Stochastic Coordinate Minimization with Progressive Precision for Stochastic Convex Optimization
This provides a scalable and parallelizable method for large-scale optimization, extending low-dimensional routines to high-dimensional problems with applications in distributed computing.
The paper tackles stochastic convex optimization by developing a framework based on iterative coordinate minimization with optimal control of minimization precision, achieving order-optimal regret performance for strongly convex and separably nonsmooth functions.
A framework based on iterative coordinate minimization (CM) is developed for stochastic convex optimization. Given that exact coordinate minimization is impossible due to the unknown stochastic nature of the objective function, the crux of the proposed optimization algorithm is an optimal control of the minimization precision in each iteration. We establish the optimal precision control and the resulting order-optimal regret performance for strongly convex and separably nonsmooth functions. An interesting finding is that the optimal progression of precision across iterations is independent of the low-dimensional CM routine employed, suggesting a general framework for extending low-dimensional optimization routines to high-dimensional problems. The proposed algorithm is amenable to online implementation and inherits the scalability and parallelizability properties of CM for large-scale optimization. Requiring only a sublinear order of message exchanges, it also lends itself well to distributed computing as compared with the alternative approach of coordinate gradient descent.