SCORE: Approximating Curvature Information under Self-Concordant Regularization

TL;DR

The paper introduces SCORE, a framework leveraging self-concordant regularization to incorporate second-order information in convex and non-convex optimization, improving convergence speed and generalization.

Contribution

It proposes the GGN-SCORE algorithm that efficiently exploits Hessian structure for faster convergence in regularized convex and non-convex problems.

Findings

GGN-SCORE accelerates convergence compared to first-order methods.

The method improves model generalization in regularized problems.

Numerical experiments confirm efficiency and fast convergence.

Abstract

Optimization problems that include regularization functions in their objectives are regularly solved in many applications. When one seeks second-order methods for such problems, it may be desirable to exploit specific properties of some of these regularization functions when accounting for curvature information in the solution steps to speed up convergence. In this paper, we propose the SCORE (self-concordant regularization) framework for unconstrained minimization problems which incorporates second-order information in the Newton-decrement framework for convex optimization. We propose the generalized Gauss-Newton with Self-Concordant Regularization (GGN-SCORE) algorithm that updates the minimization variables each time it receives a new input batch. The proposed algorithm exploits the structure of the second-order information in the Hessian matrix, thereby reducing computational overhead. GGN-SCORE demonstrates how to speed up convergence while also improving model generalization for problems that involve regularized minimization under the proposed SCORE framework. Numerical experiments show the efficiency of our method and its fast convergence, which compare favorably against baseline first-order and quasi-Newton methods. Additional experiments involving non-convex (overparameterized) neural network training problems show that the proposed method is promising for non-convex optimization.