Proximal Newton-type methods for minimizing composite functions

Abstract: We generalize Newton-type methods for minimizing smooth functions to handle a sum of two convex functions: a smooth function and a nonsmooth function with a simple proximal mapping. We show that the resulting proximal Newton-type methods inherit the desirable convergence behavior of Newton-type methods for minimizing smooth functions, even when search directions are computed inexactly. Many popular methods tailored to problems arising in bioinformatics, signal processing, and statistical learning are special cases of proximal Newton-type methods, and our analysis yields new convergence results for some of these methods.

• The paper presents a proximal Newton framework that integrates second-order information with proximal mapping to optimize composite functions.
• It proves that with accurate Hessian approximations, the method achieves global convergence and exhibits superlinear convergence under mild conditions.
• Empirical evaluations on logistic regression and inverse covariance estimation reveal that these methods can outperform traditional first-order approaches.

Proximal Newton-Type Methods for Minimizing Composite Functions

The paper presents a comprehensive analysis of proximal Newton-type methods for optimizing composite functions. Specifically, these methods deal with minimization problems where the objective function is a sum of two convex functions: a smooth, differentiable component ($g(x)$) and a non-smooth component ($h(x)$) that possesses efficient proximal mappings. This work generalizes classical Newton-type methods, traditionally reserved for smooth optimization, to encompass composite objective functions—a formulation frequently encountered in fields like bioinformatics, signal processing, and statistical learning.

Overview of Proximal Newton-Type Methods

The proximal Newton-type method extends the proximal gradient approach by incorporating second-order information to address the curvature of the smooth part of the function. The essence of the method lies in forming a local quadratic approximation of the smooth portion while handling the non-smooth portion via proximal operations. The paper discusses various strategies for defining the Hessian approximation matrix $H_k$, which is pivotal in forming these quadratic models. Exact and approximate strategies are examined, the latter of which can improve computational efficiency.

Convergence Analysis

One of the notable contributions of this study is the investigation into the convergence behaviors of these methods. It proves that proximal Newton-type methods maintain global convergence properties under mild conditions and exhibit superlinear convergence under suitable assumptions for the smooth component $g(x)$. For practical computation, the authors explore inexact proximal Newton approaches, which solve the quadratic subproblems only approximately, and they provide conditions under which inexact methods retain desirable convergence properties.

Numerical Experiments and Comparative Evaluation

Empirical evaluations are crucial for demonstrating the practical viability of theoretical methods. The authors conduct experiments on two types of optimization problems: inverse covariance estimation and logistic regression with $\ell_1$ regularization. By comparing the proximal Newton-type methods with established first-order methods like SpaRSA and FISTA, they illustrate that proximal Newton-type approaches can outperform in scenarios where function evaluation is computationally expensive or when seeking high-precision solutions.

Implications and Future Work

The implications of deploying proximal Newton-type methods are broad, with potential applications spanning various high-dimensional data analysis problems. The paper encourages further research into optimizing these methods, possibly exploring adaptive strategies for Hessian approximation or tailoring subproblem solvers to exploit specific structures of $h(x)$. Future directions could also include extending these methods to non-convex settings or integrating other advanced optimization paradigms like machine learning models.

In summation, this paper delivers a thorough theoretical foundation and practical insights into proximal Newton-type methods, reinforcing their utility as a powerful class of solvers for composite function minimization. The findings pave the way for more specialized adaptations, catering to the nuanced demands of contemporary optimization tasks in applied sciences.