The Variance Cost of Partial Reward Verification in Group-Based Policy Optimization

Partial reward verification can preserve a complete-label finite-batch surrogate gradient in expectation while increasing its error at a fixed verification cost. We study when the information purchased by an audit compensates for its cost, taking the fully verified finite-batch surrogate gradient as the target. Using established unbiased estimators, we express this comparison as a threshold on residual-gradient reduction and evaluate it with direct measurements of all 7.37 million trainable LoRA coordinates. A fixed two-stage rule has 2.20–2.77 times the cost-matched whole-group MSE on four BIRD development batches: its first audit creates gradient error in groups whose complete reward vectors are constant. A separate 128-task test shows that cost accounting can reverse the ranking of a frozen whole-group allocation, from an MSE ratio of 0.720 at SQL-body CPU to 1.076 at complete verifier-child CPU. Finally, arithmetic-risk predictors improve development comparisons but do not confirm that gain on 64 tasks from eight additional databases. Together, these fixed-gradient experiments identify residual reduction, the execution cost boundary, and budget transfer as distinct requirements for economical verification.