diff --git a/ICML19/Arxiv version/bibliography.bib b/ICML19/Arxiv version/bibliography.bib new file mode 100644 index 0000000..610cb62 --- /dev/null +++ b/ICML19/Arxiv version/bibliography.bib @@ -0,0 +1,830 @@ +%% This BibTeX bibliography file was created using BibDesk. +%% http://bibdesk.sourceforge.net/ + + +%% Saved with string encoding Unicode (UTF-8) + +% Created by Mehmet Fatih Sahin for nonconvex inexact augmented lagrangian paper +% December 14, Friday. 01:52 am + +@article{Goodfellow2014, + author = {{Goodfellow}, I.~J. and {Pouget-Abadie}, J. and {Mirza}, M. and + {Xu}, B. and {Warde-Farley}, D. and {Ozair}, S. and {Courville}, A. and + {Bengio}, Y.}, + url = {https://arxiv.org/abs/1406.2661}, + title = "{Generative Adversarial Networks}", + journal = {ArXiv e-prints}, + archivePrefix = "arXiv", + eprint = {1406.2661}, + primaryClass = "stat.ML", + keywords = {Statistics - Machine Learning, Computer Science - Machine Learning}, + year = 2014, + month = jun, + adsurl = {http://adsabs.harvard.edu/abs/2014arXiv1406.2661G}, + adsnote = {Provided by the SAO/NASA Astrophysics Data System} +} + +@ARTICLE{Ilyas2017, + author = {{Ilyas}, Andrew and {Jalal}, Ajil and {Asteri}, Eirini and {Daskalakis}, + Constantinos and {Dimakis}, Alexandros G.}, + title = "{The Robust Manifold Defense: Adversarial Training using Generative Models}", + journal = {arXiv e-prints}, + keywords = {Computer Science - Computer Vision and Pattern Recognition, Computer Science - Cryptography and Security, Computer Science - Machine Learning, Statistics - Machine Learning}, + year = 2017, + month = Dec, + eid = {arXiv:1712.09196}, + pages = {arXiv:1712.09196}, +archivePrefix = {arXiv}, + eprint = {1712.09196}, + primaryClass = {cs.CV}, + adsurl = {https://ui.adsabs.harvard.edu/\#abs/2017arXiv171209196I}, + adsnote = {Provided by the SAO/NASA Astrophysics Data System} +} + +@ARTICLE{Szegedy2013, + author = {{Szegedy}, Christian and {Zaremba}, Wojciech and {Sutskever}, Ilya and + {Bruna}, Joan and {Erhan}, Dumitru and {Goodfellow}, Ian and + {Fergus}, Rob}, + title = "{Intriguing properties of neural networks}", + journal = {arXiv e-prints}, + keywords = {Computer Science - Computer Vision and Pattern Recognition, Computer Science - Machine Learning, Computer Science - Neural and Evolutionary Computing}, + year = 2013, + month = Dec, + eid = {arXiv:1312.6199}, + pages = {arXiv:1312.6199}, +archivePrefix = {arXiv}, + eprint = {1312.6199}, + primaryClass = {cs.CV}, + adsurl = {https://ui.adsabs.harvard.edu/\#abs/2013arXiv1312.6199S}, + adsnote = {Provided by the SAO/NASA Astrophysics Data System} +} + + + +@article{tepper2018clustering, + title={Clustering is semidefinitely not that hard: Nonnegative SDP for manifold disentangling}, + author={Tepper, Mariano and Sengupta, Anirvan M and Chklovskii, Dmitri}, + journal={Journal of Machine Learning Research}, + volume={19}, + number={82}, + year={2018} +} + +@article{beck2009fast, + title={A fast iterative shrinkage-thresholding algorithm for linear inverse problems}, + author={Beck, Amir and Teboulle, Marc}, + journal={SIAM journal on imaging sciences}, + volume={2}, + number={1}, + pages={183--202}, + year={2009}, + publisher={SIAM} +} + +@article{tibshirani1996regression, + title={Regression shrinkage and selection via the lasso}, + author={Tibshirani, Robert}, + journal={Journal of the Royal Statistical Society. Series B (Methodological)}, + pages={267--288}, + year={1996}, + publisher={JSTOR} +} + + +@book{bauschke2011convex, + title={Convex analysis and monotone operator theory in Hilbert spaces}, + author={Bauschke, Heinz H and Combettes, Patrick L and others}, + volume={408}, + year={2011}, + publisher={Springer} +} + + +@article{waldspurger2018rank, + title={Rank optimality for the Burer-Monteiro factorization}, + author={Waldspurger, Ir{\`e}ne and Waters, Alden}, + journal={arXiv preprint arXiv:1812.03046}, + year={2018} +} + + +@inproceedings{jaggi2013revisiting, + title={Revisiting Frank-Wolfe: Projection-Free Sparse Convex Optimization.}, + author={Jaggi, Martin}, + booktitle={ICML (1)}, + pages={427--435}, + year={2013} +} + + + +@inproceedings{nesterov1983method, + title={A method for solving the convex programming problem with convergence rate O (1/k\^{} 2)}, + author={Nesterov, Yurii E}, + booktitle={Dokl. Akad. Nauk SSSR}, + volume={269}, + pages={543--547}, + year={1983} +} + + +@inproceedings{raghavendra2008optimal, + title={Optimal algorithms and inapproximability results for every CSP?}, + author={Raghavendra, Prasad}, + booktitle={Proceedings of the fortieth annual ACM symposium on Theory of computing}, + pages={245--254}, + year={2008}, + organization={ACM} +} + + + +@online{xiao2017/online, + author = {Han Xiao and Kashif Rasul and Roland Vollgraf}, + title = {Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms}, + date = {2017-08-28}, + year = {2017}, + eprintclass = {cs.LG}, + eprinttype = {arXiv}, + eprint = {cs.LG/1708.07747}, +} + +@article{cartis2011evaluation, + title={On the evaluation complexity of composite function minimization with applications to nonconvex nonlinear programming}, + author={Cartis, Coralia and Gould, Nicholas IM and Toint, Philippe L}, + journal={SIAM Journal on Optimization}, + volume={21}, + number={4}, + pages={1721--1739}, + year={2011}, + publisher={SIAM} +} + +@article{bolte2017error, + title={From error bounds to the complexity of first-order descent methods for convex functions}, + author={Bolte, J{\'e}r{\^o}me and Nguyen, Trong Phong and Peypouquet, Juan and Suter, Bruce W}, + journal={Mathematical Programming}, + volume={165}, + number={2}, + pages={471--507}, + year={2017}, + publisher={Springer} +} + + + +@article{obozinski2011group, + title={Group lasso with overlaps: the latent group lasso approach}, + author={Obozinski, Guillaume and Jacob, Laurent and Vert, Jean-Philippe}, + journal={arXiv preprint arXiv:1110.0413}, + year={2011} +} + +@article{tran2018smooth, + title={A smooth primal-dual optimization framework for nonsmooth composite convex minimization}, + author={Tran-Dinh, Quoc and Fercoq, Olivier and Cevher, Volkan}, + journal={SIAM Journal on Optimization}, + volume={28}, + number={1}, + pages={96--134}, + year={2018}, + publisher={SIAM} +} + +@article{tran2018adaptive, + title={An Adaptive Primal-Dual Framework for Nonsmooth Convex Minimization}, + author={Tran-Dinh, Quoc and Alacaoglu, Ahmet and Fercoq, Olivier and Cevher, Volkan}, + journal={arXiv preprint arXiv:1808.04648}, + year={2018} +} + +@article{arora2018compressed, + title={A compressed sensing view of unsupervised text embeddings, bag-of-n-grams, and LSTMs}, + author={Arora, Sanjeev and Khodak, Mikhail and Saunshi, Nikunj and Vodrahalli, Kiran}, + year={2018} +} + +@article{candes2008introduction, + title={An introduction to compressive sampling}, + author={Cand{\`e}s, Emmanuel J and Wakin, Michael B}, + journal={IEEE signal processing magazine}, + volume={25}, + number={2}, + pages={21--30}, + year={2008}, + publisher={IEEE} +} + +@article{chen2001atomic, + title={Atomic decomposition by basis pursuit}, + author={Chen, Scott Shaobing and Donoho, David L and Saunders, Michael A}, + journal={SIAM review}, + volume={43}, + number={1}, + pages={129--159}, + year={2001}, + publisher={SIAM} +} + + +@article{flores2012complete, + title={A complete characterization of strong duality in nonconvex optimization with a single constraint}, + author={Flores-Baz{\'a}n, Fabi{\'a}n and Flores-Baz{\'a}n, Fernando and Vera, Cristi{\'a}n}, + journal={Journal of Global Optimization}, + volume={53}, + number={2}, + pages={185--201}, + year={2012}, + publisher={Springer} +} + +@article{dai2002convergence, + title={Convergence properties of the BFGS algoritm}, + author={Dai, Yu-Hong}, + journal={SIAM Journal on Optimization}, + volume={13}, + number={3}, + pages={693--701}, + year={2002}, + publisher={SIAM} +} + +@article{yang2015sdpnal, + title={SDPNAL+: a majorized semismooth Newton-CG augmented Lagrangian method for semidefinite programming with nonnegative constraints}, + author={Yang, Liuqin and Sun, Defeng and Toh, Kim-Chuan}, + journal={Mathematical Programming Computation}, + volume={7}, + number={3}, + pages={331--366}, + year={2015}, + publisher={Springer} +} + + +@article{mascarenhas2004bfgs, + title={The BFGS method with exact line searches fails for non-convex objective functions}, + author={Mascarenhas, Walter F}, + journal={Mathematical Programming}, + volume={99}, + number={1}, + pages={49--61}, + year={2004}, + publisher={Springer} +} + +@inproceedings{boumal2016non, + title={The non-convex Burer-Monteiro approach works on smooth semidefinite programs}, + author={Boumal, Nicolas and Voroninski, Vlad and Bandeira, Afonso}, + booktitle={Advances in Neural Information Processing Systems}, + pages={2757--2765}, + year={2016} +} + + + + +@inproceedings{bhojanapalli2016dropping, + title={Dropping convexity for faster semi-definite optimization}, + author={Bhojanapalli, Srinadh and Kyrillidis, Anastasios and Sanghavi, Sujay}, + booktitle={Conference on Learning Theory}, + pages={530--582}, + year={2016} +} + + +@inproceedings{nesterov1983method, + title={A method for solving the convex programming problem with convergence rate O (1/k\^{} 2)}, + author={Nesterov, Yurii E}, + booktitle={Dokl. Akad. Nauk SSSR}, + volume={269}, + pages={543--547}, + year={1983} +} + + + +@article{shi2017penalty, + title={Penalty dual decomposition method for nonsmooth nonconvex optimization}, + author={Shi, Qingjiang and Hong, Mingyi and Fu, Xiao and Chang, Tsung-Hui}, + journal={arXiv preprint arXiv:1712.04767}, + year={2017} +} + + + +@article{fernandez2012local, + title={Local convergence of exact and inexact augmented Lagrangian methods under the second-order sufficient optimality condition}, + author={Fern{\'a}ndez, Dami{\'a}n and Solodov, Mikhail V}, + journal={SIAM Journal on Optimization}, + volume={22}, + number={2}, + pages={384--407}, + year={2012}, + publisher={SIAM} +} + +@article{bolte2018nonconvex, + title={Nonconvex Lagrangian-based optimization: monitoring schemes and global convergence}, + author={Bolte, J{\'e}r{\^o}me and Sabach, Shoham and Teboulle, Marc}, + journal={Mathematics of Operations Research}, + year={2018}, + publisher={INFORMS} +} + + +@article{nouiehed2018convergence, + title={Convergence to Second-Order Stationarity for Constrained Non-Convex Optimization}, + author={Nouiehed, Maher and Lee, Jason D and Razaviyayn, Meisam}, + journal={arXiv preprint arXiv:1810.02024}, + year={2018} +} + + + +@article{cartis2018optimality, + title={Optimality of orders one to three and beyond: characterization and evaluation complexity in constrained nonconvex optimization}, + author={Cartis, Coralia and Gould, Nicholas IM and Toint, Ph L}, + journal={Journal of Complexity}, + year={2018}, + publisher={Elsevier} +} + + + +@article{nesterov2009primal, + title={Primal-dual subgradient methods for convex problems}, + author={Nesterov, Yurii}, + journal={Mathematical programming}, + volume={120}, + number={1}, + pages={221--259}, + year={2009}, + publisher={Springer} +} + +@inproceedings{yurtsever2015universal, + title={A universal primal-dual convex optimization framework}, + author={Yurtsever, Alp and Dinh, Quoc Tran and Cevher, Volkan}, + booktitle={Advances in Neural Information Processing Systems}, + pages={3150--3158}, + year={2015} +} + +@article{yurtsever2018conditional, + title={A Conditional Gradient Framework for Composite Convex Minimization with Applications to Semidefinite Programming}, + author={Yurtsever, Alp and Fercoq, Olivier and Locatello, Francesco and Cevher, Volkan}, + journal={arXiv preprint arXiv:1804.08544}, + year={2018} +} + + +@article{bertsekas1976penalty, + title={On penalty and multiplier methods for constrained minimization}, + author={Bertsekas, Dimitri P}, + journal={SIAM Journal on Control and Optimization}, + volume={14}, + number={2}, + pages={216--235}, + year={1976}, + publisher={SIAM} +} + +@article{tran2018adaptive, + title={An Adaptive Primal-Dual Framework for Nonsmooth Convex Minimization}, + author={Tran-Dinh, Quoc and Alacaoglu, Ahmet and Fercoq, Olivier and Cevher, Volkan}, + journal={arXiv preprint arXiv:1808.04648}, + year={2018} +} + + +@article{hestenes1969multiplier, + title={Multiplier and gradient methods}, + author={Hestenes, Magnus R}, + journal={Journal of optimization theory and applications}, + volume={4}, + number={5}, + pages={303--320}, + year={1969}, + publisher={Springer} +} + + +@article{powell1969method, + title={A method for nonlinear constraints in minimization problems}, + author={Powell, Michael JD}, + journal={Optimization}, + pages={283--298}, + year={1969}, + publisher={Academic Press} +} + +@book{bertsekas2014constrained, + title={Constrained optimization and Lagrange multiplier methods}, + author={Bertsekas, Dimitri P}, + year={2014}, + publisher={Academic press} +} + + +@inproceedings{kulis2007fast, + title={Fast low-rank semidefinite programming for embedding and clustering}, + author={Kulis, Brian and Surendran, Arun C and Platt, John C}, + booktitle={Artificial Intelligence and Statistics}, + pages={235--242}, + year={2007} +} + + +@article{xu2017inexact, + title={Iteration complexity of inexact augmented Lagrangian methods for constrained convex programming}, + author={Xu, Yangyang}, + journal={arXiv preprint arXiv:1711.05812v2}, + year={2017} +} + + + +@article{nedelcu2014computational, + title={Computational complexity of inexact gradient augmented Lagrangian methods: application to constrained MPC}, + author={Nedelcu, Valentin and Necoara, Ion and Tran-Dinh, Quoc}, + journal={SIAM Journal on Control and Optimization}, + volume={52}, + number={5}, + pages={3109--3134}, + year={2014}, + publisher={SIAM} +} + + +@article{pataki1998rank, + title={On the rank of extreme matrices in semidefinite programs and the multiplicity of optimal eigenvalues}, + author={Pataki, G{\'a}bor}, + journal={Mathematics of operations research}, + volume={23}, + number={2}, + pages={339--358}, + year={1998}, + publisher={INFORMS} +} + + +@article{Barvinok1995problems, + title={Problems of distance geometry and convex properties of quadratic maps}, + author={Barvinok, Alexander I.}, + journal={Discrete \& Computational Geometry}, + volume={13}, + number={2}, + pages={189--202}, + year={1995}, + publisher={Springer} +} + + +@article{cartis2012complexity, + title={Complexity bounds for second-order optimality in unconstrained optimization}, + author={Cartis, Coralia and Gould, Nicholas IM and Toint, Ph L}, + journal={Journal of Complexity}, + volume={28}, + number={1}, + pages={93--108}, + year={2012}, + publisher={Elsevier} +} + + + +@article{ghadimi2016accelerated, + title={Accelerated gradient methods for nonconvex nonlinear and stochastic programming}, + author={Ghadimi, Saeed and Lan, Guanghui}, + journal={Mathematical Programming}, + volume={156}, + number={1-2}, + pages={59--99}, + year={2016}, + publisher={Springer} +} + +@article{boumal2014manopt, + Author = {Boumal, Nicolas and Mishra, Bamdev and Absil, P-A and Sepulchre, Rodolphe}, + Journal = {The Journal of Machine Learning Research}, + Number = {1}, + Pages = {1455--1459}, + Publisher = {JMLR. org}, + Title = {Manopt, a Matlab toolbox for optimization on manifolds}, + Volume = {15}, + Year = {2014}} + +@article{boumal2016global, + Author = {Boumal, Nicolas and Absil, P-A and Cartis, Coralia}, + Journal = {arXiv preprint arXiv:1605.08101}, + Title = {Global rates of convergence for nonconvex optimization on manifolds}, + Year = {2016}} + +@inproceedings{ge2016efficient, + Author = {Ge, Rong and Jin, Chi and Netrapalli, Praneeth and Sidford, Aaron and others}, + Booktitle = {International Conference on Machine Learning}, + Pages = {2741--2750}, + Title = {Efficient algorithms for large-scale generalized eigenvector computation and canonical correlation analysis}, + Year = {2016}} + +@inproceedings{boumal2016non, + Author = {Boumal, Nicolas and Voroninski, Vlad and Bandeira, Afonso}, + Booktitle = {Advances in Neural Information Processing Systems}, + Pages = {2757--2765}, + Title = {The non-convex Burer-Monteiro approach works on smooth semidefinite programs}, + Year = {2016}} + +@article{davis2011university, + Author = {Davis, Timothy A and Hu, Yifan}, + Journal = {ACM Transactions on Mathematical Software (TOMS)}, + Number = {1}, + Pages = {1}, + Publisher = {ACM}, + Title = {The University of Florida sparse matrix collection}, + Volume = {38}, + Year = {2011}} + +@article{clason2018acceleration, + Author = {Clason, Christian and Mazurenko, Stanislav and Valkonen, Tuomo}, + Journal = {arXiv preprint arXiv:1802.03347}, + Title = {Acceleration and global convergence of a first-order primal--dual method for nonconvex problems}, + Year = {2018}} + +@article{chambolle2011first, + Author = {Chambolle, Antonin and Pock, Thomas}, + Journal = {Journal of mathematical imaging and vision}, + Number = {1}, + Pages = {120--145}, + Publisher = {Springer}, + Title = {A first-order primal-dual algorithm for convex problems with applications to imaging}, + Volume = {40}, + Year = {2011}} + +@article{tran2018smooth, + Author = {Tran-Dinh, Quoc and Fercoq, Olivier and Cevher, Volkan}, + Journal = {SIAM Journal on Optimization}, + Number = {1}, + Pages = {96--134}, + Publisher = {SIAM}, + Title = {A smooth primal-dual optimization framework for nonsmooth composite convex minimization}, + Volume = {28}, + Year = {2018}} + +@article{bhojanapalli2018smoothed, + Author = {Bhojanapalli, Srinadh and Boumal, Nicolas and Jain, Prateek and Netrapalli, Praneeth}, + Journal = {arXiv preprint arXiv:1803.00186}, + Title = {Smoothed analysis for low-rank solutions to semidefinite programs in quadratic penalty form}, + Year = {2018}} + +@article{park2016provable, + Author = {Park, Dohyung and Kyrillidis, Anastasios and Bhojanapalli, Srinadh and Caramanis, Constantine and Sanghavi, Sujay}, + Journal = {arXiv preprint arXiv:1606.01316}, + Title = {Provable Burer-Monteiro factorization for a class of norm-constrained matrix problems}, + Year = {2016}} + +@article{mixon2016clustering, + Author = {Mixon, Dustin G and Villar, Soledad and Ward, Rachel}, + Journal = {arXiv preprint arXiv:1602.06612}, + Title = {Clustering subgaussian mixtures by semidefinite programming}, + Year = {2016}} + +@article{yurtsever2018conditional, + Author = {Yurtsever, Alp and Fercoq, Olivier and Locatello, Francesco and Cevher, Volkan}, + Journal = {arXiv preprint arXiv:1804.08544}, + Title = {A Conditional Gradient Framework for Composite Convex Minimization with Applications to Semidefinite Programming}, + Year = {2018}} + +@inproceedings{jin2017escape, + Author = {Jin, Chi and Ge, Rong and Netrapalli, Praneeth and Kakade, Sham M and Jordan, Michael I}, + Booktitle = {International Conference on Machine Learning}, + Pages = {1724--1732}, + Title = {How to Escape Saddle Points Efficiently}, + Year = {2017}} + +@article{birgin2016evaluation, + title={Evaluation complexity for nonlinear constrained optimization using unscaled KKT conditions and high-order models}, + author={Birgin, Ernesto G and Gardenghi, JL and Martinez, Jos{\'e} Mario and Santos, SA and Toint, Ph L}, + journal={SIAM Journal on Optimization}, + volume={26}, + number={2}, + pages={951--967}, + year={2016}, + publisher={SIAM} +} + +@article{liu2017linearized, + Author = {Liu, Qinghua and Shen, Xinyue and Gu, Yuantao}, + Journal = {arXiv preprint arXiv:1705.02502}, + Title = {Linearized admm for non-convex non-smooth optimization with convergence analysis}, + Year = {2017}} +@article{wang2015global, + Author = {Wang, Yu and Yin, Wotao and Zeng, Jinshan}, + Journal = {arXiv preprint arXiv:1511.06324}, + Title = {Global convergence of ADMM in nonconvex nonsmooth optimization}, + Year = {2015}} + +@book{bertsekas2014constrained, + title={Constrained optimization and Lagrange multiplier methods}, + author={Bertsekas, Dimitri P}, + year={2014}, + publisher={Academic press} +} +@article{lan2016iteration, + Author = {Lan, Guanghui and Monteiro, Renato DC}, + Journal = {Mathematical Programming}, + Number = {1-2}, + Pages = {511--547}, + Publisher = {Springer}, + Title = {Iteration-complexity of first-order augmented Lagrangian methods for convex programming}, + Volume = {155}, + Year = {2016}} + +@article{nedelcu2014computational, + title={Computational complexity of inexact gradient augmented Lagrangian methods: application to constrained MPC}, + author={Nedelcu, Valentin and Necoara, Ion and Tran-Dinh, Quoc}, + journal={SIAM Journal on Control and Optimization}, + volume={52}, + number={5}, + pages={3109--3134}, + year={2014}, + publisher={SIAM} +} + +@book{bertsekas1999nonlinear, + title={Nonlinear programming}, + author={Bertsekas, Dimitri P} +} + +@article{hestenes1969multiplier, + title={Multiplier and gradient methods}, + author={Hestenes, Magnus R}, + journal={Journal of optimization theory and applications}, + volume={4}, + number={5}, + pages={303--320}, + year={1969}, + publisher={Springer} +} + +@incollection{powell1978fast, + title={A fast algorithm for nonlinearly constrained optimization calculations}, + author={Powell, Michael JD}, + booktitle={Numerical analysis}, + pages={144--157}, + year={1978}, + publisher={Springer} +} + +@article{burer2005local, + title={Local minima and convergence in low-rank semidefinite programming}, + author={Burer, Samuel and Monteiro, Renato DC}, + journal={Mathematical Programming}, + volume={103}, + number={3}, + pages={427--444}, + year={2005}, + publisher={Springer} +} + +@article{burer2003nonlinear, + title={A nonlinear programming algorithm for solving semidefinite programs via low-rank factorization}, + author={Burer, Samuel and Monteiro, Renato DC}, + journal={Mathematical Programming}, + volume={95}, + number={2}, + pages={329--357}, + year={2003}, + publisher={Springer} +} + +@article{singer2011angular, + title={Angular synchronization by eigenvectors and semidefinite programming}, + author={Singer, Amit}, + journal={Applied and computational harmonic analysis}, + volume={30}, + number={1}, + pages={20}, + year={2011}, + publisher={NIH Public Access} +} +@article{singer2011three, + title={Three-dimensional structure determination from common lines in cryo-EM by eigenvectors and semidefinite programming}, + author={Singer, Amit and Shkolnisky, Yoel}, + journal={SIAM journal on imaging sciences}, + volume={4}, + number={2}, + pages={543--572}, + year={2011}, + publisher={SIAM} +} + +@inproceedings{song2007dependence, + title={A dependence maximization view of clustering}, + author={Song, Le and Smola, Alex and Gretton, Arthur and Borgwardt, Karsten M}, + booktitle={Proceedings of the 24th international conference on Machine learning}, + pages={815--822}, + year={2007}, + organization={ACM} +} + +@inproceedings{mossel2015consistency, + title={Consistency thresholds for the planted bisection model}, + author={Mossel, Elchanan and Neeman, Joe and Sly, Allan}, + booktitle={Proceedings of the forty-seventh annual ACM symposium on Theory of computing}, + pages={69--75}, + year={2015}, + organization={ACM} +} + +@incollection{lovasz2003semidefinite, + title={Semidefinite programs and combinatorial optimization}, + author={Lov{\'a}sz, L{\'a}szl{\'o}}, + booktitle={Recent advances in algorithms and combinatorics}, + pages={137--194}, + year={2003}, + publisher={Springer} +} + +@article{khot2011grothendieck, + title={Grothendieck-type inequalities in combinatorial optimization}, + author={Khot, Subhash and Naor, Assaf}, + journal={arXiv preprint arXiv:1108.2464}, + year={2011} +} + +@article{park2016provable, + title={Provable Burer-Monteiro factorization for a class of norm-constrained matrix problems}, + author={Park, Dohyung and Kyrillidis, Anastasios and Bhojanapalli, Srinadh and Caramanis, Constantine and Sanghavi, Sujay}, + journal={arXiv preprint arXiv:1606.01316}, + year={2016} +} + +@article{mixon2016clustering, + title={Clustering subgaussian mixtures by semidefinite programming}, + author={Mixon, Dustin G and Villar, Soledad and Ward, Rachel}, + journal={arXiv preprint arXiv:1602.06612}, + year={2016} +} + +@inproceedings{kulis2007fast, + title={Fast low-rank semidefinite programming for embedding and clustering}, + author={Kulis, Brian and Surendran, Arun C and Platt, John C}, + booktitle={Artificial Intelligence and Statistics}, + pages={235--242}, + year={2007} +} + +@article{Peng2007, + Author = {Peng, J. and Wei, Y.}, + Date-Added = {2017-10-26 15:22:37 +0000}, + Date-Modified = {2017-10-26 15:30:53 +0000}, + Journal = {SIAM J. Optim.}, + Number = {1}, + Pages = {186--205}, + Title = {Approximating {K}--means--type clustering via semidefinite programming}, + Volume = {18}, + Year = {2007}} + +@book{fletcher2013practical, + title={Practical methods of optimization}, + author={Fletcher, Roger}, + year={2013}, + publisher={John Wiley \& Sons} +} + +@article{Bora2017, + title = {Compressed {Sensing} using {Generative} {Models}}, + url = {http://arxiv.org/abs/1703.03208}, + urldate = {2018-10-25}, + journal = {arXiv:1703.03208 [cs, math, stat]}, + author = {Bora, Ashish and Jalal, Ajil and Price, Eric and Dimakis, Alexandros G.}, + month = mar, + year = {2017}, + note = {arXiv: 1703.03208}, + keywords = {Statistics - Machine Learning, Computer Science - Machine + Learning, Computer Science - Information Theory}, +} + + +@book{nocedal2006numerical, + title={Numerical Optimization}, + author={Nocedal, J. and Wright, S.}, + isbn={9780387400655}, + lccn={2006923897}, + series={Springer Series in Operations Research and Financial Engineering}, + url={https://books.google.ch/books?id=VbHYoSyelFcC}, + year={2006}, + publisher={Springer New York} +} + + +@article{parikh2014proximal, + title={Proximal algorithms}, + author={Parikh, Neal and Boyd, Stephen and others}, + journal={Foundations and Trends{\textregistered} in Optimization}, + volume={1}, + number={3}, + pages={127--239}, + year={2014}, + publisher={Now Publishers, Inc.} +} \ No newline at end of file diff --git a/ICML19/Arxiv version/iALM_main.tex b/ICML19/Arxiv version/iALM_main.tex new file mode 100644 index 0000000..3832091 --- /dev/null +++ b/ICML19/Arxiv version/iALM_main.tex @@ -0,0 +1,69 @@ +\documentclass[a4paper]{article} + +%% Language and font encodings +\usepackage[english]{babel} +\usepackage[utf8x]{inputenc} +\usepackage[T1]{fontenc} +\usepackage{amsfonts} + +%% Sets page size and margins +\usepackage[a4paper,top=3cm,bottom=2cm,left=3cm,right=3cm,marginparwidth=1.75cm]{geometry} + +%%% Useful packages +%\usepackage{amsmath} +%\usepackage{graphicx} +%\usepackage[colorinlistoftodos]{todonotes} +%\usepackage[colorlinks=true, allcolors=blue]{hyperref} +%\usepackage{algorithm, algorithmic} + +\usepackage{amsthm} +\usepackage{amsmath} +\usepackage{amssymb} +\usepackage{graphicx} +\usepackage{algorithm} +\usepackage{algorithmic} +\usepackage{color} +%\usepackage{cite} + + +\usepackage{enumitem} +\input{preamble} + + +\title{An Inexact Augmented Lagrangian Framework \\ for Non-Convex Optimization with Nonlinear Constraints} +\author{Authors} +\date{} +\begin{document} +\maketitle + +\input{sections/abstract.tex} +\input{sections/introduction.tex} +\input{sections/preliminaries.tex} +\input{sections/AL.tex} +\input{sections/guarantees.tex} +%\input{sections/guarantees2.tex} +\input{sections/related_works.tex} +\input{sections/experiments.tex} + + + +%\section{Conclusion and Future Work} +%We studied a non-convex inexact augmented Lagrangian framework for solving nonlinearly constrained problems and showed that when coupled with a first order method iALM finds a first-order stationary point with $\tilde{\mathcal{O}}(1/\epsilon^3)$ calls to the first-order oracle. +%Similarly, when a second-order solver is used for the inner iterates, we prove that iALM finds a second-order stationary point with $\tilde{\mathcal{O}}(1/\epsilon^5)$ calls to the second-order oracle. +%%Numerical experiments for the basis pursuit problem may lead to an interesting direction. +%Stochastic version of Algorithm $(1)$ is a natural direction. + + + + +%\bibliographystyle{alpha} +\bibliographystyle{plain} +\bibliography{bibliography.bib} + +\newpage +\appendix +\input{sections/AL_theory.tex} +\input{sections/appendix.tex} + + +\end{document} \ No newline at end of file diff --git a/ICML19/Arxiv version/preamble.tex b/ICML19/Arxiv version/preamble.tex new file mode 100644 index 0000000..16a1642 --- /dev/null +++ b/ICML19/Arxiv version/preamble.tex @@ -0,0 +1,33 @@ + + + +% AE's commands +\newcommand{\RR}{\mathbb{R}} +\newcommand{\edita}[1]{{\color{blue} #1}} +\newcommand{\notea}[1]{{\color{magenta} \textbf{Note:AE: #1}}} +\newcommand{\ol}{\overline} +\newcommand{\Null}{\operatorname{null}} +\newcommand{\relint}{\operatorname{relint}} +\newcommand{\row}{\operatorname{row}} +\newcommand{\s}{\sigma} +\newcommand{\editaa}[1]{{\color{red} #1}} +\renewcommand{\b}{\beta} +\renewcommand{\L}{\mathcal{L}} % Lagrangian +\renewcommand{\i}{\iota} +\newcommand{\g}{\gamma} +\newcommand{\cone}{\operatorname{cone}} +\newcommand{\argmin}{\operatorname{argmin}} +\newcommand{\dist}{\operatorname{dist}} + +% MFS's commands +\newcommand{\editf}[1]{{\color[rgb]{1,0.4,0.4} #1}} +%\DeclareMathOperator*{\argmax}{argmax} % thin space, limits underneath in displays + + +% Theorem style +\newtheorem{theorem}{Theorem}[section] % reset theorem numbering for each section +\newtheorem{question}[theorem]{Question} +\newtheorem{lemma}[theorem]{Lemma} +\newtheorem{corollary}[theorem]{Corollary} +\newtheorem{definition}{Definition} +\newtheorem{proposition}{Proposition} diff --git a/ICML19/Arxiv version/sections/AL.tex b/ICML19/Arxiv version/sections/AL.tex new file mode 100644 index 0000000..423ca8f --- /dev/null +++ b/ICML19/Arxiv version/sections/AL.tex @@ -0,0 +1,56 @@ +\section{Our optimization framework \label{sec:AL algorithm}} + +To solve the formulation presented in \eqref{eq:minmax}, we propose the inexact ALM (iALM), detailed in Algorithm~\ref{Algo:2}. + +At the iteration $k$, Algorithm~\ref{Algo:2} calls in Step 2 a solver that finds an approximate stationary point of the augmented Lagrangian $\L_{\b_k}(\cdot,y_k)$ with the accuracy of $\epsilon_{k+1}$, and this accuracy gradually increases in a controlled fashion. + +The increasing sequence of penalty weights $\{\b_k\}_k$ and the dual update (Steps 4 and 5) are responsible for continuously enforcing the constraints in~\eqref{prob:01}. As we will see in the convergence analysis, the particular choice of the dual step size $\s_k$ in Algorithm~\ref{Algo:2} ensures that the dual variable $y_k$ remains bounded; see~\cite{bertsekas1976penalty} for a precedent in the ALM literature where a similar choice for $\sigma_k$ is considered. + +Step 3 of Algorithm~\ref{Algo:2} removes pathological cases with divergent iterates. As an example, suppose that $g=\delta_\mathcal{X}$ in \eqref{prob:01} is the indicator function for a bounded convex set $\mathcal{X}\subset \RR^d$ and take $\rho' > \max_{x\in \mathcal{X}} \|x\|$. Then, for sufficiently large $k$, it is not difficult to verify that all the iterates of Algorithm~\ref{Algo:2} automatically satisfy $\|x_k\|\le \rho'$ without the need to execute Step 3. + +\begin{algorithm}[h!] +\begin{algorithmic} +\STATE \textbf{Input:} $\rho,\rho',\rho''>0$. A non-decreasing, positive, unbounded sequence $\{\b_k\}_{k\ge 1}$, stopping thresholds $\tau_f$ and $\tau_s$. \vspace{2pt} +\STATE \textbf{Initialization:} $x_{1}\in \RR^d$ such that $\|A(x_1)\|\le \rho$ and $\|x_1\|\le \rho'$, $y_0\in \RR^m$, $\s_1$. +%For $k=1,2,\ldots$, execute\\ +\vspace{2pt} +\FOR{$k=1,2,\dots$} +\STATE \begin{enumerate}[leftmargin=*] +\item \textbf{(Update tolerance)} $\epsilon_{k+1} = 1/\b_k$. +%\begin{align} +%\beta_k = \frac{\beta_{1}}{\log 2}\sqrt{k}\log^2(k+1) . +%\end{align} +\item \textbf{(Inexact primal solution)} Obtain $x_{k+1}\in \RR^d$ such that +\begin{equation*} +\dist(-\nabla_x \L_{\beta_k} (x_{k+1},y_k), \partial g(x_{k+1}) ) \le \epsilon_{k+1} +\end{equation*} + for first-order stationarity and, in addition, +\begin{equation*} +\lambda_{\text{min}}(\nabla _{xx}\mathcal{L}_{\beta_k}(x_{k+1}, y_k)) \ge -\epsilon_{k+1} +\end{equation*} +for second-order-stationarity. +\item \textbf{(Control)} If necessary, project $x_{k+1}$ to ensure that $\|x_{k+1}\|\le \rho'$.\\ +\item \textbf{(Update dual step size)} +\begin{align*} +\s_{k+1} & = \s_{1} \min\Big( +\frac{\|A(x_1)\| \log^2 2 }{\|A(x_{k+1})\| (k+1)\log^2(k+2)} ,1 +\Big). +\end{align*} +\item \textbf{(Dual ascent)} $y_{k+1} = y_{k} + \sigma_{k+1}A(x_{k+1})$. +\item \textbf{(Stopping criterion)} If +\begin{align*} +& \dist(-\nabla_x \L_{\b_k}(x_{k+1}),\partial g(x_{k+1})) \nonumber\\ +& \qquad + \s_{k+1} \|A(x_{k+1})\| \le \tau_f, +\end{align*} +for first-order stationarity and if also +$\lambda_{\text{min}}(\nabla _{xx}\mathcal{L}_{\beta_{k}}(x_{k+1}, y_k)) \geq -\tau_s$ for second-order stationarity, + then quit and return $x_{k+1}$ as an (approximate) stationary point of~\eqref{prob:01}. +\end{enumerate} + \ENDFOR +\end{algorithmic} +\caption{Inexact AL for solving~\eqref{prob:01}} +\label{Algo:2} +\end{algorithm} + + + diff --git a/ICML19/Arxiv version/sections/AL_theory.tex b/ICML19/Arxiv version/sections/AL_theory.tex new file mode 100644 index 0000000..b2c0660 --- /dev/null +++ b/ICML19/Arxiv version/sections/AL_theory.tex @@ -0,0 +1,143 @@ +\section{Proof of Theorem \ref{thm:main} \label{sec:theory}} + +For every $k\ge2$, recall from (\ref{eq:Lagrangian}) and Step~2 of Algorithm~\ref{Algo:2} that $x_{k}$ satisfies +\begin{align} +& \dist(-\nabla f(x_k) - DA(x_k)^\top y_{k-1} \nonumber\\ +& \qquad - \b_{k-1} DA(x_{k})^\top A(x_k) ,\partial g(x_k) ) \nonumber\\ +& = \dist(-\nabla_x \L_{\b_{k-1}} (x_k ,y_{k-1}) ,\partial g(x_k) ) \le \epsilon_{k}. +\end{align} +With an application of the triangle inequality, it follows that +\begin{align} +& \dist( -\b_{k-1} DA(x_k)^\top A(x_k) , \partial g(x_k) ) \nonumber\\ +& \qquad \le \| \nabla f(x_k )\| + \| DA(x_k)^\top y_{k-1}\| + +\epsilon_k, +\end{align} +which in turn implies that +\begin{align} +& \dist( -DA(x_k)^\top A(x_k) , \partial g(x_k)/ \b_{k-1} ) \nonumber\\ +& \le \frac{ \| \nabla f(x_k )\|}{\b_{k-1} } + \frac{\| DA(x_k)^\top y_{k-1}\|}{\b_{k-1} } + +\frac{\epsilon_k}{\b_{k-1} } \nonumber\\ +& \le \frac{\lambda'_f+\lambda'_A \|y_{k-1}\|+\epsilon_k}{\b_{k-1}} , +\label{eq:before_restriction} +\end{align} +where $\lambda'_f,\lambda'_A$ were defined in \eqref{eq:defn_restricted_lipsichtz}. +%For example, when $g=\delta_\mathcal{X}$ is the indicator , then \eqref{eq:before_restriction} reads as +%\begin{align} +%& \| P_{T_C(x_k)} DA(x_k)^\top A(x_k) \| \nonumber\\ +%& \qquad \le \frac{ \lambda'_f+ \lambda'_A \|y_{k-1}\| + \epsilon_k }{\b_{k-1} } . +%\label{eq:before} +%\end{align} +We next translate \eqref{eq:before_restriction} into a bound on the feasibility gap $\|A(x_k)\|$. Using the regularity condition \eqref{eq:regularity}, the left-hand side of \eqref{eq:before_restriction} can be bounded below as +\begin{align} +& \dist( -DA(x_k)^\top A(x_k) , \partial g(x_k)/ \b_{k-1} ) \nonumber\\ +& \ge \nu \|A(x_k) \|, +\qquad \text{(see (\ref{eq:regularity}))} +\label{eq:restrited_pre} +\end{align} +provided that $\rho,\rho'$ satisfy +\begin{align} +\max_{k\in K} \|A(x_k)\| \le \rho, +\qquad +\max_{k\in K} \|x_k\| \le \rho'. +\label{eq:to_be_checked_later} +\end{align} +By substituting \eqref{eq:restrited_pre} back into \eqref{eq:before_restriction}, we find that +\begin{align} +\|A(x_k)\| \le \frac{ \lambda'_f + \lambda'_A \|y_{k-1}\| + \epsilon_k}{\nu \b_{k-1} }. +\label{eq:before_dual_controlled} +\end{align} +In words, the feasibility gap is directly controlled by the dual sequence $\{y_k\}_k$. We next establish that the dual sequence is bounded. Indeed, for every $k\in K$, note that +\begin{align} +\|y_k\| & = \| y_0 + \sum_{i=1}^{k} \s_i A(x_i) \| +\quad \text{(Step 5 of Algorithm \ref{Algo:2})} +\nonumber\\ +& \le \|y_0\|+ \sum_{i=1}^k \s_i \|A(x_i)\| +\qquad \text{(triangle inequality)} \nonumber\\ +& \le \|y_0\|+ \sum_{i=1}^k \frac{ \|A(x_1)\| \log^2 2 }{ k \log^2(k+1)} +\quad \text{(Step 4)} +\nonumber\\ +& \le \|y_0\|+ c \|A(x_1) \| \log^2 2 =: y_{\max}, +\label{eq:dual growth} +\end{align} +where +\begin{align} +c \ge \sum_{i=1}^{\infty} \frac{1}{k \log^2 (k+1)}. +\end{align} +Substituting \eqref{eq:dual growth} back into \eqref{eq:before_dual_controlled}, we reach +\begin{align} +\|A(x_k)\| & \le \frac{ \lambda'_f + \lambda'_A y_{\max} + \epsilon_k}{\nu \b_{k-1} } \nonumber\\ +& \le \frac{ 2\lambda'_f +2 \lambda'_A y_{\max} }{\nu \b_{k-1} } , +\label{eq:cvg metric part 2} +\end{align} +where the second line above holds if $k_0$ is large enough, which would in turn guarantees that $\epsilon_k=1/\b_{k-1}$ is sufficiently small since $\{\b_k\}_k$ is increasing and unbounded. +Let us now revisit and simplify \eqref{eq:to_be_checked_later}. Note that $\rho'$ automatically satisfies the second inequality there, owing to Step~3 of Algorithm~\ref{Algo:2}. Also, $\rho$ satisfies the first inequality in \eqref{eq:to_be_checked_later} if +\begin{align} +\frac{\lambda'_f+\lambda'_A y_{\max}}{ \nu_A \b_1} \le \rho/2, +\end{align} +and $k_0$ is large enough. Indeed, this claim follows directly from \eqref{eq:cvg metric part 2}. +%\begin{align} +%\|A(x_k)\| & \le \frac{ \lambda'_f + \lambda'_A y_{\max} + \epsilon_k}{\nu \b_{k-1} } +%\qquad \text{(see \eqref{eq:cvg metric part 2})} +%\nonumber\\ +%& \le \frac{ 2( \lambda'_f + \lambda'_A y_{\max})}{\nu \b_1 } +%\qquad \text{(see \eqref{eq:dual growth})}, +%\end{align} +%where the last line above holds when $k_0$ is large enough, because $\b_k \ge \b_1$ and $\lim_{k\rightarrow\infty} \epsilon_k=0$. + + +It remains to control the first term in \eqref{eq:cvg metric}. To that end, after recalling Step 2 of Algorithm~\ref{Algo:2} and applying the triangle inequality, we can write that +\begin{align} +& \dist( -\nabla_x \L_{\b_{k-1}} (x_k,y_{k}), \partial g(x_{k}) ) \nonumber\\ +& \le \dist( -\nabla_x \L_{\b_{k-1}} (x_k,y_{k-1}) , \partial g(x_{k-1}) ) \nonumber\\ +& + \| \nabla_x \L_{\b_{k-1}} (x_k,y_{k})-\nabla_x \L_{\b_{k-1}} (x_k,y_{k-1}) \|. +\label{eq:cvg metric part 1 brk down} +\end{align} +The first term on the right-hand side above is bounded by $\epsilon_k$, by Step 5 of Algorithm~\ref{Algo:2}. +For the second term on the right-hand side of \eqref{eq:cvg metric part 1 brk down}, we write that +\begin{align} +& \| \nabla_x \L_{\b_{k-1}} (x_k,y_{k})-\nabla_x \L_{\b_{k-1}} (x_k,y_{k-1}) \| \nonumber\\ +& = \| DA(x_k)^\top (y_k - y_{k-1}) \| +\qquad \text{(see \eqref{eq:Lagrangian})} +\nonumber\\ +& \le \lambda'_A \|y_k- y_{k-1}\| +\qquad \text{(see \eqref{eq:defn_restricted_lipsichtz})} \nonumber\\ +& = \lambda'_A \s_k \|A (x_k) \| +\qquad \text{(see Step 5 of Algorithm \ref{Algo:2})} \nonumber\\ +& \le \frac{2\lambda'_A \s_k }{\nu \b_{k-1} }( \lambda'_f+ \lambda'_Ay_{\max}) . +\qquad \text{(see \eqref{eq:cvg metric part 2})} +\label{eq:part_1_2} +\end{align} +By combining (\ref{eq:cvg metric part 1 brk down},\ref{eq:part_1_2}), we find that +\begin{align} +& \dist( \nabla_x \L_{\b_{k-1}} (x_k,y_{k}), \partial g(x_{k}) ) \nonumber\\ +& \le \frac{2\lambda'_A \s_k }{\nu \b_{k-1} }( \lambda'_f+ \lambda'_Ay_{\max}) + \epsilon_k. +\label{eq:cvg metric part 1} +\end{align} +By combining (\ref{eq:cvg metric part 2},\ref{eq:cvg metric part 1}), we find that +\begin{align} +& \dist( -\nabla_x \L_{\b_{k-1}}(x_k,y_k),\partial g(x_k)) + \| A(x_k)\| \nonumber\\ +& \le \left( \frac{2\lambda'_A \s_k }{\nu \b_{k-1} }( \lambda'_f+ \lambda'_Ay_{\max}) + \epsilon_k \right) \nonumber\\ +& \qquad + 2\left( \frac{ \lambda'_f + \lambda'_A y_{\max}}{\nu \b_{k-1} } \right). +\end{align} +Applying $\s_k\le \s_1$, we find that +\begin{align} +& \dist( -\nabla_x \L_{\b_{k-1}}(x_k,y_k),\partial g(x_k)) + \| A(x_k)\| \nonumber\\ +& \le \frac{ 2\lambda'_A\s_1 + 2}{ \nu\b_{k-1}} ( \lambda'_f+\lambda'_A y_{\max}) + \epsilon_k. +\end{align} +For the second part of the theorem, we use the Weyl's inequality and Step 5 of Algorithm~\ref{Algo:2} to write +\begin{align}\label{eq:sec} +\lambda_{\text{min}} &(\nabla_{xx} \mathcal{L}_{\beta_{k-1}}(x_k, y_{k-1})) \geq \lambda_{\text{min}} (\nabla_{xx} \mathcal{L}_{\beta_{k-1}}(x_k, y_{k})) \notag \\&- \sigma_k \| \sum_{i=1}^m A_i(x_k) \nabla^2 A_i(x_k) \|. +\end{align} +The first term on the right-hand side is lower bounded by $-\epsilon_{k-1}$ by Step 2 of Algorithm~\ref{Algo:2}. We next bound the second term on the right-hand side above as +\begin{align*} +& \sigma_k \| \sum_{i=1}^m A_i(x_k) \nabla^2 A_i(x_k) \| \\ +&\le \sigma_k \sqrt{m} \max_{i} \| A_i(x_k)\| \| \nabla^2 A_i(x_k)\| \\ +&\le \sigma_k \sqrt{m} \lambda_A \frac{ 2\lambda'_f +2 \lambda'_A y_{\max} }{\nu \b_{k-1} }, +\end{align*} +where the last inequality is due to~(\ref{eq:smoothness basic},\ref{eq:cvg metric part 2}). +Plugging into~\eqref{eq:sec} gives +\begin{align*} +& \lambda_{\text{min}}(\nabla_{xx} \mathcal{L}_{\beta_{k-1}}(x_k, y_{k-1}))\nonumber\\ +& \geq -\epsilon_{k-1} - \sigma_k \sqrt{m} \lambda_A \frac{ 2\lambda'_f +2 \lambda'_A y_{\max} }{\nu \b_{k-1} }, +\end{align*} +which completes the proof of Theorem \ref{thm:main}. diff --git a/ICML19/Arxiv version/sections/Slater.pptx b/ICML19/Arxiv version/sections/Slater.pptx new file mode 100644 index 0000000..55bcaa7 Binary files /dev/null and b/ICML19/Arxiv version/sections/Slater.pptx differ diff --git a/ICML19/Arxiv version/sections/abstract.tex b/ICML19/Arxiv version/sections/abstract.tex new file mode 100644 index 0000000..93f67ba --- /dev/null +++ b/ICML19/Arxiv version/sections/abstract.tex @@ -0,0 +1,17 @@ +%!TEX root = ../main.tex + + +\begin{abstract} +%We consider a canonical nonlinear-constrained nonconvex problem template with broad applications in machine learning, theoretical computer science, and signal processing. +%This template involves the Burer-Monteiro splitting for semidefinite programming as a special case. +We propose a practical inexact augmented Lagrangian method (iALM) for nonconvex problems with nonlinear constrains. +%involving nonlinear operators. +We characterize the total computational complexity of our method subject to a verifiable geometric condition. + +In particular, when a first-order solver is used for the inner iterates, we prove that iALM finds a first-order stationary point with $\tilde{\mathcal{O}}(1/\epsilon^3)$ calls to the first-order oracle. Likewise, when a second-order solver is used for the inner iterates, we prove that iALM finds a second-order stationary point with $\tilde{\mathcal{O}}(1/\epsilon^5)$ calls to the second-order oracle. +These complexity results match the known theoretical results in the literature with a simple, implementable and versatile algorithm. + +We provide numerical evidence on large-scale machine learning problems, including the Burer-Monteiro factorization of standard form Semidefinite Programming (SDP) relaxations, for which we verify our geometric condition in specific cases. For these problems and under suitable assumptions, our algorithm in fact achieves global optimality for the underlying convex SDP. + +%\textbf{AE: we should add something about gans if we do that in time.} +\end{abstract} diff --git a/ICML19/Arxiv version/sections/appendix.tex b/ICML19/Arxiv version/sections/appendix.tex new file mode 100644 index 0000000..edec4ef --- /dev/null +++ b/ICML19/Arxiv version/sections/appendix.tex @@ -0,0 +1,327 @@ +%!TEX root = ../iALM_main.tex + +\section{Proof of Corollary~\ref{cor:first}} +%Denote $B_k = \dist( -\nabla_x \L_{\b_{k-1}}(x_k,y_k),\partial g(x_k)) + \| A(x_k)\|$ for every $k$. Using the convergence proof of the outer algorithm, we have the following bound + +Let $K$ denote the number of (outer) iterations of Algorithm~\ref{Algo:2} and let $\epsilon_{f}$ denote the desired accuracy of Algorithm~\ref{Algo:2}, see~(\ref{eq:inclu3}). Recalling Theorem~\ref{thm:main}, we can then write that +\begin{equation} + \epsilon_{f} = \frac{Q}{\b_{K}}, + \label{eq:acc_to_b} +\end{equation} +or, equivalently, $\b_{K} = Q/\epsilon_{f}$. +%where $K$ denotes the last outer iterate and $\epsilon$ is the final accuracy we would like to get for the optimality conditions given in~\eqref{eq:inclu3}. +We now count the number of total (inner) iterations $T$ of Algorithm~\ref{Algo:2} to reach the accuracy $\epsilon_{f}$. From \eqref{eq:smoothness of Lagrangian} and for sufficiently large $k$, recall that $\lambda_{\b_k}\le \lambda'' \b_k$ is the smoothness parameter of the augmented Lagrangian. Then, from \eqref{eq:iter_1storder} ad by summing over the outer iterations, we bound the total number of (inner) iterations of Algorithm~\ref{Algo:2} as +\begin{align}\label{eq: tk_bound} +T &= \sum_{k=1}^K\mathcal{O}\left ( \frac{\lambda_{\beta_{k-1}}^2 x_{\max}^2 }{\epsilon_k} \right) \nonumber\\ +& = \sum_{k=1}^K\mathcal{O}\left (\beta_{k-1}^3 x_{\max}^2 \right) +\qquad \text{(Step 1 of Algorithm \ref{Algo:2})} +\nonumber\\ +& \leq \mathcal{O} \left(K\beta_{K-1}^3 x_{\max}^2 \right) +\qquad \left( \{\b_k\}_k \text{ is increasing} \right) + \nonumber\\ + & \le \mathcal{O}\left( \frac{K Q^{{3}} x_{\max}^2}{\epsilon_{f}^{{3}}} \right). + \qquad \text{(see \eqref{eq:acc_to_b})} +\end{align} +In addition, if we specify $\beta_k=b^k$ for all $k$, we can further refine $T$. Indeed, +\begin{equation} +\beta_K = b^K~~ \Longrightarrow~~ K = \log_b \left( \frac{Q}{\epsilon_f} \right), +\end{equation} +which, after substituting into~\eqref{eq: tk_bound} gives the final bound in Corollary~\ref{cor:first}. +%\begin{equation} +%T \leq \mathcal{O}\left( \frac{Q^{\frac{3}{2}+\frac{1}{2b}} x_{\max}^2}{\epsilon_f^{\frac{3}{2}+\frac{1}{2b}}} \right), +%\end{equation} +%which completes the proof of Corollary~\ref{cor:first}. + +%\section{Analysis of different rates for $\beta_k$ and $\epsilon_k$} +% +%We check the iteration complexity analysis by decoupling $\beta_k$ and $\epsilon_k$. +%\begin{equation} +%\beta_k = k^b, ~~~~ \epsilon_k = k^{-e}. +%\end{equation} +% +%By denoting $B_k = \dist( -\nabla_x \L_{\b_{k-1}}(x_k,y_k),\partial g(x_k)) + \| A(x_k)\|$, the algorithm bound becomes, +% +%\begin{equation} +%B_k \leq \frac{1}{\beta_k} + \epsilon_k = k^{-b} + k^{-e}. +%\end{equation} +% +%Total iteration number is +% +%\begin{equation} +%T_K = \sum_{k=1}^K \frac{\beta_k^2}{\epsilon_k} \leq K^{2b+e+1}. +%\end{equation} +% +%We now consider two different relations between $b$ and $e$ to see what is going on. +% +%\textbf{Case 1:} $b\geq e$: +% +%Bound for the algorithm: +% +%\begin{equation} +%B_k \leq \frac{1}{\beta_k} + \epsilon_k = k^{-b} + k^{-e} \leq \frac{2}{k^e} = \epsilon, +%\end{equation} +% +%which gives the relation $K = \left( \frac{2}{\epsilon}\right)^{1/e}$. +%Writing down the total number of iterations and plugging in $K$, +% +%\begin{equation} +%T_K = \sum_{k=1}^K \frac{\beta_k^2}{\epsilon_k} \leq K^{2b+e+1} \leq \left(\frac{2}{\epsilon}\right)^{\frac{2b}{e}+1+\frac{1}{e}}. +%\end{equation} +% +%To get the least number of total iterations for a given accuracy $\epsilon$, one needs to pick $e$ as large as possible and $b$ as small as possible. +%Since in this case we had $b\geq e$, this suggests picking $b=e$ for the optimal iteration complexity. +% +%\textbf{Case 2:} $b\leq e$: +% +%Same calculations yield the following bound on the total number of iterations: +% +%\begin{equation} +%T_K = \sum_{k=1}^K \frac{\beta_k^2}{\epsilon_k} \leq K^{2b+e+1} \leq \left(\frac{2}{\epsilon}\right)^{2+\frac{e}{b}+\frac{1}{b}}. +%\end{equation} +% +%Given that $b\leq e$, the bound suggests picking $e$ as small as possible and $b$ as big as possible. +% +%To minimize the total number of iterations in both cases with flexible $b$ and $e$, the bounds suggest to pick $b=e=\alpha$ and take this value to be as large as possible. + + + +\section{Proof of Lemma \ref{lem:smoothness}\label{sec:proof of smoothness lemma}} +Note that +\begin{align} +\mathcal{L}_{\beta}(x,y) = f(x) + \sum_{i=1}^m y_i A_i (x) + \frac{\b}{2} \sum_{i=1}^m (A_i(x))^2, +\end{align} +which implies that +\begin{align} +& \nabla_x \mathcal{L}_\beta(x,y) \nonumber\\ +& = \nabla f(x) + \sum_{i=1}^m y_i \nabla A_i(x) + \frac{\b}{2} \sum_{i=1}^m A_i(x) \nabla A_i(x) \nonumber\\ +& = \nabla f(x) + DA(x)^\top y + \b DA(x)^\top A(x), +\end{align} +where $DA(x)$ is the Jacobian of $A$ at $x$. By taking another derivative with respect to $x$, we reach +\begin{align} +\nabla^2_x \mathcal{L}_\beta(x,y) & = \nabla^2 f(x) + \sum_{i=1}^m \left( y_i + \b A_i(x) \right) \nabla^2 A_i(x) \nonumber\\ +& \qquad +\b \sum_{i=1}^m \nabla A_i(x) \nabla A_i(x)^\top. +\end{align} +It follows that +\begin{align} +& \|\nabla_x^2 \mathcal{L}_\beta(x,y)\|\nonumber\\ + & \le \| \nabla^2 f(x) \| + \max_i \| \nabla^2 A_i(x)\| \left (\|y\|_1+\b \|A(x)\|_1 \right) \nonumber\\ +& \qquad +\beta\sum_{i=1}^m \|\nabla A_i(x)\|^2 \nonumber\\ +& \le \lambda_h+ \sqrt{m} \lambda_A \left (\|y\|+\b \|A(x)\| \right) + \b \|DA(x)\|^2_F. +\end{align} +For every $x$ such that $\|A(x)\|\le \rho$ and $\|x\|\le \rho'$, we conclude that +\begin{align} +\|\nabla_x^2 \mathcal{L}_\beta(x,y)\| +& \le \lambda_f + \sqrt{m}\lambda_A \left(\|y\| + \b\rho \right) \nonumber\\ +& \qquad + \b \max_{\|x\|\le \rho'}\|DA(x)\|_F^2, +\end{align} +which completes the proof of Lemma \ref{lem:smoothness}. + +%\section{Proof of Lemma \ref{lem:11}\label{sec:proof of descent lemma}} +% +%Throughout, let +%\begin{align} +%G = G_{\b,\g}(x,y) = \frac{x-x^+}{\g}, +%\end{align} +%for short. +%Suppose that $\|A(x)\|\le \rho$, $\|x\|\le \rho$, and similarly $\|A(x^+)\|\le \rho$, $\|x^+\|\le \rho'$. An application of Lemma \ref{lem:smoothness} yields that +%\begin{align} +%\L_\b(x^+,y)+g(x^+) & \le \L_\b(x,y)+ \langle x^+-x,\nabla_x \L_\b(x,y) \rangle +%+ \frac{\lambda_\b}{2} \|x^+ - x\|^2 + g(x^+) \nonumber\\ +%& = \L_\b(x,y)-\g \langle G ,\nabla_x \L_\b(x,y) \rangle +%+ \frac{\g^2 \lambda_\b }{2} \|G\|^2 + g(x^+) +%\label{eq:descent pr 1} +%\end{align} +%Since $x^+ = P_g(x - \g \nabla_x \L_\b(x,y))$, we also have that +%\begin{align} +%\g (G - \nabla_x \L_\b(x,y)) = \xi \in \partial g(x^+). +%\label{eq:opt of prox} +%\end{align} +%By combining (\ref{eq:descent pr 1},\ref{eq:opt of prox}), we find that +%\begin{align} +%\L_\b(x^+,y)+g(x^+) & +%\le \L_\b(x,y) -\g \|G\|^2 + \g \langle G, \xi \rangle + \frac{\g^2 \lambda_\b}{2}\|G\|^2 + g(x^+) \nonumber\\ +%& = \L_\b(x,y) -\g \|G\|^2 + \langle x- x^+ , \xi \rangle + \frac{\g^2 \lambda_\b}{2}\|G\|^2 + g(x^+) \nonumber\\ +%& \le \L_\b(x,y) + g(x) - \g\left( 1-\frac{\g\lambda_\b}{2}\right) \|G\|^2, +%\end{align} +%where the last line above uses the convexity of $g$. Recalling that $\g\le 1/\lambda_\b$ completes the proof of Lemma \ref{lem:11}. +% +% +%\section{Proof of Lemma \ref{lem:eval Lipsc cte}\label{sec:proof of linesearch}} +% +% +%Recalling $x^+_{\gamma}$ in \eqref{eq:defn of gamma line search}, we note that +%\begin{equation} +%x^+_{\gamma} - x +\gamma \nabla_x \mathcal{L}_\beta(x,y) = -\xi \in -\partial g (x^+_{\gamma}). +%\label{eq:optimality of uplus} +%\end{equation} +%Lastly, $\gamma$ by definition in \eqref{eq:defn of gamma line search} satisfies +%\begin{align} +%& \mathcal{L}_{\beta}(x^+_{\gamma},y) + g(x_\g^+) \nonumber\\ +% & \le \mathcal{L}_\beta(x,y) + \g \left\langle +%x^+_{\gamma} - x , \nabla_x \mathcal{L}_\beta (x,y) +%\right\rangle + \frac{1}{2\gamma}\|x^+_{\gamma} - x\|^2 +%+ g(x_\g^+) +%\nonumber\\ +%& = \mathcal{L}_\beta(x,y) + \left\langle +%x - x^+_{\gamma} ,\xi +%\right\rangle +%- \frac{1}{2\gamma}\|x^+_{\gamma} - x\|^2 + g(x_\g^+)\nonumber\\ +%& \le \mathcal{L}_\beta(x,y) +%- \frac{1}{2\gamma}\|x^+_{\gamma} - x\|^2 + g(x) - g(x^+_\g) +%\qquad (\text{convexity of } g) \nonumber\\ +%& = \mathcal{L}_\beta(x,y) - \frac{\gamma}{2} \|G_{\beta,\gamma}(x,y)\|^2 +g(x) - g(x^+_\g), +%\qquad \text{(see Definition \ref{def:grad map})} +%\end{align} +%which completes the proof of Lemma \ref{lem:eval Lipsc cte}. +% +% +% + + +\section{Basis Pursuit \label{sec:verification1}} + +We only verify the regularity condition in \eqref{eq:regularity} for \eqref{prob:01} with $f,A$ specified in \eqref{eq:bp-equiv}. Note that +\begin{align*} +DA(x) = 2 \overline{B} \text{diag}(x), +\end{align*} +where $\text{diag}(x)\in\RR^{2d\times 2d}$ is the diagonal matrix formed by $x$. The left-hand side of \eqref{eq:regularity} then reads as +\begin{align} +& \text{dist} \left( -DA(x_k)^\top A(x_k) , \frac{\partial g(x_k)}{\b_{k-1}} \right) \nonumber\\ +& = \text{dist} \left( -DA(x_k)^\top A(x_k) , \{0\} \right) \nonumber\\ +& = \|DA(x_k)^\top A(x_k) \| \nonumber\\ +& =2 \| \text{diag}(x_k) \overline{B}^\top ( \overline{B}x_k^{\circ 2} -b) \|. +\label{eq:cnd-bp-pre} +\end{align} +To bound the last line above, let $x_*$ be a solution of Program~\eqref{prob:01} and note that $\overline{B} x_*^{\circ 2} = b $ by definition. Let also $z_k,z_*\in \RR^d$ denote the vectors corresponding to $x_k,x_*$. Corresponding to $x_k$, also define $u_{k,1},u_{k,2}$ naturally and let $|z_k| = u_{k,1}^{\circ 2} + u_{k,2}^{\circ 2} \in \RR^d$ be the amplitudes of $z_k$. To simplify matters, let us assume also that $B$ is full-rank. +We then rewrite the last line of \eqref{eq:cnd-bp-pre} as +\begin{align} +& \| \text{diag}(x_k) \overline{B}^\top ( \overline{B}x_k^{\circ 2} -b) \|^2 \nonumber\\ +& = \| \text{diag}(x_k) \overline{B}^\top \overline{B} (x_k^{\circ 2} -x_*^{\circ 2}) \|^2 \nonumber\\ +& = \| \text{diag}(x_k)\overline{B}^\top B (x_k - x_*) \|^2 \nonumber\\ +& = \| \text{diag}(u_{k,1})B^\top B (x_k - x_*) \|^2 \nonumber\\ +& \qquad + \| \text{diag}(u_{k,2})B^\top B (x_k - x_*) \|^2 \nonumber\\ +& = \| \text{diag}(u_{k,1}^{\circ 2}+ u_{k,2}^{\circ 2}) B^\top B (x_k - x_*) \|^2 \nonumber\\ +& = \| \text{diag}(|z_k|) B^\top B (x_k - x_*) \|^2 \nonumber\\ +& \ge +\eta_n ( B \text{diag}(|z_k|) )^2 +\| B(x_k - x_*) \|^2 \nonumber\\ +& = +\eta_n ( B \text{diag}(|z_k|) )^2 +\| B x_k -b \|^2, +\end{align} +where $\eta_n(\cdot)$ returns the $n$th largest singular value of its argument. We can therefore ensure that \eqref{eq:regularity} holds by enforcing that +\begin{align} +z_k \in \left\{ z \in \RR^d: \eta_n( B \text{diag}(|z|)) > \nu \right\}, +\end{align} +for every iteration $k$. + + +\section{Clustering \label{sec:verification2}} + + +We only verify the condition in~\eqref{eq:regularity}. +Note that +\begin{align} +A(x) = VV^\top \mathbf{1}- \mathbf{1}, +\end{align} +\begin{align} +DA(x) & = +\left[ +\begin{array}{cccc} +w_{1,1} x_1^\top & \cdots & w_{1,n} x_{1}^\top\\ +%x_2^\top p& \cdots & x_{2}^\top\\ +\vdots\\ +w_{n,1}x_{n}^\top & \cdots & w_{n,n}1 x_{n}^\top +\end{array} +\right] \nonumber\\ +& = \left[ +\begin{array}{ccc} +V & \cdots & V +\end{array} +\right] ++ +\left[ +\begin{array}{ccc} +x_1^\top & \\ +& \ddots & \\ +& & x_n^\top +\end{array} +\right], +\label{eq:Jacobian clustering} +\end{align} +%where $I_n\in\RR^{n\times n}$ is the identity matrix, +where $w_{i.i}=2$ and $w_{i,j}=1$ for $i\ne j$. In the last line above, $n$ copies of $V$ appear and the last matrix above is block-diagonal. For $x_k$, define $V_k$ as in the example and let $x_{k,i}$ be the $i$th row of $V_k$. +Consequently, +\begin{align} +DA(x_k)^\top A(x_k) & = +\left[ +\begin{array}{c} +V_k^\top (V_k V_k^\top- I_n ) \mathbf{1}\\ +\vdots\\ +V_k^\top (V_k V_k^\top- I_n ) \mathbf{1} +\end{array} +\right] \nonumber\\ +& \qquad + +\left[ +\begin{array}{c} +x_{k,1} (V_k V_k^\top \mathbf{1}- \mathbf{1})_1 \\ +\vdots \\ +x_{k,n} (V_k V_k^\top \mathbf{1}- \mathbf{1})_n +\end{array} +\right], +\end{align} +where $I_n\in \RR^{n\times n}$ is the identity matrix. +Let us make a number of simplifying assumptions. First, we assume that $x_k\in \text{relint}(C)$ so that $\partial g(x_k)=\{0\}$, which can be easily enforced in the iterates. Second, we assume that $V_k$ has nearly orthonormal columns, namely, $V_k V_k^\top \approx I_n$. This can also be easily enforced in each iterate of Algorithm~\ref{Algo:2} and naturally corresponds to well-separated clusters. While a more fine-tuned argument can remove these assumptions, they will help us simplify the derivations. Under these assumptions, the squared right-hand side of \eqref{eq:regularity} becomes +\begin{align} +& \dist\left( -DA(x_k)^\top A(x_k) , \frac{\partial g(x_k)}{ \b_{k-1}} \right)^2 \nonumber\\ +& = \dist\left( -DA(x_k)^\top A(x_k) , \{0\} \right)^2 \nonumber\\ +& = \| DA(x_k)^\top A(x_k) \|^2 \nonumber\\ +& = +\left\| +\begin{array}{c} +x_{k,1} (V_k V_k^\top \mathbf{1}- \mathbf{1})_1 \\ +\vdots \\ +x_{k,n} (V_k V_k^\top \mathbf{1}- \mathbf{1})_n +\end{array} +\right\|^2 \nonumber\\ +& = \sum_{i=1}^n \| x_{k,i}\|^2 (V_kV_k^\top \mathbf{1}-\mathbf{1})_i^2 \nonumber\\ +& \ge \min_i \| x_{k,i}\|^2 +\cdot \sum_{i=1}^n (V_kV_k^\top \mathbf{1}-\mathbf{1})_i^2 \nonumber\\ +& = \min_i \| x_{k,i}\|^2 +\cdot \| V_kV_k^\top \mathbf{1}-\mathbf{1} \|^2. +\end{align} +We can enforce the iterates to satisfy $\|x_{k,i}\| \ge \nu$, which corresponds again to well-separated clusters, and guarantee \eqref{eq:regularity}. In practice, often $n$ exceeds the number of true clusters and a more fine-tuned analysis is required to establish \eqref{eq:regularity} by restricting the argument to a particular subspace of $\RR^n$. + + +\section{$\ell_\infty$ denoising with a generative prior} +The authors of \cite{Ilyas2017} have proposed to +project onto the range of a Generative Adversarial network (GAN) +\cite{Goodfellow2014}, as a way to defend against adversarial examples. For a +given noisy observation $x^* + \eta$, they consider a projection in the +$\ell_2$ norm. We instead propose to use our augmented Lagrangian method to +denoise in the $\ell_\infty$ norm, a much harder task: +\begin{align} +\begin{array}{lll} +\underset{x, z}{\text{min}} & & \|x^* + \eta - x\|_\infty \\ +\text{s.t. } && x=G(z). +\end{array} +\end{align} +We use a pretrained generator for the MNIST dataset, given by a standard +deconvolutional neural network architecture. We compare the succesful optimizer +Adam against our method. Our algorithm involves two forward/backward passes +through the network, as oposed to Adam that requires only one. For this reason +we let our algorithm run for 4000 iterations, and Adam for 8000 iterations. +For a particular example, we plot the objective value vs iteration count in figure +\ref{fig:comparison_fab}. Our method successfully minimizes the objective value, +while Adam does not succeed. + +\begin{figure*}[ht] +% \includegraphics[width=0.4\textwidth,natwidth=1300,natheight=1300]{bp_fig1.pdf} +\begin{center} +{\includegraphics[width=.7\columnwidth]{figs/example_denoising_fab.pdf}} +%\centerline{\includegraphics[width=1\columnwidth]{figs/bp_fig2_small.pdf}} +\label{fig:comparison_fab} +\caption{Augmented Lagrangian vs Adam for $\ell_\infty$ denoising (left). $\ell_2$ vs $\ell_\infty$ denoising as defense against adversarial examples} +\end{center} +\end{figure*} + diff --git a/ICML19/Arxiv version/sections/experiments.tex b/ICML19/Arxiv version/sections/experiments.tex new file mode 100644 index 0000000..333ddb2 --- /dev/null +++ b/ICML19/Arxiv version/sections/experiments.tex @@ -0,0 +1,231 @@ +%!TEX root = ../iALM_main.tex + +\section{Numerical evidence \label{sec:experiments}} + +We first begin with a caveat: It is known that quasi-Newton methods, such as BFGS and lBFGS, might not converge for non-convex problems~\cite{dai2002convergence, mascarenhas2004bfgs}. For this reason, we have used the trust region method as the second-order solver in our analysis in Section~\ref{sec:cvg rate}, which is well-studied for non-convex problems~\cite{cartis2012complexity}. + +Empirically, however, BFGS and lBGFS are extremely successful and we have also opted for those solvers in this section since the subroutine does not affect Theorem~\ref{thm:main} as long as the subsolver can perform in practice. + +\subsection{k-Means Clustering} + +Given data points $\{z_i\}_{i=1}^n $, the entries of the corresponding Euclidean distance matrix $D \in \RR^{nxn}$ are $ D_{i, j} = \left\| z_i - z_j\right\|^2 $. +Clustering is then the problem of finding a co-association matrix $Y\in \RR^{n\times n}$ such that $Y_{ij} = 1$ if points $z_i$ and $z_j$ are within the same cluster and $Y_{ij} = 0$ otherwise. In~\cite{Peng2007}, the authors provide a SDP relaxation of the clustering problem, specified as +\begin{align} +\begin{cases} +\underset{Y \in \RR^{nxn}}{\min} \text{tr}(DY) \\ +Y\mathbf{1} = \mathbf{1}, +~\text{tr}(Y) = k,~ + Y\succeq 0,~Y \geq 0, +\end{cases} +\label{eq:sdp_svx} +\end{align} +where $k$ is the number of clusters and $Y $ is both positive semidefinite and has nonnegative entries. +Standard SDP solvers do not scale well with the number of data points~$n$, since they often require projection onto the semidefinite cone with the complexity of $\mathcal{O}(n^3)$. We instead use the Burer-Monteiro splitting, sacrificing convexity to reduce the computational complexity. More specifically, we solve the program +\begin{align} +\label{eq:nc_cluster} +\begin{cases} +\underset{V \in \RR^{nxr}}{\min} \text{tr}(DVV^{\top}) \\ +VV^{\top}\mathbf{1} = \mathbf{1},~~ \|V\|_F^2 \le k, +~~V \geq 0, +\end{cases} +\end{align} +where $\mathbf{1}\in \RR^n$ is the vector of all ones. +Note that $Y \geq 0$ in \eqref{eq:sdp_svx} is replaced above by the much stronger but easier to enforce constraint $V \geq 0$ constraint above, see~\cite{kulis2007fast} for the reasoning behind this relaxation. +%. Trace constraint translates to a Frobenius norm constraint in factorized space. Semidefinite constraint is naturally removed due to factorization $Y=VV^{\top}$. +%See~\citep{kulis2007fast} for the details of the relaxation. +Now, we can cast~\eqref{eq:nc_cluster} as an instance of~\eqref{prob:01}. Indeed, for every $i\le n$, let $x_i \in \RR^r$ denote the $i$th row of $V$. We next form $x \in \RR^d$ with $d = nr$ by expanding the factorized variable $V$, namely, +\begin{align*} +x = [x_1^{\top}, \cdots, x_n^{\top}]^{\top} \in \RR^d, +\end{align*} +and then set +\begin{align*} +f(x) =\sum_{i,j=1}^n D_{i, j} \left\langle x_i, x_j \right\rangle, +\qquad g = \delta_C, +\end{align*} +\begin{align} +A(x) = [x_1^{\top}\sum_{j=1}^n x_j -1, \cdots, x_n^{\top}\sum_{j=1}^n x_j-1]^{\top}, +\end{align} +where $C$ is the intersection of the positive orthant in $\RR^d$ with the Euclidean ball of radius $\sqrt{k}$. In Appendix~\ref{sec:verification2}, we somewhat informally verify that Theorem~\ref{thm:main} applies to~\eqref{prob:01} with $f,g,A$ specified above. + + +In our simulations, we use two different solvers for Step~2 of Algorithm~\ref{Algo:2}, namely, APGM and lBFGS. APGM is a solver for non-convex problems of the form~\eqref{e:exac} with convergence guarantees to first-order stationarity, as discussed in Section~\ref{sec:cvg rate}. lBFGS is a limited-memory version of BFGS algorithm in~\cite{fletcher2013practical} that approximately leverages the second-order information of the problem. +We compare our approach against the following convex methods: +\begin{itemize} +\item HCGM: Homotopy-based Conditional Gradient Method in\cite{yurtsever2018conditional} which directly solves~\eqref{eq:sdp_svx}. +\item SDPNAL+: A second-order augmented Lagrangian method for solving SDP's with nonnegativity constraints~\cite{yang2015sdpnal}. +\end{itemize} + +As for the dataset, our experimental setup is similar to that described by~\cite{mixon2016clustering}. We use the publicly-available fashion-MNIST data in \cite{xiao2017/online}, which is released as a possible replacement for the MNIST handwritten digits. Each data point is a $28\times 28$ gray-scale image, associated with a label from ten classes, labeled from $0$ to $9$. +First, we extract the meaningful features from this dataset using a simple two-layer neural network with a sigmoid activation function. Then, we apply this neural network to 1000 test samples from the same dataset, which gives us a vector of length $10$ for each data point, where each entry represents the posterior probability for each class. Then, we form the $\ell_2$ distance matrix ${D}$ from these probability vectors. The results are depicted in Figure~\ref{fig:clustering}. +We implemented 3 algorithms on MATLAB and used the software package for SDPNAL+ which contains mex files. Convergence of the nonconvex approach will be much faster once mex implementation is used. + +\begin{figure}[] +% \includegraphics[width=0.4\textwidth,natwidth=1300,natheight=1300]{bp_fig1.pdf} +\begin{center} +{\includegraphics[width=.4\columnwidth]{figs/clustering_fig4_times_linearbeta_last.pdf}} +{\includegraphics[width=.4\columnwidth]{figs/clustering_fig4_iter_linearbeta_last.pdf}} +%\centerline{\includegraphics[width=1\columnwidth]{figs/bp_fig2_small.pdf}} +\caption{Convergence of different algorithms for k-Means clustering with fashion MNIST dataset. +%Here, we set the rank to be equal to 20 for the non-convex approaches. +The solution rank for the template~\eqref{eq:sdp_svx} is known and it is equal to number of clusters $k$ (Theorem 1. \cite{kulis2007fast}). As discussed in~\cite{tepper2018clustering}, setting rank $r>k$ leads more accurate reconstruction in expense of speed. Therefore, we set the rank to 20.} +\label{fig:clustering} +%(left:$[n=100$, $d=10^4]$, right:$[n=34$, $d=10^2$])} +\end{center} +\end{figure} + + +\subsection{Basis Pursuit} +Basis Pursuit (BP) finds sparsest solutions of an under-determined system of linear equations, namely, +\begin{align} +\begin{cases} +\min_{z} \|z\|_1 \\ +Bz = b, +\end{cases} +\label{eq:bp_main} +\end{align} +where $B \in \RR^{n \times d}$ and $b \in \RR^{n}$. BP has found many applications in machine learning, statistics and signal processing \cite{chen2001atomic, candes2008introduction, arora2018compressed}. A huge number of primal-dual convex optimization algorithms are proposed to solve BP, including, but not limited to~\cite{tran2018adaptive,chambolle2011first}. There also exists many line of works \cite{beck2009fast} to handle sparse regression problem via regularization with $\ell_1$ norm. + +%\textbf{AE: Fatih, maybe mention a few other algorithms including asgard and also say why we can't use fista and nista.} + + + +Here, we take a different approach and cast~(\ref{eq:bp_main}) as an instance of~\eqref{prob:01}. Note that any $z \in \RR^d$ can be decomposed as $z = z^+ - z^-$, where $z^+,z^-\in \RR^d$ are the positive and negative parts of $z$, respectively. Then consider the change of variables $z^+ = u_1^{\circ 2}$ and $z^-= u_2^{\circ 2} \in \RR^d$, where $\circ$ denotes element-wise power. Next, we concatenate $u_1$ and $u_2$ as $x := [ u_1^{\top}, u_2^{\top} ]^{\top} \in \RR^{2d}$ and define $\overline{B} := [B, -B] \in \RR^{n \times 2d}$. Then, \eqref{eq:bp_main} is equivalent to \eqref{prob:01} with +\begin{align} +f(x) =& \|x\|_2^2, \quad g(x) = 0\nonumber\\ +A(x) =& \overline{B}x^{\circ 2}- b. +\label{eq:bp-equiv} +\end{align} +In Appendix~\ref{sec:verification1}, we verify with minimal detail that Theorem~\ref{thm:main} indeed applies to~\eqref{prob:01} with the above $f,A$. + + + + +%Let $\mu(B)$ denote the \emph{coherence} of ${B}$, namely, +%\begin{align} +%\mu = \max_{i,j} | \langle B_i, B_j \rangle |, +%\end{align} +%where $B_i\in \RR^n$ is the $i$th column of $B$. Let also $z_k,z_*\in \RR^d$ denote the vectors corresponding to $x_k,x_*$. We rewrite the last line of \eqref{eq:cnd-bp-pre} as +%\begin{align} +%& 2 \| \text{diag}(x_k) \overline{B}^\top ( \overline{B}x^{\circ 2} -b) \| \nonumber\\ +%& = 2 \| \text{diag}(x_k) \overline{B}^\top \overline{B} (x^{\circ 2} -x_*^{\circ 2}) \| \nonumber\\ +%& = 2 \| \text{diag}(x_k) {B}^\top B(x_k- x_*) \| \nonumber\\ +%& = 2 \| \text{diag}(x_k) {B}^\top B(x_k- x_*) \|_{\infty} \nonumber\\ +%& \ge 2 \| \text{diag}(x_k) \text{diag}({B}^\top B) (x_k- x_*) \|_{\infty}\nonumber\\ +%& \qquad - 2 \| \text{diag}(x_k) \text{hollow}({B}^\top B) (x_k- x_*) \|_{\infty}, +%\end{align} +%where $\text{hollow}$ returns the hollow part of th +% +% and let $\overline{B}=U\Sigma V^\top$ be its thin singular value decomposition, where $U\in \RR^{n\times n}, V\in \RR^{d\times n}$ have orthonormal columns and the diagonal matrix $\Sigma\in \RR^{n\times n}$ contains the singular values $\{\sigma_i\}_{i=1}^r$. Let also $x_{k,i},U_i$ be the $i$th entry of $x_k$ and the $i$th row of $U_i$, respectively. Then we bound the last line above as +%\begin{align} +%& 2 \| \text{diag}(x_k) \overline{B}^\top ( \overline{B}x^{\circ 2} -b) \| \nonumber\\ +%& = 2 \| \text{diag}(x_k) U \Sigma^2 U^\top ( x^{\circ 2} - x_*^{\circ 2}) \| \nonumber\\ +%& \ge 2 \| \text{diag}(x_k) U \Sigma^2 U^\top ( x^{\circ 2} - x_*^{\circ 2}) \|_{\infty} \nonumber\\ +%& = \max_i |x_{k,i}| \cdot | U_i ^\top \Sigma \cdot \Sigma U^\top ( x^{\circ 2} - x_*^{\circ 2}) | \nonumber\\ +%& +%\end{align} +% +%where $\tilde{b}_{i,j} = (\tilde{B})_{ij},~ i\in [1:n]$ and $j \in [1:2d]$. +%\begin{align*} +%-DA(x)^\top(A(x) - b) = -2x \odot (\tilde{B}^\top (A(x) - b)), +%\end{align*} +%where $\odot$ denotes hadamard product. +%\begin{align*} +%& \text{dist} \left( -DA(x)^\top (A(x)-b), \frac{\partial g(x)}{\b} \right) \nonumber\\ +%& = \text{dist} \left( -DA(x)^\top (A(x)-b), \{0\} \right) \nonumber\\ +%& = \left\| -DA(x)^\top (A(x)-b) \right\| \nonumber\\ +%& \ge ?????, +%\end{align*} +%Hence, this completes the proof for regularity condition. + +We draw the entries of $B$ independently from a zero-mean and unit-variance Gaussian distribution. For a fixed sparsity level $k$, the support of $z_*\in \RR^d$ and its nonzero amplitudes are also drawn from the standard Gaussian distribution. +%We then pick a sparsity level $k$ and choose $k$ indexes, i.e, $\Omega \subset [1:d]$, which are corresponding to nonzero entries of $z$. +%We then assign values from normal distribution to those entries. +Then the measurement vector is created as $b = Bz + \epsilon$, where $\epsilon$ is the noise vector with entries drawn independently from the zero-mean Gaussian distribution with variance $\sigma^2=10^{-6}$. + +\begin{figure}[] +\begin{center} +{\includegraphics[width=1\columnwidth]{figs/bp_fig1_subsolvers.pdf}} +\caption{Convergence with different subsolvers for the aforementioned non-convex relaxation. +} +\label{fig:bp1} +%(left:$[n=100$, $d=10^4]$, right:$[n=34$, $d=10^2$])} +\end{center} +\end{figure} +%\vspace{-3mm} + +Figure~\ref{fig:bp1} compiles our results for the proposed relaxation. It is, indeed, interesting to see that these type of non-convex relaxations gives the solution of convex one and first order methods succeed. + +\paragraph{Discussion:} +The true potential of our reformulation is in dealing with more structured norms rather than $\ell_1$, where computing the proximal operator is often intractable. One such case is the latent group lasso norm~\cite{obozinski2011group}, defined as +\begin{align*} +\|z\|_{\Omega} = \sum_{i=1}^I \| z_{\Omega_i} \|, +\end{align*} +where $\{\Omega_i\}_{i=1}^I$ are (not necessarily disjoint) index sets of $\{1,\cdots,d\}$. Although not studied here, we believe that the non-convex framework presented in this paper can serve to solve more complicated problems, such as the latent group lasso. We leave this research direction for future work. + + +\subsection{Adversarial Denoising with GANs} +In the appendix, we provide a contemporary application example that our template applies. + +%This relaxation transformed a non-smooth objective into a smooth one while loosing the linearity on the equality constraint. + + +%\subsection{Latent Group Lasso \label{sec:latent lasso}} +% +%For a collection of subsets $\Omega=\{\Omega_i\}_{i=1}^{I}\subset [1:p]$, the latent group Lasso norm on $\RR^p$ takes $z\in \RR^p$ to +%\begin{align} +%\|z\|_{\Omega,1} = \sum_{i=1}^I \| z_{\Omega_i} \|. +%\end{align} +%Note that we do not require $\Omega_i$ to be disjoint. For $B\in \RR^{n\times p}$, $b\in \RR^n$, and $\lambda>0$, consider the latent group lasso as +%\begin{align} +%\min_{z\in \RR^d} \frac{1}{2}\| Bz - b\|_2^2+ \lambda \| z\|_{\Omega,1}. +%\label{eq:group_lasso} +%\end{align} +%Because $\Omega$ is not a partition of $[1:p]$, computing the proximal operator of $\|\cdot\|_{\Omega}$ is often intractable, ruling out proximal gradient descent and similar algorithms for solving Program~\eqref{eq:group_lasso}. Instead, often Program~\eqref{eq:group_lasso} is solved by Alternating Direction Method of Multipliers (ADMM). More concretely, ??? +% +%We take a radically different approach here and cast Program~\eqref{eq:group_lasso} as an instance of Program~\eqref{prob:01}. More specifically, let $z^+,z^-\in \RR^p$ be positive and negative parts of $z$, so that $z=z^+-z^-$. Let us introduce the nonnegative $u\in \RR^p$ such that $z^++z^- = u^{\circ 2}$, where we used $\circ$ notation to show entrywise power. We may now write that +%\begin{align} +%\| z^++z^- \|_{\Omega,1} +%& = \| u^{\circ 2} \|_{\Omega,1 } =: \|u\|_{\Omega,2}^2, +%\end{align} +%Unlike $\|\cdot\|_{\Omega,1}$, note that $\|\cdot\|_{\Omega,2}^2$ is differentiable and, in fact, there exists a positive semidefinite matrix $Q\in \RR^{d\times d}$ such that $\|u\|_{\Omega,2}^2=u^\top Q_\Omega u$. Let us form $x=[(z^+)^\top,(z^-)^\top, u^\top]^\top\in \RR^{3d}$ and set +%\begin{align*} +%f(x) = \frac{1}{2}\| Bz^+-Bz^- -b\|_1+ \|u\|_{\Omega,2}^2, +%\end{align*} +%\begin{align} +%g(x) = 0, \qquad A(x) = z^++z^--u^{\circ 2}. +%\end{align} +%We can now apply Algorithm~\ref{Algo:2} to solve Program~\eqref{prob:01} with $f,g,A$ specified above. +% +% +%\paragraph{Convergence rate.} +%Clearly, $f,A$ are strongly smooth in the sense that \eqref{eq:smoothness basic} holds with proper $\lambda_f,\lambda_A$. Likewise, both $f$ and the Jacobian $DA$ are continuous and the restricted Lipschitz constants $\lambda'_f,\lambda'_A$ in \eqref{eq:defn_lambda'} are consequently well-defined and finite for a fixed $\rho'>0$. We next verify the key regularity condition in Theorem~\ref{thm:main}, namely, \eqref{eq:regularity}. Note that +%\begin{align*} +%DA(x) & = +%\left[ +%\begin{array}{ccc} +%I_p & I_p & -2\text{diag}(u) +%\end{array} +%\right]\in \RR^{d\times 3d}, +%\end{align*} +%\begin{align*} +%-DA(x)^\top A(x) = +%\left[ +%\begin{array}{c} +%-z^+-z^-+u^{\circ 2} \\ +%-z^+-z^-+u^{\circ 2}\\ +%2\text{diag}(u)( z^++z^--u^{\circ 2}) +%\end{array} +%\right], +%\end{align*} +%\begin{align*} +%& \text{dist} \left( -DA(x)^\top A(x), \frac{\partial g(x)}{\b} \right) \nonumber\\ +%& = \text{dist} \left( -DA(x)^\top A(x), \{0\} \right) \nonumber\\ +%& = \left\| -DA(x)^\top A(x) \right\| \nonumber\\ +%& \ge \sqrt{2} \| A(x)\|, +%\end{align*} +%namely, \eqref{eq:regularity} holds with $\nu=1$. + + + + + diff --git a/ICML19/Arxiv version/sections/guarantees.tex b/ICML19/Arxiv version/sections/guarantees.tex new file mode 100644 index 0000000..d2dd7c1 --- /dev/null +++ b/ICML19/Arxiv version/sections/guarantees.tex @@ -0,0 +1,192 @@ +%!TEX root = ../iALM_main.tex + +\section{Convergence Rate \label{sec:cvg rate}} + +In this section, we detail the convergence rate of Algorithm~\ref{Algo:2} for finding first-order and second-order stationary points, along with the iteration complexity results. +All the proofs are deferred to Appendix~\ref{sec:theory} for the clarity. + +Theorem~\ref{thm:main} below characterizes the convergence rate of Algorithm~\ref{Algo:2} for finding stationary points in terms of the number of outer iterations.\\ +%<<<<<<< HEAD +%{\color{red}{Ahmet: Maybe instead of sufficiently large k0 we can say k0 such that beta is bigger than 1, it makes the proof go thru, it would be a bit more explanatory}} +%======= +%>>>>>>> b39eea61625954bc9d3858590b7b37a182a6af4f +\begin{theorem}\textbf{\emph{(convergence rate)}} +\label{thm:main} +Suppose that $f$ and $A$ are smooth in the sense specified in (\ref{eq:smoothness basic}). For $\rho'>0$, let +\begin{align} +\lambda'_f = \max_{\|x\|\le \rho'} \|\nabla f(x)\|, +~~ +\lambda'_A = \max_{\|x\| \le \rho'} \|DA(x)\|, +\label{eq:defn_restricted_lipsichtz} +\end{align} +be the (restricted) Lipschitz constants of $f$ and $A$, respectively. + For sufficiently large integers $k_0\le k_1$, consider the interval $K=[k_0:k_1]$, and let $\{x_k\}_{k\in K}$ be the output sequence of Algorithm~\ref{Algo:2} on the interval $K$. For $\nu>0$, assume that + \begin{align} +\nu \|A(x_k)\| +& \le \dist\left( -DA(x_k)^\top A(x_k) , \frac{\partial g(x_k)}{ \b_{k-1}} \right), +\label{eq:regularity} +\end{align} +for every $k\in K$. We consider two cases: +%{\color{red} Ahmet: I removed the squares and showed the rate on dist + feasibility, since it is the measure for the stationarity, using squares confuses complexity analysis...} +\begin{itemize}[leftmargin=*] +\item If a first-order solver is used in Step~2, then $x_k$ is an $(\epsilon_{k,f},\b_k)$ first-order stationary point of (\ref{prob:01}) with +%\begin{align} +%& \dist^2( -\nabla_x \L_{\b_{k-1}}(x_k,y_k),\partial g(x_k)) + \| A(x_k)\|^2 \nonumber\\ +%& = \frac{1}{\b_{k-1}^2}\left( \frac{8 \lambda'_A\s_1^2 + 4}{ \nu^2} ( \lambda'_f+\lambda'_A y_{\max})^2 + 2\right) \nonumber \\ +%&=: \frac{Q(f,g,A)}{\beta_{k-1}^2}, +%\end{align} +\begin{align} +%\epsilon_{k,f} & = \frac{1}{\beta_{k-1}} \sqrt{ \frac{8 \lambda'_A\s_1^2 + 4}{ \nu^2} ( \lambda'_f+\lambda'_A y_{\max})^2 + 2 } \nonumber \\ +%&=: \frac{Q(f,g,A,\s_1)}{\beta_{k-1}}, +\epsilon_{k,f} & = \frac{1}{\beta_{k-1}} \left(\frac{2(\lambda'_f+\lambda'_A y_{\max}) (1+\lambda_A' \sigma_k)}{\nu}+1\right) \nonumber \\ +&=: \frac{Q(f,g,A,\s_1)}{\beta_{k-1}}, +\label{eq:stat_prec_first} +\end{align} +for every $k\in K$, where the expression for $y_{\max}=y_{\max}(x_1,y_0,\s_1)$ is given in (\ref{eq:dual growth}). +\item If a second-order solver is used in Step~2, then $x_k$ is an $(\epsilon_{k,f}, \epsilon_{k,s},\b_k)$ second-order stationary point of~(\ref{prob:01}) with $\epsilon_{k,s}$ specified above and with +\begin{align} +\epsilon_{k,s} &= \epsilon_{k-1} + \sigma_k \sqrt{m} \lambda_A \frac{ 2\lambda'_f +2 \lambda'_A y_{\max} }{\nu \b_{k-1}} \nonumber\\ +&= \frac{\nu + \sigma_k \sqrt{m} \lambda_A 2\lambda'_f +2 \lambda'_A y_{\max} }{\nu \b_{k-1}} =: \frac{Q'}{\beta_{k-1}}. +\end{align} +\end{itemize} + + +\end{theorem} + +%A few remarks are in order about Theorem~\ref{thm:main}. + +%\textbf{Tradeoff.} +%Roughly speaking, Theorem~\ref{thm:main} states that the iterates of Algorithm~\ref{Algo:2} approach first-order stationarity for~\eqref{prob:01} at the rate of $1/{\b_k}$, where $\b_k$ is the penalty weight in AL in iteration $k$ as in~\eqref{eq:Lagrangian}. Note the inherent tradeoff here: For a faster sequence $\{\b_k\}_k$, the iterates approach stationarity faster but the computational cost of the inexact primal subroutine (Step 2) increases since a higher accuracy is required and Lipschitz constant of the unconstrained problem is also affected by $\beta_k$. In words, there is a tradeoff between the number of outer and inner loop iterations of Algorithm~\ref{Algo:2}. We highlight this tradeoff for a number of subroutines below. + +Loosely speaking, Theorem~\ref{thm:main} states that Algorithm~\ref{Algo:2} converges to a (first- or second-) order stationary point of \eqref{prob:01} at the rate of $1/\b_k$. + +A few remarks are in order. + + +\textbf{Regularity.} The key condition in Theorem~\ref{thm:main} is \eqref{eq:regularity} which, broadly speaking, controls the problem geometry. + + +As the penalty weight $\b_k$ grows, the primal solver in Step~2 places an increasing emphasis on reducing the feasibility gap and \eqref{eq:regularity} formalizes this intuition. +In contrast to most conditions in the nonconvex optimization literature, such as~\cite{flores2012complete}, our condition in~\eqref{eq:regularity} appears to be easier to verify, as we see in Section~\ref{sec:experiments}. + +We now argue that such a condition is necessary for controlling the feasibility gap of~\eqref{prob:01} and ensuring the success of Algorithm~\ref{Algo:2}. Consider the case where $f=0$ and $g=\delta_\mathcal{X}$, where $\mathcal{X}$ is a convex set, $A$ is a linear operator. In this case, solving \eqref{prob:01} finds a point in $\mathcal{X}\cap \text{null}(A)$, where the subspace $\text{null}(A)=\{ x\in\mathbb{R}^d: A(x) = 0 \} \subset \RR^d$ is the null space of $A$. +Here, the Slater's condition requires that +\begin{equation} +\text{relint} (\mathcal{X}) \cap \text{null}(A)\neq \emptyset. +\end{equation} +%<<<<<<< HEAD +In general, the Slater's condition plays a key role in convex optimization as a sufficient condition for strong duality and, as a result, guarantees the success of a variety of primal-dual algorithms for linearly-constrained convex programs~\cite{bauschke2011convex}. + +Intuitively, the Slater's condition here removes any pathological cases by ensuring that the subspace $\text{null}(A)$ is not tangent to $\mathcal{X}$, see Figure~\ref{fig:convex_slater}. In such pathological cases, solving~\eqref{prob:01}, namely, finding a point in $\mathcal{X}\cap \text{null}(A)$, can be particularly difficult. For instance, the alternating projection algorithm (which iteratively projects onto $\mathcal{X}$ and $\text{null}(A)$) has arbitrarily slow convergence, see Figure~\ref{fig:convex_slater}. +%{\color{red} Ahmet: a reference would be cool here} +%======= +%In general, the Slater's condition plays a key role in convex optimization as a sufficient condition for strong duality and, as a result, guarantees the success of a variety of primal-dual algorithms for linearly-constrained convex programs~\cite{bauschke2011convex}. Intuitively, the Slater's condition here removes any pathological cases by ensuring that the subspace $\text{null}(A)$ is not tangent to $\mathcal{X}$, see Figure~\ref{fig:convex_slater}. In such pathological cases, solving~\eqref{prob:01}, namely, finding a point in $\mathcal{X}\cap \text{null}(A)$, can be particularly difficult. For instance, the alternating projection algorithm (which iteratively projects onto $\mathcal{X}$ and $\text{null}(A)$) has arbitrarily slow convergence, see Figure~\ref{fig:convex_slater}. +%>>>>>>> 795311da274f2d8ab56d215c2fd481073e616732 +\begin{figure} +\begin{center} +\includegraphics[scale=.5]{Slater.pdf} +\end{center} +\caption{Solving \eqref{prob:01} can be particularly difficult, even when \eqref{prob:01} is a convex program. We present a pathological geometry where the Slater's condition does not apply. See the first remark after Theorem~\ref{thm:main} for more details. \label{fig:convex_slater}} +\end{figure} + + +\textbf{Computational complexity.} Theorem~\ref{thm:main} allows us to specify the number of iterations that Algorithm~\ref{Algo:2} requires to reach a near-stationary point of Program~\eqref{prob:01} with a prescribed precision and, in particular, specifies the number of calls made to the solver in Step~2. In this sense, Theorem~\ref{thm:main} does not fully capture the computational complexity of Algorithm~\ref{Algo:2}, as it does not take into account the computational cost of the solver in Step~2. + +To better understand the total complexity of Algorithm~\ref{Algo:2}, we consider two scenarios in the following. In the first scenario, we take the solver in Step~2 to be the Accelerated Proximal Gradient Method (APGM), a well-known first-order algorithm~\cite{ghadimi2016accelerated}. In the second scenario, we will use the second-order trust region method developed in~\cite{cartis2012complexity}. + + +\subsection{First-Order Optimality} + +%\textbf{AE: we go back and forth between "subroutine" and "solver". for consistency, I'm just using "solver" everywhere.} +Let us first consider the first-order optimality case where the solver in Step~2 is APGM~\cite{ghadimi2016accelerated}. +APGM makes use of $\nabla_x \mathcal{L}_{\beta}(x,y)$, $\text{prox}_g$ and classical Nesterov acceleration step for the iterates~\cite{nesterov1983method} to obtain first order stationarity guarantees for solving~\eqref{e:exac}. + % \textbf{AE: Ahmet to give a brief description of APGM here. } +%First, we characterize the iteration complexity of Algorithm~\ref{Algo:2} for finding a first-order stationary point of~\eqref{prob:01}. +%We propose to use the standard accelerated proximal method (APGM), guarantees of which are analyzed in~\cite{ghadimi2016accelerated} for nonconvex problems of the form~\eqref{e:exac}. +Suppose that $g=\delta_\mathcal{X}$ is the indicator function on a bounded convex set $\mathcal{X}\subset \RR^d$ and let +\begin{align} +x_{\max}= \max_{x\in \mathcal{X}}\|x\|, +\end{align} +be the radius of a ball centered at the origin that includes $\mathcal{X}$. + Then, adapting the results in~\cite{ghadimi2016accelerated} to our setup, APGM reaches $x_{k}$ in Step 2 of Algorithm~\ref{Algo:2} after +\begin{equation} +\mathcal{O}\left ( \frac{\lambda_{\beta_{k}}^2 x_{\max}^2 }{\epsilon_{k+1}} \right) +\label{eq:iter_1storder} +\end{equation} +(inner) iterations, where $\lambda_{\beta_{k}}$ denotes the Lipschitz constant of $\nabla_x{\mathcal{L}_{\beta_{k}}(x, y)}$, bounded in~\eqref{eq:smoothness of Lagrangian}. We note that for simplicity, we use a looser bound in \eqref{eq:iter_1storder} than~\cite{ghadimi2016accelerated}. +Using \eqref{eq:iter_1storder}, we derive the following corollary, describing the total computational complexity of Algorithm~\ref{Algo:2} in terms of the calls to the first-order oracle in APGM. %\textbf{AE: we haven't talked about oracle before.} + +\begin{corollary}\label{cor:first} +For $b>1$, let $\beta_k =b^k $ for every $k$. If we use APGM from~\cite{ghadimi2016accelerated} for Step~2 of Algorithm~\ref{Algo:2}, the algorithm finds an $(\epsilon_f,\b_k)$ first-order stationary point, see (\ref{eq:inclu3}), + after $T$ calls to the first-order oracle, where +% +\begin{equation} +T = \mathcal{O}\left( \frac{Q^3 x_{\max}^2}{\epsilon^{3}}\log_b{\left( \frac{Q}{\epsilon} \right)} \right) = \tilde{\mathcal{O}}\left( \frac{Q^{3} x_{\max}^2}{\epsilon^{3}} \right). +\end{equation} +\end{corollary} +For Algorithm~\ref{Algo:2} to reach a near-stationary point with an accuracy of $\epsilon_f$ in the sense of \eqref{eq:inclu3} and with the lowest computational cost, we therefore need to perform only one iteration of Algorithm~\ref{Algo:2}, with $\b_1$ specified as a function of $\epsilon_f$ by \eqref{eq:stat_prec_first} in Theorem~\ref{thm:main}. In general, however, the constants in \eqref{eq:stat_prec_first} are unknown and this approach is intractable. Instead, the homotopy approach taken by Algorithm~\ref{Algo:2} ensures achieving the desired accuracy by gradually increasing the penalty weight. This homotopy approach increases the computational cost of Algorithm~\ref{Algo:2} only by a factor logarithmic in the $\epsilon_f$, as detailed in the proof of Corollary~\ref{cor:first}. + +%\textbf{AE: if we find some time, i'll add a couple of lines for how 1st order opt implies 2nd order opt for smooth constraints.} + +\subsection{Second-Order Optimality} +%{\color{red}{Ahmet (note to myself): not sure of the constants of trust-region, check again}} \\ +Let us now consider the second-order optimality case where the solver in Step~2 is the the trust region method developed in~\cite{cartis2012complexity}. +Trust region method minimizes quadratic approximation of the function within a dynamically updated trust-region radius. +Second-order trust region method that we consider in this section makes use of Hessian (or an approximation of Hessian) of the augmented Lagrangian in addition to first order oracles. + +As shown in~\cite{nouiehed2018convergence}, finding approximate second-order stationary points of convex-constrained problems is in general NP-hard. For this reason, we focus in this section on the special case of~\eqref{prob:01} with $g=0$. + +%We first give a theorem to show the convergence rate of the algorithm for second order stationarity: \\ +%{\color{red}{Ahmet: I think that it is possible to remove sufficiently large k0 assumption here. }} \textbf{AE: not worth it really} +%\begin{corollary} +%\label{thm:main_second} +%Under the assumptions of Theorem~\ref{thm:main}, the output of Algorithm~\ref{Algo:2} satisfies +%\begin{align} +%%& \dist^2( -\nabla_x \L_{\b_{k-1}}(x_k,y_k),\partial g(x_k)) + \| A(x_k)\|^2 \nonumber\\ +%%& = \frac{1}{\b_{k-1}^2}\left( \frac{8 \lambda'_A\s_k^2 + 4}{ \nu^2} ( \lambda'_f+\lambda'_A y_{\max})^2 + 2\right) \nonumber \\ +%%&=: \frac{Q}{\beta_{k-1}^2}. +%\lambda_{\text{min}}(\nabla _{xx}\mathcal{L}_{\beta_{k-1}}(x_k, y_k)) \geq - \frac{C}{\beta_{k-1}} - \epsilon_{k-1}. +%\end{align} +%\end{corollary} +% +%We consider \textbf{AE: Ahmet to add a brief description of the algorithm.} + + + +Let us compute the total computational complexity of Algorithm~\ref{Algo:2} with the trust region method in Step~2, in terms of the number of calls made to the second-order oracle. By adapting the result in~\cite{cartis2012complexity} to our setup, we find that the number of (inner) iterations required in Step~2 of Algorithm~\ref{Algo:2} to produce $x_{k+1}$ is +% +\begin{equation} +\mathcal{O}\left ( \frac{\lambda_{\beta_{k}, H}^2 (\mathcal{L}_{\beta_{k}}(x_1, y) - \min_{x}\mathcal{L}_{\beta_k}(x, y))}{\epsilon_k^3} \right), +\label{eq:sec_inn_comp} +\end{equation} +% +%<<<<<<< HEAD +where $\lambda_{\beta, H}$ is the Lipschitz constant of the Hessian of the augmented Lagrangian, which is of the order of $\beta$, as can be proven similar to Lemma~\ref{lem:smoothness} and $x_1$ is the initial iterate of the given outer loop. +In~\cite{cartis2012complexity}, the term $\mathcal{L}_{\beta}(x_1, y) - \min_{x}\mathcal{L}_{\beta}(x, y)$ is bounded by a constant independent of $\epsilon$. +We assume a uniform bound for this quantity $\forall \beta_k$, instead of for one value of $\beta_k$ as in~\cite{cartis2012complexity}. Using \eqref{eq:sec_inn_comp} and Theorem~\ref{thm:main}, we arrive at the following: +%======= +%where $\lambda_{\beta_k, H}$ is the Lipschitz constant of the Hessian of the augmented Lagrangian, which is of the order of $\beta_k$, as can be proven similar to Lemma~\ref{lem:smoothness} and $x_1$ is the initial iterate of the given outer loop. +%In~\cite{cartis2012complexity}, the term $\mathcal{L}_{\beta_k}(x_1, y) - \min_{x}\mathcal{L}_{\beta_k}(x, y)$ is bounded by a constant independent of $\epsilon_k$. +%We assume a uniform bound for this quantity for all $\b$, instead of for one value of $\beta_k$ as in~\cite{cartis2012complexity}. Using \eqref{eq:sec_inn_comp} and Theorem~\ref{thm:main}, we arrive at the following result. The proof is very similar to that of Corollary~\ref{cor:first} and hence omitted here. +%>>>>>>> 795311da274f2d8ab56d215c2fd481073e616732 +% +\begin{corollary}\label{cor:second} +For $b>1$, let $\beta_k =b^k $ for every $k$. +We assume that +\begin{equation} +\mathcal{L}_{\beta}(x_1, y) - \min_{x}\mathcal{L}_{\beta}(x, y) \leq L_{u},~~ \forall \beta. +\end{equation} +If we use the trust region method from~\cite{cartis2012complexity} for Step~2 of Algorithm~\ref{Algo:2}, the algorithm finds an $\epsilon$-second-order stationary point of~(\ref{prob:01}) in $T$ calls to the second-order oracle where +% +\begin{equation} +T \leq \mathcal{O}\left( \frac{L_u Q'^{5}}{\epsilon^{5}} \log_b{\left( \frac{Q'}{\epsilon} \right)} \right) = \widetilde{\mathcal{O}}\left( \frac{L_u Q'^{5}}{\epsilon^{5}} \right). +\end{equation} +\end{corollary} +% +Before closing this section, we note that the remark after Corollary~\ref{cor:first} applies here as well. +% + + + + + diff --git a/ICML19/Arxiv version/sections/introduction.tex b/ICML19/Arxiv version/sections/introduction.tex new file mode 100644 index 0000000..28a7a65 --- /dev/null +++ b/ICML19/Arxiv version/sections/introduction.tex @@ -0,0 +1,91 @@ +%!TEX root = ../main.tex +\vspace{-5mm} +\section{Introduction} +\label{intro} +We study the following nonconvex optimization problem +% +\begin{equation} +\label{prob:01} +\begin{cases} +\underset{x\in \mathbb{R}^d}{\min}\,\, f(x)+g(x)\\ +A(x) = b, +\end{cases} +\end{equation} +% +where $f:\mathbb{R}^d\rightarrow\mathbb{R}$ is possibly non-convex and $A:\mathbb{R}^d\rightarrow\mathbb{R}^m$ is a nonlinear operator and $b\in\mathbb{R}^m$. +For clarity of notation, we take $b=0$ in the sequel, the extension to any $b$ is trivial. +We assume that $g:\mathbb{R}^d\rightarrow\mathbb{R}$ is proximal-friendly (possibly nonsmooth) convex function. % (not necessarily smooth). +%For instance, for a convex set $\mathcal{X}\subset\RR^d$, letting $g=\delta_\mathcal{X}$ to be the indicator function on $\mathcal{X}$, ~\eqref{prob:01} is a nonconvex optimization problem with the convex constraint $x\in \mathcal{X}$. +%Our other requirement for $g$ is that computing its proximal operator is inexpensive, see Section~\ref{sec:preliminaries}. + +A host of problems in computer science \cite{khot2011grothendieck, lovasz2003semidefinite}, machine learning \cite{mossel2015consistency, song2007dependence}, and signal processing \cite{singer2011angular, singer2011three} naturally fall under the template of ~\eqref{prob:01}, including max-cut, clustering, generalized eigenvalue, as well as community detection. % \textbf{AE: We need to add some more specifics to this paragraph to convince people that this is an important template. What are the names of the problems you're citing? Max-cut, sparse regression, statistical learning with deep networks as prior \cite{Bora2017}, what else we can think of?} + +To address these applications, this paper builds up on the vast literature on the classical inexact augmented Lagrangian framework and proposes a simple, intuitive as well as easy-to-implement algorithm with total complexity results for ~\eqref{prob:01} under an interpretable geometric condition. Before we elaborate on the results, let us first motivate ~\eqref{prob:01} with an important application to semidefinite programming (SDP): + +\vspace{-3mm} +\paragraph{Vignette: Burer-Monteiro splitting.} +A powerful convex relaxation for max-cut, clustering, and several other problems above is provided by the SDP +\begin{equation} +\label{eq:sdp} +\begin{cases} +\underset{X\in\mathbb{S}^{d \times d}}{\min} \langle C, X \rangle \\ +B(X) = b, \,\, X \succeq 0, +\end{cases} +\end{equation} +% +where $C\in \RR^{d\times d}$ and $X$ is a positive semidefinite and symmetric $d\times d$ matrix, +%$\mathbb{S}^{d \times d}$ denotes the set of real symmetric matrices, +and ${B}: \mathbb{S}^{d\times d} \to \mathbb{R}^m$ is a linear operator. If the unique-games conjecture is true, SDPs achieve the best approximation for the underlying discrete problem~\cite{raghavendra2008optimal}. + +Since $d$ is often large, many first- and second-order methods for solving such SDP's are immediately ruled out, not only due to their high computational complexity, but also due to their storage requirements, which are $\mathcal{O}(d^2)$. + +A contemporary challenge in optimization therefore is to solve SDP's in small space and in a scalable fashion. A recent algorithm, i.e., homotopy conditional gradient method (HCGM) based on Linear Minimization Oracles (LMO), can address this template in small space via sketching \cite{yurtsever2018conditional}; however, such LMO-based methods are extremely slow in obtaining accurate solutions. + + +%In practice, $d$ is often very large which makes interior point methods, with their poor scaling in $d$, an unattractive option for solving ~\eqref{eq:sdp}. Attempts to resolve this issue prompted extensive research in computationally- and memory-efficient SDP solvers. The first such solvers relied on the so-called Linear Minimization Oracle (LMO), reviewed in Section~\ref{sec:related work}, alongside other scalabe SDP solvers. + +A key approach for solving \eqref{prob:01}, dating back to~\cite{burer2003nonlinear, burer2005local}, is the so-called Burer-Monteiro (BR) splitting $X=UU^\top$, where $U\in\mathbb{R}^{d\times r}$ and $r$ is selected according to the guidelines in~\cite{pataki1998rank, barvinok1995problems}. +It has been shown that these bounds on the rank, which are shown to be optimal~\cite{waldspurger2018rank}, under some assumptions removing the spurious local minima of the nonconvex factorized problem~\cite{boumal2016non}. + +%so as to remove spurious local minima of the nonconvex factorized problem. Evidently, BR splitting has the advantage of lower storage and faster iterations. + +This splitting results in the following non-convex problem +\begin{equation} +\label{prob:nc} +\begin{cases} +\underset{U\in\mathbb{R}^{d \times r}}{\min} \langle C, UU^\top \rangle \\ +B(UU^\top) = b, +\end{cases} +\end{equation} +which can be written in the form of ~\eqref{prob:01}. + +%\textbf{Vignette 2: Latent group lasso} + +%See Section ? for several examples. \textbf{We should explain this better and give more high level examples to motivate this .} +%An example of our template in \eqref{prob:01} is semi-definite ming which provides powerful relaxations to above problems. Denote the space of $d'\times d'$ symmetric matrices by $\mathbb{S}^{d'\times d'}$ and consider +% +%\begin{equation} +% \label{sdp} +% \min_x \{h'(x): A'(x) = b', ~~x \in \mathcal{C'}~~\text{and}~~x\succeq0 \} +%\end{equation} +% +%where $h': \mathbb{S}^{d'\times d'} \to \RR$, $A'\colon\mathbb{S}^{d'\times d'}\to\RR^m$, $b'\in\RR^m$, and $C' \subseteq \mathbb{R}^{d'\times d'}$. This template clearly can be put to the form of \eqref{prob:01} by using \emph{Burer-Monteiro factorization} \cite{burer2003nonlinear, burer2005local}. + +To solve \eqref{prob:nc}, the inexact Augmented Lagrangian Method (iALM) is widely used~\cite{burer2003nonlinear, burer2005local, kulis2007fast}, due to its cheap per iteration cost and also its empirical success in practice. Every (outer) iteration of iALM calls a solver to inexactly solve an intermediate augmented Lagrangian subproblem to near stationarity, and the user has freedom in choosing this solver, which could use first-order (say,~proximal gradient descent \cite{parikh2014proximal}) or second-order information, such as BFGS \cite{nocedal2006numerical}. + +We argue that unlike its convex counterpart~\cite{nedelcu2014computational,lan2016iteration,xu2017inexact}, the convergence rate and the complexity of iALM for ~\eqref{prob:nc} are not well-understood, see Section~\ref{sec:related work} for a review of the related literature. Indeed, addressing this important theoretical gap is one of the contributions of our work. + + +{\textbf{A brief summary of our contributions:} \\[1mm] +$\circ$ Our framework is future-proof in the sense that we obtain the convergence rate of iALM for ~\eqref{prob:01} with an arbitrary solver for finding first- and second-order stationary points. \\[2mm] +$\circ$ We investigate using different solvers for augmented Lagrangian subproblems and provide overall iteration complexity bounds for finding first- and second-order stationary points of ~\eqref{prob:01}. Our complexity bounds match the best theoretical complexity results in optimization, see Section~\ref{sec:related work}. \\[2mm] +$\circ$ We propose a novel geometric condition that simplifies the algorithmic analysis for iALM. We verify the condition for key problems described in Section~\ref{sec:experiments}. + +\textbf{Roadmap.} Section~\ref{sec:preliminaries} collects the main tools and our notation. % that we utilize. +We present the iALM in Section~\ref{sec:AL algorithm} and obtain its convergence rate to first- and second-order stationary points in Section~\ref{sec:cvg rate}, alongside their iteration complexities. +We provide a comprehensive review of the literature and highlight our key differences in % with the precedents in +Section~\ref{sec:related work}. +Section~\ref{sec:experiments} presents the numerical evidence and comparisons with the state-of-the-art. % methods. + + + diff --git a/ICML19/Arxiv version/sections/preliminaries.tex b/ICML19/Arxiv version/sections/preliminaries.tex new file mode 100644 index 0000000..9577be1 --- /dev/null +++ b/ICML19/Arxiv version/sections/preliminaries.tex @@ -0,0 +1,180 @@ +\vspace{-3mm} +\section{Preliminaries \label{sec:preliminaries}} +\paragraph{\textbf{{Notation.}}} +We use the notation $\langle\cdot ,\cdot \rangle $ and $\|\cdot\|$ for the {standard inner} product and the norm on $\RR^d$. For matrices, $\|\cdot\|$ and $\|\cdot\|_F$ denote the spectral and the Frobenius norms, respectively. +%For a matrix $M$, its row and null spaces are denoted by $\row(A)$ and $\text{null}(A)$, respectively. +For a convex function $g:\RR^d\rightarrow\RR$, the subdifferential set at $x\in \RR^d$ is denoted by $\partial g(x)$ and we will occasionally use the notation $\partial g(x)/\b = \{ z/\b : z\in \partial g(x)\}$. +When presenting iteration complexity results, we often use $\widetilde{O}(\cdot)$ which suppresses the logarithmic dependencies. +%The proximal operator $P_g:\RR^d\rightarrow\RR^d$ takes $x$ to +%\begin{align} +%P_g(x) = \underset{y}{\argmin} \, g(y) + \frac{1}{2}\|x-y\|^2, +%\label{eq:defn of prox} +%\end{align} +%and, if $g=1_C$ is the indicator function of a convex set $C$, we will use the shorthand $P_C=P_{1_C}$ to show the orthogonal projection onto $C$. + +We use the indicator function $\delta_\mathcal{X}:\RR^d\rightarrow\RR$ of a set $\mathcal{X}\subset\RR^d$, which takes $x$ to +\begin{align} +\delta_\mathcal{X}(x) = +\begin{cases} +0 & x \in \mathcal{X}\\ +\infty & x\notin \mathcal{X}. +\end{cases} +\label{eq:indicator} +\end{align} + +%The tangent cone of $C$ at $x\in C$ is denoted by $T_C(x)$ and +The distance function from a point $x$ to $\mathcal{X}$ is denoted by $\dist(x,\mathcal{X}) = \min_{z\in \mathcal{X}} \|x-z\|$. +For integers $k_0 \le k_1$, we denote $[k_0:k_1]=\{k_0,\ldots,k_1\}$. + +For an operator $A:\RR^d\rightarrow\RR^m$ with components $\{A_i\}_{i=1}^m$, we let $DA(x) \in \mathbb{R}^{m\times d}$ denote the Jacobian of $A$, where the $i$th row of $DA(x)$ is the gradient vector $\nabla A_i(x) \in \RR^d$. + +%We denote the Hessian of the augmented Lagrangian with respect to $x$ as $\nabla _{xx} \mathcal{L}_{\beta}(x,y)$. \\ +%{\color{red} define $A_i$ rigorously.} + +\textbf{Smoothness.} +We require $f:\RR^d\rightarrow\RR$ and $A:\RR^d\rightarrow \RR^m$ in~\eqref{prob:01} to be smooth; i.e., there exists $\lambda_f,\lambda_A\ge 0$ such that +% +%\vspace{-.3cm} +\begin{align} +\| \nabla f(x) - \nabla f(x')\| & \le \lambda_f \|x-x'\|, \nonumber\\ +\| DA(x) - DA(x') \| & \le \lambda_A \|x-x'\|, +\label{eq:smoothness basic} +\end{align} +for every $x,x'\in \mathbb{R}^d$. + +\textbf{The augmented Lagrangian method (ALM)}. +ALM is a classical algorithm, first appeared in~\cite{hestenes1969multiplier, powell1969method} and extensively studied in~\cite{bertsekas2014constrained}. +For solving \eqref{prob:01}, ALM suggests solving the problem +% +%\vspace{-.3cm} +\begin{equation} +\min_{x} \max_y \,\,\mathcal{L}_\beta(x,y) + g(x), +\label{eq:minmax} +\end{equation} +%\vspace{-.5cm} +% +where, for $\b>0$, $\mathcal{L}_\b$ is the corresponding augmented Lagrangian, defined as +\begin{align} +\label{eq:Lagrangian} +\mathcal{L}_\beta(x,y) := f(x) + \langle A(x), y \rangle + \frac{\beta}{2}\|A(x)\|^2. +\end{align} + +%\textbf{AE: Adding the bias $b$ doesn't seem to add much except making the presentation more clunky.} +%\vspace{-.5cm} + +The minimax formulation in \eqref{eq:minmax} naturally suggests the following algorithm for solving \eqref{prob:01}. For dual step sizes $\{\s_k\}_k$, consider the iterates +\begin{equation}\label{e:exac} +x_{k+1} \in \underset{x}{\argmin} \,\, \mathcal{L}_{\beta}(x,y_k)+g(x), +\end{equation} +\begin{equation*} +y_{k+1} = y_k+\s_k A(x_{k+1}). +\end{equation*} +However, updating $x_{k+1}$ above requires solving the nonconvex problem~\eqref{e:exac} to optimality, which is typically intractable. Instead, it is often easier to find an approximate first- or second-order stationary point of~\eqref{e:exac}. +%We therefore consider an algorithm that only requires approximate stationarity in every iteration. + +Hence, we argue that by gradually improving the stationarity precision and increasing the penalty weight $\b$ above, we can reach a stationary point of the main problem in~\eqref{prob:01}, as detailed in Section~\ref{sec:AL algorithm}. + + + +{\textbf{{{Optimality conditions.}}} \label{sec:opt cnds}} +{First-order necessary optimality conditions} for \eqref{prob:01} are well-understood. {Indeed, $x\in \RR^d$ is a first-order stationary point of~\eqref{prob:01} if there exists $y\in \RR^m$ such that +\begin{align} +\begin{cases} +-\nabla f(x) - DA(x)^\top y \in \partial g(x)\\ +A(x) = 0, +\end{cases} +\label{e:inclu1} +\end{align} +where $DA(x)$ is the Jacobian of $A$ at $x$. Recalling \eqref{eq:Lagrangian}, we observe that \eqref{e:inclu1} is equivalent to +\begin{align} +\begin{cases} +-\nabla_x \mathcal{L}_\beta(x,y) \in \partial g(x)\\ +A(x) = 0, +\end{cases} +\label{e:inclu2} +\end{align} +which is in turn the necessary optimality condition for \eqref{eq:minmax}. } +%Note that \eqref{e:inclu2} is equivalent to +%\begin{align} +%\left[ +%\begin{array}{c} +%\nabla_x \L_\b(x,y) \\ +%\nabla_y \L_\b(x,y) +%\end{array} +%\right] +%\in +%\left\{ +%\left[ +%\begin{array}{c} +% v\\ +%0 +%\end{array} +%\right] +%: v\in \partial g(x) \right\} +%, +%\end{align} +%which rephrases the stationarity condition in terms of the gradient of the duality gap of Program~\eqref{eq:Lagrangian}. +Inspired by this, we say that $x$ is an $(\epsilon_f,\b)$ first-order stationary point if +\begin{align} +\begin{cases} +\dist(-\nabla_x \mathcal{L}_\beta(x,y), \partial g(x)) \leq \epsilon_f \\ +\| A(x) \| \leq \epsilon_f, +\end{cases} +\label{eq:inclu3} +\end{align} +for $\epsilon_f\ge 0$. +In light of \eqref{eq:inclu3}, a suitable metric for evaluating the stationarity of a pair $(x,y)\in \RR^d\times \RR^m$ is +\begin{align} +\dist\left(-\nabla_x \mathcal{L}_\beta(x,y), \partial g(x) \right) + \|A(x)\| , +\label{eq:cvg metric} +\end{align} +which we use as the first-order stopping criterion. +%When $g=0$, it is also not difficult to verify that the expression in \eqref{eq:cvg metric} is the norm of the gradient of the duality gap in \eqref{eq:minmax}. +As an example, for a convex set $\mathcal{X}\subset\RR^d$, suppose that $g = \delta_\mathcal{X}$ is the indicator function on $\mathcal{X}$.} +Let also $T_\mathcal{X}(x) \subseteq \RR^d$ denote the tangent cone to $\mathcal{X}$ at $x$, and with $P_{T_\mathcal{X}(x)}:\RR^d\rightarrow\RR^d$, we denote the orthogonal projection onto this tangent cone. Then, for $u\in\RR^d$, it is not difficult to verify that +\begin{align}\label{eq:dist_subgrad} +\dist\left(u, \partial g(x) \right) = \| P_{T_\mathcal{X}(x)}(u) \|. +\end{align} +% +When $g = 0$, a first-order stationary point $x\in \RR^d$ of \eqref{prob:01} is also second-order stationary if +% +%\vspace{-.5cm} +\begin{equation} +\lambda_{\text{min}}(\nabla _{xx} \mathcal{L}_{\beta}(x,y)) > 0, +\end{equation} +%\vspace{-.5cm} +% +where $\nabla_{xx}\mathcal{L}_\b$ is the Hessian with respect to $x$, and $\lambda_{\text{min}}(\cdot)$ returns the smallest eigenvalue of its argument. +Analogously, $x$ is an $(\epsilon_f, \epsilon_s,\b)$ second-order stationary point if, in addition to \eqref{eq:inclu3}, it holds that +% +%\vspace{-.5cm} +\begin{equation}\label{eq:sec_opt} +\lambda_{\text{min}}(\nabla _{xx} \mathcal{L}_{\beta}(x,y)) \ge -\epsilon_s, +\end{equation} +for $\epsilon_s>0$. +%\vspace{-.5cm} +% +Naturally, for second-order stationarity, we use $\lambda_{\text{min}}(\nabla _{xx} \mathcal{L}_{\beta}(x,y))$ as the stopping criterion. + + +\paragraph{{\textbf{Smoothness lemma.}}} This next result controls the smoothness of $\L_\b(\cdot,y)$ for a fixed $y$. The proof is standard but nevertheless included in Appendix~\ref{sec:proof of smoothness lemma} for completeness. +\begin{lemma}[\textbf{smoothness}]\label{lem:smoothness} + For fixed $y\in \RR^m$ and $\rho,\rho'\ge 0$, it holds that + \begin{align} +\| \nabla_x \mathcal{L}_{\beta}(x, y)- \nabla_x \mathcal{L}_{\beta}(x', y) \| \le \lambda_\b \|x-x'\|, + \end{align} + for every $x,x' \in \{ x'': \|A(x'') \|\le \rho, \|x''\|\le \rho'\}$, where + \begin{align} +\lambda_\beta +& \le \lambda_f + \sqrt{m}\lambda_A \|y\| + (\sqrt{m}\lambda_A\rho + d \lambda'^2_A )\b \nonumber\\ +& =: \lambda_f + \sqrt{m}\lambda_A \|y\| + \lambda''(A,\rho,\rho') \b. +\label{eq:smoothness of Lagrangian} +\end{align} +Above, $\lambda_f,\lambda_A$ were defined in (\ref{eq:smoothness basic}) and +\begin{align} +\lambda'_A := \max_{\|x\|\le \rho'}\|DA(x)\|. +\end{align} +\end{lemma} + + + diff --git a/ICML19/Arxiv version/sections/related_works.tex b/ICML19/Arxiv version/sections/related_works.tex new file mode 100644 index 0000000..562a826 --- /dev/null +++ b/ICML19/Arxiv version/sections/related_works.tex @@ -0,0 +1,87 @@ +%!TEX root = ../main.tex + + + +\section{Related works \label{sec:related work}} + +ALM has a long history in the optimization literature, dating back to~\cite{hestenes1969multiplier, powell1969method}. +In the special case of~\eqref{prob:01} with a convex function $f$ and a linear operator $A$, standard, inexact and linearized versions of ALM have been extensively studied~\cite{lan2016iteration,nedelcu2014computational,tran2018adaptive,xu2017inexact}. + +Classical works on ALM focused on the general template of~\eqref{prob:01} with nonconvex $f$ and nonlinear $A$, with arguably stronger assumptions and required exact solutions to the subproblems of the form \eqref{e:exac}, which appear in Step 2 of Algorithm~\ref{Algo:2}, see for instance \cite{bertsekas2014constrained}. + +A similar analysis was conducted in~\cite{fernandez2012local} for the general template of~\eqref{prob:01}. +The authors considered inexact ALM and proved convergence rates for the outer iterates, under specific assumptions on the initialization of the dual variable. +However, unlike our results, the authors did not analyze how to solve the subproblems inexactly and they did not provide total complexity results and verifiable conditions. +%\textbf{AE: Mention if this paper was convex or nonconvex?} + +Problem~\eqref{prob:01} with similar assumptions to us is also studied in~\cite{birgin2016evaluation} and~\cite{cartis2018optimality} for first-order and second-order stationarity, respectively, with explicit iteration complexity analysis. +As we have mentioned in Section~\ref{sec:cvg rate}, our iteration complexity results matches these theoretical algorithms with a simpler algorithm and a simpler analysis. +In addition, these algorithms require setting final accuracies since they utilize this information in the algorithm. In contrast to \cite{birgin2016evaluation,cartis2018optimality}, Algorithm~\ref{Algo:2} does not set accuracies a priori. + + +\cite{cartis2011evaluation} also considers the same template~\eqref{prob:01} for first-order stationarity with a penalty-type method instead of ALM. +Even though the authors show $\mathcal{O}(1/\epsilon^2)$ complexity, this result is obtained by assuming that the penalty parameter remains bounded. +We note that such an assumption can also be used to match our complexity results. + + + +\cite{bolte2018nonconvex} studies the general template~\eqref{prob:01} with specific assumptions involving local error bound conditions for the~\eqref{prob:01}. +These conditions are studied in detail in~\cite{bolte2017error}, but their validity for general SDPs~\eqref{eq:sdp} has never been established. This work also lacks the total iteration complexity analysis presented here. % that we can compare. + +Another work~\cite{clason2018acceleration} focused on solving~\eqref{prob:01} by adapting the primal-dual method of Chambolle and Pock~\cite{chambolle2011first}. +The authors proved the convergence of the method and provided convergence rate by imposing error bound conditions on the objective function that do not hold for standard SDPs.%~\eqref{eq:sdp}. + +%Some works also considered the application of ALM/ADMM to nonconvex problems~\cite{wang2015global, liu2017linearized}. +%These works assume that the operator in~\eqref{prob:01} is linear, therefore, they do not apply to our setting. +%since we have a nonlinear constraint in addition to a nonconvex objective function. + +\cite{burer2003nonlinear, burer2005local} is the first work that proposes the splitting $X=UU^\top$ for solving SDPs of the form~\eqref{eq:sdp}. +Following these works, the literature on Burer-Monteiro (BM) splitting for the large part focused on using ALM for solving the reformulated problem~\eqref{prob:nc}. + +However, this approach has a few drawbacks: First, it requires exact solutions in Step 2 of Algorithm~\ref{Algo:2} in theory, which in practice is replaced with inexact solutions. Second, their results only establish convergence without providing the rates. In this sense, our work provides a theoretical understanding of the BM splitting with inexact solutions to Step 2 of Algorithm~\ref{Algo:2} and complete iteration complexities. + +%\textbf{AE: Ahmet you discuss global optimality below but are you sure that's relevant to our work? in the sense that we just give an algorithm to solve to local optimality.} +%Their practical stopping condition is also not analyzed theoretically. +%In addition, Burer and Monteiro in~\cite{burer2005local} needed to bound the primal iterates they have to put an artificial bound to the primal domain which will be ineffective in practice; which is impossible to do without knowing the norm of the solution. +% and their results do not extend to the case where projection in primal domain are required in each iteration. +\cite{bhojanapalli2016dropping, park2016provable} are among the earliest efforts to show convergence rates for BM splitting, focusing on the special case of SDPs without any linear constraints. +For these specific problems, they prove the convergence of gradient descent to global optima with convergence rates, assuming favorable initialization. +These results, however, do not apply to general SDPs of the form~\eqref{eq:sdp} where the difficulty arises due to the linear constraints. +%{\color{red} Ahmet: I have to cite this result bc it is very relevant.} +%\textbf{AE: also if you do want to talk about global optimality you could also talk about the line of works that show no spurious local optima.} +%In their followup work~, the authors considered projected gradient descent, but only for restricted strongly convex functions. +%Their results are not able to extend to linear constraints and general convex functions. +%, therefore not applicable to general SDPs. + +%They do not apply to the general semidefinite programming since $f(U)=\langle C, UU^\ast \rangle$. %, these conditions are not satisfied. +%{\color{red} Ahmet: I have to cite this result bc it is relevant and the reviewers will complain if I don't} +\cite{bhojanapalli2018smoothed} focused on the quadratic penalty formulation of~\eqref{prob:01}, namely, +\begin{equation}\label{eq:pen_sdp} +\min_{X\succeq 0} \langle C, X\rangle + \frac{\mu}{2} \| B(x)-b\|^2, +\end{equation} +which after BM splitting becomes +\begin{equation}\label{eq:pen_nc} +\min_{U\in\mathbb{R}^{d\times r}} \langle C, UU^\top \rangle + \frac{\mu}{2} \|B(UU^\top)-b\|^2, +\end{equation} +for which they study the optimality of the second-order stationary points. These results are for establishing a connection between the stationary points of~\eqref{eq:pen_nc} and global optima of~\eqref{eq:pen_sdp}. +In contrast, we focus on the relation of the stationary points of~\eqref{eq:minmax} to the constrained problem~\eqref{prob:01}. +%\textbf{AE: so i'm not sure why we are comparing with them. their work establish global optimality of local optima but we just find local optima. it seems to add to our work rather compare against it. you also had a paragraph for global optimality with close initialization earlier. could you put the two next to each other?} + + +Another popular method for solving SDPs are due to~\cite{boumal2014manopt, boumal2016global, boumal2016non}, focusing on the case where the constraints in~\eqref{prob:01} can be written as a Riemannian manifold after BM splitting. +In this case, the authors apply the Riemannian gradient descent and Riemannian trust region methods for obtaining first- and second-order stationary points, respectively. +They obtain~$\mathcal{O}(1/\epsilon^2)$ complexity for finding first-order stationary points and~$\mathcal{O}(1/\epsilon^3)$ complexity for finding second-order stationary points. + +While these complexities appear better than ours, the smooth manifold requirement in these works is indeed restrictive. In particular, this requirement holds for max-cut and generalized eigenvalue problems, but it is not satisfied for other important SDPs such as quadratic programming (QAP), optimal power flow and clustering with general affine constraints. +In addition, as noted in~\cite{boumal2016global}, per iteration cost of their method for max-cut problem is an astronomical~$\mathcal{O}({d^6})$. % which is astronomical. %ly larger than our cost of $\mathcal{O}(n^2r)$ where $r \ll n$. + + +%\cite{birgin2016evaluation} can handle the same problem but their algorithm is much more complicated than ours. + +Lastly, there also exists a line of work for solving SDPs in their original convex formulation, in a storage efficient way~\cite{nesterov2009primal, yurtsever2015universal, yurtsever2018conditional}. These works have global optimality guarantees by their virtue of directly solving the convex formulation. On the downside, these works require the use of eigenvalue routines and exhibit significantly slower convergence as compared to nonconvex approaches~\cite{jaggi2013revisiting}. + + + + + + diff --git a/ICML19/Arxiv version/sections/slater.tex b/ICML19/Arxiv version/sections/slater.tex new file mode 100644 index 0000000..81da799 --- /dev/null +++ b/ICML19/Arxiv version/sections/slater.tex @@ -0,0 +1,106 @@ +\section{Nonconvex Slater's Condition \label{sec:slater}} + +The (convex) Slater's condition (CSC) plays a key role in convex optimization as a sufficient condition for strong duality. As a result, CSC guarantees the success of a variety of primal-dual algorithms for convex and constrained programming. As a visual example, in Program~\eqref{prob:01}, when $f=0$, $g=1_C$ is the indicator function of a convex set $C$, and $A$ is an affine operator, CSC removes any pathological cases by ensuring that the affine subspace is not tangent to $C$, see Figure~\ref{fig:convex_slater}. \textbf{We should add a figure here with circle and line.} + +Likewise, to successfully solve Program~\eqref{prob:01} with nonlinear constraints, we require the following condition which, loosely speaking, extends CSC to the nonconvex setting, as clarified shortly afterwards. +\begin{definition}\label{defn:nonconvex slater} \textbf{\emph{(Nonconvex Slater's condition)}} +For $\rho,\rho'>0$ and subspace $S\subset \RR^{d}$, let +\begin{align} +\nu(g,A,S) := +\begin{cases} +\underset{v,x}{\min} \, \frac{\left\| \left( \operatorname{id} - P_S P_{\partial g(x)/\b} \right) ( DA(x)^\top v) \right\|}{\|v\|} \\ +\|v\|\le \rho\\ +\|x\|\le \rho', +\end{cases} +\label{eq:new slater defn} +\end{align} +where $\operatorname{id}$ is the identity operator, $P_{\partial g(x)/\b}$ projects onto the set $\partial g(x)/\b$, and $DA(x)$ is the Jacobian of $A$. We say that Program (\ref{prob:01}) satisfies the nonconvex Slater's condition if $\nu(g,A,S) >0$. +\end{definition} +A few remarks about the nonconvex Slater's condition (NSC) is in order. + +\paragraph{Jacobian $DA$.} +As we will see later, $DA(x)^\top \overset{\operatorname{QR}}{=} Q(x) R(x)$ in NSC might be replaced with its orthonormal basis, namely, $Q(x)$. For simplicity, we will avoid this minor change and instead, whenever needed, assume that $DA(x)$ is nonsingular; otherwise a simple projection can remove any redundancy from $A(x)=0$ in Program~\eqref{prob:01}. + +\paragraph{Subspace $S$.} The choice of subspace $S$ helps broaden the generality of NSC. In particular, when $S = \RR^{d}$, \eqref{eq:new slater defn} takes the simpler form of +\begin{align} +\nu(g,A,S) := +\begin{cases} +\underset{v,x}{\min} \, \frac{ \dist( DA(x)^\top v , \partial g(x)/\b) }{\|v\|} \\ +\|v\|\le \rho\\ +\|x\|\le \rho', +\end{cases} +\label{eq:defn_nu_A} +\end{align} +where $\dist(\cdot,\partial g(x)/\b)$ returns the Euclidean distance to the set $\partial g(x)/\b$. + +\paragraph{Convex case.} +To better parse Definition \ref{defn:nonconvex slater}, let us assume that $f:\RR^d\rightarrow\RR$ is convex, $g = 1_C:\RR^d\rightarrow\RR$ is the indicator function for a convex and bounded set $C$, and $A$ is a nonsingular linear operator represented with the full-rank matrix $A\in \RR^{m\times d}$. +%We also let $T_C(x)$ denote the tangent cone to $C$ at $x$, and reserve $P_{T_C(x)}:\RR^d\rightarrow\RR^d$ for the orthogonal projection onto this cone. +%We also set $S_K = \RR^d$ to simplify the discussion. +We can now study the geometric interpretation of $\nu()$. +Assuming that $S=\RR^d$ and using the well-known Moreau decomposition, it is not difficult to rewrite \eqref{eq:new slater defn} as +\begin{align} +\nu(g,A,S) & := +\begin{cases} +\underset{v,x}{\min} \, \, \frac{\left\| P_{T_C(x)} A^\top v \right\|}{\|v\|} \\ +\|v\|\le \rho\\ +\|x\|\le \rho' +\end{cases} \nonumber\\ +& = \begin{cases} +\underset{x}{\min} \,\, \eta_{m}\left( P_{T_C(x)} A^\top \right) \\ +\|x\|\le \rho', +\end{cases} +\label{eq:nu for cvx} +\end{align} +where $P_{T_C(x)}$ is the projection onto the tangent cone of $C$ at $x$, and $\eta_{m}(\cdot)$ returns the $m$th largest singular value of its input matrix. Intuitively then, NSC ensures that the the row span of $A$ is not tangent to $C$, similar to CSC. This close relationship between NSC and CSC is formalized next and proved in Appendix~\ref{sec:proof of prop}. + +\begin{proposition}\label{prop:convex} +In Program (\ref{prob:01}), suppose that +\begin{itemize} +\item $f:\RR^d\rightarrow\RR$ is convex, +\item $g=1_C$ is the indicator on a convex and bounded set $C\subset\RR^d$, +\item $A:\RR^d\rightarrow\RR^m$ is a nonsingular linear operator, represented with the full-rank matrix $A\in \RR^{m\times d}$.\footnote{As mentioned earlier, it is easy to remove the full-rank assumption by replacing $DA(x)$ in \eqref{eq:new slater defn} with its orthonormal basis. We assume $A$ to be full-rank for clarity at the cost of a simple projection to remove any ``redundant measurements'' from Program~\eqref{prob:01}.} +\end{itemize} +Assume also that Program (\ref{prob:01}) is feasible, namely, there exists $x\in C$ such that $Ax=0$. Then CSC holds if NSC holds. +Moreover, suppose that $S$ is the affine hull of $C$. Then CSC holds if and only if NSC holds. +\end{proposition} + +\paragraph{Boundedness of $C$.} +Let us add that the boundedness of $C$ in Proposition~\ref{prop:convex} is necessary. For example, suppose that $A\in \RR^{1\times d}$ is the vector of all ones and, for a small perturbation vector $\epsilon\in \RR^d$, let $C=\{x\in \RR^d: (A+\epsilon) x\ge 0\}$ be a half space. Then CSC holds but $\nu(g,A,S)$ (with $S=\RR^d$) can be made arbitrarily small by making $\epsilon$ small. + + +\section{Proof of Proposition \ref{prop:convex} \label{sec:proof of prop}} +To prove the first claim of the proposition, suppose that CSC does not hold, namely, that +\begin{equation} +\relint(\Null(A) \cap C) = \Null(A)\cap \relint(C) = \emptyset, +\label{eq:no feas} +\end{equation} +where $\Null(A)$ and $\relint(C)$ denote the null space of the matrix $A$ and the relative interior of $C$, respectively. +By assumption, there exists $x_0\in C$ such that $Ax_0=0$. It follows from \eqref{eq:no feas} that $x_0\in \text{boundary}(C)$ and that $\Null(A)$ supports $C$ at $x_0$, namely, +$A x\ge 0$, for every $x\in C$. (The inequality applies to each entry of the vector $Ax$.) +Consequently, $\Null(A) \cap T_C(x_0) \ne \{0\}$, where $T_C(x_0)$ is the tangent cone of the set $C$ at $x_0$. +Equivalently, it holds that +$\row(A)\cap N_C(x_0) \ne \{0\}$, where $\row(A)$ is the row space of the matrix $A$. That is, there exists a unit-norm vector $v$ such that +$P_{T_C(x_0)}A^\top v=0$ +and, consequently, $P_S P_{T_C(x_0)}A^\top v=0$. Let us take $\rho'=\|x_0\|$ in \eqref{eq:nu for cvx}. We then conclude that $\nu(g,A,S)=0$, namely, NSC also does not hold, which proves the first claim in Proposition \ref{prop:convex}. + +For the converse, suppose that NSC does not hold with $\rho'=0$, namely, there exists $x\in \RR^d$ such that +\begin{align*} +\eta_m(P_S P_{T_C(x)} A^\top) & = \eta_m( P_{T_C(x)} A^\top) =0, +\end{align*} +\begin{align*} +\quad A(x) = 0, \quad x\in C, +\end{align*} +where the first equality above holds because $S$ is the affine hull of $C$. + Then, thanks to the boundedness of $C$, it must be the case that $x\in \text{boundary}(C)$. Indeed, if $x\in \text{relint}(C)$, we have that $T_C(x)=S$ and thus +\begin{equation*} +\eta_m(P_{S}P_{T_C(x)} A^\top)= \eta_m( P_S A^\top)>0, +\end{equation*} +where the inequality holds because, by assumption, $A$ is full-rank. +Since $x\in\text{boundary}(C)$, it follows that $\text{dim}(T_C(x)) \ge m-1$. That is, $\text{row}(A)$ contains a unique direction orthogonal to $T_C(x)$. In particular, it follows that $\text{null}(A) \subset T_C(x)$ and, consequently, $\text{int}(\text{null}(A)\cap C) =\emptyset $, namely, CSC does not hold. This proves the second (and last) claim in Proposition \ref{prop:convex}. + + + + + +