Entropy-Guided Multiplicative Updates: KL Projections for Multi-Factor Target Exposures
Yimeng Qiu
TL;DR
This work formulates multi-factor target-exposure construction as a KL-information projection on the simplex: minimize $D_{ m KL}(\mathbf{w}\|\boldsymbol{b})$ subject to linear factor constraints, yielding an exponential-family solution and a convex dual. It provides two provably convergent solvers—a damped dual Newton method and KL-projection schemes (IPF/Bregman--Dykstra)—and extends the framework to elastic targets, robust target sets, and solution-path tracing via a sensitivity ODE. Theoretical guarantees establish existence, uniqueness, and explicit dual sensitivities, while implementation guidelines ensure stability and scalability for large $N$ and small $K$. The approach unifies exact and approximate feasibility, robustness, and turnover considerations under a single dual-moment core, enabling reproducible, principled baseline portfolios for diverse factor-exposure views with minimal empirical tuning. Its practical impact lies in providing a rigorous, efficient, and extensible toolkit for disciplined target-exposure design in portfolio engineering.
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.