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

ArXiv ID: 2510.24607 “View on arXiv”

Authors: Yimeng Qiu

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.

Keywords: Kullback-Leibler divergence, convex optimization, multi-factor portfolio, Newton method, Bregman-Dykstra projections, Portfolio Management

Complexity vs Empirical Score

• Math Complexity: 9.0/10
• Empirical Rigor: 2.0/10
• Quadrant: Lab Rats
• Why: The paper is dense with advanced mathematical derivations, including convex optimization, sensitivity analysis, and ODE-based path-following, while explicitly stating that empirical benchmarking is optional and providing only theoretical algorithms without real-world data or backtests.

  flowchart TD
    A["Research Goal: Construct multi-factor target-exposure portfolios"]
    B["Methodology: Entropy-Guided Multiplicative Updates EGMU Framework"]
    C["Input: Benchmark & Target Exposures"]
    D["Solver 1: Damped Dual Newton Method"]
    E["Solver 2: Bregman-Dykstra KL Projections"]
    F["Generalization: Elastic Targets & Robust Sets"]
    G["Key Findings: Uniqueness, Convergence, Stability"]

    A --> B
    C --> B
    B --> D
    B --> E
    B --> F
    D & E & F --> G