New Algorithms for Contextual Bandits

Transcription

1 New Algorithms for Contextual Bandits Lev Reyzin Georgia Institute of Technology Work done at Yahoo! 1

2 S A. Beygelzimer, J. Langford, L. Li, L. Reyzin, R.E. Schapire Contextual Bandit Algorithms with Supervised Learning Guarantees (AISTATS 2011) S M. Dudik, D. Hsu, S. Kale, N. Karampatziakis, J. Langford, L. Reyzin, T. Zhang Efficient Optimal Learning for Contextual Bandits (UAI 2011) S S. Kale, L. Reyzin, R.E. Schapire Non-Stochastic Bandit Slate Problems (NIPS 2010) 2

3 Serving Content to Users Query, IP address, browser properties, etc. 3

4 Serving Content to Users Query, IP address, browser properties, etc. result (ie. ad, news story) 4

5 Serving Content to Users Query, IP address, browser properties, etc. result (ie. ad, news story) click or not 5

6 Serving Content to Users Query, IP address, browser properties, etc. result (ie. ad, news story) click or not 6

7 Serving Content to Users Query, IP address, browser properties, etc. result (ie. ad, news story) click or not 7

8 Serving Content to Users 8

9 Outline S The setting and some background S Show ideas that fail S Give a high probability optimal algorithm S Dealing with VC sets S An efficient algorithm S Slates 9

10 Multiarmed Bandits [Robbins 52] T $ k 10

11 Multiarmed Bandits [Robbins 52] T $0.50 $ k 11

12 Multiarmed Bandits [Robbins 52] T $0.50 $ $0 k 12

13 Multiarmed Bandits [Robbins 52] T $ $0.50 $ $0 k 13

14 Multiarmed Bandits [Robbins 52] T $0.20 $0.4T 2 $0.50 $ $0 k $

15 Multiarmed Bandits [Robbins 52] 1 $0.5T $0.4T 2 $0.2T 3 $0.33T k $0.1T 15

16 Multiarmed Bandits [Robbins 52] 1 $0.5T $0.4T 2 $0.2T 3 $0.33T regret = $0.1T k $0.1T 16

17 Contextual Bandits [Auer-CesaBianchi-Freund-Schapire 02] context: T $ k N experts/policies/functions think of N >> K 17

18 Contextual Bandits [Auer-CesaBianchi-Freund-Schapire 02] context: x T $ k K 3 N experts/policies/functions think of N >> K 18

19 Contextual Bandits [Auer-CesaBianchi-Freund-Schapire 02] context: x T $ $ k K 3 N experts/policies/functions think of N >> K 19

20 Contextual Bandits [Auer-CesaBianchi-Freund-Schapire 02] context: x 1 x 2 x 3 x 4 x T T $0.2T 1 $0.15 $0.4 2 $.12T $.22T 3 $0.5 $0.2 $.1T 0 k $0.1 $.2T $.17T N experts/policies/functions think of N >> K 20

21 Contextual Bandits [Auer-CesaBianchi-Freund-Schapire 02] context: x 1 x 2 x 3 x 4 x T T $0.2T 1 2 $.12T $.22T 3 regret = $0.02T $.1T 0 k $.2T $.17T N experts/policies/functions think of N >> K 21

22 Contextual Bandits [Auer-CesaBianchi-Freund-Schapire 02] context: x 1 x 2 x 3 x 4 x T T $0.2T the rewards can come i.i.d. from a distribution or be arbitrary stochastic / adversarial The experts can be present or not. contextual / non-contextual k 22

23 The Setting S T rounds, K possible actions, N policies π in Π (context à actions) S for t=1 to T S world commits to rewards r(t)=r 1 (t),r 2 (t),,r K (t) (adversarial or iid) S world provides context x t S learner s policies recommend π 1 (x t ), π 2 (x t ),, π N (x t ) S learner chooses action j t S learner receives reward r jt (t) S want to compete with following the best policy in hindsight 23

24 Regret S reward of algorithm A: S expected reward of policy i: G A T ( ) T t=1 G i T ( ) = r jt t T t=1 ( ) = π i x t ( ) r( t) S algorithm A s regret: max i G i T ( ) G A (T) 24

25 Regret S algorithm A s regret: max i G i T ( ) G A (T) S bound on expected regret: S high probability bound: max i G i P[max i ( T ) E[G A (T )] < ε G i T ( ) G A (T) > ε] δ 25

26 u Harder than supervised learning: u In the bandit setting we do not know the rewards of actions not taken. u Many applications u Ad auctions, medicine, finance, u Exploration/Exploitation u Can exploit expert/arm you ve learned to be good. u Can explore expert/arm you re not sure about. 26

27 Some Barriers Ω(kT) 1/2 (non-contextual) and ~ Ω(TK ln N) 1/2 (contextual) are known lower bounds [Auer et al. 02] on regret, even in the stochastic case. Any algorithm achieving regret Õ(KT polylog N) 1/2 is said to be optimal. ε-greedy algorithms that first explore (act randomly) and then exploit (follow the best policy) cannot be optimal. Any optimal algorithm must be adaptive. 27

28 Two Types of Approaches 1 UCB [Auer 02] EXP3 Exponential Weights [Littlestone-Warmuth 94] [Auer et al. 02] t=1 t=2 t=3 Algorithm: at every time step 1) pull arm with highest UCB 2) update confidence bound of the arm pulled. Algorithm: at every time step 1) sample from distribution defined by weights (mixed w/ uniform) 2) update weights exponentially 28

29 UCB vs EXP3 A Comparison u Pros UCB [Auer 02] u Optimal for the stochastic setting. u Succeeds with high probability. u Cons u Does not work in the adversarial setting. u Is not optimal in the contextual setting. EXP3 & Friends [Auer-CesaBianchi-Freund-Schapire 02] u Pros u Optimal for both the adversarial and stochastic settings. u Can be made to work in the contextual setting u Cons u Does not succeed with high probability in the contextual setting (only in expectation). 29

30 Algorithm Regret High Prob? Context? Exp4 [ACFS 02] Õ(KT ln(n)) 1/2 No Yes ε-greedy, epochgeedy [LZ 07] Exp3.P[ACFS 02] UCB [Auer 00] Õ((K ln(n) 1/3 )T 2/3 ) Yes Yes Õ(KT) 1/2 Yes No Ω( KT ) lower bound [ACFS 02] 30

31 Algorithm Regret High Prob? Context? Exp4 [ACFS 02] Õ(KT ln(n)) 1/2 No Yes ε-greedy, epochgeedy [LZ 07] Exp3.P[ACFS 02] UCB [Auer 00] Õ((K ln(n) 1/3 )T 2/3 ) Yes Yes Õ(KT) 1/2 Yes No Exp4.P [BLLRS 10] Õ(K ln(n/δ)t) 1/2 Yes Yes Ω( KT ) lower bound [ACFS 02] 31

32 EXP4P [Beygelzimer-Langford-Li-R-Schapire 11] Main Theorem [Beygelzimer-Langford-Li-R-Schapire 11]: For any δ>0, with probability at least 1-δ, EXP4P has regret at most O(KT ln (N/δ)) 1/2 in the adversarial contextual bandit setting. EXP4P combines the advantages of Exponential Weights and UCB. optimal for both the stochastic and adversarial settings works for the contextual case (and also the non-contextual case) a high probability result 32

33 Outline S The setting and some background S Show ideas that fail S Give a high probability optimal algorithm S Dealing with VC sets S An efficient algorithm S Slates 33

34 Some Failed Approaches S Bad idea 1: Maintain a set of plausible hypotheses and randomize uniformly over their predicted actions. S Adversary has two actions, one always paying off 1 and the other 0. Hypothesis generally agree on correct action, except for a different one which defects each round. This incurs regret of ~T/2. S Bad idea 2: Maintain a set of plausible hypotheses and randomize uniformly among the hypothesis. S Adversary has two actions, one always paying off 1 and the other 0. If all but one of > 2T hypothesis always predict wrong arm, and only 1 hypothesis always predicts good arm, with probability > ½ it is never picked and algorithm incurs regret of T. 34

35 Some Failed Approaches S Bad idea 1: Maintain a set of plausible hypotheses and randomize uniformly over their predicted actions. S Adversary has two actions, one always paying off 1 and the other 0. Hypothesis generally agree on correct action, except for a different one which defects each round. This incurs regret of ~T/2. S Bad idea 2: Maintain a set of plausible hypotheses and randomize uniformly among the hypothesis. S Adversary has two actions, one always paying off 1 and the other 0. If all but one of > 2T hypothesis always predict wrong arm, and only 1 hypothesis always predicts good arm, with probability > ½ it is never picked and algorithm incurs regret of T. 35

36 Some Failed Approaches S Bad idea 1: Maintain a set of plausible hypotheses and randomize uniformly over their predicted actions. S Adversary has two actions, one always paying off 1 and the other 0. Hypothesis generally agree on correct action, except for a different one which defects each round. This incurs regret of ~T/2. S Bad idea 2: Maintain a set of plausible hypotheses and randomize uniformly among the hypothesis. S Adversary has two actions, one always paying off 1 and the other 0. If all but one of > 2T hypothesis always predict wrong arm, and only 1 hypothesis always predicts good arm, with probability > ½ it is never picked and algorithm incurs regret of T. 36

37 Some Failed Approaches S Bad idea 1: Maintain a set of plausible hypotheses and randomize uniformly over their predicted actions. S Adversary has two actions, one always paying off 1 and the other 0. Hypothesis generally agree on correct action, except for a different one which defects each round. This incurs regret of ~T/2. S Bad idea 2: Maintain a set of plausible hypotheses and randomize uniformly among the hypothesis. S Adversary has two actions, one always paying off 1 and the other 0. If all but one of > 2T hypothesis always predict wrong arm, and only 1 hypothesis always predicts good arm, with probability > ½ it is never picked and algorithm incurs regret of T. 37

38 epsilon-greedy S Rough idea of ε-greedy (or ε-first): act randomly for ε rounds, then go with best (arm or expert). S Even if we know the number of rounds in advance, ε-first won t get us regret O(T) 1/2, even in the non-contextual setting. S Rough analysis: even for just 2 arms, we suffer regret of ε+(t-ε)/(ε 1/2 ). S ε T 2/3 is optimal tradeoff. S gives regret T 2/3 38

39 Outline S The setting and some background S Show ideas that fail S Give a high probability optimal algorithm S Dealing with VC sets S An efficient algorithm S Slates 39

40 Ideas Behind Exp4.P (all appeared in previous algorithms) S exponential weights S keep a weight on each expert that drops exponentially in the expert s (estimated) performance S upper confidence bounds S use an upper confidence bound on each expert s estimated reward S ensuring exploration S make sure each action is taken with some minimum probability S importance weighting S give rare events more importance to keep estimates unbiased 40

41 (slide from Beygelzimer & Langford ICML 2010 tutorial) 41

42 (Exp4.P) [Beygelzimer, Langford, Li, R, Schapire 10] 42

43 Lemma 1 43

44 Lemma 2 44

45 EXP4P [Beygelzimer-Langford-Li-R-Schapire 11] Main Theorem [Beygelzimer-Langford-Li-R-Schapire 11]: For any δ>0, with probability at least 1-δ, EXP4P has regret at most O(KT ln (N/δ)) 1/2 in the adversarial contextual bandit setting. key insights (on top of UCB/ EXP) 1) exponential weights and upper confidence bounds stack 2) generalized Bernstein s inequality for martingales t=1 t=2 t=3 45

46 Efficiency Algorithm Regret High Prob? Context? Efficient? Exp4 [ACFS 02] epoch-geedy [LZ 07] Exp3.P/UCB [ACFS 02][A 00] Exp4.P [BLLRS 10] Õ(T 1/2 ) No Yes No Õ(T 2/3 ) Yes Yes Yes Õ(T 1/2 ) Yes No Yes Õ(T 1/2 ) Yes Yes No 46

47 EXP4P Applied to Yahoo! 47

48 Experiments on Yahoo! Data u We chose a policy class for which we could efficiently keep track of the weights. u Created 5 clusters, with users (at each time step) getting features based on their distances to clusters. u Policies mapped clusters to article (action) choices. u Ran on personalized news article recommendations for Yahoo! front page. u We used a learning bucket on which we ran the algorithms and a deployment bucket on which we ran the greedy (best) learned policy. 48

49 Experimental Results Reported estimated (normalized) click-through rates on front page news. Over 41M user visits. 253 total articles. 21 candidate articles per visit. Learning ectr Deployment ectr EXP4P EXP4 ε-greedy

50 Experimental Results Reported estimated (normalized) click-through rates on front page news. Over 41M user visits. 253 total articles. 21 candidate articles per visit. Learning ectr Deployment ectr EXP4P EXP4 ε-greedy Why does this work in practice? 50

51 Outline S The setting and some background S Show ideas that fail S Give a high probability optimal algorithm S Dealing with VC sets S An efficient algorithm S Slates 51

52 Infinitely Many Policies S What if we have an infinite number of policies? S Our bound of Õ(K ln(n)t) 1/2 becomes vacuous. S If we assume our policy class has a finite VC dimension d, then we can tackle this problem. S Need i.i.d. assumption. We will also assume k=2 to illustrate the argument. 52

53 VC Dimension S The VC dimension of a hypothesis class captures the class s expressive power. S It is the cardinality of the largest set (in our case, of contexts) the class can shatter. S Shatter means to label in all possible configurations. 53

54 VE, an Algorithm for VC Sets The VE algorithm: S Act uniformly at random for τ rounds. S This partitions our policies Π into equivalence classes according to their labelings of the first τ examples. S Pick one representative from each equivalence class to make Π. S Run Exp4.P on Π. 54

55 Outline of Analysis of VE S Sauer s lemma bounds the number of equivalence classes to (eτ/d) d. S Hence, using Exp4.P bounds, VE s regret to Π is τ+ O (Td ln(τ)) S We can show that the regret of Π to Π is (T/τ)(d lnt) S by looking at the probability of disagreeing on future data given agreement for τ steps. S τ (Td ln 1/δ) 1/2 achieves the optimal trade-off. S Gives Õ(Td) 1/2 regret. S Still inefficient! 55

56 Outline S The setting and some background S Show ideas that fail S Give a high probability optimal algorithm S Dealing with VC sets S An efficient algorithm S Slates 56

57 Hope for an Efficient Algorithm? [Dudik-Hsu-Kale-Karampatziakis-Langford-R-Zhang 11] For EXP4P, the dependence on N in the regret is logarithmic. this suggests We could compete with a large, even super-polynomial number of policies! (e.g. N=K 100 becomes 10 log 1/2 K in the regret) however All known contextual bandit algorithms explicitly keep track of the N policies. Even worse, just reading in the N would take too long for large N. 57

58 Idea: Use Supervised Learning S Competing with a large (even exponentially large) set of policies is commonplace in supervised learning. S Targets: e.g. linear thresholds, CNF, decision trees (in practice only) S Methods: e.g. boosting, SVM, neural networks, gradient descent S The recommendations of the policies don t need to be explicitly read in when the policy class has structure! x 1 x 2 x 3 x 4 x 5 x 6 Policy class Π Supervised Learning Oracle A good policy in Π idea originates with [Langford-Zhang 07] 58

59 Idea: Use Supervised Learning S Competing with a large (even exponentially large) set of policies is commonplace in supervised learning. S Targets: e.g. linear thresholds, CNF, decision trees (in practice only) S Methods: e.g. boosting, SVM, neural networks, gradient descent S The recommendations of the policies don t need to be explicitly read in when the policy class has structure! x 1 x 2 x 3 x 4 x 5 x 6 Policy class Π Warning: NP-Hard in Supervised Learning General Oracle A good policy in Π idea originates with [Langford-Zhang 07] 59

60 Back to Contextual Bandits context: x 1 x 2 x T $ $ $ k k 3 N experts/policies/functions think of N >> K 60

61 Back to Contextual Bandits context: x 1 x 2 x T $ $ $0.50 made-up data Supervised Learning Oracle k k 3 N experts/policies/functions think of N >> K 61

62 Back to Contextual Bandits context: x 1 x 2 x T $ $ $0.50 made-up data Supervised Learning Oracle k k 3 N experts/policies/functions think of N >> K 62

63 Randomized-UCB Main Theorem [Dudik-Hsu-Kale-Karampatziakis-Langford-R-Zhang 11]: For any δ>0, w.p. at least 1-δ, given access to a supervised learning oracle, Randomized-UCB has regret at most O((KT ln (NT/δ)) 1/2 +K ln(nk/δ)) in the stochastic contextual bandit setting and runs in time poly(k,t, ln N). 63

64 Randomized-UCB Main Theorem [Dudik-Hsu-Kale-Karampatziakis-Langford-R-Zhang 11]: For any δ>0, w.p. at least 1-δ, given access to a supervised learning oracle, Randomized-UCB has regret at most O((KT ln (NT/δ)) 1/2 +K ln(nk/δ)) in the stochastic contextual bandit setting and runs in time poly(k,t, ln N). if arms are chosen among only good policies s.t. all have variance < approx 2K, we win can prove this exists via a minimax theorem this condition can be softened to occasionally allow choosing of bad policies via randomized upper confidence bounds creates a problem of how to choose arms as to satisfy the constraints expressed as convex optimization problem solvable by ellipsoid algorithm can implement a separation oracle with the supervised learning oracle 64

65 Randomized-UCB Main Theorem [Dudik-Hsu-Kale-Karampatziakis-Langford-R-Zhang 11]: For any δ>0, w.p. at least 1-δ, given access to a supervised learning oracle, Randomized-UCB has regret at most O((KT ln (NT/δ)) 1/2 +K ln(nk/δ)) in the stochastic contextual bandit setting and runs in time poly(k,t, ln N). Not practical if arms are chosen among only good policies s.t. all have variance < approx 2K, we win can prove this exists via a minimax theorem to implement! this condition can be softened to occasionally allow choosing of bad policies via randomized upper confidence bounds (yet) creates a problem of how to choose arms as to satisfy the constraints expressed as convex optimization problem solvable by ellipsoid algorithm can implement a separation oracle with the supervised learning oracle 65

66 Outline S The setting and some background S Show ideas that fail S Give a high probability optimal algorithm S Dealing with VC sets S An efficient algorithm S Slates 66

67 Bandit Slate Problems [Kale-R-Schapire 11] Problem: Instead of selecting one arm, we need to select s 1, arms (possibly ranked). The motivation is web ads where a search engine shows multiple ads at once. 67

68 Slates Setting S On round t algorithm selects a sate S t of s arms S Unordered or Ordered S No context or Contextual S Algorithm sees r j (t) for all j in S. S Algorithm gets reward j S r j (t) S Obvious solution is to reduce to the regular bandit problem, but we can do much better. 68

69 Algorithm Idea 69

70 Algorithm Idea

71 Algorithm Idea 71

72 Algorithm Idea multiplicative update 72

73 Algorithm Idea relative entropy projection Also Component Hedge, independently by Koolen et al

74 Algorithm Idea 74

75 Slate Results Unordered Slates Ordered, with Positional Factors No Policies Õ(sKT) 1/2 * Õ(s(KT) 1/2 ) N Policies Õ(sKT ln N) 1/2 Õ(s(KT ln N) 1/2 ) *Independently obtained by Uchiya et al. 10, using different methods. 75

76 Discussion S The contextual bandit setting captures many interesting real-world problems. S We presented the first optimal, high-probability, contextual algorithm. S We showed how one could possibly make it efficient. S Not fully there yet S We discussed slates a more real-world setting. S How to make those efficient? 76