# The Sample Complexity of Exploration in the Multi-Armed Bandit Problem

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7 EXPLORATION IN MULTI-ARMED BANDITS where the equaity above made use of the defiitio of t. Therefore, 1 ε) T 1 K 1 ε) /ε c)og1/θ) e 4d/ c)og1/θ) = θ 4d/ c. Substitutig the above i Eq. 3), we obtai L 1 W) L 0 W) c) θ16d/c)+4d/. By pickig c arge eough c = 100 suffices), we obtai that L 1 W)/L 0 W) is arger tha θ = 4δ wheever the evet S occurs. More precisey, we have L 1 W) L 0 W) 1 S 4δ1 S, where 1 S is the idicator fuctio of the evet S. The, [ ] L1 W) P 1 B) P 1 S) = E 1 [1 S ] = E 0 L 0 W) 1 S E 0 [4δ1 S ] = 4δP 0 S) > δ, where we used the fact that P 0 S) > 1/4. To summarize, we have show that whe c 100, if E 0 [T 1 ] 1/cε )og1/4δ)), the P 1 B) > δ. Therefore, if we have a ε/,δ)-correct poicy, we must have E 0 [T ] > 1/cε )og1/4δ)), for every > 0. Equivaety, if we have a ε,δ)-correct poicy, we must have E 0 [T ] > /4cε ))og1/4δ)), which is of the desired form. 4. A Lower Boud o the Sampe Compexity - Geera Probabiities I Theorem 1, we worked with a particuar ufavorabe vector p the oe correspodig to hypothesis H 0 ), uder which a ot of exporatio is ecessary. This eaves ope the possibiity that for other, more favorabe choices of p, ess exporatio might suffice. I this sectio, we refie Theorem 1 by deveopig a ower boud that expicity depeds o the actua though ukow) vector p. Of course, for ay give vector p, there is a optima poicy, which seects the best coi without ay exporatio: e.g., if p 1 for a, the poicy that immediatey seects coi 1 is optima. However, such a poicy wi ot be ε,δ)-correct for a possibe vectors p. We start with a ower boud that appies whe a coi biases p i ie i the rage [0,1/]. We wi ater use a reductio techique to exted the resut to a geeric rage of biases. I the rest of the paper, we use the otatioa covetio x) + = max0,x}. Theorem 5 Fix some p 0,1/). There exists a positive costat δ 0, ad a positive costat c 1 that depeds oy o p, such that for every ε 0,1/), every δ 0,δ 0 ), every p [0,1/], ad every ε,δ)-correct poicy, we have } Mp,ε) 1) + 1 E p [T ] c 1 ε + p ) og 1 8δ, 69 Np,ε)

10 MANNOR AND TSITSIKLIS As i the proof of Lemma 4, we defie the ikeihood fuctio L by ettig L w) = P W = w), for every possibe history w, ad use agai L W) to defie the correspodig radom variabe. Let K be a shorthad otatio for K T, the tota umber of uit rewards heads ) obtaied from coi. We have L W) L 0 W) = p 1 + ε) K 1 p 1 ε) T K p K 1 ) T K p1 = + ε ) K 1 p1 ε 1 1 = 1 + ε + ) K 1 ε + ) T K, 1 ) T K where we have used the defiitio = p 1. It foows that L W) L 0 W) = = = 1 + ε + ) K 1 ε + ) ) ε + K 1 1 ε + ) K 1 ε + ) ) ε + K 1 1 ε + ) K 1 ε + 1 ) K 1 ε + 1 ) K 1 ε + 1 ) T K ) K1 p )/ 1 ε + 1 ) T K ) p T K)/. 7) We wi ow proceed to ower boud the right-had side of Eq. 7) for histories uder which evet S occurs. If evet S has occurred, the A has occurred, ad we have K T 4t, so that for every Nε, p), we have ) ) ε + K 1 = a b ) ) ε + 4t 1 ) ) ε + 4/c )og1/θ) 1 exp exp = θ 16d/p c. ε/ ) + 1 d 4 c d 16 cp og1/θ) ) } og1/θ) I step a), we have used Lemma 3 which appies because of Eq. 6); i step b), we used the fact ε/ 1, which hods because Nε, p). } 63

11 EXPLORATION IN MULTI-ARMED BANDITS Simiary, for Mε, p), we have ) ) ε + K 1 = a b ) ) ε + 4t 1 ) ) ε + 4/cε )og1/θ) 1 exp exp = θ 16d/p c. 1 + /ε) d 4 c d 16 cp og1/θ) ) } og1/θ) I step a), we have agai used Lemma 3; i step b), we used the fact /ε 1, which hods because Mε, p). We ow boud the product of the secod ad third terms i Eq. 7). If b 1, the the mappig y 1 y) b is covex for y [0,1]. Thus, 1 y) b 1 by, which impies that 1 ε + ) 1 p )/ 1 ε + ), 1 so that the product of the secod ad third terms ca be ower bouded by 1 ε + ) K 1 ε + ) K1 p )/ 1 ε + ) K 1 ε + ) K = 1. 1 We sti eed to boud the fourth term of Eq. 7). We start with the case where Np,ε). We have 1 ε + ) p T K)/ a 1 ε b = c d e 1 ε + 1 } ) 1/p ) t og1/θ) 8) ) 1/p c )og1/θ) exp d } ε + og1/θ) c 1 ) } d exp og1/θ) c1 p ) exp 4d } og1/θ) cp = θ 4d/ c). Here, a) hods because we are assumig that the evets A ad C occurred; b) uses the defiitio of t for Np,ε); c) foows from Eq. 6) ad Lemma 3; d) foows because > ε; ad e) hods because 0 1/, which impies that 1/1 ). 9) 10) 633

12 MANNOR AND TSITSIKLIS Cosider ow the case where M 0 p,ε). Equatio 8) hods for the same reasos as whe Np,ε). The oy differece from the above cacuatio is i step b), where t shoud be repaced with 1/cε )og1/θ). The, the right-had side i Eq. 9) becomes exp d } ε + og1/θ). c ε1 ) For M 0 p,ε), we have ε, which impies that ε + )/ε, which the eads to the same expressio as i Eq. 10). The rest of the derivatio is idetica. Summarizig the above, we have show that if M 0 p,ε) Np,ε), ad evet S has occurred, the L W) L 0 W) θ4d/ c)+16d/p c). For M 0 p,ε) Np,ε), we have p <. We ca choose c arge eough so that L W)/L 0 W) θ = 8δ; the vaue of c depeds oy o the costat p. Simiar to the proof of Theorem 1, we have L W) L 0 W) 1 S 8δ1 S, where 1 S is the idicator fuctio of the evet S. It foows that [ ] P B c ) P L W) S ) = E [1 S ] = E 0 L 0 W) 1 S E 0 [8δ1 S ] = 8δP 0 S ) > δ, where the ast iequaity reies o the aready estabished fact P 0 S ) > 1/8. Sice the poicy is ε,δ)-correct, we must have P B c ) δ, for every. Lemma 6 the impies that E 0 [T ] > t for every M 0p,ε) Np,ε). We sum over a M 0 p,ε) Np,ε), use the defiitio of t, together with the fact M 0p,ε) Mp,ε) 1) +, to cocude the proof of the theorem. Remark 7 A cose examiatio of the proof reveas that the depedece of c 1 o p is captured by a requiremet of the form c 1 c p, for some absoute costat c. This suggests that there is a tradeoff i the choice of p. By choosig a arge p, the costat c 1 is made arger, but the sets M ad N become smaer, ad vice versa. The precedig resut may give the impressio that the sampe compexity is high oy whe the p i are bouded by 1/. The ext resut shows that simiar ower bouds hod with a differet costat) wheever the p i ca be assumed to be bouded away from 1. However, the ower boud becomes weaker i.e., the costat c 1 is smaer) whe the upper boud o the p i approaches 1. I fact, the depedece of a ower boud o ε caot be Θ1/ε ) whe max i p i = 1. To see this, cosider the foowig poicy π. Try each coi O1/ε)og/δ)) times. If oe of the cois aways resuted i heads, seect it. Otherwise, use some ε,δ)-correct poicy π. It ca be show that the poicy π is ε,δ)-correct for every p [0,1] ), ad that if max i p i = 1, the E p [T ] = O/ε)og/δ)). 634

17 EXPLORATION IN MULTI-ARMED BANDITS We assume from ow o that P 0 D) 3/4. Rearragig Eq. 15), ad omittig the third term, we have E[R t ] ε 4 E 0 [T ] + 1 ) E [T 0 ]. Sice E [T 0 ] t/)p D), we have E[R t ] ε 4 For every 0, et us defie δ by =1 =1 E 0 [T ] + t ) P D). 16) E 0 [T ] = 1 cε og 1 4δ. Such a δ exists because of the mootoicity of the mappig x og1/x).) Let δ 0 = e 4 /4. If δ < δ 0, we argue exacty as i Lemma 4, except that the evet B i that emma is repaced by evet D. Sice P 0 D) 3/4, the same proof appies ad shows that P D) δ, so that E 0 [T ] + t P D) 1 cε og 1 4δ + t δ. If o the other had, δ δ 0, the E 0 [T ] 1/cε )og1/4δ 0 ), which impies by the earier aaogy with Lemma 4) that P D) δ 0, ad E 0 [T ] + t P D) 1 cε og 1 4δ + t δ 0. Usig the above bouds i Eq. 16), we obtai E[R t ] ε 4 =1 1 cε og 1 + hδ ) t ), 17) 4δ where hδ) = δ if δ < δ 0, ad hδ) = δ 0 otherwise. We ca ow view the δ as free parameters, ad cocude that E[R t ] is ower bouded by the miimum of the right-had side of Eq. 17), over a δ. Whe optimizig, a the δ wi be set to the same vaue. The miimizig vaue ca be δ 0, i which case we have E[R t ] 4cε og 1 ε + δ 0 4δ 0 8 t. Otherwise, the miimizig vaue is δ = /ctε, i which case we have 1 E[R t ] 16cε + 1 ) 4cε ogcε /) + 1 4cε og1/) + 4cε ogt. Thus, the theorem hods with c = 1/4cε)og1/4δ 0 ), c 3 = δ 0 ε/8, c 4 = 1/4cε, ad c 5 = 1/4) + ogcε /). 639

19 EXPLORATION IN MULTI-ARMED BANDITS Iput: Accuracy ad cofidece parameters ε 0,1) ad δ 0,1); the bias of the best coi q. Parameter: δ 1/. 0. k = 1; 1. Ru the media eimiatio agorithm to fid a coi I k whose bias is withi ε/3 of q, with probabiity at east 1 δ.. Try coi I k for m k = 9/ε )og k /δ) times. Let ˆp k be the fractio of these trias that resut i heads. 3. If ˆp k q ε/3 decare that coi I k is a ε-optima coi ad termiate. 4. Set k := k + 1 ad go back to Step 1. Tabe 1: A poicy for fidig a ε-optima coi whe the bias of the best coi is kow. Let K be the umber of iteratios uti the poicy termiates. Give that K > k 1 i.e., the poicy did ot termiate i the first k 1 iteratios), there is probabiity at east 1 δ 1/ that p Ik q ε/3), i which case, from Eq. 18), there is probabiity at east 1 δ/ k ) 1/ that ˆp k q ε/3). Thus, PK > k K > k 1) 1 η, with η = 1/4. Cosequety, the probabiity that the poicy does ot termiate by the kth iteratio, PK > k), is bouded by 1 η) k. Thus, the probabiity that the poicy ever termiates is bouded above by 3/4) k for a k, ad is therefore 0. We ow boud the expected umber of trias. Let c be such that the umber of trias i oe executio of the media eimiatio agorithm is bouded by c/ε )og1/δ ). The, the umber of trias, tk), durig the kth iteratio is bouded by c/ε )og1/δ ) + m k. It foows that the expected tota umber of trias uder our poicy is bouded by k)tk) k=1pk 1 ) ε 1 η) k 1 cog1/δ ) + 9/)og k /δ) + 1 k=1 = 1 ε 1 η) k 1 cog1/δ ) + 9/)og1/δ) + 9k/)og + 1 ) k=1 1 ε c 1 + c og1/δ)), for some positive costats c 1 ad c. We fiay argue that the poicy is q,ε,δ)-correct. For the poicy to seect a coi I with bias p I q ε, it must be that at some iteratio k, a coi I k with p Ik q ε was obtaied, but ˆp k came out arger tha q ε/3. From Eq. 18), for ay fixed k, the probabiity of this occurrig is bouded by δ/ k. By the uio boud, the probabiity that p I q ε is bouded by k=1 δ/k = δ. Remark 1 The kowedge of q turs out to be sigificat: it eabes the poicy to termiate as soo as there is high cofidece that a coi has bee foud whose bias is arger tha q ε, without havig to check the other cois. A poicy of this type woud ot work for the hypotheses 641

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