Recent Advances in Randomized Methods for Big Data Optimization

In this thesis, we discuss and develop randomized algorithms for big data problems. In particular, we study the finite-sum optimization with newly emerged variance- reduction optimization methods (Chapter 2), explore the efficiency of second-order information applied to both convex and non-convex finite-sum objectives (Chapter 3) and employ the fast first-order method in power system problems (Chapter 4).In Chapter 2, we propose two variance-reduced gradient algorithms – mS2GD and SARAH. mS2GD incorporates a mini-batching scheme for improving the theoretical complexity and practical performance of SVRG/S2GD, aiming to minimize a strongly convex function represented as the sum of an average of a large number of smooth con- vex functions and a simple non-smooth convex regularizer. While SARAH, short for StochAstic Recursive grAdient algoritHm and using a stochastic recursive gradient, targets at minimizing the average of a large number of smooth functions for both con- vex and non-convex cases. Both methods fall into the category of variance-reduction optimization, and obtain a total complexity of O((n+κ)log(1/ε)) to achieve an ε-accuracy solution for strongly convex objectives, while SARAH also maintains a sub-linear convergence for non-convex problems. Meanwhile, SARAH has a practical variant SARAH+ due to its linear convergence of the expected stochastic gradients in inner loops.In Chapter 3, we declare that randomized batches can be applied with second- order information, as to improve upon convergence in both theory and practice, with a framework of L-BFGS as a novel approach to finite-sum optimization problems. We provide theoretical analyses for both convex and non-convex objectives. Meanwhile, we propose LBFGS-F as a variant where Fisher information matrix is used instead of Hessian information, and prove it applicable to a distributed environment within the popular applications of least-square and cross-entropy losses.In Chapter 4, we develop fast randomized algorithms for solving polynomial optimization problems on the applications of alternating-current optimal power flows (ACOPF) in power system field. The traditional research on power system problem focuses on solvers using second-order method, while no randomized algorithms have been developed. First, we propose a coordinate-descent algorithm as an online solver, applied for solving time-varying optimization problems in power systems. We bound the difference between the current approximate optimal cost generated by our algorithm and the optimal cost for a relaxation using the most recent data from above by a function of the properties of the instance and the rate of change to the instance over time. Second, we focus on a steady-state problem in power systems, and study means of switching from solving a convex relaxation to Newton method working on a non-convex (augmented) Lagrangian of the problem.