Forward-backward truncated Newton methods for convex composite optimization

This paper proposes two proximal Newton-CG methods for convex nonsmooth optimization problems in composite form. The algorithms are based on a a reformulation of the original nonsmooth problem as the unconstrained minimization of a continuously differentiable function, namely the forward-backward envelope (FBE). The first algorithm is based on a standard line search strategy, whereas the second one combines the global efficiency estimates of the corresponding first-order methods, while achieving fast asymptotic convergence rates. Furthermore, they are computationally attractive since each Newton iteration requires the approximate solution of a linear system of usually small dimension

A Semismooth Newton Stochastic Proximal Point Algorithm with Variance Reduction

We develop an implementable stochastic proximal point (SPP) method for a class of weakly convex, composite optimization problems. The proposed stochastic proximal point algorithm incorporates a variance reduction mechanism and the resulting SPP updates are solved using an inexact semismooth Newton framework. We establish detailed convergence results that take the inexactness of the SPP steps into account and that are in accordance with existing convergence guarantees of (proximal) stochastic variance-reduced gradient methods. Numerical experiments show that the proposed algorithm competes favorably with other state-of-the-art methods and achieves higher robustness with respect to the step size selection

Newton-type methods under generalized self-concordance and inexact oracles

Many modern applications in machine learning, image/signal processing, and statistics require to solve large-scale convex optimization problems. These problems share some common challenges such as high-dimensionality, nonsmoothness, and complex objectives and constraints. Due to these challenges, the theoretical assumptions for existing numerical methods are not satisfied. In numerical methods, it is also impractical to do exact computations in many cases (e.g. noisy computation, storage or time limitation). Therefore, new approaches as well as inexact computations to design new algorithms should be considered. In this thesis, we develop fundamental theories and numerical methods, especially second-order methods, to solve some classes of convex optimization problems, where first-order methods are inefficient or do not have a theoretical guarantee. We aim at exploiting the underlying smoothness structures of the problem to design novel Newton-type methods. More specifically, we generalize a powerful concept called \mbox{self-concordance} introduced by Nesterov and Nemirovski to a broader class of convex functions. We develop several basic properties of this concept and prove key estimates for function values and its derivatives. Then, we apply our theory to design different Newton-type methods such as damped-step Newton methods, full-step Newton methods, and proximal Newton methods. Our new theory allows us to establish both global and local convergence guarantees of these methods without imposing unverifiable conditions as in classical Newton-type methods. Numerical experiments show that our approach has several advantages compared to existing works. In the second part of this thesis, we introduce new global and local inexact oracle settings, and apply them to develop inexact proximal Newton-type schemes for optimizing general composite convex problems equipped with such inexact oracles. These schemes allow us to measure errors theoretically and systematically and still lead to desired convergence results. Moreover, they can be applied to solve a wider class of applications arising in statistics and machine learning.Doctor of Philosoph