Scalable Nonconvex Inexact Proximal Splitting. However, to obtain the desired conclusion, an envelope theorem should be invoked in the proof.
In fact, just having a unique solution to displayed equation 18 of the above paper is not sufficient to guarantee the differentiability of the function E g. The envelope theorem mentioned on p. Manuscript, Envelope theorems have many applications in economics; see the above paper and the pointers therein for details. The statistical nature of regularized loss minimization problems can be exploited to develop an alternative convergence analysis of first-order methods.
This is done, e. Annals of Statistics 40 5 : , Supplementary Material The above paper shows that for several classes of regularized loss minimization problems, certain first-order methods enjoy global linear convergence up to the statistical precision of the model. Modulo the global vs. Nevertheless, the approach developed in the above paper can be used to tackle problems that are not known to possess an error bound.
Week 6 - Feb 13, Algorithmic aspects of regularized loss minimization problems - Error bounds. Reading: The lecture material is based on Zhou, So. Lecture Notes A streamlined proof of Hoffman's error bound can be found in Section Foundations of Optimization. The notion of linear regularity used in Corollary 1 of the lecture notes and other related notions are discussed at length in Bauschke, Borwein. SIAM Review 38 3 : , The outer Lipschitz continuity property used in the lecture is discussed in Sections 3C and 3D particularly Theorem 3D.
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Implicit Functions and Solution Mappings. Week 7 - Feb 20, Design and analysis of a successive quadratic approximation method for regularized loss minimization problems. Reading: The lecture material is based on Yue, Zhou, So.
Lecture Notes Week 8 - Feb 27, Sketching and its applications to the design and analysis of second-order methods. Reading: The lecture material is based on Pilanci, Wainwright. As noted in the lecture, the approach in the above paper does not offer any advantage for unconstrained problems, as the sketching dimension could be as high as the dimension of the decision vector.
In fact, the same is true for a constrained problem if the optimal solution lies in the interior of the constraint set; see displayed equation 3.
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Lecture Notes Week 9 - Mar 6, 7: Non-convex regularized loss minimization for sparse linear regression. Reading: The lecture material is based on Loh, Wainwright. Annals of Statistics 40 3 : , Journal of the American Statistical Association 96 : , Annals of Statistics 38 2 : , For an in-depth analysis of the role of various non-convex regularizers in sparse estimation problems, see Zhang, Zhang.
Supplementary Material The lecture material is based on Loh, Wainwright. Journal of Machine Learning Research 16 Mar : , Lecture Notes Week 11 - Mar 20, Bound on optimization error in the non-convex setting. Lecture Notes The restricted strong convexity property for sparse linear regression with additively corrupted observations is established in Corollary 1 of Loh, Wainwright. Big Data Notes from Guy Harrison. Emerging Technologies from Guy Harrison.
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