What Fixed-Rollout pass@k Evaluations Can Identify

Estimating how often an ML model will fail at deployment scale is central to pre-deployment safety assessment, but a feasible evaluation set is rarely large enough to observe the failures that matter. Jones et al. (2025) address this by extrapolating from the largest k failure scores in an evaluation set to predict deployment-scale failure rates. We give a finite-k decomposition of this estimator's forecast error and show that it has a built-in bias toward over-prediction in the typical case, which is the safety-favorable direction. This bias is offset when the evaluation set misses a rare high-failure mode that the deployment set contains, leaving the forecast to under-predict at deployment scale. We propose a fine-tuning objective, the forecastability loss, that addresses this failure mode. In two proof-of-concept experiments, a language-model password game and an RL gridworld, fine-tuning substantially reduces held-out forecast error while preserving primary-task capability and achieving safety similar to that of supervised baselines.

Math and science reasoning benchmarks rely on pass@k, the fraction of sampled chains that reach gold, as the canonical per-example difficulty signal. The same signal drives RL with verifiable rewards, math data curation, synthetic curricula, and verifier training. We show this proxy has a persistent blind spot on its hardest stratum: on the eight free-form math cells we test (GSM8K and MATH across four open-weight models), 10.3-22.9% of the examples that no sampling seed solves in six tries are instead solved at matched compute by a six-chain deterministic regime. These are greedy decoding plus five cheap residual-stream perturbations applied via activation grafting, while greedy alone solves at most 6% on these math cells. Recovery scales with the additional budget, across perturbations whose mechanistic distinctness we verify across all twelve cells (cross-kind fix-set Jaccard <= 0.47 in every setup). Activation grafting is used as an intervention on internal representations, not a decoding method; we use it purely as a diagnostic and diversification tool, and our recovered items show that the pass@k= 0 % stratum is structurally identifiable in the residual stream rather than that the unmodified model reaches them under ordinary inference.

While existing benchmarks demonstrate the near-perfect performance of large language models (LLMs) on various tasks, this apparent saturation often obscures the need for rigorous evaluation of their reliability. In real-world deployment, however, achieving extremely high reliability (e.g., "five-nines" (99.999%) vs. "three-nines" (99.9%)) is fundamentally critical, as this gap results in an order-of-magnitude increase in failures, which is catastrophic in reliability-critical applications. Still, estimating such a rare failure probability with tight confidence bounds requires prohibitively large LLM inference sizes, making standard Monte Carlo evaluation infeasible under limited compute budgets. In this paper, we observe that LLM failures exhibit strong systematic patterns: across broad parameterized input spaces, a small subset of inputs disproportionately accounts for the majority of failures. Leveraging this observation, we propose to learn a sampling distribution concentrated on failure-prone inputs via the cross-entropy method (CEM). We evaluate our framework on three LLMs, Qwen2.5-Math-7B-Instruct, gpt-oss-20b-low, and Gemini 2.5 Flash Lite, across parameterized GSM8K templates and achieve up to 156.22x reduction in required inferences compared to naive uniform sampling. Our estimates reveal that models with indistinguishable accuracy on standard benchmarks can differ substantially in estimated failure rates, underscoring that reliability is a distinct and measurable axis of model quality. Our simple yet practical framework enables the evaluation of extreme reliability in LLMs, a distinct and underexplored dimension of evaluation beyond existing benchmarks, for their growing use in reliability-sensitive applications.