Toward Robust In-Context Learning: Leveraging Out-of-distribution Proxies for Target Inaccessible Demonstration Retrieval

Rejecting inputs outside the defined in-distribution (IND) service scope is critical for large language model (LLM) services, where unsupported requests should be filtered before full generation. Existing out-of-distribution (OOD) detectors often rely on final outputs or final-layer representations, leaving unclear where service-boundary signals are most clearly encoded inside the model; they also lack a theoretical guarantee for held-out inputs. In this paper, we introduce SCOPE (Sequential Conformal OOD Probing and Evaluation), a framework that selects a readable hidden layer, constructs a conformal gate with IND calibration, and uses a supermartingale e-process to certify persistent service-boundary evidence. Experiments across multiple LLM backbones and six carefully designed boundary conditions show that SCOPE improves gate-level rejection over standard final-layer detectors, while revealing how different OOD boundaries take different geometric forms in hidden space.

Knowledge distillation (KD) aims to compress high-performance teacher LLMs into lightweight students. However, distilled students often exhibit substantial performance degradation in out-of-distribution (OOD) settings, a critical gap that remains underexplored. We identify two compounding mechanisms causing OOD performance degradation: (1) data spuriousness: students can learn spurious correlations in the distillation dataset over genuine causal relationships; and (2) teacher capability: standard KD treats all samples uniformly, ignoring whether the teacher is guided by causal features or misled by spurious shortcuts on a given sample. To address these challenges, we propose Invariance-Weighted Distillation (IWD), a theoretically grounded framework that dynamically reweights training samples using an estimate of the teacher's causal reliance derived from prediction invariance across multiple synthetic environments. IWD perturbs spurious cues while preserving core semantics, assigning higher distillation weights to samples whose teacher predictions remain invariant, indicating greater reliance on causal rather than spurious features. We theoretically show that IWD reduces the student's Spurious-to-Causal (S2C) gradient ratio compared to standard uniformly weighted KD, driving the student toward more invariant representations. Experiments on four NLP benchmarks (MNLI, SQuAD-v2, CoNLL-2003 NER, and SST-2) across two model families (DeBERTa-v3 and Qwen-2.5) demonstrate that IWD consistently outperforms strong KD baselines on OOD evaluations while maintaining competitive in-distribution (ID) performance. Specifically, IWD achieves the highest accuracy in 15 out of 16 OOD benchmarks and improves average OOD performance over standard KD by 4.34 percentage points on NLI and 14.94 percentage points on QA.

In this work, we investigate how Large Language Models (LLMs) adapt their internal representations when encountering inputs of increasing difficulty, quantified as the degree of out-of-distribution (OOD) shift. We reveal a consistent and quantifiable phenomenon: as task difficulty increases, whether through harder reasoning questions, longer contexts, or adding answer choices, the last hidden states of LLMs become substantially sparser. In short, the farther the shift, the sparser the representations. This sparsity--difficulty relation is observable across diverse models and domains, suggesting that language models respond to unfamiliar or complex inputs by concentrating computation into specialized subspaces in the last hidden state. Through a series of controlled analyses with a learning dynamic explanation, we demonstrate that this sparsity is not incidental but an adaptive mechanism for stabilizing reasoning under OOD. Leveraging this insight, we design Sparsity-Guided Curriculum In-Context Learning (SG-ICL), a strategy that explicitly uses representation sparsity to schedule few-shot demonstrations, leading to considerable performance enhancements. Our study provides new mechanistic insights into how LLMs internalize OOD challenges. The source code is available at the URL: https://github.com/MingyuJ666/sparsityLLM.