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 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 chi2 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.