Whitened Advantage Summation Policy Optimization: Correlation-Aware Aggregation for Multi-Reward Reinforcement Learning

Reinforcement learning for language models often combines several rewards by normalizing each one and adding them with fixed weights. When the rewards are correlated, the variation they share is counted once for every reward that carries it. In terms of the ZCA-decorrelated components of the reward signal, the weighted sum applies weights $Σ^{1/2} w$ rather than the chosen weights $w$, where $Σ$ is the reward correlation matrix. We study WAS-PO, which whitens the normalized advantages with $Σ^{-1/2}$ before adding them, so that the chosen weights apply to those components. We derive the scalarized objective whose gradient it follows when normalization statistics are held fixed, show that its weights on the raw rewards can be negative, give an exact two-reward condition under which the lighter-weighted reward is reversed, and give a loose bound on the error from estimating $Σ$ during training. On tool calling with Qwen2.5-1.5B-Instruct, WAS-PO reached a higher mean training reward than GDPO in all seven seeds with three rewards (3.8% higher) and all three seeds with four (3.9% higher). WAS-PO also replaces GDPO's per-prompt normalization with batch-level normalization, and we did not separate the two changes with three or four rewards. The training-reward gains did not carry over to held-out accuracy, which was slightly lower than GDPO's with three rewards (not significant) and 0.03 lower in absolute terms with four (two seeds evaluated). With two rewards, WAS-PO trailed GDPO in training reward, and a three-seed control weakly suggests that the gap comes from batch-level normalization rather than whitening. In a math setting where the two rewards were nearly identical by construction, WAS-PO drove down the lighter-weighted reward, as the sign condition predicts.