Statistical Optimization in High Dimensions

References

• Ansari A, Essegaier S, Kohli R (2000) Internet recommendation systems. J. Marketing Res. 37(3):363–375.Crossref, Google Scholar
• Anthony M, Bartlett PL (1999) Neural Network Learning: Theoretical Foundations (Cambridge University Press, Cambridge, UK).Crossref, Google Scholar
• Ben-Tal A, Nemirovski A (1999) Robust solutions of uncertain linear programs. Oper. Res. Lett. 25(1):1–13.Crossref, Google Scholar
• Ben-Tal A, El Ghaoui L, Nemirovski A (2009) Robust Optimization (Princeton University Press, Princeton, NJ).Crossref, Google Scholar
• Bertsimas D, Brown DB (2009) Constructing uncertainty sets for robust linear optimization. Oper. Res. 57(6):1483–1495.Link, Google Scholar
• Bertsimas D, Sim M (2004) The price of robustness. Oper. Res. 52(1):35–53.Link, Google Scholar
• Bertsimas D, Brown DB, Caramanis C (2011) Theory and applications of robust optimization. SIAM Rev. 53(3):464–501.Crossref, Google Scholar
• Bertsimas D, Gupta V, Kallus N (2015) Data-driven robust optimization. Submitted.Google Scholar
• Birge JR, Louveaux F (1997) Introduction to Stochastic Programming (Springer, New York).Google Scholar
• Bodapati A (2008) Recommendation systems with purchase data. J. Marketing Res. 45(1):77–93.Crossref, Google Scholar
• Bottou L, Lin C-J (2007) Support vector machine solvers. Bottou L, Chapelle O, DeCoste D, Weston J, eds. Large Scale Kernel Machines (MIT Press, Cambridge, MA), 1–28.Crossref, Google Scholar
• Cai J-F, Candès E, Shen Z (2008) A singular value thresholding algorithm for matrix completion. SIAM J. Optim. 20:1956–1982.Crossref, Google Scholar
• Calafiore G, Campi M (2005) Uncertain convex programs: Randomized solutions and confidence levels. Math. Programming 102(1):25–46.Crossref, Google Scholar
• Calafiore G, El Ghaoui L (2006) On distributionally robust chance-constrained linear programs. J. Optim. Theory Appl. 130(1):1–22.Crossref, Google Scholar
• Candès EJ, Tao T (2007) The Dantzig selector: Statistical estimation when p is much larger than n. Ann. Statist. 35(6):2313–2351.Crossref, Google Scholar
• Cortes C, Vapnik VN (1995) Support vector networks. Machine Learn. 20(3):273–297.Crossref, Google Scholar
• Delage E, Ye Y (2010) Distributionally robust optimization under moment uncertainty with applications to data-driven problems. Oper. Res. 58(3):596–612.Link, Google Scholar
• Donoho DL (2000) High-dimensional data analysis: The curses and blessings of dimensionality. Lecture, Math Challenges of the 21st Century, American Math. Society Conference.Google Scholar
• Grant M, Boyd S (2011) CVX: Matlab software for disciplined convex programming, version 1.21. http://cvxr.com/cvx.Google Scholar
• Jiang R, Guan Y (2015) Data-driven chance constrained stochastic program. Math. Programming 158(1):291–327.Google Scholar
• Jolliffe IT (1986) Principal Component Analysis, Springer Series in Statistics (Springer, Berlin).Crossref, Google Scholar
• Kleywegt A, Shapiro A, de Mello TH (2002) The sample average approximation method for stochastic discrete optimization. SIAM J. Optim. 12(2):479–502.Crossref, Google Scholar
• Levi R, Perakis G, Uichanco J (2015) The data-driven newsvendor problem: New bounds and insights. Oper. Res. 63(6):1294–1306.Link, Google Scholar
• Levi R, Roundy R, Shmoys D (2007) Provably near-optimal sampling-based policies for stochastic inventory control models. Math. Oper. Res. 32(4):821–839.Link, Google Scholar
• Li JLL, Zhang T (2009) Sparse online learning via truncated gradient. J. Machine Learn. Res. 10:777–801.Google Scholar
• Louviere J, Hensher D, Swait J (2000) Stated Choice Methods—Analysis and Application (Cambridge University Press, Cambridge, UK).Crossref, Google Scholar
• Mohajerin Esfahani P, Kuhn D (2015) Data-driven distributionally robust optimization using the Wasserstein metric: Performance guarantees and tractable reformulations. arXiv preprint arXiv:1505.05116.Google Scholar
• Natarajan K, Sim M, Uichanco J (2010) Tractable robust expected utility and risk models for portfolio optimization. Math. Finance 20(4):695–731.Crossref, Google Scholar
• Netflix (2009) Netflix prize: FAQ. Accessed September 1, 2015, http://www.netflixprize.com/faq.Google Scholar
• Platt J (1998) Sequential minimal optimization: A fast algorithm for training support vector machines. Technical report, Microsoft Research, Redmond, WA.Google Scholar
• Prékopa A (1995) Stochastic Programming (Kluwer, Amsterdam).Crossref, Google Scholar
• Rockafellar RT (2002) Conditional value-at-risk for general loss distributions. J. Banking and Finance 26(7):1443–1471.Crossref, Google Scholar
• Rockafellar RT, Uryasev ST (2000) Optimization of conditional value-at-risk. J. Risk 2(3):21–41.Crossref, Google Scholar
• Shalev-Shwartz S, Singer Y (2007) A primal-dual perspective of online learning algorithms. Machine Learn. 69(2–3):115–142.Crossref, Google Scholar
• Shalev-Shwartz S, Srebro N (2008) SVM optimization: Inverse dependence on training set size. Cohen WW, McCallum A, Roweis ST, eds. Proc. 25nd Internat. Conf. Machine Learn., ICML ’08 (ACM, New York), 928–935.Crossref, Google Scholar
• Shapiro A, Homem-de Mello T (2000) On the rate of convergence of optimal solutions of Monte Carlo approximations of stochastic programs. SIAM J. Optim. 11(1):70–86.Crossref, Google Scholar
• Shivaswamy PK, Bhattacharyya C, Smola AJ (2006) Second order cone programming approaches for handling missing and uncertain data. J. Machine Learn. Res. 7(July):1283–1314.Google Scholar
• Steinwart I, Christmann A (2008) Support Vector Machines (Springer, New York).Crossref, Google Scholar
• Sturm J (1999) Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones. Optim. Methods and Software 11(1–4):625–653.Crossref, Google Scholar
• Tropp JA (2006) Just relax: Convex programming methods for identifying sparse signals in noise. IEEE Trans. Inform. Theory 51(3):1030–1051.Crossref, Google Scholar
• van der Vaart AW, Wellner JA (2000) Weak Convergence and Empirical Processes (Springer, New York).Google Scholar
• Vapnik VN (1998) Statistical Learning Theory (John Wiley & Sons, New York).Google Scholar
• Vershynin R (2011) Introduction to the non-asymptotic analysis of random matrices. arXiv: 1011.3027v6.Google Scholar
• Xu H, Caramanis C, Mannor S (2009) Robustness and regularization of support vector machines. J. Machine Learn. Res. 10(July):1485–1510.Google Scholar
• Xu H, Caramanis C, Mannor S (2010) Robust regression and Lasso. IEEE Trans. Inform. Theory 56(7):3561–3574.Crossref, Google Scholar
• Xu H, Caramanis C, Mannor S (2012) Statistical optimization in high dimensions. Lawrence ND, Girolami MA, eds. Proc. Fifteenth Internat. Conf. Artificial Intelligence and Statist. AISTATS ’12 (JMLR), 1332–1340.Google Scholar