The Frank-Wolfe Algorithm:

Transcription

1 The Frank-Wolfe Algorithm: New Results, and Connections to Statistical Boosting Paul Grigas, Robert Freund, and Rahul Mazumder Massachusetts Institute of Technology May

2 Modified Title Original title: The Frank-Wolfe Algorithm: New Results, Connections to Statistical Boosting, and Computation 2

3 Problem of Interest Our problem of interest is: h := max λ s.t. h(λ) λ Q Q E is convex and compact h( ) : Q R is concave and differentiable with Lipschitz gradient on Q Assume it is easy to solve linear optimization problems on Q 3

4 Frank-Wolfe (FW) Method Frank-Wolfe Method for maximizing h(λ) on Q Initialize at λ 1 Q, (optional) initial upper bound B 0, k 1. 1 Compute h(λ k ). 2 Compute λ k arg max λ Q {h(λ k) + h(λ k ) T (λ λ k )}. B w k h(λ k) + h(λ k ) T ( λ k λ k ). G k h(λ k ) T ( λ k λ k ). 3 (Optional: compute other upper bound Bk o ), update best bound B k min{b k 1, Bk w, Bo k }. 4 Set λ k+1 λ k + ᾱ k ( λ k λ k ), where ᾱ k [0, 1). Note the condition ᾱ k [0, 1) 4

5 Wolfe Bound Wolfe Bound is B w k := h(λ k) + h(λ k ) T ( λ k λ k ) B w k h Suppose h( ) arises from minmax structure: h(λ) = min x P φ(x, λ) P is convex and compact φ(x, λ) : P Q R is convex in x and concave λ Define f (x) := max λ Q φ(x, λ) Then a dual problem is min x P f (x) In FW method, let B m k := f (x k) where x k arg min x P {φ(x, λ k)} Then B w k Bm k 5

6 Wolfe Gap Wolfe Gap is G k := B w k h(λ k) = h(λ k ) T ( λ k λ k ) Recall B m k Bw k Unlike generic duality gaps, the Wolfe gap is determined completely by the current iterate λ k. It arises in: Khachiyan s analysis of FW for rounding of polytopes, FW for boosting methods in statistics and learning, perhaps elsewhere.... 6

7 Pre-start Step for FW Pre-start Step of Frank-Wolfe method given λ 0 Q and (optional) upper bound B 1 1 Compute h(λ 0 ). 2 Compute λ 0 arg max λ Q {h(λ 0) + h(λ 0 ) T (λ λ 0 )}. B w 0 h(λ 0) + h(λ 0 ) T ( λ 0 λ 0 ). G 0 h(λ 0 ) T ( λ 0 λ 0 ). 3 (Optional: compute other upper bound B0 o ), update best bound B 0 min{b 1, B0 w, Bo 0 }. 4 Set λ 1 λ 0. This is the same as a regular FW step at λ 0 with ᾱ 0 = 1 7

8 Curvature Constant C h,q and Classical Metrics Following Clarkson, define C h,q to be the minimal value of C for which λ, λ Q and α [0, 1] implies: h(λ + α( λ λ)) h(λ) + h(λ) T (α( λ λ)) Cα 2 Let Diam Q := max { λ λ, λ Q λ } Let L := L h,q be the smallest constant L for which λ, λ Q implies: h(λ) h( λ) L λ λ It is straightforward to bound C h,q 1 2 L h,q(diam Q ) 2 8

9 Auxiliary Sequences {α k } and {β k } Define the following two auxiliary sequences as functions of the step-sizes {α k } from the Frank-Wolfe method: β k = 1 (1 ᾱ j ) k 1 j=1 and α k = β kᾱ k 1 ᾱ k, k 1 (By convention 0 j=1 = 1 and 0 i=1 = 0 ) 9

10 Two Technical Theorems Theorem Consider the iterate sequences of the Frank-Wolfe Method {λ k } and { λ k } and the sequence of upper bounds {B k } on h, using the step-size sequence {ᾱ k }. For the auxiliary sequences {α k } and {β k }, and for any k 1, the following inequality holds: B k h(λ k+1 ) B k h(λ 1 ) β k+1 + C h,q k i=1 β k+1 α 2 i β i+1 The summation expression appears also in the dual averages method of Nesterov. This is not a coincidence, indeed it is by design, see Grigas. We will henceforth refer to the sequences {α k } and {β k } as the dual averages sequences associated with the FW step-sizes. 10

11 Second Technical Theorem Theorem Consider the iterate sequences of the Frank-Wolfe Method {λ k } and { λ k } and the sequence of Wolfe gaps {G k } from Step (2.), using the step-size sequence {ᾱ k }. For the auxiliary sequences {α k } and {β k }, and for any k 1 and l k + 1, the following inequality holds: min G i i {k+1,...,l} 1 l i=k+1 ᾱi B k h(λ 1) β k+1 + C k h,q i=1 β k+1 α 2 i β i+1 + C h,q l i=k+1 ᾱ2 i l i=1 ᾱi 11

12 A Well-Studied Step-size Sequence Suppose we initiate the Frank-Wolfe method with the Pre-start step from a given value λ 0 Q (which by definition assigns the step-size ᾱ 0 = 1 as discussed earlier), and then use the step-sizes: ᾱ i = 2 i + 2 for i 0 (1) Guarantee Under the step-size sequence (1), the following inequalities hold for all k 1: B k h(λ k+1 ) 4C h,q k + 4 and min i {1,...,k} G i 8.7C h,q k 12

13 Simple Averaging Suppose we initiate the Frank-Wolfe method with the Pre-start step, and then use the following step-size sequence: ᾱ i = 1 i + 1 for i 1 (2) This has the property that λ k+1 is the simple average of λ 0, λ 1, λ 2,..., λ k Guarantee Under the step-size sequence (2), the following inequalities holds for all k 1: B k h(λ k+1 ) C h,q(1 + ln(k + 1)) k + 1 and min i {1,...,k} G i 2.9 C h,q(2 + ln(k + 1)) k

14 Constant Step-size Given ᾱ (0, 1), suppose we initiate the Frank-Wolfe method with the Pre-start step, and then use the following step-size sequence: ᾱ i = ᾱ for i 1 (3) (This step-size rule arises in the analysis of the Incremental Forward Stagewise Regression algorithm FS ε ) Guarantee Under the step-size sequence (3), the following inequality holds for all k 1: B k h(λ k+1 ) C h,q [ (1 ᾱ) k+1 + ᾱ ] 14

15 Constant Step-size, continued If we decide a priori to run the Frank-Wolfe method for k iterations, then the optimized value of ᾱ in the previous guarantee is: which yields the following: Guarantee ᾱ = 1 1 k k + 1 (4) For any k 1, using the constant step-size (4) for all iterations, the following inequality holds: B k h(λ k+1 ) C h,q [1 + ln(k + 1)] k Furthermore, after l = 2k + 2 iterations it also holds that: min i {1,...,l} G i 2C h,q [ ln(l)] l 2 15

16 On the Need for Warm Start Analysis Recall the well-studied step-size sequence initiated with a Pre-start step from λ 0 Q: ᾱ i = 2 for i 0 i + 2 and computational guarantee: This guarantee is independent of λ 0 If h(λ 0 ) h this is good B k h(λ k+1 ) 4C h,q k + 4 But if h(λ 0 ) h ε then this is not good Let us see how we can take advantageous of a warm start λ

17 A Warm Start Step-size Rule Let us start the Frank-Wolfe method at an initial point λ 1 and an upper bound B 0 Let C 1 be a given estimate of the curvature constant C h,q 2 Use the step-size sequence: ᾱ i = 4C 1 B + i + 1 for i 1 (5) 1 h(λ 1) One can think of this as if the Frank-Wolfe method had run for 4C 1 B 1 h(λ 1) iterations before arriving at λ 1 Guarantee Using (5), the following inequality holds for all k 1: B k h(λ k+1 ) 4 max{c 1, C h,q } 4C 1 + k B 1 h(λ 1 ) 17

18 Warm Start Step-size Rule, continued λ 1 Q is initial value C 1 is a given estimate of C h,q Guarantee Using (5), the following inequality holds for all k 1: B k h(λ k+1 ) 4 max{c 1, C h,q } 4C 1 + k B 1 h(λ 1 ) Easy to see C 1 C h,q optimizes the above guarantee If B 1 h(λ 1 ) is small, the incremental decrease in the guarantee from an additional iteration is lessened. This is different from first-order methods that use prox functions and/or projections 18

19 Dynamic Version of Warm-Start Analysis The warm-start step-sizes: 2 ᾱ i = 4C 1 B + i + 1 for i 1 1 h(λ 1) are based-on two pieces of information at λ 1 : B 1 h(λ 1 ), and C 1 This is a static warm-start step-size strategy Let us see how we can improve the computational guarantee by treating every iterate as if it were the initial iterate... 19

20 Dynamic Version of Warm-Starts, continued At a given iteration k of FW, we will presume that we have: λ k Q, an upper bound B k 1 on h (from previous iteration), and an estimate C k 1 of C h,q (also from the previous iteration) Consider ᾱ k of the form: 2 ᾱ k := 4C k B k h(λ k ) + 2 (6) where ᾱ k will depend explicitly on the value of C k. Let us now discuss how C k is computed... 20

21 Updating the Estimate of the Curvature Constant We require that C k (and ᾱ k which depends explicitly on C k ) satisfy C k C k 1 and: h(λ k + ᾱ k ( λ k λ k )) h(λ k ) + ᾱ k (B k h(λ k )) C k ᾱ 2 k (7) We first test if C k := C k 1 satisfies (7), and if so we set C k C k 1 If not, perform a standard doubling strategy, testing values C k 2C k 1, 4C k 1, 8C k 1,..., until (7) is satisfied Alternatively, say if h( ) is quadratic, C k can be determined analytically It will always hold that C k max{c 0, 2C h,q } 21

22 Frank-Wolfe Method with Dynamic Step-sizes FW Method with Dynamic Step-sizes for maximizing h(λ) on Q Initialize at λ 1 Q, initial estimate C 0 of C h,q, (optional) initial upper bound B 0, k 1. 1 Compute h(λ k ). 2 Compute λ k arg max λ Q {h(λ k) + h(λ k ) T (λ λ k )}. B w k h(λ k ) + h(λ k ) T ( λ k λ k ). G k h(λ k ) T ( λ k λ k ). 3 (Optional: compute other upper bound B o k ), update best bound B k min{b k 1, B w k, B o k }. 4 Compute C k for which the following conditions hold: C k C k 1, and h(λ k + ᾱ k ( λ k λ k )) h(λ k ) + ᾱ k (B k h(λ k )) C k ᾱk 2, where ᾱ k := 2 4C k B k h(λ k ) +2 5 Set λ k+1 λ k + ᾱ k ( λ k λ k ), where ᾱ k [0, 1). 22

23 Dynamic Warm Start Step-size Rule, continued Guarantee Using Frank-Wolfe method with Dynamic Step-sizes, the following inequality holds for all k 1 and l 1: B k+l h(λ k+l ) 4C k+l 4C k+l B k h(λ k ) + l Furthermore, if the doubling strategy is used to update the estimates C k of C h,q, it holds that C k+l max{c 0, 2C h,q } 23

24 Frank-Wolfe with Inexact Gradient Consider the case when the gradient h( ) is given inexactly d Aspremont s δ-oracle for the gradient: given λ Q, the δ-oracle returns g δ ( λ) that satisfies: ( h( λ) g δ ( λ)) T (ˆλ λ) δ for all ˆλ, λ Q Devolder/Glineur/Nesterov s (δ, L)-oracle for the the function value and the gradient: given λ Q, the (δ, L)-oracle returns h (δ,l) ( λ) and g (δ,l) ( λ) that satisfies for all λ Q: and h(λ) h (δ,l) ( λ) + g (δ,l) ( λ) T (λ λ) h(λ) h (δ,l) ( λ) + g (δ,l) ( λ) T (λ λ) L 2 λ λ δ 24

25 Frank-Wolfe with d Aspremont δ-oracle for Gradient Suppose that we use a δ-oracle for the gradient. Then the main technical theorem is amended as follows: Theorem Consider the iterate sequences of the Frank-Wolfe Method {λ k } and { λ k } and the sequence of upper bounds {B k } on h, using the step-size sequence {ᾱ k }. For the auxiliary sequences {α k } and {β k }, and for any k 1, the following inequality holds: B k h(λ k+1 ) B k h(λ 1 ) β k+1 + C h,q k i=1 β k+1 α 2 i β i+1 + 2δ The error δ does not accumulate All bounds presented earlier are amended by 2δ 25

26 Frank-Wolfe with DGN (δ, L)-oracle for Functions and Gradient Suppose that we use a (δ, L)-oracle for the function and gradient. Then the errors accumulate. (δ, L)-oracle for the function and gradient Guarantee Guarantee Sparsity Method without Errors with Errors of Iterates ( ) ( ) 1 1 Frank-Wolfe O O + O(δk) YES ( k ) ( k ) 1 1 Prox Gradient Acclerated Prox Grad. O ( k ) 1 O k 2 O O + O(δ) NO ( k ) 1 k 2 + O(δk) NO No method dominates all three criteria 26

27 Applications to Statistical Boosting We consider two applications in statistical boosting: 1 Linear Regression 2 Binary Classification / Supervised Learning 27

28 Linear Regression Consider linear regression: y = Xβ + e y R n is the response, β R p are the coefficients, X R n p is the model matrix and e R n are the errors In high-dimension statistical regime, especially with p n, we desire: good predictive performance, good performance on samples ( residuals r := y Xβ are small ), an interpretable model β coefficients are not excessively large ( β δ), and a sparse solution (β has few non-zero coefficients) 28

29 LASSO The form of the LASSO we consider is: L δ = min β L(β) := 1 2 y Xβ 2 2 s.t. β 1 δ δ > 0 is the regularization parameter As is well-known, the 1 -norm regularizer induces sparse solutions Let β 0 denote the number of non-zero coefficients of the vector β 29

30 LASSO with Frank-Wolfe Consider using Frank-Wolfe to solve the LASSO: L δ = min β L(β) := 1 2 y Xβ 2 2 s.t. β 1 δ the linear optimization subproblem any c: let j arg max c j j {1,...,p} then arg min β ct β δsgn(c j )e j 1 δ min β ct β is trivial to solve for 1 δ extreme points of feasible region are ±δe 1,..., ±δe p therefore β k+1 0 β k

31 Frank-Wolfe for LASSO Adaptation of Frank-Wolfe Method for LASSO Initialize at β 0 with β 0 1 δ. At iteration k: 1 Compute: 2 Set: r k y Xβ k j k arg max (r k ) T X j j {1,...,p} β k+1 j k (1 ᾱ k )β k j k + ᾱ k δsgn((r k ) T X jk ) β k+1 j (1 ᾱ k )β k j for j j k, and where ᾱ k [0, 1] 31

32 Computational Guarantees for Frank-Wolfe on LASSO Guarantees Suppose that we use the Frank-Wolfe Method to solve the LASSO problem, with either the fixed step-size rule ᾱ i = 2 i+2 or a line-search to determine ᾱ i for i 0. Then after k iterations, there exists an i {0,..., k} satisfying: 1 2 y Xβi 2 2 L δ 17.4 X 2 1,2 δ2 k X T (y Xβ i ) 1 2δ Xβ LS X 2 1,2 δ k β i 0 k β i 1 δ where β LS is the least-squares solution, and so Xβ LS 2 y 2 32

33 Frank-Wolfe on LASSO, and FS ε There are some structural connections between Frank-Wolfe on LASSO and Incremental Forward Stagewise Regression FS ε There are even more connections in the case of Frank-Wolfe with constant step-size 33

34 Binary Classification / Supervised Learning The set-up of the binary classification boosting problem consists of: Data/training examples (x 1, y 1 ),..., (x m, y m ) where each x i X and each y i [ 1, 1] A set of n base classifiers H = {h 1 ( ),..., h n ( )} where each h j ( ) : X [ 1, 1] h j (x i ) is classifier j s score of example x i y i h j (x i ) > 0 if and only if classifier j correctly classifies example x i We would like to construct a classifier H = λ 1 h 1 ( ) + + λ n h n ( ) that performs significantly better than any individual classifier in H 34

35 Binary Classification, continued We construct a new classifier H = H λ := λ 1 h 1 ( ) + + λ n h n ( ) from nonnegative multipliers λ R n + H(x) = H λ (x) := λ 1 h 1 (x) + + λ n h n (x) (associate sgnh H for ease of notation) In the high-dimensional case where n m 0, we desire: good predictive performance, good performance on the training data (y i H λ (x i ) > 0 for all examples x i ), a good interpretable model λ coefficients are not excessively large ( λ δ), and λ has few non-zero coefficients ( λ 0 is small) 35

36 Weak Learner Form the matrix A R m n by A ij = y i h j (x i ) A ij > 0 if and only if classifier h j ( ) correctly classifies example x i We suppose we have a weak learner W( ) that, for any distribution w on the examples (w m := {w R m : e T w = 1, w 0 }), returns the base classifier h j ( ) in H that does best on the weighted example determined by w: m w i y i h j (x i ) = i=1 max j=1,...,n m w i y i h j (x i ) = i=1 max j=1,...,n w T A j 36

37 Performance Objectives: Maximize the Margin The margin of the classifier H λ is defined by: p(λ) := min y ih λ (x i ) = min (Aλ) i i {1,...,m} i {1,...,m} One can then solve: p = max λ s.t. p(λ) λ n 37

38 Performance Objectives: Minimize the Exponential Loss The (logarithm of the) exponential loss of the classifier H λ is defined by: ( ) 1 m L exp (λ) := ln exp ( (Aλ) i ) m One can then solve: i=1 L exp,δ = min λ s.t. L exp (λ) λ 1 δ λ 0 38

39 Minimizing Exponential Loss with Frank-Wolfe Consider using Frank-Wolfe to minimize the exponential loss problem: L exp,δ = min λ s.t. L exp (λ) λ 1 δ λ 0 the linear optimization subproblem min λ 1 δ, λ 0 ct λ is trivial to solve for any c: arg min λ 1 δ, λ 0 ct λ 0 if c 0 else let j arg max j {1,...,p} c j then arg min λ 1 δ, λ 0 ct λ δe j extreme points of feasible region are 0, δe 1,..., δe n therefore λ k+1 0 λ k

40 Frank-Wolfe for Exponential Loss Minimization Adaptation of Frank-Wolfe Method for Minimizing Exponential Loss Initialize at λ 0 0 with λ 0 1 δ. Set w 0 i = exp( (Aλ0 ) i ) m l=1 exp( (Aλ0 ) l ) i = 1,..., m. Set k = 0. At iteration k: 1 Compute j k W(w k ) 2 Choose ᾱ [0, 1] and set: (1 ᾱ k )λ k j k + ᾱ k δ λ k+1 j (1 ᾱ k )λ k j for j j k, and where ᾱ k [0, 1] λ k+1 j k (wj k)1 ᾱ k exp( ᾱ k δa i,jk ), i = 1,..., m and re-normalize w k+1 so that e T w k+1 = 1 w k+1 j 40

41 Computational Guarantees for FW for Binary Classification Guarantees Suppose that we use the Frank-Wolfe Method to solve the exponential loss minimization problem, with either the fixed step-size rule ᾱ i = 2 i+2 or a line-search to determine ᾱ i, using ᾱ 0 = 1. Then after k iterations: L exp (λ k ) L exp,δ p p( λ k ) λ k 0 k λ k 1 δ 8δ2 k + 3 8δ k ln(m) δ where λ k is the normalization of λ k, namely λ k = λk e T λ k 41

42 Frank-Wolfe for Binary Classification, and AdaBoost There are some structural connections between Frank-Wolfe for binary classification, and AdaBoost L exp (λ) is a (Nesterov-) smoothing of the margin using smoothing parameter µ = 1 : write the margin as p(λ) := min (Aλ) i = min (w T Aλ) i {1,...,m} w m do µ-smoothing of the margin using entropy function e(w): then p µ (λ) = µl exp (λ/µ) p µ (λ) := min w m (w T Aλ + µe(w)) it easily follows that 1 µ p(λ) L exp(λ/µ) 1 µ p(λ) + ln(m) 42

43 Comments Computational evaluation of Frank-Wolfe for boosting: LASSO binary classification Structural connections in G-F-M: AdaBoost is equivalent to Mirror Descent for maximizing the margin Incremental Forward Stagewise Regression (FS ε ) is equivalent to subgradient descent for minimizing X T r 43