The alternating gradient descent (AGD) is a simple but popular algorithm which has been applied to problems in optimization, machine learning, data ming, and signal processing, etc. The algorithm updates two blocks of variables in an alternating manner, in which a gradient step is taken on one block, while keeping the remaining block fixed. When the objective function is nonconvex, it is well-known the AGD converges to the first-order stationary solution with a global sublinear rate. In this paper, we show that a variant of AGD-type algorithms will not be trapped by "bad" stationary solutions such as saddle points and local maximum points. In particular, we consider a smooth unconstrained optimization problem, and propose a perturbed AGD (PA-GD) which converges (with high probability) to the set of second-order stationary solutions (SS2) with a global sublinear rate. To the best of our knowledge, this is the first alternating type algorithm which takes @math iterations to achieve SS2 with high probability [where polylog @math is polynomial of the logarithm of dimension @math of the problem].
A central challenge to using first-order methods for optimizing nonconvex problems is the presence of saddle points. First-order methods often get stuck at saddle points, greatly deteriorating their performance. Typically, to escape from saddles one has to use second-order methods. However, most works on second-order methods rely extensively on expensive Hessian-based computations, making them impractical in large-scale settings. To tackle this challenge, we introduce a generic framework that minimizes Hessian based computations while at the same time provably converging to second-order critical points. Our framework carefully alternates between a first-order and a second-order subroutine, using the latter only close to saddle points, and yields convergence results competitive to the state-of-the-art. Empirical results suggest that our strategy also enjoys a good practical performance. We consider the minimization of non-convex quadratic forms regularized by a cubic term, which exhibit multiple saddle points and poor local minima. Nonetheless, we prove that, under mild assumptions, gradient descent approximates the @math to within @math accuracy in @math steps for large @math and @math steps for small @math (compared to a condition number we define), with at most logarithmic dependence on the problem dimension. When we use gradient descent to approximate the Nesterov-Polyak cubic-regularized Newton step, our result implies a rate of convergence to second-order stationary points of general smooth non-convex functions.  Preface 1. Introduction Part I. Preliminaries: 2. Basic Concepts 3. Basic Analysis and Optimality Conditions 4. Basic Linear Algebra 5. Krylov Subspace Methods Part II. Trust-Region Methods for Unconstrained Optimization: 6. Global Convergence of the Basic Algorithm 7.The Trust-Region Subproblem 8. Further Convergence Theory Issues 9. Conditional Models 10. Algorithmic Extensions 11. Nonsmooth Problems Part III. Trust-Region Methods for Constrained Optimization with Convex Constraints: 12. Projection Methods for Convex Constraints 13. Barrier Methods for Inequality Constraints Part IV. Trust-Region Mewthods for General Constained Optimization and Systems of Nonlinear Equations: 14. Penalty-Function Methods 15. Sequential Quadratic Programming Methods 16. Nonlinear Equations and Nonlinear Fitting Part V. Final Considerations: Practicalities Afterword Appendix: A Summary of Assumptions Annotated Bibliography Subject and Notation Index Author Index.