Correlation Structure Governs Scalarization Bias in a Group-Relative Reinforcement Learning System for Portfolio Optimization

This paper introduces Pareto-GRPO, a group-relative reinforcement learning system with verifiable rewards for portfolio rebalancing under four objectives: Sharpe ratio, negative maximum drawdown, diversification ratio, and negative turnover. At each rebalancing step the policy samples a group of candidate weight vectors, scores them with a closed-form auditable objective, and reduces the resulting objective matrix to a per-candidate advantage through one of two interchangeable schemes: a Minkowski-weighted (MW) linear combination, or a one-time computation of an in-group rank-and-crowding-distance statistic adapted from NSGA-II. Working under an equicorrelation asset model, this paper applies a differential-geometric treatment to the mean-variance frontier and a spectral decomposition to the asset covariance matrix. Using this framework, this paper derives a closed-form relationship between average pairwise asset correlation and the curvature of the Sharpe-drawdown trade-off surface. This paper demonstrates that as asset correlation falls, the region of this surface that cannot be reached via linear reduction grows monotonically. This analysis is extended to a finite-sample statement, built from Chernoff-type concentration bounds on a two-component sampling mixture, of what an in-group rank statistic can actually recover of that region. Further results establish an unbiasedness and variance-reduction property of the group-relative baseline, a trust-region bound on the clipped surrogate, and a consistency result for the Bradley-Terry pairwise variant as a maximum-likelihood estimator of the group's latent dominance ranking. Together these results yield a testable law: linear reduction should track rank-based reduction in concentrated, high-correlation markets, and increasingly lag it as the universe diversifies. The law is tested with a walk-forward protocol across six equity universes spanning several continents, using Wilcoxon signed-rank tests with Holm-Bonferroni correction and a Friedman rank test across markets. The empirical pattern matches the theoretical prediction on all six markets (S&P 500 tech subset, NASDAQ 100 subset, NIFTY 50, Russell 1000 Value, Dow 30, STOXX Europe 50): linear reduction leads on the four higher-correlation panels and rank-based reduction leads on the two lowest-correlation panels, though the cross-market Friedman statistic, while directionally consistent, remains underpowered at this sample size. A secondary job-scheduling instance, analyzed via a submodularity argument for the hypervolume indicator, indicates the reduction machinery is not specific to finance.