Free Everywhere, Exact on Trees: PPO's Dropped Correction Buys Sample Efficiency Under Aggressive Reuse

Nima H. Siboni

Common policy improvement methods, including TRPO, PPO, and GRPO, estimate policy improvement under the behavioral policy's state-visitation distribution rather than the improved policy's own. The substitution makes the objective estimable from the behavioral policy's rollouts but adds a bias growing with policy divergence, hence the trust region or clip, and hence no reuse of a batch far off-policy. We show that under history-injective dynamics, where each state is reached by exactly one history, the dropped state-visitation ratio equals the product of per-step policy ratios along the sampled prefix, on every trajectory and not only in expectation. The ratio is therefore restored exactly, from log-probabilities PPO already computes. Autoregressive generation and canonical-order constructive optimization are both history-injective. The exact correction pays importance-sampling variance that grows with the horizon, so we generalize it to a one-parameter family with PPO ($α{=}0$) and the full correction ($α{=}1$) as endpoints: a single bias--variance knob. A gradient-level analysis of the unclipped surrogate identifies two channels the correction acts through and three conditions…