Abstract
Two accounts of how reinforcement learning with verifiable rewards (RLVR) interacts with an imperfect verifier coexist: in one, reward hacking is inherited from the base model; in the other, RL amplifies it. We study both in a controlled environment small enough to compute the entire response distribution exactly. We vary the universality of a leak, or the fraction of prompts that accept the same wrong response, and the verifier's static false-positive rate. At its KL-regularized optimum, RLVR acts as a filter that reweights the accepted set without altering its composition, so the hack rate is inherited exactly. Taking that as a null model, a known decomposition writes the drift from it as a sum over prompts of covariances between reward and the empirical tangent kernel. The terms from other prompts vanish without parameter sharing. With sharing, every prompt that accepts a wrong answer pushes it up and every prompt that rejects it pushes it down through the baseline, in proportion to the model's probability of producing it there. A shared-parameter policy thus drives a leak that few prompts accept below the base rate and amplifies one that most prompts accept. At every convergent learning rate under Adam, SGD, and sampled GRPO, hacking ends higher at high universality than at low, but whether either end crosses the base rate depends on the step size. Independent per-prompt copies remove the dependence. The direction of this push depends on neither the verifier's error rate nor universality alone, but on how much of the model's kernel-weighted probability of a wrong answer falls on prompts that accept it, which frozen base model controls support. Verifier fixes should target the mistakes the model repeats across prompts: two patches that remove equally many errors differ by up to 38 points of true accuracy. We propose this alignment as a hypothesis for why earlier studies disagree.
Abstract (translated)
URL
https://arxiv.org/abs/2611.13218
