50 th Anniversary of The Curse of Dimensionality

1. 50th Anniversary of The Curse of Dimensionality • Continuous States: Storage cost:  resolutiondx Computational cost:  resolutiondx • Continuous Actions: Computational cost:  resolutiondu

2. Beating The Curse Of Dimensionality • Reduce dimensionality (biped examples) • Use primitives (Poincare section) • Parameterize V, policy (future lecture) • Reduce volume of state space explored • Use greater depth search • Adaptive/Problem-specific grid/sampling • Split where needed • Random sampling – add where needed • Random action search • Random state search • Hybrid Approaches: combine local and global opt.

3. Use Brute Force • Deal with computational cost by using cluster supercomputer. • Main issue is minimizing communication between nodes.

4. Cluster Supercomputing • (8) Cores w/ small local memory (cache) • (100) Nodes w/ shared memory (16GB) • (4-16Gb/s) Network • (100T) Disks

5. Q(x,u) = L(x,u) + V(f(x,u)) • c = L(x,u): as in desktop case • x_next = f(x,u): as in desktop case • V(x_next) • Uniform grid • Multilinear interpolation if all values available, distance weighted averaging if bad values

6. Allocate grid to cores/nodes

7. Handle Overlap

8. Push Updated V’s To Users

9. So what does this all mean for programming? • On a node, split grid cells among threads, which execute on cores. • Share updates of V(x) and u(x) within node almost for free using shared memory. • Pushing updated V(x) and u(x) to other nodes uses the network which is relatively slow…..

10. Dealing with the slow network • Organize grid cells into packet-sized blocks. Send them as a unit. • Threshold updates: too small, don’t send it. • Only do 1/N updates for each block (maximum skip time). • Tolerate packet loss (UDP) vs. verification (TCP/MPI)

11. Use Adaptive Grid • Reduce computational and storage costs by using adaptive grid. • Generate adaptive grid using random sampling.

12. Trajectory-Based Dynamic Programming

13. Full Trajectories Helps Reduce Resolution Needed SIDP Trajectory Based

14. Reducing the Volume Explored

15. An Adaptive Grid Approach

20. Global PlanningPropagate Value Function Across Trajectoriesin Adaptive Grid

21. Growing the Explored Region: Adaptive Grids

22. Bidirectional Search

23. Bidirectional Search Closeup

24. Spine Representation

25. Growing the Explored Region: Spine Representation

26. Comparison

28. One Link Swing Up Needed Only 63 Points

29. Trajectories For Each Point

30. Random Sampling of States • Initialize with a point at the goal with local models based on LQR. • Choose a random new state x. • Use the nearest stored point’s local model of the value function to predict the value of the new point (VP). • Optimize a trajectory from x to the goal. At each step use the nearest stored point’s local model of the policy to create an action. Use DDP to refine this trajectory. VT is cost of trajectory starting from x. • Store point at start of trajectory if |VT - VP |> λ(surprise), VT < Vlimit and VP < Vlimit, otherwise discard. • Interleave re-optimization of all stored points. Only update if Vnew < V (V is upper bound on value). • Gradually increase Vlimit.

31. Two Link Pendulum • Criterion:

33. Ankle Angle Hip Angle Ankle Torque Hip Torque

36. Four Links

37. Four Links: 8 dimensional system

39. Convergence? • Because we create trajectories to the goal, each value function estimate at a point is an upper bound for the value at that point. • Eventually all value function entries will be consistent with their nearest neighbor’s local model, and no new points can be added. • We are using more aggressive acceptance tests for new points: VB < λVP, λ < 1, and VP < Vlimit vs. |VB – VP| < ε and VB < Vlimit • Not clear if needed new points can be blocked.

40. Use Local Models • Try to achieve a sparse representation using local models.

41. Linear Quadratic Regulators

42. Learning From Observation

43. Regulator tasks • Examples: balance a pole, move at a constant velocity • A reasonable starting point is a Linear Quadratic Regulator (LQR controller) • Might have nonlinear dynamics xk+1 = f(xk,uk), but since stay around xd, can locally linearize xk+1 = Axk + Buk • Might have complex scoring function c(x,u), but can locally approximate with a quadratic model c  xTQx + uTRu • dlqr() in matlab

44. Linearization Example • Iθdd = -mgl sin(θ) – μθd + τ • Linearize • Discretize time • Vectorize • (θθd)k+1T = (1 T; -mglT/I 1-μT/I) (θθd)kT + (0 T/I)Tτk

45. LQR Derivation • Assume V() quadratic: Vk+1(x) = xTVxx:k+1x • C(x,u) = xTQx + uTRu + (Ax+Bu)TVxx:k+1 (Ax+Bu) • Want C/u = 0 • BTVxx:k+1Ax = -(BTVxx:k+1B + R)u • u = Kx (linear controller) • K = - (BTVxx:k+1B + R)-1BTVxx:k+1A • Vxx:k= ATVxx:k+1A + Q + ATVxx:k+1BK

46. Trajectory Optimization (closed loop) • Differential Dynamic Programming (local approach to DP).

47. Learning Trajectories

48. Q function • x: state, u: control or action • Dynamics: xk+1 = f(xk, uk) • Cost function: L(x,u) • Value function V(x) = ∑L(x,u) • Q function Q(x,u) = L(x,u) + V(f(x,u)) • Bellman’s Equation V(x) = minu Q(x,u) • Policy/control law: u(x) = argminu Q(x,u)

49. Local Models About

50. Propagating Local Models Along a Trajectory: Differential Dynamic ProgrammingGradient version • Vx:k-1 = Qx = Lx + Vxfx • Δu = Qu = Lu + Vxfu