linear total regret Is it possible to achieve sublinear total regret? Credit: David Silver (DeepMind) Multi-Armed Bandits Presenter: Tianlu Wang 14 / 27

16 Outline 1 Introduction Exploration vs. Exploitation Dilemma How to do Exploration 2 Multi-Armed Bandits The Multi-Armed Bandit Regret Greedy and ɛ-greedy algorithms Lower Bound Upper Confidence Bound Credit: David Silver (DeepMind) Multi-Armed Bandits Presenter: Tianlu Wang 15 / 27

17 Greedy Algorithm We consider algorithms that estimate ˆQ t (a) Q(a) Estimate the value of each action by Monte-Carlo evaluation: ˆQ t (a) = 1 N t (a) ΣT t=1r t 1(a t = a) The greedy algorithm selects action with highest value: a t = argmax a A ˆQ t (a) Greedy can lock onto a suboptimal action forever Greedy has linear total regret Credit: David Silver (DeepMind) Multi-Armed Bandits Presenter: Tianlu Wang 16 / 27

18 ɛ-greedy Algorithm The ɛ-greedy algorithm continues to explore forever With probability 1 ɛ select a = argmax a A ˆQ(a) With probability ɛ select a random action Constant ɛ ensures minimum regret: ɛ-greedy has linear total regret I t ɛ A Σ a A a Credit: David Silver (DeepMind) Multi-Armed Bandits Presenter: Tianlu Wang 17 / 27

19 Optimistic Initialisation Initialise Q(a) to high value Update action value by incremental Monte-Carlo evaluation: ˆQ t (a t ) = ˆQ t N t (a t ) (r t ˆQ t 1 ) Encourages systematic exploration early on But can still lock onto suboptimal action greedy(ɛ-greedy) + optimistic initialisation has linear total regret Credit: David Silver (DeepMind) Multi-Armed Bandits Presenter: Tianlu Wang 18 / 27

20 Decaying ɛ t -Greedy Algorithm Pick a decay schedule for ɛ 1, ɛ 2,... Consider the following schedule: c > 0 d = min a a>0 a ɛ t = min{1, c A d 2 t } Logarithmic asymptotic total regret Requires advance knowledge of gaps Goal: find an algorithm with sublinear regret for any multi-armed bandit (without knowledge of R) Credit: David Silver (DeepMind) Multi-Armed Bandits Presenter: Tianlu Wang 19 / 27

21 Outline 1 Introduction Exploration vs. Exploitation Dilemma How to do Exploration 2 Multi-Armed Bandits The Multi-Armed Bandit Regret Greedy and ɛ-greedy algorithms Lower Bound Upper Confidence Bound Credit: David Silver (DeepMind) Multi-Armed Bandits Presenter: Tianlu Wang 20 / 27

22 Lower Bound The performance of any algorithm is determined by similarity between optimal arm and other arms Hard problems have similar-looking arms with different means This is described formally by the gap a and the similarity in distributions KL(R a R a ) Theorem (Lai and Robbins) Asymptotic total regret is at least logarithmic in number of steps a lim L t log tσ a a>0 t KL(R a R a ) Credit: David Silver (DeepMind) Multi-Armed Bandits Presenter: Tianlu Wang 21 / 27

23 Outline 1 Introduction Exploration vs. Exploitation Dilemma How to do Exploration 2 Multi-Armed Bandits The Multi-Armed Bandit Regret Greedy and ɛ-greedy algorithms Lower Bound Upper Confidence Bound Credit: David Silver (DeepMind) Multi-Armed Bandits Presenter: Tianlu Wang 22 / 27

24 Upper Confidence Bound Which action should we pick? The more uncertain we are about an action-value The more important it is to explore that action It could turn out to be the best action Credit: David Silver (DeepMind) Multi-Armed Bandits Presenter: Tianlu Wang 23 / 27

25 Upper Confidence Bound Estimate an upper confidence Û t (a) for each action value Such that Q(a) ˆQ t (a) + Û t (a) with high probability This depends on the number of times N(a) has been selected Small N t (a) largeû t (a)(estimated value is uncertain) Large N t (a) smallût(a)(estimated value is accurate) Select action maximising Upper Confidence Bound (UCB) a t = argmax a A ˆQt (a) + Ût(a) Credit: David Silver (DeepMind) Multi-Armed Bandits Presenter: Tianlu Wang 24 / 27

26 Upper Confidence Bound Theorem (Hoeffding s Inequality) Let X 1,..., X t be i.i.d. random variables in [0,1], and let X t = 1 τ Σt τ=1 X τ be the sample mean. Then P[E[X ] > X t + u] e 2tu2 We will apply Hoeffdings Inequality to rewards of the bandit conditioned on selecting action a P[Q(a) > ˆQ t (a) + U t (a)] e 2Nt(a)Ut(a)2 Credit: David Silver (DeepMind) Multi-Armed Bandits Presenter: Tianlu Wang 25 / 27

27 Calculating Upper Confidence Bounds Pick a probability p that true value exceeds UCB Now solve for U t (a): e 2Nt(a)Ut(a)2 = p log p U t (a) = 2N t (a) Reduce p as we observe more rewards, e.g. p = t 4 Ensures we select optimal action as t Credit: David Silver (DeepMind) Multi-Armed Bandits Presenter: Tianlu Wang 26 / 27

28 UCB1 This leads to the UCB1 algorithm a t = argmax a A Q(a) + 2 log t N t (a) Theorem The UCB algorithm achieves logarithmic asymptotic total regret lim t L t 8 log tσ a a>0 a Credit: David Silver (DeepMind) Multi-Armed Bandits Presenter: Tianlu Wang 27 / 27

COS 402 Machine Learning and Artificial Intelligence Fall Lecture 22. Exploration & Exploitation in Reinforcement Learning: MAB, UCB, Exp3

COS 402 Machine Learning and Artificial Intelligence Fall 2016 Lecture 22 Exploration & Exploitation in Reinforcement Learning: MAB, UCB, Exp3 How to balance exploration and exploitation in reinforcement