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This work concerns the local convergence theory of Newton and quasi-Newton methods for convex-composite optimization: where one minimizes an objective that can be written as the composition of a convex function with one that is continuiously differentiable. We focus on the case in which the convex function is a potentially infinite-valued piecewise linear-quadratic function. Such problems include nonlinear programming, mini-max optimization, and estimation of nonlinear dynamics with non-Gaussian noise as well as many modern approaches to large-scale data analysis and machine learning. Our approach embeds the optimality conditions for convex-composite optimization problems into a generalized equation. We establish conditions for strong metric subregularity and strong metric regularity of the corresponding set-valued mappings. This allows us to extend classical convergence of Newton and quasi-Newton methods to the broader class of nonfinite valued piecewise linear-quadratic convex-composite optimization problems. In particular, we establish local quadratic convergence of the Newton method under conditions that parallel those in nonlinear programming.
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LSEMINK: a modified Newton–Krylov method for Log-Sum-Exp minimization
This paper introduces LSEMINK, an effective modified Newton–Krylov algorithm geared toward minimizing the log-sum-exp function for a linear model. Problems of this kind arise commonly, for example, in geometric programming and multinomial logistic regression. Although the log-sum-exp function is smooth and convex, standard line-search Newton-type methods can become inefficient because the quadratic approximation of the objective function can be unbounded from below. To circumvent this, LSEMINK modifies the Hessian by adding a shift in the row space of the linear model. We show that the shift renders the quadratic approximation to be bounded from below and that the overall scheme converges to a global minimizer under mild assumptions. Our convergence proof also shows that all iterates are in the row space of the linear model, which can be attractive when the model parameters do not have an intuitive meaning, as is common in machine learning. Since LSEMINK uses a Krylov subspace method to compute the search direction, it only requires matrix-vector products with the linear model, which is critical for large-scale problems. Our numerical experiments on image classification and geometric programming illustrate that LSEMINK considerably reduces the time-to-solution and increases the scalability compared to geometric programming and natural gradient descent approaches. It has significantly faster initial convergence than standard Newton–Krylov methods, which is particularly attractive in applications like machine learning. In addition, LSEMINK is more robust to ill-conditioning arising from the nonsmoothness of the problem. We share our MATLAB implementation at a GitHub repository (https://github.com/KelvinKan/LSEMINK).
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- PAR ID:
- 10594556
- Publisher / Repository:
- RICAM
- Date Published:
- Journal Name:
- ETNA - Electronic Transactions on Numerical Analysis
- Volume:
- 60
- ISSN:
- 1068-9613
- Page Range / eLocation ID:
- 618 to 635
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1553/etna_vol60s618
