Entropy-Guided Multiplicative Updates: KL Projections for Multi-Factor Target Exposures

TL;DR

This paper presents EGMU, a convex optimization framework for constructing multi-factor portfolios using KL divergence minimization, with proven convergence, scalability, and extensions for robustness and elastic targets.

Contribution

The paper introduces EGMU, a novel convex optimization method for multi-factor portfolio construction with explicit solutions, convergence guarantees, and scalable algorithms, extending prior approaches.

Findings

Proven feasibility and uniqueness of solutions under convex-hull conditions.

Development of two convergent solvers: dual Newton and KL-projection schemes.

Extensions for elastic targets, robust sets, and solution path tracing.

Abstract

We introduce Entropy-Guided Multiplicative Updates (EGMU), a convex optimization framework for constructing multi-factor target-exposure portfolios by minimizing Kullback-Leibler divergence from a benchmark under linear factor constraints. We establish feasibility and uniqueness of strictly positive solutions when the benchmark and targets satisfy convex-hull conditions. We derive the dual concave formulation with explicit gradient, Hessian, and sensitivity expressions, and provide two provably convergent solvers: a damped dual Newton method with global convergence and local quadratic rate, and a KL-projection scheme based on iterative proportional fitting and Bregman-Dykstra projections. We further generalize EGMU to handle elastic targets and robust target sets, and introduce a path-following ordinary differential equation for tracing solution trajectories. Stable and scalable implementations are provided using LogSumExp stabilization, covariance regularization, and half-space KL projections. Our focus is on theory and reproducible algorithms; empirical benchmarking is optional.