Expanding the reach of optimal methods
|
|
- Cleopatra Brooks
- 5 years ago
- Views:
Transcription
1 Expanding the reach of optimal methods Dmitriy Drusvyatskiy Mathematics, University of Washington Joint work with C. Kempton (UW), M. Fazel (UW), A.S. Lewis (Cornell), and S. Roy (UW) BURKAPALOOZA! WCOM SPRING 2016 AFOSR YIP
2 Notation Function f : R n R is α-convex and β-smooth if where q x f Q x Q x (y) = f (x) + f (x), y x + β y x 2 2 q x (y) = f (x) + f (x), y x + α y x 2 2 Q x f q x x 2/15
3 Notation Function f : R n R is α-convex and β-smooth if where q x f Q x Q x (y) = f (x) + f (x), y x + β y x 2 2 q x (y) = f (x) + f (x), y x + α y x 2 2 Q x f q x Condition number: x κ = β α 2/15
4 Complexity of first-order methods Gradient descent: x k+1 = x k 1 β f (x k) 3/15
5 Complexity of first-order methods Gradient descent: x k+1 = x k 1 β f (x k) Majorization view: x k+1 = argmin Q xk ( ) 3/15
6 Complexity of first-order methods Gradient descent: x k+1 = x k 1 β f (x k) Majorization view: x k+1 = argmin Q xk ( ) Gradient Descent β-smooth β ɛ β-smooth & α-convex κ ln 1 ɛ Table: Iterations until f (x k ) f < ɛ 3/15
7 Complexity of first-order methods Gradient descent: x k+1 = x k 1 β f (x k) Majorization view: x k+1 = argmin Q xk ( ) Gradient Descent Optimal Methods β-smooth β ɛ β ɛ β-smooth & α-convex κ ln 1 ɛ κ ln 1 ɛ Table: Iterations until f (x k ) f < ɛ 3/15
8 Complexity of first-order methods Gradient descent: x k+1 = x k 1 β f (x k) Majorization view: x k+1 = argmin Q xk ( ) Gradient Descent Optimal Methods β-smooth β ɛ β ɛ β-smooth & α-convex κ ln 1 ɛ κ ln 1 ɛ Table: Iterations until f (x k ) f < ɛ (Nesterov 83, Yudin-Nemirovsky 83) 3/15
9 Complexity of first-order methods Gradient descent: x k+1 = x k 1 β f (x k) Majorization view: x k+1 = argmin Q xk ( ) Gradient Descent Optimal Methods β-smooth β ɛ β ɛ β-smooth & α-convex κ ln 1 ɛ κ ln 1 ɛ Table: Iterations until f (x k ) f < ɛ (Nesterov 83, Yudin-Nemirovsky 83) Optimal methods have downsides: Not intuitive Non-monotone Difficult to augment with memory 3/15
10 Optimal method by optimal averaging Idea: Maximize lower models of f. 4/15
11 Optimal method by optimal averaging Idea: Maximize lower models of f. Notation: x + = x 1 β f (x) and x++ = x 1 f (x) α 4/15
12 Optimal method by optimal averaging Idea: Maximize lower models of f. Notation: x + = x 1 β f (x) and x++ = x 1 f (x) α Convexity bound f q x in canonical form: ( ) f (x) 2 f (y) f (x) + α 2α 2 y x++ 2 4/15
13 Optimal method by optimal averaging Idea: Maximize lower models of f. Notation: x + = x 1 β f (x) and x++ = x 1 f (x) α Convexity bound f q x in canonical form: ( ) f (x) 2 f (y) f (x) + α 2α 2 y x++ 2 Lower models: Q A (x) = v A + α 2 x x A 2 Q B (x) = v B + α 2 x x B 2 4/15
14 Optimal method by optimal averaging Idea: Maximize lower models of f. Notation: x + = x 1 β f (x) and x++ = x 1 f (x) α Convexity bound f q x in canonical form: ( ) f (x) 2 f (y) f (x) + α 2α 2 y x++ 2 Lower models: Q A (x) = v A + α 2 x x A 2 Q B (x) = v B + α 2 x x B 2 = for any λ [0, 1] new lower-model Q λ := λq A + (1 λ)q B = v λ + α 2 x λ 2 4/15
15 Optimal method by optimal averaging Idea: Maximize lower models of f. Notation: x + = x 1 β f (x) and x++ = x 1 f (x) α Convexity bound f q x in canonical form: ( ) f (x) 2 f (y) f (x) + α 2α 2 y x++ 2 Lower models: Q A (x) = v A + α 2 x x A 2 Q B (x) = v B + α 2 x x B 2 = for any λ [0, 1] new lower-model Q λ := λq A + (1 λ)q B = v λ + α 2 x λ 2 Key observation: v λ f 4/15
16 Optimal method by optimal averaging The minimum v λ is maximized when ( 1 λ = proj [0,1] 2 + v ) A v B α x A x B 2. The quadratic Q λ is the optimal averaging of (Q A, Q B ). 5/15
17 Optimal method by optimal averaging The minimum v λ is maximized when ( 1 λ = proj [0,1] 2 + v ) A v B α x A x B 2. The quadratic Q λ is the optimal averaging of (Q A, Q B ). 5/15
18 Optimal method by optimal averaging The minimum v λ is maximized when ( 1 λ = proj [0,1] 2 + v ) A v B α x A x B 2. The quadratic Q λ is the optimal averaging of (Q A, Q B ). Related to cutting plane, bundle methods, geometric descent (Bubeck-Lee-Singh 15) 5/15
19 Optimal method by optimal averaging for k = 1,..., K do Set Q(x) = (f (x k ) f (x k) 2 end ) 2α + α 2 x x k ++ 2 Let Q k (x) = v k + α 2 x c k 2 be optim. average of (Q, Q k 1 ). Set x k+1 = line_search ( c k, x + k Algorithm: Optimal averaging The line-search is due to (Bubeck-Lee-Singh 15) ) 6/15
20 Optimal method by optimal averaging for k = 1,..., K do Set Q(x) = (f (x k ) f (x k) 2 end ) 2α + α 2 x x k ++ 2 Let Q k (x) = v k + α 2 x c k 2 be optim. average of (Q, Q k 1 ). Set x k+1 = line_search ( c k, x + k Algorithm: Optimal averaging The line-search is due to (Bubeck-Lee-Singh 15) ) Optimal Linear Rate (D-Fazel-Roy 16): f (x k + ) v k ɛ after O β α ln 1 ɛ iterations. 6/15
21 Optimal method by optimal averaging for k = 1,..., K do Set Q(x) = (f (x k ) f (x k) 2 end ) 2α + α 2 x x k ++ 2 Let Q k (x) = v k + α 2 x c k 2 be optim. average of (Q, Q k 1 ). Set x k+1 = line_search ( c k, x + k Algorithm: Optimal averaging The line-search is due to (Bubeck-Lee-Singh 15) ) Optimal Linear Rate (D-Fazel-Roy 16): f (x k + ) v k ɛ after O β α ln 1 ɛ iterations. Intuitive Monotone in f (x + k ) and in v k. Memory by optimally averaging (Q, Q k 1,..., Q k t ). 6/15
22 Optimal method by optimal averaging Figure: Logistic regression with regularization α = /15
23 Nonsmooth & Nonconvex minimization Convex composition min g(x) + h(c(x)) x where g : R n R {+ } is closed, convex. h : R m R is convex and L-Lipschitz. c : R n R m is C 1 -smooth and c is β-lipschitz. 8/15
24 Nonsmooth & Nonconvex minimization Convex composition min g(x) + h(c(x)) x where g : R n R {+ } is closed, convex. h : R m R is convex and L-Lipschitz. c : R n R m is C 1 -smooth and c is β-lipschitz. For convenience, set µ = Lβ. 8/15
25 Nonsmooth & Nonconvex minimization Convex composition min g(x) + h(c(x)) x where g : R n R {+ } is closed, convex. h : R m R is convex and L-Lipschitz. c : R n R m is C 1 -smooth and c is β-lipschitz. For convenience, set µ = Lβ. Main examples: Additive composite minimization: min g(x) + c(x) x Nonlinear least squares: min { c(x) : l i x i u i for i = 1,..., m} x Exact penalty subproblem: min g(x) + dist K (c(x)) x (Burke 85, 91, Fletcher 82, Powell 84, Wright 90, Yuan 83) 8/15
26 Prox-linear algorithm Prox-linear mapping x + = argmin y g(y) + h and the prox-gradient ( ) c(x) + c(x)(y x) + µ y x 2 2 G(x) = µ(x x + ). 9/15
27 Prox-linear algorithm Prox-linear mapping x + = argmin y g(y) + h and the prox-gradient ( ) c(x) + c(x)(y x) + µ y x 2 2 G(x) = µ(x x + ). Prox-linear method (Burke, Fletcher, Osborne, Powell,... 80s): x k+1 = x + k. 9/15
28 Prox-linear algorithm Prox-linear mapping x + = argmin y g(y) + h and the prox-gradient ( ) c(x) + c(x)(y x) + µ y x 2 2 G(x) = µ(x x + ). Prox-linear method (Burke, Fletcher, Osborne, Powell,... 80s): x k+1 = x + k. Eg: proximal gradient, Levenberg-Marquardt methods 9/15
29 Prox-linear algorithm Prox-linear mapping x + = argmin y g(y) + h and the prox-gradient ( ) c(x) + c(x)(y x) + µ y x 2 2 G(x) = µ(x x + ). Prox-linear method (Burke, Fletcher, Osborne, Powell,... 80s): x k+1 = x + k. Eg: proximal gradient, Levenberg-Marquardt methods Convergence rate: G(x k ) 2 < ɛ after ( ) µ 2 O ɛ iterations 9/15
30 Stopping criterion What does G(x) 2 < ɛ actually mean? 10/15
31 Stopping criterion What does G(x) 2 < ɛ actually mean? Stationarity for target problem: 0 g(x) + c(x) h(c(x)) Stationarity for prox-subproblem: G(x) g(x + ) + c(x) h ( c(x) + c(x)(x + x) ) 10/15
32 Stopping criterion What does G(x) 2 < ɛ actually mean? Stationarity for target problem: 0 g(x) + c(x) h(c(x)) Stationarity for prox-subproblem: G(x) g(x + ) + c(x) h ( c(x) + c(x)(x + x) ) Thm: (D-Lewis 16) x + is nearly stationary because (ˆx, ˆv) with where ˆv g(ˆx) + c(ˆx) h(c(ˆx)) ˆx x + µ G(x) and ˆv 5 G(x) 10/15
33 Stopping criterion What does G(x) 2 < ɛ actually mean? Stationarity for target problem: 0 g(x) + c(x) h(c(x)) Stationarity for prox-subproblem: G(x) g(x + ) + c(x) h ( c(x) + c(x)(x + x) ) Thm: (D-Lewis 16) x + is nearly stationary because (ˆx, ˆv) with where ˆv g(ˆx) + c(ˆx) h(c(ˆx)) ˆx x + µ G(x) and ˆv 5 G(x) (pf: Ekeland s variational principle) 10/15
34 Local linear convergence Error bound property (Luo-Tseng 93) dist(x, {soln. set}) 1 α G(x) for x near x 11/15
35 Local linear convergence Error bound property (Luo-Tseng 93) dist(x, {soln. set}) 1 α G(x) for x near x = local linear convergence ( F(x k+1 ) F 1 α2 µ 2 ) (F(x k ) F ) 11/15
36 Local linear convergence Error bound property (Luo-Tseng 93) dist(x, {soln. set}) 1 α G(x) for x near x = local linear convergence ( F(x k+1 ) F 1 α2 µ 2 ) (F(x k ) F ) The following are essentially equivalent (D-Lewis 16): EB property Subregularity: dist(x; {soln. set}) 1 dist(0; F(x)) α for x near x Quadratic growth: F(x) F( x) + α 2 dist2 (x, {soln. set}) for x near x 11/15
37 Local linear convergence Error bound property (Luo-Tseng 93) dist(x, {soln. set}) 1 α G(x) for x near x = local linear convergence ( F(x k+1 ) F 1 α2 µ 2 ) (F(x k ) F ) The following are essentially equivalent (D-Lewis 16): EB property Subregularity: dist(x; {soln. set}) 1 dist(0; F(x)) α for x near x Quadratic growth: F(x) F( x) + α 2 dist2 (x, {soln. set}) for x near x Rate becomes α µ under tilt-stability (Poliquin-Rockafellar 98) 11/15
38 Acceleration For nonconvex problems, the rate ( ) µ G(x k ) 2 2 < ɛ after O ɛ iterations is essentially best possible. 12/15
39 Acceleration For nonconvex problems, the rate ( ) µ G(x k ) 2 2 < ɛ after O ɛ iterations is essentially best possible. Measuring non-convexity: h c(x) = sup w { w, c(x) h (w)} 12/15
40 Acceleration For nonconvex problems, the rate ( ) µ G(x k ) 2 2 < ɛ after O ɛ iterations is essentially best possible. Measuring non-convexity: h c(x) = sup w { w, c(x) h (w)} Fact: x w, c(x) + µ 2 x 2 convex w dom h 12/15
41 Acceleration For nonconvex problems, the rate ( ) µ G(x k ) 2 2 < ɛ after O ɛ iterations is essentially best possible. Measuring non-convexity: h c(x) = sup w { w, c(x) h (w)} Fact: x w, c(x) + µ 2 x 2 convex w dom h Defn: Parameter ρ [0, 1] such that x w, c(x) + ρ µ 2 x 2 is convex w dom h 12/15
42 Acceleration Accelerated prox-linear method! 13/15
43 Acceleration Accelerated prox-linear method! Thm: (D-Kempton 16) min G(x i) 2 O i=1,...,k where R = diam (dom g) ( ) ( µ 2 µ 2 R 2 ) k 3 + ρ O k 13/15
44 Acceleration Accelerated prox-linear method! Thm: (D-Kempton 16) min G(x i) 2 O i=1,...,k where R = diam (dom g) ( ) ( µ 2 µ 2 R 2 ) k 3 + ρ O k Generalizes (Ghadimi, Lan 16) for additively composite. 13/15
45 Conclusions Balanced approach: Computational complexity Acceleration Variational analysis 14/15
46 Happy Birthday, Jim! 15/15
Composite nonlinear models at scale
Composite nonlinear models at scale Dmitriy Drusvyatskiy Mathematics, University of Washington Joint work with D. Davis (Cornell), M. Fazel (UW), A.S. Lewis (Cornell) C. Paquette (Lehigh), and S. Roy (UW)
More informationTaylor-like models in nonsmooth optimization
Taylor-like models in nonsmooth optimization Dmitriy Drusvyatskiy Mathematics, University of Washington Joint work with Ioffe (Technion), Lewis (Cornell), and Paquette (UW) SIAM Optimization 2017 AFOSR,
More informationAn accelerated algorithm for minimizing convex compositions
An accelerated algorithm for minimizing convex compositions D. Drusvyatskiy C. Kempton April 30, 016 Abstract We describe a new proximal algorithm for minimizing compositions of finite-valued convex functions
More informationEfficiency of minimizing compositions of convex functions and smooth maps
Efficiency of minimizing compositions of convex functions and smooth maps D. Drusvyatskiy C. Paquette Abstract We consider global efficiency of algorithms for minimizing a sum of a convex function and
More informationAccelerated Proximal Gradient Methods for Convex Optimization
Accelerated Proximal Gradient Methods for Convex Optimization Paul Tseng Mathematics, University of Washington Seattle MOPTA, University of Guelph August 18, 2008 ACCELERATED PROXIMAL GRADIENT METHODS
More informationDual Proximal Gradient Method
Dual Proximal Gradient Method http://bicmr.pku.edu.cn/~wenzw/opt-2016-fall.html Acknowledgement: this slides is based on Prof. Lieven Vandenberghes lecture notes Outline 2/19 1 proximal gradient method
More informationA Proximal Method for Identifying Active Manifolds
A Proximal Method for Identifying Active Manifolds W.L. Hare April 18, 2006 Abstract The minimization of an objective function over a constraint set can often be simplified if the active manifold of the
More informationBlock Coordinate Descent for Regularized Multi-convex Optimization
Block Coordinate Descent for Regularized Multi-convex Optimization Yangyang Xu and Wotao Yin CAAM Department, Rice University February 15, 2013 Multi-convex optimization Model definition Applications Outline
More informationOptimization methods
Lecture notes 3 February 8, 016 1 Introduction Optimization methods In these notes we provide an overview of a selection of optimization methods. We focus on methods which rely on first-order information,
More informationMcMaster University. Advanced Optimization Laboratory. Title: A Proximal Method for Identifying Active Manifolds. Authors: Warren L.
McMaster University Advanced Optimization Laboratory Title: A Proximal Method for Identifying Active Manifolds Authors: Warren L. Hare AdvOl-Report No. 2006/07 April 2006, Hamilton, Ontario, Canada A Proximal
More informationInexact alternating projections on nonconvex sets
Inexact alternating projections on nonconvex sets D. Drusvyatskiy A.S. Lewis November 3, 2018 Dedicated to our friend, colleague, and inspiration, Alex Ioffe, on the occasion of his 80th birthday. Abstract
More informationOptimization methods
Optimization methods Optimization-Based Data Analysis http://www.cims.nyu.edu/~cfgranda/pages/obda_spring16 Carlos Fernandez-Granda /8/016 Introduction Aim: Overview of optimization methods that Tend to
More informationSIAM Conference on Imaging Science, Bologna, Italy, Adaptive FISTA. Peter Ochs Saarland University
SIAM Conference on Imaging Science, Bologna, Italy, 2018 Adaptive FISTA Peter Ochs Saarland University 07.06.2018 joint work with Thomas Pock, TU Graz, Austria c 2018 Peter Ochs Adaptive FISTA 1 / 16 Some
More informationProximal Newton Method. Ryan Tibshirani Convex Optimization /36-725
Proximal Newton Method Ryan Tibshirani Convex Optimization 10-725/36-725 1 Last time: primal-dual interior-point method Given the problem min x subject to f(x) h i (x) 0, i = 1,... m Ax = b where f, h
More informationAccelerated primal-dual methods for linearly constrained convex problems
Accelerated primal-dual methods for linearly constrained convex problems Yangyang Xu SIAM Conference on Optimization May 24, 2017 1 / 23 Accelerated proximal gradient For convex composite problem: minimize
More informationSubgradient Method. Ryan Tibshirani Convex Optimization
Subgradient Method Ryan Tibshirani Convex Optimization 10-725 Consider the problem Last last time: gradient descent min x f(x) for f convex and differentiable, dom(f) = R n. Gradient descent: choose initial
More informationAgenda. Fast proximal gradient methods. 1 Accelerated first-order methods. 2 Auxiliary sequences. 3 Convergence analysis. 4 Numerical examples
Agenda Fast proximal gradient methods 1 Accelerated first-order methods 2 Auxiliary sequences 3 Convergence analysis 4 Numerical examples 5 Optimality of Nesterov s scheme Last time Proximal gradient method
More informationGeometric Descent Method for Convex Composite Minimization
Geometric Descent Method for Convex Composite Minimization Shixiang Chen 1, Shiqian Ma 1, and Wei Liu 2 1 Department of SEEM, The Chinese University of Hong Kong, Hong Kong 2 Tencent AI Lab, Shenzhen,
More informationProximal Gradient Descent and Acceleration. Ryan Tibshirani Convex Optimization /36-725
Proximal Gradient Descent and Acceleration Ryan Tibshirani Convex Optimization 10-725/36-725 Last time: subgradient method Consider the problem min f(x) with f convex, and dom(f) = R n. Subgradient method:
More informationSequential convex programming,: value function and convergence
Sequential convex programming,: value function and convergence Edouard Pauwels joint work with Jérôme Bolte Journées MODE Toulouse March 23 2016 1 / 16 Introduction Local search methods for finite dimensional
More informationarxiv: v1 [math.oc] 9 Oct 2018
Cubic Regularization with Momentum for Nonconvex Optimization Zhe Wang Yi Zhou Yingbin Liang Guanghui Lan Ohio State University Ohio State University zhou.117@osu.edu liang.889@osu.edu Ohio State University
More informationLinear Convergence under the Polyak-Łojasiewicz Inequality
Linear Convergence under the Polyak-Łojasiewicz Inequality Hamed Karimi, Julie Nutini and Mark Schmidt The University of British Columbia LCI Forum February 28 th, 2017 1 / 17 Linear Convergence of Gradient-Based
More informationA Multilevel Proximal Algorithm for Large Scale Composite Convex Optimization
A Multilevel Proximal Algorithm for Large Scale Composite Convex Optimization Panos Parpas Department of Computing Imperial College London www.doc.ic.ac.uk/ pp500 p.parpas@imperial.ac.uk jointly with D.V.
More informationStochastic model-based minimization under high-order growth
Stochastic model-based minimization under high-order growth Damek Davis Dmitriy Drusvyatskiy Kellie J. MacPhee Abstract Given a nonsmooth, nonconvex minimization problem, we consider algorithms that iteratively
More informationarxiv: v1 [math.oc] 1 Jul 2016
Convergence Rate of Frank-Wolfe for Non-Convex Objectives Simon Lacoste-Julien INRIA - SIERRA team ENS, Paris June 8, 016 Abstract arxiv:1607.00345v1 [math.oc] 1 Jul 016 We give a simple proof that the
More informationLecture 17: October 27
0-725/36-725: Convex Optimiation Fall 205 Lecturer: Ryan Tibshirani Lecture 7: October 27 Scribes: Brandon Amos, Gines Hidalgo Note: LaTeX template courtesy of UC Berkeley EECS dept. Disclaimer: These
More informationOptimizing Nonconvex Finite Sums by a Proximal Primal-Dual Method
Optimizing Nonconvex Finite Sums by a Proximal Primal-Dual Method Davood Hajinezhad Iowa State University Davood Hajinezhad Optimizing Nonconvex Finite Sums by a Proximal Primal-Dual Method 1 / 35 Co-Authors
More informationShiqian Ma, MAT-258A: Numerical Optimization 1. Chapter 9. Alternating Direction Method of Multipliers
Shiqian Ma, MAT-258A: Numerical Optimization 1 Chapter 9 Alternating Direction Method of Multipliers Shiqian Ma, MAT-258A: Numerical Optimization 2 Separable convex optimization a special case is min f(x)
More informationSparse Optimization Lecture: Dual Methods, Part I
Sparse Optimization Lecture: Dual Methods, Part I Instructor: Wotao Yin July 2013 online discussions on piazza.com Those who complete this lecture will know dual (sub)gradient iteration augmented l 1 iteration
More informationBlock stochastic gradient update method
Block stochastic gradient update method Yangyang Xu and Wotao Yin IMA, University of Minnesota Department of Mathematics, UCLA November 1, 2015 This work was done while in Rice University 1 / 26 Stochastic
More informationStochastic Gradient Descent. Ryan Tibshirani Convex Optimization
Stochastic Gradient Descent Ryan Tibshirani Convex Optimization 10-725 Last time: proximal gradient descent Consider the problem min x g(x) + h(x) with g, h convex, g differentiable, and h simple in so
More informationarxiv: v1 [math.oc] 24 Mar 2017
Stochastic Methods for Composite Optimization Problems John C. Duchi 1,2 and Feng Ruan 2 {jduchi,fengruan}@stanford.edu Departments of 1 Electrical Engineering and 2 Statistics Stanford University arxiv:1703.08570v1
More informationStochastic and online algorithms
Stochastic and online algorithms stochastic gradient method online optimization and dual averaging method minimizing finite average Stochastic and online optimization 6 1 Stochastic optimization problem
More informationSubgradient Method. Guest Lecturer: Fatma Kilinc-Karzan. Instructors: Pradeep Ravikumar, Aarti Singh Convex Optimization /36-725
Subgradient Method Guest Lecturer: Fatma Kilinc-Karzan Instructors: Pradeep Ravikumar, Aarti Singh Convex Optimization 10-725/36-725 Adapted from slides from Ryan Tibshirani Consider the problem Recall:
More informationConvex Optimization. (EE227A: UC Berkeley) Lecture 15. Suvrit Sra. (Gradient methods III) 12 March, 2013
Convex Optimization (EE227A: UC Berkeley) Lecture 15 (Gradient methods III) 12 March, 2013 Suvrit Sra Optimal gradient methods 2 / 27 Optimal gradient methods We saw following efficiency estimates for
More informationProximal Newton Method. Zico Kolter (notes by Ryan Tibshirani) Convex Optimization
Proximal Newton Method Zico Kolter (notes by Ryan Tibshirani) Convex Optimization 10-725 Consider the problem Last time: quasi-newton methods min x f(x) with f convex, twice differentiable, dom(f) = R
More information6. Proximal gradient method
L. Vandenberghe EE236C (Spring 2016) 6. Proximal gradient method motivation proximal mapping proximal gradient method with fixed step size proximal gradient method with line search 6-1 Proximal mapping
More informationActive sets, steepest descent, and smooth approximation of functions
Active sets, steepest descent, and smooth approximation of functions Dmitriy Drusvyatskiy School of ORIE, Cornell University Joint work with Alex D. Ioffe (Technion), Martin Larsson (EPFL), and Adrian
More informationConvex Optimization Lecture 16
Convex Optimization Lecture 16 Today: Projected Gradient Descent Conditional Gradient Descent Stochastic Gradient Descent Random Coordinate Descent Recall: Gradient Descent (Steepest Descent w.r.t Euclidean
More informationCoordinate Descent and Ascent Methods
Coordinate Descent and Ascent Methods Julie Nutini Machine Learning Reading Group November 3 rd, 2015 1 / 22 Projected-Gradient Methods Motivation Rewrite non-smooth problem as smooth constrained problem:
More informationMath 273a: Optimization Convex Conjugacy
Math 273a: Optimization Convex Conjugacy Instructor: Wotao Yin Department of Mathematics, UCLA Fall 2015 online discussions on piazza.com Convex conjugate (the Legendre transform) Let f be a closed proper
More informationTame variational analysis
Tame variational analysis Dmitriy Drusvyatskiy Mathematics, University of Washington Joint work with Daniilidis (Chile), Ioffe (Technion), and Lewis (Cornell) May 19, 2015 Theme: Semi-algebraic geometry
More informationNonsmooth optimization: conditioning, convergence, and semi-algebraic models
Nonsmooth optimization: conditioning, convergence, and semi-algebraic models Adrian Lewis ORIE Cornell International Congress of Mathematicians Seoul, August 2014 1/16 Outline I Optimization and inverse
More informationOn the Local Quadratic Convergence of the Primal-Dual Augmented Lagrangian Method
Optimization Methods and Software Vol. 00, No. 00, Month 200x, 1 11 On the Local Quadratic Convergence of the Primal-Dual Augmented Lagrangian Method ROMAN A. POLYAK Department of SEOR and Mathematical
More informationIdentifying Active Constraints via Partial Smoothness and Prox-Regularity
Journal of Convex Analysis Volume 11 (2004), No. 2, 251 266 Identifying Active Constraints via Partial Smoothness and Prox-Regularity W. L. Hare Department of Mathematics, Simon Fraser University, Burnaby,
More informationFast proximal gradient methods
L. Vandenberghe EE236C (Spring 2013-14) Fast proximal gradient methods fast proximal gradient method (FISTA) FISTA with line search FISTA as descent method Nesterov s second method 1 Fast (proximal) gradient
More informationOn the acceleration of augmented Lagrangian method for linearly constrained optimization
On the acceleration of augmented Lagrangian method for linearly constrained optimization Bingsheng He and Xiaoming Yuan October, 2 Abstract. The classical augmented Lagrangian method (ALM plays a fundamental
More informationI P IANO : I NERTIAL P ROXIMAL A LGORITHM FOR N ON -C ONVEX O PTIMIZATION
I P IANO : I NERTIAL P ROXIMAL A LGORITHM FOR N ON -C ONVEX O PTIMIZATION Peter Ochs University of Freiburg Germany 17.01.2017 joint work with: Thomas Brox and Thomas Pock c 2017 Peter Ochs ipiano c 1
More informationECE G: Special Topics in Signal Processing: Sparsity, Structure, and Inference
ECE 18-898G: Special Topics in Signal Processing: Sparsity, Structure, and Inference Sparse Recovery using L1 minimization - algorithms Yuejie Chi Department of Electrical and Computer Engineering Spring
More informationLecture 23: November 21
10-725/36-725: Convex Optimization Fall 2016 Lecturer: Ryan Tibshirani Lecture 23: November 21 Scribes: Yifan Sun, Ananya Kumar, Xin Lu Note: LaTeX template courtesy of UC Berkeley EECS dept. Disclaimer:
More informationLinear Convergence under the Polyak-Łojasiewicz Inequality
Linear Convergence under the Polyak-Łojasiewicz Inequality Hamed Karimi, Julie Nutini, Mark Schmidt University of British Columbia Linear of Convergence of Gradient-Based Methods Fitting most machine learning
More information5. Subgradient method
L. Vandenberghe EE236C (Spring 2016) 5. Subgradient method subgradient method convergence analysis optimal step size when f is known alternating projections optimality 5-1 Subgradient method to minimize
More informationSelected Topics in Optimization. Some slides borrowed from
Selected Topics in Optimization Some slides borrowed from http://www.stat.cmu.edu/~ryantibs/convexopt/ Overview Optimization problems are almost everywhere in statistics and machine learning. Input Model
More information1. Gradient method. gradient method, first-order methods. quadratic bounds on convex functions. analysis of gradient method
L. Vandenberghe EE236C (Spring 2016) 1. Gradient method gradient method, first-order methods quadratic bounds on convex functions analysis of gradient method 1-1 Approximate course outline First-order
More informationALGORITHMS FOR MINIMIZING DIFFERENCES OF CONVEX FUNCTIONS AND APPLICATIONS
ALGORITHMS FOR MINIMIZING DIFFERENCES OF CONVEX FUNCTIONS AND APPLICATIONS Mau Nam Nguyen (joint work with D. Giles and R. B. Rector) Fariborz Maseeh Department of Mathematics and Statistics Portland State
More informationLECTURE 25: REVIEW/EPILOGUE LECTURE OUTLINE
LECTURE 25: REVIEW/EPILOGUE LECTURE OUTLINE CONVEX ANALYSIS AND DUALITY Basic concepts of convex analysis Basic concepts of convex optimization Geometric duality framework - MC/MC Constrained optimization
More informationOn the Quadratic Convergence of the Cubic Regularization Method under a Local Error Bound Condition
On the Quadratic Convergence of the Cubic Regularization Method under a Local Error Bound Condition Man-Chung Yue Zirui Zhou Anthony Man-Cho So Abstract In this paper we consider the cubic regularization
More informationOn the Quadratic Convergence of the Cubic Regularization Method under a Local Error Bound Condition
On the Quadratic Convergence of the Cubic Regularization Method under a Local Error Bound Condition Man-Chung Yue Zirui Zhou Anthony Man-Cho So Abstract In this paper we consider the cubic regularization
More informationA PROXIMAL METHOD FOR COMPOSITE MINIMIZATION. 1. Problem Statement. We consider minimization problems of the form. min
A PROXIMAL METHOD FOR COMPOSITE MINIMIZATION A. S. LEWIS AND S. J. WRIGHT Abstract. We consider minimization of functions that are compositions of prox-regular functions with smooth vector functions. A
More information14. Nonlinear equations
L. Vandenberghe ECE133A (Winter 2018) 14. Nonlinear equations Newton method for nonlinear equations damped Newton method for unconstrained minimization Newton method for nonlinear least squares 14-1 Set
More informationOptimization for Machine Learning
Optimization for Machine Learning (Problems; Algorithms - A) SUVRIT SRA Massachusetts Institute of Technology PKU Summer School on Data Science (July 2017) Course materials http://suvrit.de/teaching.html
More informationAccelerated Block-Coordinate Relaxation for Regularized Optimization
Accelerated Block-Coordinate Relaxation for Regularized Optimization Stephen J. Wright Computer Sciences University of Wisconsin, Madison October 09, 2012 Problem descriptions Consider where f is smooth
More informationConditional Gradient (Frank-Wolfe) Method
Conditional Gradient (Frank-Wolfe) Method Lecturer: Aarti Singh Co-instructor: Pradeep Ravikumar Convex Optimization 10-725/36-725 1 Outline Today: Conditional gradient method Convergence analysis Properties
More informationAn Accelerated Hybrid Proximal Extragradient Method for Convex Optimization and its Implications to Second-Order Methods
An Accelerated Hybrid Proximal Extragradient Method for Convex Optimization and its Implications to Second-Order Methods Renato D.C. Monteiro B. F. Svaiter May 10, 011 Revised: May 4, 01) Abstract This
More informationMaster 2 MathBigData. 3 novembre CMAP - Ecole Polytechnique
Master 2 MathBigData S. Gaïffas 1 3 novembre 2014 1 CMAP - Ecole Polytechnique 1 Supervised learning recap Introduction Loss functions, linearity 2 Penalization Introduction Ridge Sparsity Lasso 3 Some
More informationShiqian Ma, MAT-258A: Numerical Optimization 1. Chapter 3. Gradient Method
Shiqian Ma, MAT-258A: Numerical Optimization 1 Chapter 3 Gradient Method Shiqian Ma, MAT-258A: Numerical Optimization 2 3.1. Gradient method Classical gradient method: to minimize a differentiable convex
More informationDownloaded 09/27/13 to Redistribution subject to SIAM license or copyright; see
SIAM J. OPTIM. Vol. 23, No., pp. 256 267 c 203 Society for Industrial and Applied Mathematics TILT STABILITY, UNIFORM QUADRATIC GROWTH, AND STRONG METRIC REGULARITY OF THE SUBDIFFERENTIAL D. DRUSVYATSKIY
More informationNewton s Method. Ryan Tibshirani Convex Optimization /36-725
Newton s Method Ryan Tibshirani Convex Optimization 10-725/36-725 1 Last time: dual correspondences Given a function f : R n R, we define its conjugate f : R n R, Properties and examples: f (y) = max x
More informationKey words. prox-regular functions, polyhedral convex functions, sparse optimization, global convergence, active constraint identification
A PROXIMAL METHOD FOR COMPOSITE MINIMIZATION A. S. LEWIS AND S. J. WRIGHT Abstract. We consider minimization of functions that are compositions of convex or proxregular functions (possibly extended-valued
More informationIntroduction. New Nonsmooth Trust Region Method for Unconstraint Locally Lipschitz Optimization Problems
New Nonsmooth Trust Region Method for Unconstraint Locally Lipschitz Optimization Problems Z. Akbari 1, R. Yousefpour 2, M. R. Peyghami 3 1 Department of Mathematics, K.N. Toosi University of Technology,
More informationSparse Covariance Selection using Semidefinite Programming
Sparse Covariance Selection using Semidefinite Programming A. d Aspremont ORFE, Princeton University Joint work with O. Banerjee, L. El Ghaoui & G. Natsoulis, U.C. Berkeley & Iconix Pharmaceuticals Support
More informationHigher-Order Methods
Higher-Order Methods Stephen J. Wright 1 2 Computer Sciences Department, University of Wisconsin-Madison. PCMI, July 2016 Stephen Wright (UW-Madison) Higher-Order Methods PCMI, July 2016 1 / 25 Smooth
More informationGRADIENT = STEEPEST DESCENT
GRADIENT METHODS GRADIENT = STEEPEST DESCENT Convex Function Iso-contours gradient 0.5 0.4 4 2 0 8 0.3 0.2 0. 0 0. negative gradient 6 0.2 4 0.3 2.5 0.5 0 0.5 0.5 0 0.5 0.4 0.5.5 0.5 0 0.5 GRADIENT DESCENT
More informationLasso: Algorithms and Extensions
ELE 538B: Sparsity, Structure and Inference Lasso: Algorithms and Extensions Yuxin Chen Princeton University, Spring 2017 Outline Proximal operators Proximal gradient methods for lasso and its extensions
More informationLecture 5: September 12
10-725/36-725: Convex Optimization Fall 2015 Lecture 5: September 12 Lecturer: Lecturer: Ryan Tibshirani Scribes: Scribes: Barun Patra and Tyler Vuong Note: LaTeX template courtesy of UC Berkeley EECS
More informationContraction Methods for Convex Optimization and Monotone Variational Inequalities No.16
XVI - 1 Contraction Methods for Convex Optimization and Monotone Variational Inequalities No.16 A slightly changed ADMM for convex optimization with three separable operators Bingsheng He Department of
More informationPrimal-dual Subgradient Method for Convex Problems with Functional Constraints
Primal-dual Subgradient Method for Convex Problems with Functional Constraints Yurii Nesterov, CORE/INMA (UCL) Workshop on embedded optimization EMBOPT2014 September 9, 2014 (Lucca) Yu. Nesterov Primal-dual
More informationRandomized Block Coordinate Non-Monotone Gradient Method for a Class of Nonlinear Programming
Randomized Block Coordinate Non-Monotone Gradient Method for a Class of Nonlinear Programming Zhaosong Lu Lin Xiao June 25, 2013 Abstract In this paper we propose a randomized block coordinate non-monotone
More informationNon-smooth Non-convex Bregman Minimization: Unification and new Algorithms
Non-smooth Non-convex Bregman Minimization: Unification and new Algorithms Peter Ochs, Jalal Fadili, and Thomas Brox Saarland University, Saarbrücken, Germany Normandie Univ, ENSICAEN, CNRS, GREYC, France
More informationCoordinate Update Algorithm Short Course Operator Splitting
Coordinate Update Algorithm Short Course Operator Splitting Instructor: Wotao Yin (UCLA Math) Summer 2016 1 / 25 Operator splitting pipeline 1. Formulate a problem as 0 A(x) + B(x) with monotone operators
More informationHomework 3 Conjugate Gradient Descent, Accelerated Gradient Descent Newton, Quasi Newton and Projected Gradient Descent
Homework 3 Conjugate Gradient Descent, Accelerated Gradient Descent Newton, Quasi Newton and Projected Gradient Descent CMU 10-725/36-725: Convex Optimization (Fall 2017) OUT: Sep 29 DUE: Oct 13, 5:00
More informationConvex Analysis Notes. Lecturer: Adrian Lewis, Cornell ORIE Scribe: Kevin Kircher, Cornell MAE
Convex Analysis Notes Lecturer: Adrian Lewis, Cornell ORIE Scribe: Kevin Kircher, Cornell MAE These are notes from ORIE 6328, Convex Analysis, as taught by Prof. Adrian Lewis at Cornell University in the
More informationLecture 3. Optimization Problems and Iterative Algorithms
Lecture 3 Optimization Problems and Iterative Algorithms January 13, 2016 This material was jointly developed with Angelia Nedić at UIUC for IE 598ns Outline Special Functions: Linear, Quadratic, Convex
More informationc Copyright 2017 Scott Roy
c Copyright 2017 Scott Roy Algorithms for convex optimization with applications to data science Scott Roy A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of
More informationSome Inexact Hybrid Proximal Augmented Lagrangian Algorithms
Some Inexact Hybrid Proximal Augmented Lagrangian Algorithms Carlos Humes Jr. a, Benar F. Svaiter b, Paulo J. S. Silva a, a Dept. of Computer Science, University of São Paulo, Brazil Email: {humes,rsilva}@ime.usp.br
More informationUses of duality. Geoff Gordon & Ryan Tibshirani Optimization /
Uses of duality Geoff Gordon & Ryan Tibshirani Optimization 10-725 / 36-725 1 Remember conjugate functions Given f : R n R, the function is called its conjugate f (y) = max x R n yt x f(x) Conjugates appear
More informationMath 273a: Optimization Subgradient Methods
Math 273a: Optimization Subgradient Methods Instructor: Wotao Yin Department of Mathematics, UCLA Fall 2015 online discussions on piazza.com Nonsmooth convex function Recall: For ˉx R n, f(ˉx) := {g R
More informationarxiv: v3 [math.oc] 19 Oct 2017
Gradient descent with nonconvex constraints: local concavity determines convergence Rina Foygel Barber and Wooseok Ha arxiv:703.07755v3 [math.oc] 9 Oct 207 0.7.7 Abstract Many problems in high-dimensional
More informationConvergence of Cubic Regularization for Nonconvex Optimization under KŁ Property
Convergence of Cubic Regularization for Nonconvex Optimization under KŁ Property Yi Zhou Department of ECE The Ohio State University zhou.1172@osu.edu Zhe Wang Department of ECE The Ohio State University
More informationarxiv: v1 [math.oc] 13 Dec 2018
A NEW HOMOTOPY PROXIMAL VARIABLE-METRIC FRAMEWORK FOR COMPOSITE CONVEX MINIMIZATION QUOC TRAN-DINH, LIANG LING, AND KIM-CHUAN TOH arxiv:8205243v [mathoc] 3 Dec 208 Abstract This paper suggests two novel
More informationBeyond Heuristics: Applying Alternating Direction Method of Multipliers in Nonconvex Territory
Beyond Heuristics: Applying Alternating Direction Method of Multipliers in Nonconvex Territory Xin Liu(4Ð) State Key Laboratory of Scientific and Engineering Computing Institute of Computational Mathematics
More informationKey words. constrained optimization, composite optimization, Mangasarian-Fromovitz constraint qualification, active set, identification.
IDENTIFYING ACTIVITY A. S. LEWIS AND S. J. WRIGHT Abstract. Identification of active constraints in constrained optimization is of interest from both practical and theoretical viewpoints, as it holds the
More informationStochastic Composition Optimization
Stochastic Composition Optimization Algorithms and Sample Complexities Mengdi Wang Joint works with Ethan X. Fang, Han Liu, and Ji Liu ORFE@Princeton ICCOPT, Tokyo, August 8-11, 2016 1 / 24 Collaborators
More informationOptimization: an Overview
Optimization: an Overview Moritz Diehl University of Freiburg and University of Leuven (some slide material was provided by W. Bangerth and K. Mombaur) Overview of presentation Optimization: basic definitions
More informationARock: an algorithmic framework for asynchronous parallel coordinate updates
ARock: an algorithmic framework for asynchronous parallel coordinate updates Zhimin Peng, Yangyang Xu, Ming Yan, Wotao Yin ( UCLA Math, U.Waterloo DCO) UCLA CAM Report 15-37 ShanghaiTech SSDS 15 June 25,
More information10. Unconstrained minimization
Convex Optimization Boyd & Vandenberghe 10. Unconstrained minimization terminology and assumptions gradient descent method steepest descent method Newton s method self-concordant functions implementation
More informationConvex Optimization. Ofer Meshi. Lecture 6: Lower Bounds Constrained Optimization
Convex Optimization Ofer Meshi Lecture 6: Lower Bounds Constrained Optimization Lower Bounds Some upper bounds: #iter μ 2 M #iter 2 M #iter L L μ 2 Oracle/ops GD κ log 1/ε M x # ε L # x # L # ε # με f
More information8 Numerical methods for unconstrained problems
8 Numerical methods for unconstrained problems Optimization is one of the important fields in numerical computation, beside solving differential equations and linear systems. We can see that these fields
More informationGradient Descent. Ryan Tibshirani Convex Optimization /36-725
Gradient Descent Ryan Tibshirani Convex Optimization 10-725/36-725 Last time: canonical convex programs Linear program (LP): takes the form min x subject to c T x Gx h Ax = b Quadratic program (QP): like
More informationarxiv: v2 [math.oc] 21 Nov 2017
Unifying abstract inexact convergence theorems and block coordinate variable metric ipiano arxiv:1602.07283v2 [math.oc] 21 Nov 2017 Peter Ochs Mathematical Optimization Group Saarland University Germany
More information