Leanng Theoy and Algoths fo Revenue Optzaton n Second-Pce Auctons wth Reseve A. Poofs fo leanng guaantees A.. Revenue foula The sple expesson of the expected evenue (2) can be obtaned as follows: E b Revenue(, b) E b (2) <b (2) + Pb (2) b () b (2) + Pb (2) <b (2) > t dt + Pb (2) b () P < b (2) dt + + Pb (2) b () Pb (2) > t dt Pb (2) > tdt + (Pb (2) > + Pb (2) > Pb () < ) Pb (2) > t dt + Pb (). A.2. Contacton lea The followng s a veson of Talagand s contacton lea (Ledoux & Talagand, 2). Snce ou defnton of Radeache coplexty does not use absolute values, we gve an explct poof below. Lea 8. Let H be a hypothess set of functons appng X to R and Ψ,..., Ψ, µ-lpschtz functons fo soe µ >. Then, fo any saple S of ponts x,..., x X, the followng nequalty holds E σ σ (Ψ h)(x ) µ E σ µ R S (H). σ h(x ) Poof. The poof s sla to the case whee the functons Ψ ae all equal. Fx a saple S (x,..., x ). Then, we can ewte the epcal Radeache coplexty as follows: E σ σ (Ψ h)(x ) E E σ,...,σ σ u (h)+σ (Ψ h)(x ), whee u (h) σ (Ψ h)(x ). Assue that the ea can be attaned and let h, h 2 H be the hypotheses satsfyng u (h ) + Ψ (h (x )) u (h) + Ψ (h(x )) u (h 2 ) Ψ (h 2 (x )) u (h) Ψ (h(x )). When the ea ae not eached, a sla aguent to what follows can be gven by consdeng nstead hypotheses that ae ɛ-close to the ea fo any ɛ >. By defnton of expectaton, snce σ unfo dstbuted ove {, +}, we can wte E u (h) + σ (Ψ h)(x ) σ 2 u (h) + (Ψ h)(x ) + 2 u (h) (Ψ h)(x ) 2 u (h ) + (Ψ h )(x ) + 2 u (h 2 ) (Ψ h 2 )(x ). Let s sgn(h (x ) h 2 (x )). Then, the pevous equalty ples E u (h) + σ (Ψ h)(x ) σ 2 u (h ) + u (h 2 ) + sµ(h (x ) h 2 (x )) 2 u (h ) + sµh (x ) + 2 u (h 2 ) sµh 2 (x ) 2 u (h) + sµh(x ) + 2 u (h) sµh(x ) E u (h) + σ µh(x ), σ whee we used the µ Lpschtzness of Ψ n the fst equalty and the defnton of expectaton ove σ fo the last equalty. Poceedng n the sae way fo all othe σ s ( ) poves the lea. A.3. Bounds on Radeache coplexty Poposton 9. Fo any hypothess set H and any saple S ((x, b ),..., (x, b )), the epcal Radeache coplexty of l H can be bounded as follows: R S (l H ) R S (H). Poof. By defnton of the epcal Radeache coplexty, we can wte R S (l H ) E σ l (h(x ), b ) σ E σ (ψ h)(x ), σ
Leanng Theoy and Algoths fo Revenue Optzaton n Second-Pce Auctons wth Reseve whee, fo all,, ψ s the functon defned by ψ : l (, b ). Fo any,, ψ s - Lpschtz, thus, by the contacton lea 8, we have the nequalty R S (l H ) E σ σ h(x ) R S (H). Poposton. Let M b B b (). Then, fo any hypothess set H wth pseudo-denson d Pd(H) and any saple S ((x, b ),..., (x, b )), the epcal Radeache coplexty of l 2H can be bounded as follows: R S (l 2H ) 2d log e d. Poof. By defnton of the epcal Radeache coplexty, we can wte R S (l 2H ) E σ E σ σ b () h(x)>b () σ Ψ ( h(x)>b ), () whee fo all,, Ψ s the M-Lpschtz functon x b () x. Thus, by Lea 8 cobned wth Massat s lea (see fo exaple (Moh et al., 22)), we can wte R S (l 2H ) M E σ σ h(x)>b () 2d log e d M, whee d VCd({(x, b) h(x) b () > : (x, b) X B}). Snce the second bd coponent b (2) plays no ole n ths defnton, d concdes wth VCd({(x, b () ) h(x) b () > : (x, b () ) X B }), whee B s the pojecton of B R 2 onto ts fst coponent, and s uppe-bounded by VCd({(x, t) h(x) t> : (x, t) X R}), that s the pseudo-denson of H. A.4. Calbaton Theoe 2 (convex suogates). Thee exsts no nonconstant functon L c : R R + R convex wth espect to ts fst aguent and satsfyng the followng condtons: fo any b R +, l b b L c (b, b) L c (b, b ). fo any dstbuton D on R +, thee exsts a nonnegatve nze agn E b D L(, b) such that n E b D L c (, b) E b D L c (, b). Poof. Fo any loss L c satsfyng the assuptons, we can defne a loss L c by L c(, b) L c (, b) L c (b, b). L c then also satsfes the assuptons. Thus, wthout loss of genealty, we can assue that L c (b, b). Futheoe, snce L(, b) s nzed at b we ust have L c (, b) L c (b, b). Notce that fo any b R+, b < R + and µ,, the nze of E µ ( L(, b)) µ L(, b )+( µ) L(, ) s ethe b o. In fact, by defnton of L, the soluton s b as long as b ( µ), that s, when µ b2 b. Snce the nzng popety of L c should hold fo evey dstbuton we ust have µl c (b, b ) + ( µ)l c (b, ) µl c (, b ) + ( µ)l c (, ) () when µ b2 b and the evese nequalty othewse. Ths ples that () ust hold as an equalty when µ b2 b. Ths, cobned wth the equalty L c (b, b) vald fo all b, yelds b L c (b, ) ( b )L c (, b ). (2) Dvdng by b and takng the lt b esult n L c (b, ) l b l L c (, b ). (3) b b b 2 2 b b b 2 By convexty of L c wth espect to the fst aguent, we know that the left-hand sde s well-defned and s equal to b D L c (, ), whee D L c denotes the left devatve of L c wth espect to the fst coodnate. By assupton, the ght-hand sde s equal to L c (, ). Snce b >, ths ples that D L c (, ). Let µ < b2 b. Fo ths choce of µ, E µ (L c (, b)) s nzed at. Ths ples: µd L c (, b ) + ( µ)d L c (, ). (4) Howeve, convexty ples that D L c (, b ) D L c (b, b ) fo b. Thus, nequalty (4) can only be satsfed f D L c (, b ). Let D + L c denote the ght devatve of L c wth espect to the fst coodnate. The convexty of L c ples that D L c (b, b ) D + L c (b, b ) D L c (, b ) fo > b. Hence, D + L c (b, b ). If we let µ > b2 b then b s a nze fo E µ (L c (, b)) and µd + (b, b ) + ( µ)d + L c (b, ). As befoe, snce b <, D + (b, ) D + (, ) and we ust have D + L c (b, ) fo ths nequalty to hold. We have theefoe poven that fo evey b, f b, then D L c (, b), wheeas f b then D + L c (, b). It s not had to see that ths ples D L c (, b) fo all (, b) and thus that L c (, b) ust be a constant. In patcula, snce L c (b, b), we have L c.
Leanng Theoy and Algoths fo Revenue Optzaton n Second-Pce Auctons wth Reseve Lea 4. Let H be a closed, convex subset of a lnea space of functons contanng. Denote by h γ the soluton of n L γ (h). If b B b () M <, then E h γ(x) I2 (x) γ E h γ(x) I3 (x) Poof. Let < λ <, because λh γ H by convexty and h γ s a nze we ust have: E L γ (h γ(x), b) E L γ (λh γ(x), b). (5) If h γ(x) <, then L γ (h γ(x), b) L γ (λh γ(x)) b (2) by defnton. If on the othe hand h γ(x) >, because λh γ(x) < h γ(x) we ust have that fo (x, b) I L γ (h γ(x), b) L γ (λh γ(x), b) b (2) too. Moeove, because L γ and L γ (h γ(x), b) fo (x, b) I 4 t s edate that L γ (h γ(x), b) L γ (λh γ(x), b) fo (x, b) I 4. The followng nequalty holds tvally: E L γ (h γ(x), b)( I (x) + I4 (x)) E L γ (λh γ(x), b)( I (x) + I4 (x)). (6) Subtactng (6) fo (5) we obtan E L γ (h γ(x), b)( I2 (x) + I3 (x)) E L γ (λh γ(x), b)( I2 (x) + I3 (x)). By eaangng tes we can see ths nequalty s equvalent to (L γ (λh γ(x), b) L γ (h γ(x), b)) I2 (x) E E (L γ (h γ(x), b) L γ (λh γ(x), b)) I3 (x) (7) Notce that f (x, b) I 2, then L γ (h γ(x), b) h γ(x). If λh γ(x) > b (2) too then L γ (λh γ(x), b) λh γ(x). On the othe hand f λh γ(x) b (2) then L γ (λh γ(x), b) b (2) λh γ(x). Thus E(L γ (λh γ(x), b) L γ (h γ(x), b)) I2 (x)) ( λ) E(h γ(x) I2 (x)) (8) Ths gves an uppe bound fo the left-hand sde of nequalty (7). We now seek to deve a lowe bound on the ghthand sde. To do that, we analyze two dffeent cases:. λh γ(x) b () ; 2. λh γ(x) > b (). In the fst case, we know that L γ (h γ(x), b) γ (h γ(x) ( + γ)b () ) > b () (snce h γ(x) > b () fo (x, b) I 3 ). Futheoe, f λh γ(x) b (), then, by defnton L γ (λh γ(x), b) n( b (2), λh γ(x)) λh γ(x). Thus, we ust have: L γ (h γ(x), b) L γ (λh γ(x), b) > λh γ(x) b () > (λ )b () (λ )M, (9) whee we used the fact that h γ(x) > b () fo the second nequalty. We analyze the second case now. If λh γ(x) > b (), then fo (x, b) I 3 we have L γ (h γ(x), b) L γ (λh γ(x), b) γ ( λ)h γ(x). Thus, lettng (x, b) L γ (h γ(x), b) L γ (λh γ(x), b), we can lowe bound the ght-hand sde of (7) as: E (x, b) I3 (x) E (x, b) I3 (x) {λh γ (x)>b () } + E (x, b) I3 (x) {λh γ (x) b () } λ E h γ γ(x) I3 (x) {λh γ (x)>b () } + (λ )M P h γ(x) > b () λh γ(x), (2) whee we have used (9) to bound the second suand. Cobnng nequaltes (7), (8) and (2) and dvdng by ( λ) we obtan the bound E h γ(x) I2 (x) γ E h γ(x) I3 (x) {λh γ (x)>b () } M P h γ(x) > b () λh γ(x). Fnally, takng the lt λ, we obtan E h γ(x) I2 (x) γ E h γ(x) I3 (x). Takng the lt nsde the expectaton s justfed by the bounded convegence theoe and Ph γ(x) > b () λh γ(x) holds by the contnuty of pobablty easues. A.5. Magn bounds Theoe 5. Fx γ (, and let S denotes a saple of sze. Then, fo any δ >, wth pobablty at least δ ove the choce of the saple S, fo all h H, the followng holds: L γ (h) L γ (h) + 2 γ R log δ (H) + M 2. (2)
Leanng Theoy and Algoths fo Revenue Optzaton n Second-Pce Auctons wth Reseve Poof. Let L γ,h denote the faly of functons {(x, b) L γ (h(x), b): h H}. The loss functon L γ s γ -Lpschtz snce the slope of the lnes defnng t s at ost γ. Thus, usng the contacton lea (Lea 8) as n the poof of Poposton 9 gves R (L γ,h ) γ R (H). The applcaton of a standad Radeache coplexty bound to the faly of functons L γ,h then shows that fo any δ >, wth pobablty at least δ, fo any h H, the followng holds: L γ (h) L γ (h) + 2 γ R log δ (H) + M 2. 2 3 4 b () b () 2 V Ω() V 2 We conclude ths secton by pesentng a stonge fo of consstency esult. We wll show that we can lowe bound the genealzaton eo of the best hypothess n class L : L(h ) n tes of that of the epcal nze of L γ, ĥγ : agn Lγ (h). Theoe. Let M b B b () and let H be a hypothess set wth pseudo-denson d Pd(H). Then fo any δ > and a fxed value of γ >, wth pobablty at least δ ove the choce of a saple S of sze, the followng nequalty holds: L(ĥγ) L + 2γ + 2 R (H) + γm γ 2d log ɛ d log 2 δ 2M + 2M 2. Poof. By Theoe, wth pobablty at least δ/2, the followng holds: L(ĥγ) L S (ĥγ) + 2R (H)+ 2d log ɛ d log 2 δ 2M + M 2. (22) Futheoe, applyng Lea 4 wth the epcal dstbuton nduced by the saple, we can bound L S (ĥγ) by L γ (ĥγ) + γm. The fst te of the pevous expesson s less than L γ (h γ) by defnton of ĥγ. Fnally, the sae analyss as the one used n the poof of Theoe 5 shows that wth pobablty δ/2, L γ (h γ) L γ (h γ) + 2 γ R log 2 δ (H) + M 2. Agan, by defnton of h γ and usng the fact that L s an uppe bound on L γ, we can wte L γ (h γ) L γ (h ) L(h ). Thus, L S (ĥγ) L(h ) + γ R log 2 δ (H) + M 2 + γm. Fgue 8. Illustaton of the egon Ω(). The functons V ae onotonc and concave when estcted to ths egon. Cobnng ths wth (22) and applyng the unon bound yelds the esult. Ths bound can be extended to hold unfoly ove all γ at the pce of a te n O ( q log log 2 γ ). Thus, fo appopate choces of γ and (fo nstance γ / /4 ) t would guaantee the convegence of L(ĥγ) to L, a stonge fo of consstency. B. Cobnatoal algoth B.. Popety of the soluton We wll show that poble (8) adts a soluton b () fo soe. We wll need the followng defnton. Defnton 2. Fo any R, defne the followng subset of R: Ω() {ɛ < b () + ɛ b () } We wll dop the dependency on when t s undestood what value of we ae efeng to. Lea 3. Let b () fo all. If ɛ > s such that ɛ, ɛ Ω() then F ( + ɛ) < F () o F ( ɛ) F (). The condton that b () fo all ples that thee exsts ɛ sall enough that satsfes ɛ Ω(). Poof. Let v V (, b ) and v (ɛ) V ( + ɛ, b ). Fo ɛ Ω() defne the sets D(ɛ) { v (ɛ) v } and I(ɛ) { v (ɛ) > v }. If D(ɛ) v + I(ɛ) v > D(ɛ) v (ɛ) + v (ɛ), I(ɛ)
Leanng Theoy and Algoths fo Revenue Optzaton n Second-Pce Auctons wth Reseve then, by defnton, we have F () > F ( + ɛ) and the esult s poven. If ths nequalty s not satsfed, then, by goupng ndces n D(ɛ) and I(ɛ) we ust have v v (ɛ) v (ɛ) v (23) D(ɛ) I(ɛ) Notce that v (ɛ) v f and only f v ( ɛ) v. Indeed, the functon V ( +η, b ) s onotone fo η ɛ, ɛ as long as ɛ, ɛ Ω whch s tue by the choce of ɛ. Ths fact can easly be seen n Fgue 8. Hence D(ɛ) I( ɛ), slaly I(ɛ) D( ɛ) Futheoe, because V ( + η, b ) s also concave fo η ɛ, ɛ. We ust have 2 (v ( ɛ) + v (ɛ)) v. (24) Usng (24), we can obtan the followng nequaltes: v ( ɛ) v v v (ɛ) fo D(ɛ) (25) v (ɛ) v v v ( ɛ) fo I(ɛ). (26) Cobnng nequaltes (25), (23) and (26) we obtan v ( ɛ) v v v ( ɛ) D(ɛ) I( ɛ) v ( ɛ) v I(ɛ) D( ɛ) v v ( ɛ). By eaangng back the tes n the nequalty we can easly see that F ( ɛ) F (). Lea 4. Unde the condtons of Lea 3, f F ( + ɛ) F () then F ( +λɛ) F () fo evey λ that satsfes λɛ Ω f and only f ɛ Ω. Poof. The poof follows the sae deas as those used n the pevous lea. By assupton, we can wte v v (ɛ) v (ɛ) v. (27) D(ɛ) I(ɛ) It s also clea that I(ɛ) I(λɛ) and D(ɛ) D(λɛ). Futheoe, the sae concavty aguent of Lea 3 also yelds: v (ɛ) λ λ v + λ v (λɛ), whch can be ewtten as λ (v v (λɛ)) v v (ɛ). (28) Applyng nequalty (28) n (27) we obtan v v (λɛ) v (λɛ) v. λ λ D(λɛ) I(λɛ) Snce λ >, we can ultply the nequalty by λ to deve an nequalty sla to (27) whch ples that F (+λɛ) F (). Poposton 7. Poble (8) adts a soluton that satsfes b () fo soe,. Poof. Let b () fo evey. By Lea 3, we can choose ɛ sall enough wth F (+ɛ) F (). Futheoe f λ n () b ɛ then λ satsfes the hypotheses of Lea 4. Hence, F () F ( + λɛ) F (b ), whee s the nze of b() B.2. Algoth ɛ. We now pesent a cobnatoal algoth to solve the optzaton poble (8) n O( log ). Let N {b(), b (2), ( + η)b () } denote the set of all bounday ponts assocated wth the functons V (, b ). The algoth poceeds as follows: fst, sot the set N to obtan the odeed sequence (n,..., n 3 ), whch can be acheved n O( log ) usng a copason-based sotng algoth. Next, evaluate F (n ) and copute F (n k+ ) fo F (n k ) fo all k. The an dea of the algoth s the followng: snce the defnton of V (, b ) can only change at bounday ponts (see also Fgue 4(b)), coputng F (n k+ ) fo F (n k ) can be acheved n constant te. Snce between n k and n k+ thee ae only two bounday ponts, we can copute V (n k+, b ) fo V (n k, b ) by calculatng V fo only two values of b, whch can be done n constant te. We now gve a oe detaled descpton and poof of coectness fo the algoth. Poposton 5. Thee exsts an algoth to solve the optzaton poble (8) n O( log ). Poof. The pseudocode fo the desed algoth s pesented n Algoth. Whee a (),..., a (4) denote the paaetes defnng the functons V (, b ). We wll pove that afte unnng Algoth we can copute F (n j ) n constant te usng: F (n j ) c () j + c (2) j n j + c (3) j n j + c (4) j. (29) Ths holds tvally fo n snce by constucton n b (2) fo all and by defnton then F (n ) a(). Now, assue that (29) holds fo j, we pove that then t ust also hold fo j +. Suppose n j fo soe (the cases n j b () and n j ( + η)b () can be handled n the sae way). Then V (n j, b ) a () and we can wte V k (n j, b k ) F (n j ) V (n j, b ) k (c () j + c (2) j n j + c (3) j n j + c (4) j ) + a ().
Algoth Sotng Leanng Theoy and Algoths fo Revenue Optzaton n Second-Pce Auctons wth Reseve N : {b(), b (2), ( + η)b () }; (n,..., n 3 ) Sot(N ); Set c : (c (), c (2), c (3), c (4) ) fo,..., 3; Set c () a() fo j 2,..., 3 do Set c j c j ; f n j b (2) fo soe then c () j c () j + a () c (2) j c (2) j a (2) else f n j b () fo soe then c (2) j c () j + a (2) c (3) j c (3) j + a (3) c (4) j c () j a (4) else c (3) j c (3) j a (3) c (4) j c () j + a (4) end f end fo Thus, by constucton we would have: c () j+ + c(2) j+ n j+ + c (3) j+ n j+ + c (4) j+ c () j + a () + (c (2) j a (2) )n j+ + c (3) j n j+ + c (4) j (c () j + c (2) j n j+ + c (3) j n j+ + c (4) j ) + a () a (2) n j+ k V k (n j+, b k ) a (2) n j+, whee the last equalty holds snce the defnton of V k (, b k ) does not change fo n j, n j+. Fnally, snce n j was a bounday pont, the defnton of V (, b ) ust change fo a () to a (2), thus the last equaton s ndeed equal to F (n j+ ). A sla aguent can be gven f n j b () o n j ( + η)b (). Let us analyze the coplexty of the algoth: sotng the set N can be pefoed n O( log ) and each teaton takes only constant te. Thus the evaluaton of all ponts can be done n lnea te. Once all evaluatons ae done, fndng the nu can also be done n lnea te. Thus, the oveall te coplexty of the algoth s O( log ).