Global Optimization & Learning Algorithms

1. CS B553: Algorithms for Optimization and Learning Global optimization

2. Agenda: Global Optimization • Local search, optimization • Branch and bound search • Online search

3. f Global Optimization • min f(x), x in S  Rn • Want guarantees on a global optimum x2 S x1

4. Exhaustive Search S e

5. Exhaustive Search S

6. Lower-bound function • LowerBound(A): for any set A  S, returns a lower bound on f(x) for all xA f x2 S A fL(A) x1

7. 1 2 4 2.5 3 Values of LowerBound(A) f(x) = 3.2 Pruning the Search Tree

8. What order? Branch-and-bound Algorithm • Let f* be the best value seen so far • Init: f* = f(point in S0) • Q = {S0} • While Q not empty, repeat: • S = remove an item from Q • If LowerBound(S)  f* or |S|<e, then discard S • f* = min(f*,f(point in S)) • Split S and add subregions to FRINGE Pruning step

9. Performance • Works well when • When LowerBoundis relatively tight • When n isn’t too large • Methods for generating LowerBound • Interval arithmetic • Solving “relaxed” versions of f • Problem-specific ways

10. Example: 3D Needle Steering Feedback Controller Closed-loop feedback rule: Sense current position/orientation of needle Twist at the speed such that the predicted helix path minimizes the distance to target

11. Constant-twist-rate helices

12. Reachable points under constant-twist-rates

13. Find small initial domain Finding closest point Use cylindrical lower bound BnB takes < 1ms on average

14. Collision Checking • Check whether objects overlap

15. Hierarchical Collision Checking • Enclose objects into bounding volumes (spheres or boxes) • Check the bounding volumes

16. Hierarchical Collision Checking • Enclose objects into bounding volumes (spheres or boxes) • Check the bounding volumes first • Decompose an object into two

17. Hierarchical Collision Checking • Enclose objects into bounding volumes (spheres or boxes) • Check the bounding volumes first • Decompose an object into two • Proceed hierarchically

18. Hierarchical Collision Checking • Enclose objects into bounding volumes (spheres or boxes) • Check the bounding volumes first • Decompose an object into two • Proceed hierarchically

19. Bounding Volume Hierarchy (BVH) A BVH (~ balanced binary tree) is pre-computed for each object (obstacle, robot link)

20. BVH of a 3D Triangulated Cat

21. A A C C B B Collision Checking Between Two Objects BVH of object 1 BVH of object 2 [Usually, the two trees have different sizes]  Search for a collision

22. pruning Search for a Collision Search tree AA A A

23. Search for a Collision Search tree AA Heuristic: Break the largest BV A A

24. BA CA B C Search for a Collision Search tree AA Heuristic: Break the largest BV A

25. BA CA CB CC C C B Search for a Collision Search tree AA

26. C B BA CA CB CC C B Search for a Collision Search tree AA If two leaves of the BVH’s overlap(here, C and B) check their content for collision

27. rY d rX Y X Search Strategy • If there is no collision, all paths must eventually be followed down to pruning or a leaf node • But if there is collision, one may try to detect it as quickly as possible •  Greedy best-first search strategy with f(N) = h(N) = d/(rX+rY) [Expand the node XY with largest relative overlap (most likely to contain a collision)]

28. Performance • On average, over 10,000 collision checks per second for two 3-D objects each described by 500,000 triangles, on a contemporary PC • Checks are much faster when the objects are either neatly separated ( early pruning) or neatly overlapping ( quick detection of collision)

29. Review of Optimization Unit • Descent vs root finding • Gradient descent, Newton’s method, Quasi-newton methods • Constraints: Lagrange multipliers and KKT conditions • Convex optimization • LP, QP • Interior point methods • Metaheuristic methods

30. Putting Optimization into Practice • Being able to formulate optimization problems is often just as important as choosing the right algorithm • Well conditioned functions • Constraints • Auxiliary variables • Transformations • Analysis vs. computation

31. What else is out there? • Special techniques for sparse problems • Special problem formulations • e.g., SDP, SOCP • Mixed-integer programming • Semi-infinite programming • In-depth analysis suitable for “industrial strength” optimization

32. Modern Trends • Learning with big data • Combinatorial / hybrid problems • Online or distributed optimization • Specific function classes: • Stochastic functions • Robot/biomechanical motions • Feedback control policies

33. Readings for Next Class • K&F, Chapter 2