Non-Convex Portfolio Optimization via Energy-Based Models: A Comparative Analysis Using the Thermodynamic HypergRaphical Model Library (THRML) for Index Tracking

TL;DR

This paper presents a novel probabilistic approach to index tracking portfolio optimization using energy-based models and the THRML library, demonstrating superior tracking and return performance over traditional methods.

Contribution

It introduces a new energy-based modeling framework with adaptive coupling, bias reweighting, and sector-aware post-processing for portfolio optimization, validated through backtesting.

Findings

Achieves lower tracking error (4.31%) compared to baselines (5.66-6.30%).

Generates higher total return (128.63%) than the index (79.61%).

Statistically significant improvements confirmed by Diebold-Mariano test.

Abstract

Portfolio optimization under cardinality constraints transforms the classical Markowitz mean-variance problem from a convex quadratic problem into an NP-hard combinatorial optimization problem. This paper introduces a novel approach using THRML (Thermodynamic HypergRaphical Model Library), a JAX-based library for building and sampling probabilistic graphical models that reformulates index tracking as probabilistic inference on an Ising Hamiltonian. Unlike traditional methods that seek a single optimal solution, THRML samples from the Boltzmann distribution of high-quality portfolios using GPU-accelerated block Gibbs sampling, providing natural regularization against overfitting. We implement three key innovations: (1) dynamic coupling strength that scales inversely with market volatility (VIX), adapting diversification pressure to market regimes; (2) rebalanced bias weights prioritizing tracking quality over momentum for index replication; and (3) sector-aware post-processing ensuring institutional-grade diversification. Backtesting on a 100-stock S and P 500 universe from 2023 to 2025 demonstrates that THRML achieves 4.31 percent annualized tracking error versus 5.66 to 6.30 percent for baselines, while simultaneously generating 128.63 percent total return against the index total return of 79.61 percent. The Diebold-Mariano test confirms statistical significance with p less than 0.0001 across all comparisons. These results position energy-based models as a promising paradigm for portfolio construction, bridging statistical mechanics and quantitative finance.