Convex Optimization CMU-10725
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1 Convex Optimization CMU Newton Method Barnabás Póczos & Ryan Tibshirani
2 Administrivia Scribing Projects HW1 solutions Feedback about lectures / solutions on blackboard 2
3 Books to read Boyd and Vandenberghe: Convex Optimization, Chapters 9.5 Nesterov: Introductory lectures on convex optimization Bazaraa, Sherali, Shetty: Nonlinear Programming Dimitri P. Bestsekas: Nonlinear Programming Wikipedia 3
4 Goal of this lecture Newton method Finding a root Unconstrained minimization Motivation with quadratic approximation Rate of Newton s method Newton fractals Next lectures: Conjugate gradients Quasi Newton Methods 4
5 Newton method for finding a root 5
6 Newton method for finding a root Newton method: originally developed for finding a root of a function also known as the Newton Raphson method 6
7 History Babylonian method: This is a special case of Newton s method S=100. starting values x 0 = 50, x 0 = 1, and x 0 = 5. 7
8 History 1690, Joseph Raphson: 1669, Isaac Newton: finding roots of polynomials finding roots of polynomials 1740, Thomas Simpson: 1879, Arthur Cayley: solving general nonlinear equation generalization to systems of two equations solving optimization problems (gradient = zero) generalizing the Newton's method to finding complex roots of polynomials 8
9 Goal: Newton Method for Finding a Root Linear Approximation (1 st order Taylor approx): Therefore, 9
10 Illustration of Newton s method Goal: finding a root In the next step we will linearize here in x 10
11 Example: Finding a Root 11
12 Newton Method for Finding a Root This can be generalized to multivariate functions Therefore, [Pseudo inverse if there is no inverse] 12
13 Newton method for minimization 13
14 Newton method for minimization Iterate until convergence, or max number of iterations exceeded (divergence, loops, division by zero might happen ) 14
15 How good is the Newton method? 15
16 Descent direction Lemma [Descent direction] Proof: We know that if a vector has negative inner product with the gradient vector, then that direction is a descent direction 16
17 Newton method properties Quadratic convergence in the neighborhood of a strict local minimum [under some conditions]. It can break down if f (x k ) is degenerate. [no inverse] It can diverge. It can be trapped in a loop. It can converge to a loop 17
18 Motivation with Quadratic Approximation 18
19 Motivation with Quadratic Approximation Second order Taylor approximation: Assume that Newton step: 19
20 Motivation with Quadratic Approximation 20
21 Convergence rate (f: R R case) 21
22 Rates Example: Superlinear rate: Example: Sublinear rate: Example: Quadratic rate: Example: 22
23 Finding a root: Convergence rate Goal: Assumption: Taylor theorem: Therefore, 23
24 Finding a root: Convergence rate We have seen that 24
25 Problematic cases 25
26 Finding a root: chaotic behavior Let f(x)=x^3-2x^2-11x+12 Goal: find the roots, (-3, 1, 4), using Newton s method converges to 4; converges to 3; converges to 4; converges to 3; converges to 1. 26
27 Finding a root: Cycle Goal: find its roots Stating point is important! 27
28 Finding a root: divergence everywhere (except in the root) Newton s method might never converge (except in the root)! 28
29 Finding a root: Linear convergence only If the first derivative is zero at the root, then convergence might be only linear (not quadratic) Linear convergence only! 29
30 Difficulties in minimization range of quadratic convergence 30
31 Generalizations Newton method in Banach spaces Newton method on the Banach space of functions We need Frechet derivatives Newton method on curved manifolds E.g. on space of otrthonormal matrices Newton method on complex numbers 31
32 Newton Fractals 32
33 Gradient descent 33
34 f(z)= z 4-1, Roots: -1, +1, -i, +i Compex functions color the starting point according to which root it ended up 34
35 f(z)= z 4-1, Roots: -1, +1, -i, +i Basins of attraction Shading according to how many iterations it needed till convergence 35
36 Basins of attraction f (z)=(z-1) 4 (z+1) 4 36
37 No convergence polynomial f, having the roots at +i, -i, -2.3and +2.3 Black circles: no convergence, attracting cycle with period 2 37
38 Avoiding divergence 38
39 Damped Newton method In order to avoid possible divergence do line search with back tracking We already know that the Newton direction is descent direction 39
40 Classes of differentiable functions 40
41 Classes of Differentiable functions Any continuous (but otherwise ugly, nondifferentiable) function can be approximated arbitrarily well with smooth (k times differentiable) function Assuming smoothness only is not enough to talk about rates We need stronger assumptions on the derivatives. [e.g. their magnitudes behaves nicely] 41
42 The C L k,p (Q) Class Definition [Lipschitz continuous pth order derivative] Notation 42
43 Lemma [Linear combinations] Trivial Lemmas Lemma [Class hierarchy] 43
44 Relation between 1 st and 2 nd derivatives Lemma Proof Therefore, Special case, Q.E.D 44
45 C L 2,1 (R n ) and the norm of f Lemma [C L 2,1 and the norm of f ] Proof Q.E.D 45
46 C L 2,1 (R n ) and the norm of f Proving the other direction Q.E.D 46
47 Examples 47
48 Error of 1 st ordertaylor approx. in C L 1,1 Lemma [1 st ordertaylorapproximation in C L 1,1 ] Proof 48
49 Error of 1 st ordertaylor approx. in C L 1,1 Q.E.D 49
50 Sandwiching with quadratic functions in C L 1,1 We have proved: Corollary [Sandwiching C L 1,1 functions with quadratic functions] Function f can be lower and upper bounded with quadratic functions 50
51 Lemma [Properties of C L 2,2 Class] C L 2,2 (R n ) Class [Error of the 1st order approximation of f ] [Error of the 2nd order approximation of f] Proof (*1) Definition (*2) Same as previous lemma [f instead of f] (*3) Similar [Homework] 51
52 By definition Sandwiching f (y) in C L 2,2 (R n ) Corollary [Sandwiching f (y) matrix] Proof Q.E.D 52
53 Convergence rate of Newton s method 53
54 Convergence rate of Newton s method Assumptions [Local convergence only] 54
55 Convergence rate of Newton s method Newton step: We already know: Therefore, Trivial identity: 55
56 Convergence rate of Newton s method 56
57 Convergence rate of Newton s method We have already proved: Therefore, and thus, 57
58 Convergence rate of Newton s method We already know: 58
59 Convergence rate of Newton s method Now, we have that The error doesn t increase! 59
60 Convergence rate of Newton s method We have proved the following theorem Theorem [Rate of Newton s method] Quadratic rate! 60
61 Summary Newton method Finding a root Unconstrained minimization Motivation with quadratic approximation Rate of Newton s method Newton fractals Classes of differentiable functions 61
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