Blocks of Coordnates, Stoch Progr, and Markets Sjur Ddrk Flåm Dep. Informatcs, Unversty of Bergen, Norway Bergamo June 2017 jur Ddrk Flåm (Dep. Informatcs, Unversty Blocks of Bergen, of Coordnates, Norway) Stoch Progr, and Markets Bergamo June 2017 1 / 21
Sup-convoluton π I (x I ) := sup ( 2I ) π (x ) : x = x I. 2I Fnte, xed set I of ndependent/ ndvdual agents. π : X! R[ f g payo or pro t of ndvdual 2 I. x I 2 X s a xed aggregate endowment. Sjur Ddrk Flåm (Dep. Informatcs, Unversty Blocks of Bergen, of Coordnates, Norway) Stoch Progr, and Markets Bergamo June 2017 2 / 21
Presumptons X s Eucldean (or Hlbert) wth nner product h,. There exsts an optmal soluton - alas e cent allocaton. Each π : X! R[ f g concave usc wth nt(domπ ) 6=?. For each 2 I and x 2 domπ d 7! π 0 (x ; d) := lm s!0 + π (x + sd) π (x ) s s usc. Sjur Ddrk Flåm (Dep. Informatcs, Unversty Blocks of Bergen, of Coordnates, Norway) Stoch Progr, and Markets Bergamo June 2017 3 / 21
Motvatng nstances represents π I (x I ) := sup ( 2I ) π (x ) : x = x I 2I e cent allocaton of scarce resources, or mutual sharng of aggregate rsk. Partcularty here: Just (repeated) blateral reallocatons! Sjur Ddrk Flåm (Dep. Informatcs, Unversty Blocks of Bergen, of Coordnates, Norway) Stoch Progr, and Markets Bergamo June 2017 4 / 21
Classc case: The co ee house Lloyds of London agent 2 I s a "name" = accredted underwrter of rsk. C = nte "complete" lst of ver able contngences, X = R C, and π (x ) := E Π [x ] := Π [c, x (c)]µ (c). c2c Probablty measure (ndvdual belef) µ over C. Agents stroll n the house. They meet and dscuss par-wse =) repeated blateral trade. Double stoch: 1) random encounters, 2) exchange of random labltes. Sjur Ddrk Flåm (Dep. Informatcs, Unversty Blocks of Bergen, of Coordnates, Norway) Stoch Progr, and Markets Bergamo June 2017 5 / 21
Central versus dstrbuted computaton. Query: Knowledge about π I (x I ) := sup ( s dspersed. No omnscent center. 2I ) π (x ) : x = x I 2I Followng Nesterov et al, Tseng et al. use Block-coordnate methods exchange! construed as blateral drect No center! No coordnatng agent! Sjur Ddrk Flåm (Dep. Informatcs, Unversty Blocks of Bergen, of Coordnates, Norway) Stoch Progr, and Markets Bergamo June 2017 6 / 21
Blateral exchange Agent 2 I meets agent j, holdng x 2 X and x j 2 X respectvely. After drect trade, wth no LOG: where x +1 := x + and x +1 j := x j := sd features step-sze s > 0 and drecton d 2 X. Frst ssue: Secondary ssue: Thrd, ssue: whch drecton? whch step-sze? who meets whom? Sjur Ddrk Flåm (Dep. Informatcs, Unversty Blocks of Bergen, of Coordnates, Norway) Stoch Progr, and Markets Bergamo June 2017 7 / 21
Frst ssue: Whch drectons are feasble? Recall that x 2 domπ and x +1 := x + = x + sd 2 domπ wth s > 0 =) d 2 D (x ) := fd 2 X : x + sd 2 domπ for s 0 su cently smallg. Assump: any such cone s closed wth non-empty nteror. Smlarly, x j 2 domπ j and x +1 j := x j = x sd 2 domπ j wth s > 0 =) d 2 D j (x j ). Hence d 2 D j (x, x j ) := D (x ) \ D j (x j ). Sjur Ddrk Flåm (Dep. Informatcs, Unversty Blocks of Bergen, of Coordnates, Norway) Stoch Progr, and Markets Bergamo June 2017 8 / 21
Second ssue: whch drectons appear reasonable? Proposal: choose supgradents x d = x 2 π (x ), x j x j. Economc ratonale: When X = R C and D j (x, x j ) = X, d c > 0 () x c > x jc. 2 π j (x j ) and post Mathematcal ratonale, stll wth X = R C and D j (x, x j ) = X: If x = π 0(x ) and xj = πj 0(x j ) are d erentable, then d = π 0 (x ) π 0 j (x j ) =) π 0 (x ; d) + π 0 j (x j ; d) = x In that case, x 6= x j =) system π (x +1 ) + m > π (x ) and π j (x +1 j ) + m j > π j (x j ) s solvable wth sde payments m + m j = 0. xj 2. Sjur Ddrk Flåm (Dep. Informatcs, Unversty Blocks of Bergen, of Coordnates, Norway) Stoch Progr, and Markets Bergamo June 2017 9 / 21
Synthess: Drectons ought be feasble and "reasonable" Assumpton: for any agent, holdng x 2 domπ, π (x ) = X (x ) N (x ), N (x ) := N(domπ, x ), where x 2 domπ X (x ) s outer semcontnuous, wth non-empty, unformly bounded, closed convex values. Proposton: Any supgradent par x 2 π (x ), x j 2 π j (x j ) admts x = x n 2 X (x ) N (x ) & x j = x j n j 2 X j (x j ) N (x j ) such that projecton P j [] of supgradent d erence onto D j (x, x j ) yelds d = P j x xj wth d x x j. Upshot: appears feasble and reasonable. d 2 π (x ) π j (x j ) Sjur Ddrk Flåm (Dep. Informatcs, Unversty Blocks of Bergen, of Coordnates, Norway) Stoch Progr, and Markets Bergamo June 2017 10 / 21
On projectons Projecton onto own doman domπ done of by the agent hmself: x 2 π (x ) = X (x ) N (x ). Projecton onto allocaton space ( ) X := x = (x ) : x = x I 2I rendered super uous by blateral trade. Sjur Ddrk Flåm (Dep. Informatcs, Unversty Blocks of Bergen, of Coordnates, Norway) Stoch Progr, and Markets Bergamo June 2017 11 / 21
On best choce of drecton The steepest slope S j (x, x j ) := sup π 0 (x ; d) + π 0 j (x j ; d) : d 2 D j (x, x j ) and kdk 1 Proposton (steepest slope and supgrad d erence). Recall P j [] := projecton onto D j (x, x j ). Then S j (x, x j ) = dst[ π (x ), π j (x j )] = mn kp j [ π (x ) π j (x j )]k = mn P j [X (x ) X j (x j )] Moreover, (x, x j )! S j (x, x j ) s lsc. Obs: No trade f the nterlocutors see a common prce: S j (x, x j ) = 0 () π (x ) \ π j (x j ) 6=?. How could a common prce emerge? Tentatve answer: By repeated blateral exchanges! Sjur Ddrk Flåm (Dep. Informatcs, Unversty Blocks of Bergen, of Coordnates, Norway) Stoch Progr, and Markets Bergamo June 2017 12 / 21
Thrd ssue, the protocol: who meets next whom? Assumpton: Completely random matchng: Pck any agent par ndependently - and wth unform probablty =) Agent meets agent j wth probablty n 1/ = 2 2 n(n 1) where n := #I 2. Sjur Ddrk Flåm (Dep. Informatcs, Unversty Blocks of Bergen, of Coordnates, Norway) Stoch Progr, and Markets Bergamo June 2017 13 / 21
Algorthm: Repeated blateral deals Exchange construed as algorthm: Agent starts wth some endowment x 2 domπ 2I x = x I. such that Select any agent par, j n equprobable, ndependent manner. Update ther actual holdngs x, x j : x +1 := x + sd and x +1 k := x j sd wth step-sze s 0 and feasble drecton d 2 π (x ) π j (x j ). Contnue untl convergence. Sjur Ddrk Flåm (Dep. Informatcs, Unversty Blocks of Bergen, of Coordnates, Norway) Stoch Progr, and Markets Bergamo June 2017 14 / 21
Convergence of dstance to opt. Assume ^X := foptmal solutonsg non-empty. Let x k := (x k ). Proposton (Convergence of dstance to optmalty). Suppose sk 2 < +. k2k Then, for any ^x 2 ^X, the sequence k 7! x k to a nte lmt. ^x converges almost surely Lkewse for k 7! dst(x k, ^X). Sjur Ddrk Flåm (Dep. Informatcs, Unversty Blocks of Bergen, of Coordnates, Norway) Stoch Progr, and Markets Bergamo June 2017 15 / 21
Man argument When, j trade, let d 2 X I have d = d 2 π (x ) π j (x j ) & d j = d and all other d-comp = 0. Updatng takes the form, x +1 = x + sd. For any optmal ^x : x +1 ^x 2 = kx ^xk 2 + 2s[hx ˆx, d + hx j ˆx j, d j ] + 2s 2 kdk 2. Here 2 kdk 2 D. Take E wrt random matchng: E x +1 ^x 2 kx ^xk 2 + 8s n 1,j x ˆx, n 1 (x x j ) + s 2 D. Obs: Projecton P[x ] onto fx : 2I x = 0g has -th component n 1 j (x ). Hence above sum x j hx ˆx, P[x ] = hx ^x, P[x ] = hx ^x, x. Contnue wth sem-martngale (Robbns-Segmund) arguments. Sjur Ddrk Flåm (Dep. Informatcs, Unversty Blocks of Bergen, of Coordnates, Norway) Stoch Progr, and Markets Bergamo June 2017 16 / 21
Traps and queres A possble trap: It may happen that an accumulaton pont x just sats es par-wse optmalty: π (x ) \ π j (x j ) 6=? 8, j. (1) Shortcut: From here onwards, suppose (1) holds at each accumulaton pont. Broadly, for ths: s k : agent meets agent j = + 8, j, and mprovement π j n jont payo π + π j π j s ( xed fracton of steepest slope) = sσ j S j (x, x j ). A query:? Passng from par-wse optmalty to overall optmalty? Sjur Ddrk Flåm (Dep. Informatcs, Unversty Blocks of Bergen, of Coordnates, Norway) Stoch Progr, and Markets Bergamo June 2017 17 / 21
On e cency/optmalty and shadow prces Recall π I (x) := sup ( 2I ) π (x ) : x = x. 2I Each p 2 π I (x I ) s called a shadow prce. Proposton (optmalty $ shadow prcng). (x ) s optmal allocaton of x I =) any shadow prce s common: Conversely, f 2I x = x I, then π I (x I ) \ 2I π (x ). \ 2I π (x ) π I (x I ). and non-empty lhs =) (x ) optmal allocaton. Sjur Ddrk Flåm (Dep. Informatcs, Unversty Blocks of Bergen, of Coordnates, Norway) Stoch Progr, and Markets Bergamo June 2017 18 / 21
Exchange markets drven by d erental valuatons steepest slope S j (x, x j ) > 0 () π (x ) \ π j (x j ) =?. In every cluster pont (x ) each agent par see a common prce; that s: π (x ) \ π j (x j ) 6=? 8, j. Standng assumpton: at least one agent has x 2 ntx and π s d erentable there. ) p 2 \ 2I π (x ). Common prces emerge nally! They are results, not prerequstes! Proposton (market equlbrum). Any optmal allocaton (x ) supported by shadow prce p 2 π I (x I ) s an exchange market equlbrum n that x 2 arg max fπ hp, g 8. Conversely, then (x ) s optmal and p a shadow prce. Sjur Ddrk Flåm (Dep. Informatcs, Unversty Blocks of Bergen, of Coordnates, Norway) Stoch Progr, and Markets Bergamo June 2017 19 / 21
Economc summary Prces need not come from somewhere; they rather emerge. Prce-takng or maxmzaton s nether necessary nor qute realstc. Agents can do wthout posted (common) prces. Agents merely seek own mprovements. Everybody can contend wth dosyncratc, local nformaton. No coordnaton, central agency, or global knowledge s ever requred. Sjur Ddrk Flåm (Dep. Informatcs, Unversty Blocks of Bergen, of Coordnates, Norway) Stoch Progr, and Markets Bergamo June 2017 20 / 21
References Flåm, Blateral exchange and compettve equlbrum, Set-Valued and Varatonal Analyss (2015) Flåm, Monotoncty and market equlbrum, Set-Valued and Varatonal Analyss (2016) Necoara, Nesterov & Glneur, A random coordnate descent method on large-scale optmzaton problems wth lnear constrants, (2015) Tseng & Yun, Block-coordnate gradent descent method for lnearly constraned nonsmooth separable optmzaton, Journal of Optmzaton Theory and Applcatons 140, 513-535 (2009) Xao & Boyd, Optmal scalng of a gradent method for dstrbuted resource allocaton, Journal of Optmzaton Theory and Applcatons 129, 3, 469-488 (2006) Sjur Ddrk Flåm (Dep. Informatcs, Unversty Blocks of Bergen, of Coordnates, Norway) Stoch Progr, and Markets Bergamo June 2017 21 / 21