Trust Region Methods. Lecturer: Pradeep Ravikumar Co-instructor: Aarti Singh. Convex Optimization /36-725

Size: px

Start display at page:

Download "Trust Region Methods. Lecturer: Pradeep Ravikumar Co-instructor: Aarti Singh. Convex Optimization /36-725"

Paul Harvey
6 years ago
Views:

1 Trust Region Methods Lecturer: Pradeep Ravikumar Co-instructor: Aarti Singh Convex Optimization /36-725

2 Trust Region Methods min p m k (p) f(x k + p) s.t. p 2 R k Iteratively solve approximations to objective function that are accurate only in trust region restrict step to lie in trust region R_k

3 A Popular Approximation for the Objective Function Recall Taylor s Theorem: for some scalar t in (0,1) f (x k + p) f k + f T k p pt 2 f (x k + tp)p, So: m k (p) f k + f T k p pt B k p, for some positive-definite symmetric B_k satisfies: m k (p) f(x k + p) =O(kpk 2 ) so the approx. error is small when p is small

4 A Popular Approximation for the Objective Function Recall Taylor s Theorem: for some scalar t in (0,1) f (x k + p) f k + f T k p pt 2 f (x k + tp)p, So: m k (p) f k + f T k p pt B k p, for some positive-definite symmetric B_k satisfies: m k (p) f(x k + p) =O(kpk 2 ) so the approx. error is small when p is small {p : kpk apple k } is the trust-region k is known as the trust-region radius

5 Quadratic Trust Region Method ( ) min p IR n m k(p) f k + f T k p pt B k p s.t. p k, kpk = p p T p is the `2 or Euclidean norm

6 Line Search vs Trust Region Trust region Line search direction contours of m k Trust region step contours of f

7 Solution of Trust Region Problem for Different Radii contours of m k p 1 * p 2 * p 3 *

8 Adaptive Trust Region Radius Algorithm 4.1 Given > 0, 0 (0, ), and η [ 0, 4) 1 : for k 0, 1, 2,... Obtain p k by (approximately) solving trust region solving problem (4.3); Evaluate ρ k (reduction from (4.4); ratio) 1 f (x k) f (x k + p k ) ; m k (0) m k (p k )

9 Adaptive Trust Region Radius Algorithm 4.1 Given > 0, 0 (0, ), and η [ 0, 4) 1 : for k 0, 1, 2,... Obtain p k by (approximately) solving trust region solving problem (4.3); Evaluate ρ k (reduction from (4.4); ratio) if ρ k < 1 4 k p k else if ρ k > 3 4 and p k k k+1 min(2 k, ) else k+1 k ; f (x k) f (x k + p k ) ; m k (0) m k (p k ) reduce trust region radius increase trust region radius same trust region radius

10 Adaptive Trust Region Radius Algorithm 4.1 Given > 0, 0 (0, ), and η [ 0, 4) 1 : for k 0, 1, 2,... Obtain p k by (approximately) solving trust region solving problem (4.3); Evaluate ρ k (reduction from (4.4); ratio) if ρ k < 1 4 k p k else if ρ k > 3 4 and p k k k+1 min(2 k, ) else k+1 k ; if ρ k > η x k+1 x k + p k else x k+1 x k ; end (for). f (x k) f (x k + p k ) ; m k (0) m k (p k ) reduce trust region radius increase trust region radius same trust region radius take step only if relative reduction is large

11 How to solve trust region problem?

12 ( ) Unconstrained Optimum Trust Region Problem: min p IR n m k(p) f k + f T k p pt B k p s.t. p k, So the unconstrained optimum can be written as: p B k = B 1 k rf k So if unconstrained optimum lies within trust region, it is also the constrained optimum: p B k is the solution to the trust region problem when kpb k kapple k

13 Unconstrained vs Constrained Optimum But the unconstrained optimum will typically not be the solution to trust region problem Solving exactly might be too expensive recall that in large scale iterative methods, we do not want to spend too much computation per iteration Solve trust region problem approximately

14 Approximate Solutions to Trust Region Problem Cauchy Dogleg Two-Dim Subspace Minimization One-dimensional root finding

15 Cauchy Point Trust region contours of m k p k C - g k

16 Cauchy Point Solve just the linear approximation: p S k arg min p IR n f k + f T k p s.t. p k;

17 Cauchy Point Solve just the linear approximation: p S k arg min p IR n f k + f T k p s.t. p k; Calculate the scalar τ k > 0 that minimizes m k (τp S k )subjectto satisfying the trust-region bound, that is, τ k arg min τ>0 m k(τp S k ) s.t. τps k k;

18 Cauchy Point Solve just the linear approximation: p S k arg min p IR n f k + f T k p s.t. p k; Calculate the scalar τ k > 0 that minimizes m k (τp S k )subjectto satisfying the trust-region bound, that is, τ k arg min τ>0 m k(τp S k ) s.t. τps k k; Set p C k τ kp S k. These steps have a closed form

19 Cauchy Point Cauchy Direction: p S k arg min p IR n f k + f T k p s.t. p k; ) p S k k f k f k.

20 Cauchy Point Cauchy Direction: p S k k f k f k. Cauchy Point: τ k arg min τ>0 m k(τp S k ) s.t. τps k k;

21 Cauchy Point Cauchy Direction: p S k k f k f k. Cauchy Point: p C k τ k k f k f k, τ k { 1 if f T k B k f k 0; min ( f k 3 /( k fk T B k f k ), 1 ) otherwise.

22 Dogleg Method Trust region Optimal trajectory p( ) p U ( unconstrained min along g ) p B ( full step) Dogleg path - g

23 Dogleg p B = B 1 k rf k... unconstrained minimum p U (rf k ) T (rf k ) = (rf k ) T B k (rf k ) rf k... steepest descent Dogleg path: p(τ) { τpu, 0 τ 1, p U + (τ 1)(p B p U ), 1 τ 2.

24 Dogleg p B = B 1 k rf k... unconstrained minimum p U (rf k ) T (rf k ) = (rf k ) T B k (rf k ) rf k... steepest descent Dogleg path: p(τ) { τpu, 0 τ 1, p U + (τ 1)(p B p U ), 1 τ 2. Dogleg Step: = arg p D = p( ) inf m k( p( )) 2[0,2]

25 Two-dimensional Subspace Minimization min p m(p) f + g T p pt Bp s.t. p, p span[g, B 1 g]. Note that entire dogleg path lies in span[g, B^-1 g] Note also that Cauchy point is feasible

26 Characterization of Solution The vector p is a global solution of the trust-region problem min m(p) f + p IR gt p + 1 n 2 pt Bp, s.t. p, (4.18) if and only if p is feasible and there is a scalar λ 0 such that the following conditions are satisfied: (B + λi)p g, (4.19a) λ( p ) 0, (4.19b) (B + λi) ispositivesemidefinite. (4.19c)

27 One-dim. root finding Define: p(λ) (B + λi) 1 g + \lambda large enough s.t. B + \lambda I is positive definite Solve: p(λ). one-dimensional root finding problem Approaches include Newton Raphson

28 Convergence Analyses Loosely: the gradients converge to zero under mild regularity conditions Requires adaptive adjusting of trust region radius as discussed earlier

Suppose that the approximate solutions of Eq. (1) satisfy the condition (3). Then (1) if η = 0 in the algorithm Trust Region, then lim inf.

Suppose that the approximate solutions of Eq. (1) satisfy the condition (3). Then (1) if η = 0 in the algorithm Trust Region, then lim inf. Maria Cameron 1. Trust Region Methods At every iteration the trust region methods generate a model m k (p), choose a trust region, and solve the constraint optimization problem of finding the minimum of