Composite convex minimization involving self-concordant-like cost functions
This work addresses convex optimization for applications with self-concordant-like properties, offering incremental improvements in convergence for specific domains.
The paper tackles the problem of composite convex minimization involving self-concordant-like functions, a new analytical structure, by developing a variable metric framework and proving that a basic gradient algorithm achieves improved convergence guarantees compared to fast Lipschitz-based methods, with numerical tests on real datasets confirming theoretical results.
The self-concordant-like property of a smooth convex function is a new analytical structure that generalizes the self-concordant notion. While a wide variety of important applications feature the self-concordant-like property, this concept has heretofore remained unexploited in convex optimization. To this end, we develop a variable metric framework of minimizing the sum of a "simple" convex function and a self-concordant-like function. We introduce a new analytic step-size selection procedure and prove that the basic gradient algorithm has improved convergence guarantees as compared to "fast" algorithms that rely on the Lipschitz gradient property. Our numerical tests with real-data sets shows that the practice indeed follows the theory.