A Sign Law for Entropy Drift under Sampler Mismatch in RLVR
Zhiyuan Wang, Jacob Steeves, Chang Liu
In reinforcement learning with verifiable rewards (RLVR), whether entropy rises or falls can depend on choices that matter little under exact sampling, such as whether over-length responses are masked or counted as failures. We trace this to sampler mismatch: rollouts are drawn with top-k/top-p truncation or at a temperature the learner does not apply. We prove that a constant advantage shift then changes the entropy of a softmax policy, at first order and in expectation, with a sign fixed by the sampler. Positive shifts sharpen the policy under any sampler whose likelihood ratio to the policy increases with token probability, including top-k, top-p, min-p, and temperatures below one, and flatten it under flatter samplers. In group-based RLVR with token-level losses, the baseline-dependent part of this drift is set by the token-weighted net advantage, which masking and failing can push to opposite signs. The rest comes from a covariance between advantages and the sampler's sharpness gap and feeds back on itself: in a two-level model, a push away from a sharper sampler takes off in finite time, while a pull toward it saturates. On a 2B model, the entropy slope is linear in shifts of the net advantage under truncated sampling and indistinguishable from flat under exact sampling. Token-level importance ratios leave 98% of the drift, and four clipped minibatch updates per batch amplify it sevenfold. In 100-step runs, a net-zero baseline only delays an entropy explosion, and entropy raised at the start keeps rising only under truncation; cancelling the first-order drift at every step keeps entropy bounded without changing the learner. On a 9B reasoning model forked from an uncorrected run, zeroing the net advantage or renormalizing the learner raises held-out accuracy after ten steps by 8.2 and 8.8 points over the uncorrected rule, with no difference under exact sampling.
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