Safety Hacking in Constrained Best-of-$N$ Inference-time Scaling
Inference-time pipelines often sample multiple outputs, filter them with a learned safety model, and return the proxy-feasible output with the highest learned reward. We show that this composition creates a two-stage failure: an imperfect safety proxy first contaminates the feasible set with unsafe outputs, and reward maximization can then amplify this residual contamination. We define \emph{safety hacking} as selecting an output that passes the learned constraint but violates the true safety criterion. For constrained Best-of-$N$ sampling, we derive finite-$N$ bounds governed by the joint upper reward tails of safe and unsafe outputs within the proxy-feasible set. If unsafe-but-feasible outputs have the heavier tail, safety hacking becomes asymptotically certain as $N$ grows, even when false-positive mass and average safety- and reward-proxy errors are arbitrarily small. We also show that policies within a bounded $\chi^2$ divergence from the proxy-feasible reference distribution admit an $N$-independent safety-hacking bound, and instantiate this general coverage-control principle with constrained pessimistic sampling. Coverage control limits amplification but cannot repair a contaminated feasible set: admitted unsafe outputs may still be favored, and regularized selection is not necessarily safer than constrained Best-of-$N$ for every reward proxy. Toy and language-model experiments characterize both contamination and its reward-tail amplification, which exposes an inherent difficulty in inference-time scaling with learned safety models.