Optimization II: Unconstrained Multivariable
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1 Optimization II: Unconstrained Multivariable CS 205A: Mathematical Methods for Robotics, Vision, and Graphics Doug James (and Justin Solomon) CS 205A: Mathematical Methods Optimization II: Unconstrained Multivariable 1 / 24
2 Announcements Today s class: Unconstrained optimization: Newton s method (uses Hessians) BFGS method (no Hessians) Automatic differentiation Homework 5 (due Thursday); polynomial differentiation Midterm (update) CS 205A: Mathematical Methods Optimization II: Unconstrained Multivariable 2 / 24
3 Unconstrained Multivariable Problems minimize f : R n R CS 205A: Mathematical Methods Optimization II: Unconstrained Multivariable 3 / 24
4 Recall f( x) Direction of steepest ascent CS 205A: Mathematical Methods Optimization II: Unconstrained Multivariable 4 / 24
5 Recall f( x) Direction of steepest descent CS 205A: Mathematical Methods Optimization II: Unconstrained Multivariable 5 / 24
6 Observation If f( x) 0, for sufficiently small α > 0, f( x α f( x)) f( x) CS 205A: Mathematical Methods Optimization II: Unconstrained Multivariable 6 / 24
7 Gradient Descent Algorithm Iterate until convergence: 1. g k (t) f( x k t f( x k )) 2. Find t 0 minimizing (or decreasing) g k 3. x k+1 x k t f( x k ) CS 205A: Mathematical Methods Optimization II: Unconstrained Multivariable 7 / 24
8 Stopping Condition f( x k ) 0 Don t forget: Check optimality! CS 205A: Mathematical Methods Optimization II: Unconstrained Multivariable 8 / 24
9 Line Search g k (t) f( x k t f( x k )) One-dimensional optimization Don t have to minimize completely: Wolfe conditions Constant t: Learning rate CS 205A: Mathematical Methods Optimization II: Unconstrained Multivariable 9 / 24
10 Line Search g k (t) f( x k t f( x k )) One-dimensional optimization Don t have to minimize completely: Wolfe conditions Constant t: Learning rate Worth reading about: Nesterov s Accelerated Gradient Descent CS 205A: Mathematical Methods Optimization II: Unconstrained Multivariable 9 / 24
11 Gradient Descent (book) CS 205A: Mathematical Methods Optimization II: Unconstrained Multivariable 10 / 24
12 Newton s Method (again!) f( x) f( x k ) + f( x k ) ( x x k ) ( x x k) H f ( x k )( x x k ) CS 205A: Mathematical Methods Optimization II: Unconstrained Multivariable 11 / 24
13 Newton s Method (again!) f( x) f( x k ) + f( x k ) ( x x k ) ( x x k) H f ( x k )( x x k ) = x k+1 = x k [H f ( x k )] 1 f( x k ) CS 205A: Mathematical Methods Optimization II: Unconstrained Multivariable 11 / 24
14 Newton s Method (again!) f( x) f( x k ) + f( x k ) ( x x k ) ( x x k) H f ( x k )( x x k ) = x k+1 = x k [H f ( x k )] 1 f( x k ) Consideration: What if H f is not positive (semi-)definite? CS 205A: Mathematical Methods Optimization II: Unconstrained Multivariable 11 / 24
15 Motivation f might be hard to compute but H f is harder H f might be dense: n 2 CS 205A: Mathematical Methods Optimization II: Unconstrained Multivariable 12 / 24
16 Quasi-Newton Methods Approximate derivatives to avoid expensive calculations e.g. secant, Broyden,... CS 205A: Mathematical Methods Optimization II: Unconstrained Multivariable 13 / 24
17 Common Optimization Assumption f known H f unknown or hard to compute CS 205A: Mathematical Methods Optimization II: Unconstrained Multivariable 14 / 24
18 Quasi-Newton Optimization x k+1 = x k α k B 1 k f( x k) B k H f ( x k ) CS 205A: Mathematical Methods Optimization II: Unconstrained Multivariable 15 / 24
19 Warning <advanced material> See Nocedal & Wright CS 205A: Mathematical Methods Optimization II: Unconstrained Multivariable 16 / 24
20 Broyden-Style Update B k+1 ( x k+1 x k ) = f( x k+1 ) f( x k ) CS 205A: Mathematical Methods Optimization II: Unconstrained Multivariable 17 / 24
21 Additional Considerations B k should be symmetric B k should be positive (semi-)definite CS 205A: Mathematical Methods Optimization II: Unconstrained Multivariable 18 / 24
22 Davidon-Fletcher-Powell (DFP) min B k+1 B k B k+1 s.t. B k+1 = B k+1 B k+1 ( x k+1 x k ) = f( x k+1 ) f( x k ) CS 205A: Mathematical Methods Optimization II: Unconstrained Multivariable 19 / 24
23 Observation B k+1 B k small does not mean B 1 is small k+1 B 1 k CS 205A: Mathematical Methods Optimization II: Unconstrained Multivariable 20 / 24
24 Observation B k+1 B k small does not mean B 1 is small k+1 B 1 k Idea: Try to approximate directly B 1 k CS 205A: Mathematical Methods Optimization II: Unconstrained Multivariable 20 / 24
25 BFGS Update min H k+1 H k HB k+1 s.t. H k+1 = H k+1 x k+1 x k = H k+1 ( f( x k+1 ) f( x k )) State of the art! CS 205A: Mathematical Methods Optimization II: Unconstrained Multivariable 21 / 24
26 BFGS (book - typo) CS 205A: Mathematical Methods Optimization II: Unconstrained Multivariable 22 / 24
27 Lots of Missing Details Choice of Limited-memory alternative Next CS 205A: Mathematical Methods Optimization II: Unconstrained Multivariable 23 / 24
28 Automatic Differentiation Techniques to numerically evaluate the derivative of a function specified by a computer program. Different from finite differences (approximation) and symbolic differentiation. In Julia: Example: ForwardDiff. (uses dual numbers) Next CS 205A: Mathematical Methods Optimization II: Unconstrained Multivariable 24 / 24
Optimization II: Unconstrained Multivariable
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