Appendix to Online Clustering with Experts

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1 A Appendx o Onlne Cluserng wh Expers Furher dscusson of expermens. Here we furher dscuss expermenal resuls repored n he paper. Ineresngly, we observe ha OCE (and n parcular Learn- ) racks he bes exper much more e ecvely on all he real daa ses han on he 5 Gaussan expermen. hs s a smulaed daa se from a mxure of Gaussans ha s fxed, and where k s known o be 5. Whle OCE makes a favorable showng on he 3-expers expermen, n he 6-exper expermen, expers ncur loss ha s orders of magnude lower han he remanng ones. We drlled down on he performance of he OCE algorhms wh low values, ncludng Sac-Exper, as well as Learn-, and observed ha he means were smply hur by he algorhms unform prors over expers. ha s, n a few early eraons, OCE ncurred coss from cluserngs gvng nonrval weghs o all he expers predcons, so n hose eraons coss could be orders of magnude hgher han hose of expers Moreover, as menoned above, our regre bounds nsead upper bound loss. Addonal expermenal deals. Expers are 3 varans of k-means# algorhm [3]. In parcular hey are: 4. k-means# ha oupus 3 k log k ceners. 5. k-means# ha oupus.5 k log k ceners. 6. k-means# ha oupus.5 k log k ceners. Alhough OCE can sar ranng from he frs observaon, va our analyss reang smaller wndow szes (encounered a he begnnng and he end of he sequence), n he expermens we used he frs bach of pons as npu o all he cluserng algorhms, and sared ranng OCE varans afer ha. he parameer R was no uned bu was se as follows: R = for all daa ses excep Inruson and Spambase, n whch R =. Addonal expermenal resuls are provded n Secon C.

2 Anna Choromanska, Clare Moneleon B Addonal proofs Frs we provde some lemmas ha wll be used n subsequen proofs. hese follow he approach n [6]. We use he shor-hand noaon L(, ) for L(x,c ), whch s vald snce our loss s symmerc wh respec o s argumens. Lemma 6. P log P n p()e L(,) = log[ P,..., p ( )e L(,) Q e L(,) P (, )] Proof. Frs noe ha followng [5], we desgn he HMM such ha we equae our loss funcon, L(, ), wh he negave log-lkelhood of he observaon gven ha exper s he curren value of he hdden varable. In he unsupervsed seng, he observaon s x. hus L(, ) = log P (x a,x,...,x ). herefore, we can expand he lef hand sde of hep clam as follows. log P n p()e L(,) = P log P n p()p (x a,x,...,x ) = log P (x x,...,x ) = log p (x ) Y P (x x,...,x ) = log P (x,...,x ) = log[ P (x,...,x,..., )P (,..., )],..., = log[ = log[ = log[,..., p ( )P (x,..., ),..., p ( )P (x ),..., p ( )e L(,) Y P (x,...,,x,...,x )P (,...,, )] Y P (x,x,...,x )P (, )] Y e L(,) P (, )] Lemma 7. n D( k )=D( k ) when j = for = j and j = n for 6= j, [, ], P n =. Proof. n D( k )= n n j= j log j j = n [ log + n j6= j log j j ] = n [( )log n + n log j6= n n ]= n [( )log + log ] n n = D( k ) =D( k ) = D( k ) B. Proof of heorem Proof. Usng Defnons 3 and 4, we can express he loss of he algorhm on a pon x as L(x, clus()) = k P n p()(x c )k 4R

3 hen he followng chan of nequales are equvalen o eachoher. c L(x, clus()) log k P n p()(x c )k 4R log e k P n p()(x c )k 4R n p()e kx n c k 4R e n p()e L(x,c ) n p()e kx c k 4R! p()e kx c k 4R k P n p()(x c )k 4R () Le v = x.sncekxk R and R kc kr hen v [ ] d.equaon()sequvaleno n p()e kv k e k P n p()v k hs nequaly holds by Jensen s heorem snce he funcon f(v )=e kv k s concave when v [ ] d. B. Proof of heorem Proof. We can proceed by applyng heorem o bound he cumulave loss of he algorhm and hen use Lemma 6. As we proceed we follow he approach n he proof of heorem.. n [6]. L (alg) = L(x, clus()) n log p ()e L(,) = logp (x,...,x ) n = log P (x,...,x a,,...,a, )P (a,,...,a, ) n Y = log p ()P (x a,) P (x a,x,...,x ) = log n = log n = log n n Y e L(,) e n e P L(,) log L (a )+logn he las nequaly holds for any a, so n parcular for a. B.3 Proof of heorem 3 n L(,) e P L(,) Proof. By applyng frs heorem and hen Lemma 6 and followng he proof of heorem 3 (Man heorem) n he [6] we oban: L (alg) = L (alg) n log p ()e L(,)

4 Anna Choromanska, Clare Moneleon Y Y = log[ p ( ) P (, ) e L(,) ],..., Y = log[ P (,..., ) e L(,) ],..., where Noce ha also: Y P (,..., ) = p ( ) P (, ) ny ny P (,..., ) = p ( ) j= ( j) n j (,..., ) where n j(,..., ) s he number of ransons from sae o sae j, n a sequence,..., and P j nj(,..., ), s he number of mes he sequence was n sae, excep a he fnal me-sep. hus P j nj(,..., )=( )ˆ (,..., ), where ˆ (,..., ) s he emprcal esmae, from he sequence,...,, of he margnal probably of beng n sae, a any me-sep excep he fnal one. I follows ha: n j(,..., )=( )ˆ (,..., )ˆ j(,..., ) where ˆ j(,..., )= s he emprcal esmae of he probably of ha parcular sae ranson, on he bass of,...,. n j (,..., ) Pj n j (,..., ) hus: P (,..., ) = p ( ) ny ny j= ( j) ( )ˆ (,..., )ˆ j (,..., ) = p ( )e ( ) P n P nj= ˆ (,..., )ˆ j (,..., )log j hus: L (alg) Y log[ P (,..., ) e L(,) ],..., = log[,..., p ( )e ( ) P n P nj= ˆ (,..., )ˆ j (,..., )log j Y e L(,) ] Le,..., correspond o he bes segmenaon of he sequence no s segmens meanng he s-paronng wh mnmal cumulave loss. Obvously hen he hndsgh-opmal (cumulave loss mnmzng) seng of swchng rae parameer, gvens, s: = s and snce we are n he fxed-share seng: j = for = j and j = n can connue as follows: log[p ( )e ( ) P n P nj= ˆ (,..., )ˆ j (,..., )log j Y e L(,) ] for 6= j. We n n = logp ( ) ( ) ˆ (,..., )]ˆ j(,..., )log j + L(,) j= n n =logn ( ) ˆ (,..., )ˆ j(,..., )log j + L(,) j= n = L(,)+logn ( ) ˆ (,..., )ˆ j(,..., )log j,j=

5 n n ( ) ˆ (,..., )ˆ j(,..., )log j j=,j6= n = L(,)+logn ( )( )log( ) ˆ (,..., ),j= n ( ) n log( n ) j=,j6= n ˆ (,..., ) = L(,)+logn ( )( )log( ) ( ) log( n ) = L(,)+logn ( )( )log( ) ( ) log +( ) log(n ) = L(,)+logn ( )( )log( ) ( ) log +slog(n ) = L(,)+logn +slog(n ) ( )(( )log( )+ log ) = L(,)+logn +slog(n ) + ( )(H( )+D( k )) B.4 Proof of Lemma Proof. Gven any b, wll su ce o provde a sequence such ha any b-approxmaon algorhm canno oupu x,he curren pon n he sream, as one of s ceners. We wll provde a couner example n he seng where k =. Gven any b, consder a sequence such ha he sream before he curren pon consss enrely of n daa pons locaed a some poson A, and n daa pons locaed a some poson B, where n >n >b. Le x be he curren pon n he sream, and le be locaed a a poson ha les on he lne segmen connecng A and B, bucloseroa. ha s, le ka k = a and kb k = c such ha <a< n n c. hs s refleced by he fgure, whch ncludes some + addonal pons D and E: A D E -B - where A,D,,E,B are lyng on he same lne. Le D [A, ] ande [, B]. Le for parcular locaon of D and E, ka Dk = a, kd k = a, k Ek = c and ke Bk = c,suchhaa + a = a and c + c = c. We wll frs reason abou he opmal k-means cluserng of he sream ncludng x.wewllconsdercases: ) Case : opmal ceners le nsde he nerval (A, ). Any such se canno be opmal snce by mrror-reflecng he cener closes o wh respec o (such ha now les n he nerval (B,) andhashesamedsanceo as before) we can decrease he cos. In parcular he cos of pons n B wll only decrease, leavng he cos of pons n A, plus he cos of, unchanged. ) Case : opmal ceners le nsde he nerval (B,). In hs case we can alernaely mrror-reflec he closes cener o wh respec o and hen wh respec o A (reducng he cos wh each reflecon) unl wll end up n nerval [A, ). he cos of he fnal soluon s smaller han when boh ceners were lyng n (B,), because whle mrror-reflecng wh respec o, hecosofponsna can only decrease, leavng he cos of pons n B and pon unchanged and whle mrror-reflecng wh respec o A, hecosofpon can only decrease, leavng he cos of pons n A and n B unchanged.

6 Anna Choromanska, Clare Moneleon hus he opmal se of ceners (le s call hem: C,C )musbesuchha: C [A, ] andc [B,]. he fgure above reflecs hs suaon and s su cenly general; hus D = C and E = C. We wll now consder all possble locaons of such ceners (C,C )andhercoss: ) (A,): cos = n (c + c ).) (A,E) where E (, B): cos = (a + a ) + n c when c a.) (A,E) where E (, B): cos = c + n c when c <a 3) (A,B): cos = (a + a ) 4) (D,) where D (A, ): cos = n a + n (c + c ) 5.) (D,E) where D (A, ) ande (, B): cos = n a + a + n c when a c 5.) (D,E) where D (A, ) ande (, B): cos = n a + c + n c when a >c 6) (D,B) where D (A, ): cos = n a + a 7) (,E) where E (, B): cos = n (a + a ) + n c 8) (,B) where E (, B): cos = n (a + a ) Noce: cos() > cos(.) hus opmal confguraon of ceners can no be as n case ) cos(7) > cos(8) hus opmal confguraon of ceners can no be as n case 7) cos(4) > cos(5.) hus opmal confguraon of ceners can no be as n case 4) cos(8) > cos(6) hus opmal confguraon of ceners can no be as n case 8) cos(5.) > cos(.) hus opmal confguraon of ceners can no be as n case 5.) cos(5.) > cos(6) hus opmal confguraon of ceners can no be as n case 5.) cos(.) > cos(3) hus opmal confguraon of ceners can no be as n case.) Consder case (.): cos = c + n c = c + n (c c ) and c (,a). Keepng n mnd ha <a< n c,seasy n + o show ha cos >a + n (c a) >n a >n a >a =cos(case(3)). hus he opmal confguraon of ceners can no be as n case.). herefore, he only cases we are lef o consder are cases 3 and 6. In boh cases, one of he opmal ceners les a B. Le hs cener be C. Snce a<c, he remanng pons (n A and ) are assgned o cener C whose locaon can be compued as follows (please see fgure below for noaon smplcy): A C B = C Le ka C k = and ka k = a (as defned above). Snce we proved ha C mus be fxed a B, heponsab wll conrbue o he objecve, and we can solve for C by mnmzng he cos from pons n A, plus he cos from : mn{n +(a ) }.hesoluons = a.husheoalopmalcoss: n + OP = n +(a ) = na (n +) + n a (n +) We wll now consder any -cluserng of se A,, B when one cluser cener s x, whch s herefore locaed a. We can lower bound he k-means cos of any wo se of ceners ha conan, as follows: cos({, ĉ}) n a for any ĉ; he mnmal cos s acheved when he oher cener s locaed a B (when one of he ceners s locaed n, helocaonof he oher cener ha gves he smalles possble k-means cos can be eher A (case ), D (case 4), E (case 7) or B (case 8), where case 8 has he smalles cos from among hem). Volang he b-approxmaon assumpon occurs when cos({, ĉ}) >b OP. Gvenheabove, wouldsu n a >b OP. has: n a na >b (n +) + n a, b<(n (n +) +) ce o show hs holds, as we chose n >b n he begnnng. herefore he b-approxmaon assumpon s volaed. B.5 Proof of Lemma 4 Proof. For ease of noaon, we denoe by ( W,> he k-means cos of algorhm a on he daa seen n he wndow ( W, > (omng he argumen, whch s he se of ceners oupu by algorhm a a me ). Snce a s b-approxmae hen: 8 W ( W,> b OP ( W,> (3) For any W such ha W<,wecandecompose such ha = mw + r where m N and r W. Noce hen ha

7 for any j {,...,W } he followng chan of nequales holds as he drec consequence of Equaon 3: ( W +j, > + ( W +j, W +j> + ( 3W +j, W +j> ( mw +j, (m )W +j> + <, mw +j> b OP ( W +j, > + b OP ( W +j, W +j> + b OP ( 3W +j, W +j> b OP ( mw +j, (m )W +j> + b OP <, mw +j> b OP <, > (4) where he las nequaly s he drec consequence of Lemma 3. Noce ha d eren value of j refers o d eren paronng of he me span <, >. Fgure llusraes an example when = and W = 3. Fgure : D eren paronng of me span = no me wndows, W = 3, wh respec o d eren values of j. Le W refer o he daa chunk seen n me span < max(, W +),>.Wefnshbyshowngha, W W { ( W +j, > (5) j= + ( W +j, W +j> + ( 3W +j, W +j> ( mw +j, (m )W +j> + <, mw +j>} b W OP <, > he lef hand sde of he frs nequaly (5) sums he losses over only a subse of all he wndows ha are nduced by paronng he me span usng all possble values of j. he fnal nequaly follows by applyng (4). o llusrae some of he deas used n hs proof, Fgures and 3 provde schemacs. Fgure 3 shows he wndows over whch he loss s compued on he lef hand sde of nequaly (5), whch s a subse of he se of all wndows nduced by all possble paronng of me span usng all possble values of j, whch s shown n Fgure.

8 Anna Choromanska, Clare Moneleon Fgure 3: Illusraon of he me spans over whch he loss s beng compued n Equaon 5. =, W = 3. Colors red, blue and black correspond o d eren paronngs, wh respec o j, of me span llusraed n Fgure. B.6 Proof of Lemma 5 Proof. he loss of exper a me s defned n Defnon 3 as he scaled componen of he k-means cos of algorhm a s cluserng a me. ha s, (C )= P x W mncc kx ck =mn cc kx ck + P x W \x mn cc kx ck = 4R L (C )+ P x W \x mn cc kx ck. herefore, snce all erms n he sum are posve, 4R L (C ) (C ), and L (a). where n he second 4R nequaly we subsue n our smplfyng noaon, where C are he se of clusers generaed by algorhm a a me, and P s algorhm a s k-means cos on W. Now, summng over eraons, and applyng Lemma 4, we oban b W L(a) OP 4R <, >. B.7 Proof of heorem 7 Proof. he heorem drecly follows from heorem 3, Lemma 4 and Lemma 5. Noce ha boh Lemma 4 and Lemma 5 hold n a more general seng when n each me wndow he deny of he exper (b-approxmae algorhm) may change n an arbrarly way. However we dd no provde he generalzed proofs of hose lemmas snce would only complcae he noaon. B.8 Proof of heorem 4, Corollary, and heorem 8 Lemma 8. L log ( ) = L log (p,)= log 4 p ( )e,..., L(,) 3 Y e L(,) P (, ) 5 Proof. I follows drecly from Lemma 6. Lemma 9. " ( L log ( ) Llog ( )= log Q(~s )exp ~s n n j= ˆ (~s )ˆ j(~s )log( j ) j )# where ~s =,..., and Q(~s ) s he poseror probably over he choces of expers along he sequence, nduced by hndsgh-opmal. Proof. I follows drecly from applyng proof of Lemma A.. n [6] wh redefned (~s )suchha (~s )= Q e L(,).

9 Lemma. L log ( ) Llog ( ) ( ) n D( k ) ( ) max D( k ) Proof. I holds by heorem 3 n [6]. Lemma. L log ( ) Llog ( )+( )D( k ) Proof. I holds by Lemma 7, Lemma. Lemma. L ( ) L log ( )+( )D( k ) Proof. I holds by Lemma and Lemma. Lemma 3. L ( ) bw OP +( R ) n D( ) Proof. L log ( )= L log (p,) =( n log p ()e L(,) ) = =( n log p ()e k x c R k ) Defne as prevously v = x and noce R v [ ; ] d. c hus we connue: n =( log p ()e kv k ) By he proof of Lemma 4 kv k 4R where s he k-means cos of algorhm h cluserng ( h exper) a me. hus we can connue as follows (we om condonng by snce holds for any ~p ): n log p ()e 4R n log p ()e max 4R log e max 4R = max = 4R 8R max bw 8R OP

10 Anna Choromanska, Clare Moneleon he las nequaly follows from he proof of heorem 7. hus fnally: L ( ) L log n ( ) Llog ( )+( ) D( ) bw OP +( 4R ) n D( ) B.9 Proof of heorem 5 Proof. he logloss per me sep of he op-level algorhm (for he ease of noaon we skp ndex log), whch updaes s dsrbuon over - expers (Fxed-Share ( j algorhms) s: L op (p op,)= log m j= p op (j)e Llog (j,) where L log (j, ) = n log p j()e L(,) hus: L op (p op,)= log m j= p op (j) n p j()e L(,) he updae s done va he Sac Exper algorhm. When runnng Learn- (m), m possble values j are esed. Followng [5, 6], he probablsc predcon of he algorhm s defned as: m j= p op (j) n p j()e L(,) Le us frs show ha hs loss-predcon par are (, )-realzable, hus hey sasfy: hus we have o prove ha followng holds: log m j= L op (p op,) log p op m p op(j)e Llog (j,) j= n m (j) p j()e L(,) log j= p op (j)e Llog (j,) hus we have o prove: whch s equvalen o log m j= p op n m (j) p j()e L(,) log j= p op (j)e log P n p j ()e L(,) log m j= p op n m (j) p j()e L(,) log j= p op (j) n p j()e L(,) where he las nequaly holds. Now, snce our logloss-predcon par are (,)-realzable, by Lemma n [9] we have: L op mn L log ( j)+logm { j }

11 where { j} s he dscrezaon of he parameer ha he Learn- algorhm akes as npu. Corollary we ge: L log (alg) Llog ( )+( ) mn D( k j)+logm { j } Now, by applyng C Addonal Expermenal Deals o provde qualave resuls on OCE s performance, n Fgures 4-5 we show cluserng analogs o learnng curves from our expermens n he predcve seng. hese curves generaed he sascs for able. We plo he bach k-means cos of each exper, and he OCE algorhms, on all he daa seen so far, versus. Whle Fxed-Share algorhms wh hgh values of su er large oscllaons n cos, Learn- s performance racks, and ofen surpasses ha of he Fxed-Share algorhms. We also demonsrae he evoluon of he weghs mananed by he Fxed-Share and Learn- OCE algorhms. In Fgures 6-7 we show he evoluon over me of he weghs mananed by he OCE Fxed-Share algorhm over he expers (cluserng algorhms). For smaller values of we observe an nverson n he wegh orderng beween expers 4 and 5 a around eraon 48 for hs parcular expermen. For values closer o / and here s a hgher amoun of shfng of weghs among expers. In Fgure 8 we show he evoluon over me of he weghs mananed by he OCE Learn- algorhm over -expers (Fxed-Share algorhms run wh a d eren seng of he parameer). he expermen was performed wh 45 -expers (Fxed-Share algorhms wh d eren values of he parameer). Lower values receved hgher weghs. One value of (he lowes) receves an ncreasng share of he wegh, whch s conssen wh he fac ha he Sac-Exper algorhm s used o updae weghs over -expers.

12 Anna Choromanska, Clare Moneleon 4.5 x Exper Exper Exper 3 Sac Exper Fxed Share Fxed Share Fxed Share 3 Fxed Share 4 Learn alpha Cloud daa 3 k means cos x 8 Spambase daa 5 4 k means cos x Inruson daa.5 k means cos x 7 Fores fres daa 3.5 k means cos Fgure 4: Cluserng analogs o learnng curves; k-means cos vs. ; op o boom: Cloud, Spambase, Inruson, Fores fres. Legend n upper lef.

13 6 x 4 Robo daa 5 4 k means cos x 7 Mxure of 5 Gaussans 8 k means cos x 6 Mxure of 5 Gaussans k means cos Fgure 5: Cluserng analogs o learnng curves; k-means cos vs. ; op o boom: Robo daa, 5 Gaussans daa, Zoomng n on y-axs of 5 Gaussans. Legend n Fgure 5.

14 Anna Choromanska, Clare Moneleon exper exper exper3 exper4 exper5 exper6 Fores fres daa. Fxed share. alpha =.57 Expers weghs Fores fres daa. Fxed share. alpha = Expers weghs Fores fres daa. Fxed share. alpha = Expers weghs Fgure 6: Evoluon of weghs over expers for Fxed-Share algorhms usng d eren (low) values of he parameer; Fores fres daa. 6-expers. Legend n op lef.

15 .6 Fores fres daa. Fxed share. alpha = Expers weghs Fores fres daa. Fxed share. alpha = Expers weghs Fgure 7: Evoluon of weghs over expers for Fxed-Share algorhms usng d eren values of he parameer (op: close o /; Boom: close o ); Fores fres daa. 6-expers. Legend n Fgure 6..9 Fores fres daa. Learn alpha. W =.k = Expers weghs Fgure 8: Evoluon of weghs mananed by Learn-, overs -expers, n he 6-exper Fores-Fres expermen. Lowes values of receve hghes wegh.