1 / 28

Simulated annealing for convex optimization

Simulated annealing for convex optimization. Adam  . Kalai: TTI-Chicago Santosh Vempala: MIT. Bar Ilan University 2004. 100-million dollar endowment (thanks, Toyoda!) 12 tenure -track slots, 18 visitors On University of Chicago campus Optional teaching Advising graduate students.

susies
Download Presentation

Simulated annealing for convex optimization

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Simulated annealing for convex optimization Adam . Kalai: TTI-Chicago Santosh Vempala: MIT Bar Ilan University 2004

  2. 100-million dollar endowment (thanks, Toyoda!) • 12 tenure-track slots, 18 visitors • On University of Chicago campus • Optional teaching • Advising graduate students

  3. Outline Simulated annealing gives the best known run-time guarantees for this problem. It is optimal among a class of random search techniques. • Simulated annealing • A method for blind search: • f:X!, minx2X f(x) • Neighbor structure N(x) µ X • Useful in practice • Difficult to analyze • A generalization of linear programming • Minimize a linear function over a convex set S ½ n • Example: min 2x1+5x2-11x3with x12+5x22+3x32· 1 • Set S specified by membership oracle M: n! {0,1} • M(x) = 1 $ x 2 S • Difficult, cannot use most linear programming techniques [GLS81,BV02] In high dimensions

  4. Steepest descent

  5. Random Search

  6. Simulated Annealing [KGV83] Phase 1: Hot (Random) Phase 2: Warm (Bias down) Phase 3: Cold (Descend) Phase 1: Hot (Random) Phase 2: Warm (Bias down) Phase 3: Cold (Descent)

  7. Simulated Annealing • f:X!, minx2X f(x) • Proceed in phases i=0,1,2,…,m • Temperature Ti = T0(1-)i • In phase i, do a random walk with stationary distributioni:i(x) / e-f(x)/Ti • i=0: near uniform ! i=m: near optima Geometric temperature schedule Boltzmann distribution Metropolis filter for stationary dist : From x, pick random neighbor y. If (y)>(x), move to y. If (y)·(x) move to y with prob. (y)/(x)

  8. Simulated Annealing • Great blind search technique • Works well in practice • Little theory • Exponential time • Planted graph bisection [JS93] • Fractal functions [S91]

  9. Convex optimization minimize f(x) = c ¢ x = height x 2 S = hill Find the bottom of the hill using few pokes (membership queries) Convex and linear slope

  10. Convex optimization minimize f(x) = c ¢ x = height x 2 S ½n = hill Find the bottom of the hill using few pokes (membership queries) • Ellipsoid method: O*(n10) queries • Random walks [BV02] O*(n5) queries Convex and linear slope n=# dimensions

  11. Walking in a convex set Metropolis filter for stationary dist: From x, pick random neighbor y. If (y)>(x), move to y. If (y)·(x), move to y with prob. (y)/(x)

  12. Walking in a high-dimensional convex set

  13. Hit and run • To sample with stationary dist. • Pick a random direction through the point • C = S Å line in direction • Take a random point from|C C S

  14. Hit and run • Start from a point x, random from dist. • After O*(n3) steps, you have a new random point, “almost independent” from x [LV03] • Difficult analysis C S

  15. Random walks for optimization [BV02] • Each phase, volume decreases by¼ 2/3 • In n dimensions, O(n) phases to halve distance to opt.

  16. Annealing is slightly faster • minx 2 S c ¢ x • Use distributions: • i(x) / e-c¢x/Ti • . • After O( ) phases, halve distance to opt. • That’s compared to O(n) phases [BV02] Boltzmann distribution Geometric temperature schedule

  17. Annealing Optimality • Assumptions: • Sequence of distributions1,2,… • Each density diis log-concave: • Consecutive densities di, di+1overlap: • Requires at least*( ) phases • Simulated Annealing does it in O*( ) phases

  18. Lower bound idea • mean mi = Ei[c ¢ x] • variancei2 = Ei[(c ¢ x – mi)2] • overlap • lemma: mi – mi+1· (i+i+1)ln(2P) • follows from log-concavity ofi • log-concave ! P(t std dev’s from mean) < e-t • In worst case, e.g. cone, small std dev • i· (mi - min c ¢ x)/

  19. Worst case: a cone • minx 2 S x0 • S = { x2n | -x0· x1,x2,…,xn-1· x0 · 10} • Uniform dist. on S|x0 <  • mean ¼ – /n • std dev ¼/n • Boltzmann dist. e- x/ • mean ¼ n • std dev ¼ linear program

  20. Any convex shape • Fix convex set S and direction c. • Fix mean m = E[c ¢ x] • d(x)=f(c¢x), log-concave • Conjecture:The log-concave distributionover S with largest variancei2 = Ei[(c ¢ x – mi)2] is a Boltzmann dist. (exponential dist.)

  21. Upper bound basics • Dist i/ e-c¢x/Ti • Lemma: Ei[c ¢ x] · (minx 2 S c ¢ x ) + n|c|Ti

  22. Upper bound difficulties • Not sufficient that distributions overlap • An expected warm start: Shape may change

  23. Shape estimation Estimate covariance with O*(n) samples Similar issues with hit and run

  24. Shape re-estimation • Shape estimate is covariance matrix (normalized) • OK as long as relative estimates are accurate within a constant factor • In most cases shape changes little • No need for re-estimation • Cube, ball, cone, … • In worst case, shape may change every phase • Increase run-time by factor of n • Differs from simulated annealing

  25. Run-time guarantees • Annealing: O*(n0.5) phases • State-of-the-art walks [LV03] • Worst case: O*(n) samples per phase(for shape) • O*(n3) steps per sample • Total: O*(n4.5) (compare to O*(n10) [GLS81], O*(n5) [BV02])

  26. Conclusions • Random search is useful for convex optimization [BV02] • Simulated annealing can be analyzed for convex optimization [KV04] • It’s opt among random search procedures • Annoying shape re-estimation • Difficult analyses of random walks [LV02] • Weird: no local minima! • Analyzed for other problems?

  27. Reverse annealing [LV03] • Start near single point v • Idea • Sample from density / e-|x-v|/Ti in phase i • Temperature increases • Move from single point to uniform dist • Estimate volume increase each time • Able to do in O*(n4) rather than O(n4.5) • Similar algorithm analysis

More Related