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CS885 Reinforcement Learning Lecture 8b: May 25, 2018 Bayesian and Contextual Bandits [SutBar] Sec. 2.9 University of Waterloo CS885 Spring 2018 Pascal Poupart 1 Outline Bayesian bandits Thompson sampling Contextual bandits

  1. CS885 Reinforcement Learning Lecture 8b: May 25, 2018 Bayesian and Contextual Bandits [SutBar] Sec. 2.9 University of Waterloo CS885 Spring 2018 Pascal Poupart 1

  2. Outline • Bayesian bandits – Thompson sampling • Contextual bandits University of Waterloo CS885 Spring 2018 Pascal Poupart 2

  3. Multi-Armed Bandits • Problem: – ! bandits with unknown average reward "($) – Which arm $ should we play at each time step? – Exploitation/exploration tradeoff • Common frequentist approaches: – & -greedy – Upper confidence bound (UCB) • Alternative Bayesian approaches – Thompson sampling – Gittins indices University of Waterloo CS885 Spring 2018 Pascal Poupart 3

  4. Bayesian Learning • Notation: – ! " : random variable for # ’s rewards – Pr ! " ; ' : unknown distribution (parameterized by ' ) – ( # = *[! " ] : unknown average reward • Idea: – Express uncertainty about ' by a prior Pr(') " , ! " , … , ! " ) based on – Compute posterior Pr('|! 0 2 4 " observed for # so far. " , ! " , … , ! samples ! 0 2 4 • Bayes theorem: " ∝ Pr ' Pr(! " , ! " , … , ! " , ! " , … , ! " |') Pr ' ! 0 2 4 0 2 4 University of Waterloo CS885 Spring 2018 Pascal Poupart 4

  5. Distributional Information • Posterior over ! allows us to estimate – Distribution over next reward " # # = - # 0! Pr " # |" Pr " # ; ! Pr ! " # , " # , … , " # , " # , … , " + + ' ) ' ) . – Distribution over 1(3) when ! includes the mean # = Pr ! " # if ! = 1(3) # , " # , … , " # , " # , … , " Pr 1(3) " ' ) + ' ) + • To guide exploration: # , " ) # , … , " + # ) ≥ 1 − = – UCB : Pr 1 3 ≤ 67890( " ' # , " ) # , … , " + # – Bayesian techniques: Pr 1 3 | " ' University of Waterloo CS885 Spring 2018 Pascal Poupart 5

  6. Coin Example • Consider two biased coins ! " and ! # $ ! " = Pr ! " = ℎ)*+ $ ! # = Pr ! # = ℎ)*+ • Problem: – Maximize # of heads in , flips – Which coin should we choose for each flip? University of Waterloo CS885 Spring 2018 Pascal Poupart 6

  7. Bernoulli Variables • ! " # , ! " $ are Bernoulli variables with domain {0,1} • Bernoulli dist. are parameterized by their mean – i.e. Pr ! " # ; - . = - . = 0 1 . Pr ! " $ ; - 2 = - 2 = 0(1 2 ) University of Waterloo CS885 Spring 2018 Pascal Poupart 7

  8. Beta distribution • Let the prior Pr($) be a Beta distribution &'() $; +, - ∝ $ /01 1 − $ 401 &'() $; 1, 1 &'() $; 2, 8 • + − 1: # of heads &'()($; 20, 80) • - − 1 : # of tails Pr($) • 6 $ = +/(+ + -) $ University of Waterloo CS885 Spring 2018 Pascal Poupart 8

  9. Belief Update • Prior: Pr # = %&'( #; *, , ∝ # ./0 1 − # 3/0 • Posterior after coin flip: Pr # ℎ&(5 ∝ Pr # Pr ℎ&(5 # ∝ # ./0 1 − # 3/0 # = # .60 /0 1 − # 3/0 ∝ %&'((#; * + 1, ,) Pr # '(:; ∝ Pr # Pr '(:; # ∝ # ./0 1 − # 3/0 (1 − #) = # ./0 1 − # (360)/0 ∝ %&'((#; *, , + 1) University of Waterloo CS885 Spring 2018 Pascal Poupart 9

  10. Thompson Sampling • Idea: – Sample several potential average rewards: . , … , - . ) for each # ! " # , … ! & # ~ Pr(!(#)|- " / – Estimate empirical average ! # = " & 0 & ∑ 34" ! 3 (#) – Execute #-56#7 . 0 ! # • Coin example . = Beta < . ; > . , ? . . , … , - – Pr !(#) - " / where > . − 1 = #ℎF#GH and ? . − 1 = #I#JKH University of Waterloo CS885 Spring 2018 Pascal Poupart 10

  11. Thompson Sampling Algorithm Bernoulli Rewards ThompsonSampling( ℎ ) " ← 0 For % = 1 to ℎ Sample * + , , … , * / (,) ~ Pr(* , ) ∀, * , ← + 6 / / ∑ 89+ * 8 (,) ∀, , ∗ ← argmax ? 6 * , Execute , ∗ and receive @ " ← " + @ Update Pr(*(, ∗ )) based on @ Return " University of Waterloo CS885 Spring 2018 Pascal Poupart 11

  12. Comparison Thompson Sampling Greedy Strategy • Action Selection • Action Selection ! ∗ = argmax ) * ! ∗ = argmax ) < + ! + ! • Empirical mean • Empirical mean + ! = , + ! = , - 9 7 * < - ∑ /0, 9 ∑ /0, 6 / + / (!) • Samples • Samples 7 … 6 7 ~ Pr(6 7 ; ;) 7 ) 6 + / ! ~ Pr(+ / (!)|6 9 , / 7 ~ Pr(6 7 ; ;) 6 / • Some exploration • No exploration University of Waterloo CS885 Spring 2018 Pascal Poupart 12

  13. Sample Size • In Thompson sampling, amount of data ! and sample size " regulate amount of exploration • As ! and " increase, # $ % becomes less stochastic, which reduces exploration . … , . ) becomes more peaked – As ! ↑ , Pr($(%)|, 0 - . … , – As " ↑ , # . ] $ % approaches 1[$(%)|, - 0 • The stochasticity of # $(%) ensures that all actions are chosen with some probability University of Waterloo CS885 Spring 2018 Pascal Poupart 13

  14. Analysis • Thompson sampling converges to best arm • Theory: – Expected cumulative regret: !(log &) – On par with UCB and ( -greedy • Practice: – Sample size ) often set to 1 University of Waterloo CS885 Spring 2018 Pascal Poupart 14

  15. Contextual Bandits • In many applications, the context provides additional information to select an action – E.g., personalized advertising, user interfaces – Context: user demographics (location, age, gender) • Actions can also be characterized by features that influence their payoff – E.g., ads, webpages – Action features: topics, keywords, etc. University of Waterloo CS885 Spring 2018 Pascal Poupart 15

  16. Contextual Bandits • Contextual bandits: multi-armed bandits with states (corresponding to contexts) and action features • Formally: – ! : set of states where each state " is defined by a vector of features # $ = (' ( $ , ' * $ , … , ' , $ ) – . : set of actions where each action a is associated with a vector of features # / = (' ( / , ' * / , … , ' 0 / ) – Space of rewards (often ℝ ) • No transition function since the states at each step are independent • Goal find policy 2: # $ → 5 that maximizes expected rewards 6 7 ", 5 = 6(7|# $ , # / ) University of Waterloo CS885 Spring 2018 Pascal Poupart 16

  17. Approximate Reward Function • Common approach: – learn approximate average reward function " #, % = ! ! "(() (where ( = (( * , ( + ) ) by regression • Linear approximation: ! " , ( = , - ( • Non-linear approximation: ! " , ( = ./01%23/4((; ,) University of Waterloo CS885 Spring 2018 Pascal Poupart 17

  18. Bayesian Linear Regression • Consider a Gaussian prior: !"# $ = & $ ', ) * + ∝ -.! − $ 0 $ 2) * • Consider also a Gaussian likelihood: !"# 2|4, $ = & 2 $ 0 4, 5 * ∝ -.! − 2 − $ 0 4 * 25 * • The posterior is also Gaussian: !"# $|2, 4 ∝ !"# $ Pr 2 4, $ ∝ -.! − $ 0 $ -.! − 2 − $ 0 4 * 2) * 25 * = &($|9, :) where 9 = 5 <* :42 and : = 5 <* 44 0 + ) <* + <> University of Waterloo CS885 Spring 2018 Pascal Poupart 18

  19. Predictive Posterior • Consider a state-action pair (" # , " % ) = " for which we would like to predict the reward ( • Predictive posterior: . / ( . 0 ", 1 2 / . 3, 4 *. )*+ (|" = ∫ = /((|1 2 " 0 3, " 0 4") • UCB: Pr ( < 1 2 " 0 3 + 9 " 0 4" > 1 − = where 9 = 1 + ln 2/= /2 ( ~ /((|1 2 " 0 3, " 0 4") • Thomson sampling: B University of Waterloo CS885 Spring 2018 Pascal Poupart 19

  20. Upper Confidence Bound (UCB) Linear Gaussian UCB( ℎ ) " ← 0 , %&' (|*, , = . ( /, 0 1 2 Repeat until 3 = ℎ Receive state 4 5 For each action 4 6 where 4 = (4 5 , 4 6 ) do 9:3';&<39<=:>3&(?) = @ 1 4 A * + 9 4 A ,4 ? ∗ ← argmax I 9:3';&<39<=:>3&(?) Execute ? ∗ and receive J " ← " + J update * and , based on 4 = (4 5 , 4 6 ∗ ) and J Return " University of Waterloo CS885 Spring 2018 Pascal Poupart 20

  21. Thompson Sampling Algorithm Linear Gaussian ThompsonSampling( ℎ ) " ← 0 ; %&' (|*, , = . ( /, 0 1 2 For 3 = 1 to ℎ Receive state 7 8 For each action 7 9 where 7 = (7 8 , 7 9 ) do Sample < = > , … , < @ (>) ~ .(B|C 1 7 D *, 7 D ,7) = E @ @ ∑ GH= < > ← < G (>) > ∗ ← argmax O E < > Execute > ∗ and receive B " ← " + B update * and , based on 7 = (7 8 , 7 9 ∗ ) and B Return " University of Waterloo CS885 Spring 2018 Pascal Poupart 21

  22. Industrial Use • Contextual bandits are now commonly used for – Personalized advertising – Personalized web content • MSN news: 26% improvement in click through rate after adoption of contextual bandits (https://www.microsoft.com/en- us/research/blog/real-world-interactive-learning- cusp-enabling-new-class-applications/) University of Waterloo CS885 Spring 2018 Pascal Poupart 22

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