Large language models (LLMs) can exhibit undesirable and unexpected behavior in the blink of an eye. In a recent Anthropic demo, Claude switched from coding to Googling pictures of Yellowstone, and these sudden shifts in behavior have also been observed in reasoning patterns and jailbreaks. This phenomenon is not unique to autoregressive models: in diffusion models, key features of the final output are decided in narrow ``critical windows'' of the generation process. In this work we develop a simple, unifying theory to explain this phenomenon. We show that it emerges generically as the generation process localizes to a sub-population of the distribution it models. While critical windows have been studied at length in diffusion models, existing theory heavily relies on strong distributional assumptions and the particulars of Gaussian diffusion. In contrast to existing work our theory (1) applies to autoregressive and diffusion models; (2) makes no distributional assumptions; (3) quantitatively improves previous bounds even when specialized to diffusions; and (4) requires basic tools and no stochastic calculus or statistical physics-based machinery. We also identify an intriguing connection to the all-or-nothing phenomenon from statistical inference. Finally, we validate our predictions empirically for LLMs and find that critical windows often coincide with failures in problem solving for various math and reasoning benchmarks.

The profound success of transformer-based language models can largely be attributed to their ability to integrate relevant contextual information from an input sequence in order to generate a response or complete a task. However, we know very little about the algorithms that a model employs to implement this capability, nor do we understand their failure modes. For example, given the prompt "John is going fishing, so he walks over to the bank. Can he make an ATM transaction?", a model may incorrectly respond "Yes" if it has not properly contextualized "bank" as a geographical feature, rather than a financial institution. We propose the LLM Race Conditions Hypothesis as an explanation of contextualization errors of this form. This hypothesis identifies dependencies between tokens (e.g., "bank" must be properly contextualized before the final token, "?", integrates information from "bank"), and claims that contextualization errors are a result of violating these dependencies. Using a variety of techniques from mechanistic intepretability, we provide correlational and causal evidence in support of the hypothesis, and suggest inference-time interventions to address it.

We develop theory to understand an intriguing property of diffusion models for image generation that we term critical windows. Empirically, it has been observed that there are narrow time intervals in sampling during which particular features of the final image emerge, e.g. the image class or background color (Ho et al., 2020b; Georgiev et al., 2023; Raya & Ambrogioni, 2023; Sclocchi et al., 2024; Biroli et al., 2024). While this is advantageous for interpretability as it implies one can localize properties of the generation to a small segment of the trajectory, it seems at odds with the continuous nature of the diffusion. We propose a formal framework for studying these windows and show that for data coming from a mixture of strongly log-concave densities, these windows can be provably bounded in terms of certain measures of inter- and intra-group separation. We also instantiate these bounds for concrete examples like well-conditioned Gaussian mixtures. Finally, we use our bounds to give a rigorous interpretation of diffusion models as hierarchical samplers that progressively "decide" output features over a discrete sequence of times. We validate our bounds with synthetic experiments. Additionally, preliminary experiments on Stable Diffusion suggest critical windows may serve as a useful tool for diagnosing fairness and privacy violations in real-world diffusion models.

Pollutant exposure during gestation is a known and adverse factor for birth and health outcomes. However, the links between prenatal air pollution exposures and educational outcomes are less clear, in particular the critical windows of susceptibility during pregnancy. Using a large cohort of students in North Carolina, we study the link between prenatal daily $\mbox{PM}_{2.5}$ exposure and 4th end-of-grade reading scores. We develop and apply a locally adaptive and highly scalable Bayesian regression model for scalar responses with functional and scalar predictors. The proposed model pairs a B-spline basis expansion with dynamic shrinkage priors to capture both smooth and rapidly-changing features in the regression surface. The model is accompanied by a new decision analysis approach for functional regression that extracts the critical windows of susceptibility and guides the model interpretations. These tools help to identify and address broad limitations with the interpretability of functional regression models. Simulation studies demonstrate more accurate point estimation, more precise uncertainty quantification, and far superior window selection than existing approaches. Leveraging the proposed modeling, computational, and decision analysis framework, we conclude that prenatal $\mbox{PM}_{2.5}$ exposure during early and late pregnancy is most adverse for 4th end-of-grade reading scores.

Children's health studies support an association between maternal environmental exposures and children's birth and health outcomes. A common goal in such studies is to identify critical windows of susceptibility -- periods during gestation with increased association between maternal exposures and a future outcome. The associations and timings of critical windows are likely heterogeneous across different levels of individual, family, and neighborhood characteristics. However, the few studies that have considered effect modification were limited to a few pre-specified subgroups. We propose a statistical learning method to estimate critical windows at the individual level and identify important characteristics that induce heterogeneity. The proposed approach uses distributed lag models (DLMs) modified by Bayesian additive regression trees to account for effect heterogeneity based on a potentially high-dimensional set of modifying factors. We show in a simulation study that our model can identify both critical windows and modifiers responsible for DLM heterogeneity. We estimate the relationship between weekly exposures to fine particulate matter during gestation and birth weight in an administrative Colorado birth cohort. We identify maternal body mass index (BMI), age, Hispanic designation, and education as modifiers of the distributed lag effects and find non-Hispanics with increased BMI to be a susceptible population.