Learning Perturbations to Extrapolate Your LLM

We introduce a simple yet powerful framework for training large language models. In contrast to the standard autoregressive next-token prediction based on an exact prefix, we propose a perturbation-based procedure that first transforms the prefix into a semantic neighbor and then conditions on this perturbed variant for next-token prediction. This yields a hierarchical model with a pre-post-additive noise structure. Within this framework, we develop a rigorous theory of extrapolability, namely, the capacity of a model class to make reliable predictions for token sequences that lie outside the empirical support of the training corpus. We evaluate the finite-sample performance of the proposed procedure using both synthetic and real-world language data. Results show that the proposed method consistently improves out-of-support prediction while maintaining competitive in-support performance, demonstrating that perturbation offers a practical route to language modeling.

The mechanisms behind LLMs' broad over-generalization beyond training examples remain unclear. Emergent misalignment (EM) offers a striking case study: finetuning on narrow tasks induces broad misalignment to semantically-unrelated test domains. In this work, we propose the Piggyback Hypothesis: the chat-template tokens can piggyback the finetuned behaviour onto out-of-domain queries. We validate this hypothesis by showing that subtle perturbations to the prefix (tokens preceding all user queries), or patching the prefix representations with those from the unfinetuned model, can restore alignment without changing the user query. Building on this finding, we propose Token-Regularized Finetuning (TReFT), which regularizes specific token representations during training to mitigate EM. Across different models and multiple EM-inducing datasets, TReFT reduces EM while preserving in-domain learning. On Llama-3.1-8B finetuned on the legal domain, TReFT achieves 33.5% more EM reduction than data interleaving with a retain set of aligned examples. We further show that TReFT extends to other narrow-finetuning settings, including abstention, tool use, and refusal (off-topic generalization is reduced by 54.3% on average), supporting the Piggyback Hypothesis. Broadly, our work highlights that LLMs may learn and generalize in unintended ways and suggests a path toward more constrained finetuning. It also calls for further study of how shared input features can piggyback model behavior across domains.

Modern learning systems excel at interpolation but struggle to generalize to unseen tasks outside the training distribution's support. This failure occurs even in simple settings, such as handling task parameters beyond the training range, and persists despite advances in foundation models. To this end, we develop the Relational Task Extrapolator (RTE), an algorithm designed to enable systematic extrapolation to novel tasks. The key observation is that extrapolation is inherently relational: extrapolating to unseen tasks requires learning how tasks transform into one another. If a model learns the transformation between tasks A and B during training, it can apply that same transformation to relate known tasks to unseen ones at test time. RTE operationalizes this idea by decomposing each target task into a known anchor task and a transformation linking the anchor and target. It then learns a relational operator, mapping an anchor-transformation pair to predictions for the target task. We instantiate RTE across multiple task extrapolation regimes in function prediction, e.g. where target tasks use out-of-range parameters (parameter extrapolation), have greater compositional depth (length extrapolation), and/or recombine function primitives in unseen ways (compositional extrapolation). We further extend RTE to sequence prediction, integrating it into fine-tuning algorithms for foundation models. Across empirical studies, we find that RTE substantially outperforms existing approaches on extrapolation to novel, unseen tasks.