An Algorithm for Unconstrained Quadratically Penalized Convex Optimization

TL;DR

The paper introduces QQMM, a new descent algorithm tailored for unconstrained convex optimization problems involving a convex function plus a quadratic, demonstrating faster performance than BFGS in relevant statistical applications.

Contribution

The paper presents QQMM, a novel optimization algorithm specifically designed for convex problems with a quadratic penalty, featuring improved stopping control and trial minimization methods.

Findings

QQMM outperforms BFGS in speed for targeted problems.

QQMM effectively controls stopping criteria in statistical contexts.

Hybrid QQMM-BFGS further enhances optimization speed.

Abstract

A descent algorithm, "Quasi-Quadratic Minimization with Memory" (QQMM), is proposed for unconstrained minimization of the sum, $F$, of a non-negative convex function, $V$, and a quadratic form. Such problems come up in regularized estimation in machine learning and statistics. In addition to values of $F$, QQMM requires the (sub)gradient of $V$. Two features of QQMM help keep low the number of evaluations of the objective function it needs. First, QQMM provides good control over stopping the iterative search. This feature makes QQMM well adapted to statistical problems because in such problems the objective function is based on random data and therefore stopping early is sensible. Secondly, QQMM uses a complex method for determining trial minimizers of $F$. After a description of the problem and algorithm a simulation study comparing QQMM to the popular BFGS optimization algorithm is described. The simulation study and other experiments suggest that QQMM is generally substantially faster than BFGS in the problem domain for which it was designed. A QQMM-BFGS hybrid is also generally substantially faster than BFGS but does better than QQMM when QQMM is very slow.