By far the most common way to estimate an expected loss in machine learning is to draw samples, compute the loss on each one, and take the empirical average. However, sampling is not necessarily optimal. Given an MLP at initialization, we show how to estimate its expected output over Gaussian inputs without running samples through the network at all. Instead, we produce approximate representations of the distributions of activations at each layer, leveraging tools such as cumulants and Hermite expansions. We show both theoretically and empirically that for sufficiently wide networks, our estimator achieves a target mean squared error using substantially fewer FLOPs than Monte Carlo sampling. We find moreover that our methods perform particularly well at estimating the probabilities of rare events, and additionally demonstrate how they can be used for model training. Together, these findings suggest a path to producing models with a greatly reduced probability of catastrophic tail risks.
Quantifying the risk of rare failures in language models, such as those triggered by adversarial distribution shifts or very large-scale deployments, requires estimating probabilities far too small for random sampling. While recent work has formalized Low Probability Estimation, existing pipelines remain fragile in the rarest regimes: estimators can suffer zero-estimate collapse or systematic bias, and standard evaluation losses can become unstable or poorly matched to asymmetric safety costs. In this work, we introduce Gradient Activation Adaptive Multi-Level Splitting (GA-AMLS), which adapts rare-event Monte Carlo methods to the continuous activation space of language models. Specifically, GA-AMLS uses a gradient-based MCMC kernel to navigate activation space, eliminating the zero-estimate collapse of input-space search and replacing the independence assumptions of prior activation-space estimators with conditional sampling under an explicit, heavier-tailed activation prior. We also propose the Shifted-Power Bregman (SPB) Loss, a proper scoring rule that remains finite for zero-estimates and offers tunable asymmetry between underestimation and overestimation penalties. Experiments on small transformer models reveal a bias-variance tradeoff: GA-AMLS achieves the lowest loss under symmetric evaluation, reducing average log-space squared error relative to the strongest baseline across model sizes, while methods with overestimation bias prevail under asymmetric penalties. Our findings highlight that estimator choice should be matched to deployment context. More broadly, our work establishes activation space as a tractable domain for rare-event estimation in language models, circumventing the brittleness of discrete input-space search.
Language models are increasingly capable and are being rapidly deployed on a population-level scale. As a result, the safety of these models is increasingly high-stakes. Fortunately, advances in alignment have significantly reduced the likelihood of harmful model outputs. However, when models are queried billions of times in a day, even rare worst-case behaviors will occur. Current safety evaluations focus on capturing the distribution of inputs that yield harmful outputs. These evaluations disregard the probabilistic nature of models and their tail output behavior. To measure this tail risk, we propose a method to efficiently estimate the probability of harmful outputs for any input query. Instead of naive brute-force sampling from the target model, where harmful outputs could be rare, we operationalize importance sampling by creating unsafe versions of the target model. These unsafe versions enable sample-efficient estimation by making harmful outputs more probable. On benchmarks measuring misuse and misalignment, these estimates match brute-force Monte Carlo estimates using 10-20x fewer samples. For example, we can estimate probability of harmful outputs on the order of 10^-4 with just 500 samples. Additionally, we find that these harmfulness estimates can reveal the sensitivity of models to perturbations in model input and predict deployment risks. Our work demonstrates that accurate rare-event estimation is both critical and feasible for safety evaluations. Code is available at https://github.com/rangell/LMTailRisk
As agents are deployed with increased autonomy, even extremely rare events along their stochastic output trajectories can occur and prove catastrophic. Safe deployment therefore does not depend on whether these events can occur, but on how often they might. We study the problem of estimating the probability of rare events that arise from stochastic variation in the agent's own actions. Estimating this type of risk requires searching over the combinatorially vast space of trajectories. Naive Monte Carlo is computationally prohibitive in this regime, and constructing effective importance sampling (IS) proposals requires coordinated changes to a context-dependent chain of conditional distributions. We develop a new IS method that perturbs the original model's weights to construct the proposal. The proposal is itself a differentiably parameterized language model, enabling gradient-based search over weight space. We formulate an objective that combines a differentiable surrogate for event amplification and an adaptive regularization scheme that dynamically balances amplification against estimator stability. We evaluate our approach on ∼120M and ∼2.6B models across three event families spanning 300+ rare events as rare as 10−9, with reference probabilities computed with <10% relative standard error. In our most verifiable settings, we observe that our IS estimator achieves over 800× compute-weighted efficiency gains over naive Monte Carlo for events with probabilities lower than 10−7. Our implementation is available at https://github.com/namkoong-lab/iterative-unalignment.