Kazuhiro Hishinuma | Searcharxiv

arXiv2019

Fixed Point Quasiconvex Subgradient Method

Constrained quasiconvex optimization problems appear in many fields, such as economics, engineering, and management science. In particular, fractional programming, which models ratio indicators such as the profit/cost ratio as fractional objective functions, is an important instance. Subgradient methods and their variants are useful ways for solving these problems efficiently. Many complicated constraint sets onto which it is hard to compute the metric projections in a realistic amount of time appear in these applications. This implies that the existing methods cannot be applied to quasiconvex optimization over a complicated set. Meanwhile, thanks to fixed point theory, we can construct a computable nonexpansive mapping whose fixed point set coincides with a complicated constraint set. This paper proposes an algorithm that uses a computable nonexpansive mapping for solving a constrained quasiconvex optimization problem. We provide convergence analyses for constant and diminishing step-size rules. Numerical comparisons between the proposed algorithm and an existing algorithm show that the proposed algorithm runs stably and quickly even when the running time of the existing algorithm exceeds the time limit.

math.OC↗

arXiv2019

Incremental and Parallel Machine Learning Algorithms with Automated Learning Rate Adjustments

The existing machine learning algorithms for minimizing the convex function over a closed convex set suffer from slow convergence because their learning rates must be determined before running them. This paper proposes two machine learning algorithms incorporating the line search method, which automatically and algorithmically finds appropriate learning rates at run-time. One algorithm is based on the incremental subgradient algorithm, which sequentially and cyclically uses each of the parts of the objective function; the other is based on the parallel subgradient algorithm, which uses parts independently in parallel. These algorithms can be applied to constrained nonsmooth convex optimization problems appearing in tasks of learning support vector machines without adjusting the learning rates precisely. The proposed line search method can determine learning rates to satisfy weaker conditions than the ones used in the existing machine learning algorithms. This implies that the two algorithms are generalizations of the existing incremental and parallel subgradient algorithms for solving constrained nonsmooth convex optimization problems. We show that they generate sequences that converge to a solution of the constrained nonsmooth convex optimization problem under certain conditions. The main contribution of this paper is the provision of three kinds of experiment showing that the two algorithms can solve concrete experimental problems faster than the existing algorithms. First, we show that the proposed algorithms have performance advantages over the existing ones in solving a test problem. Second, we compare the proposed algorithms with a different algorithm Pegasos, which is designed to learn with a support vector machine efficiently, in terms of prediction accuracy, value of the objective function, and computational time. Finally, we use...

math.OC↗