While Large Language Models (LLMs) excel at in-context learning (ICL) using just a few demonstrations, their performances are sensitive to demonstration orders. The reasons behind this sensitivity remain poorly understood. In this paper, we investigate the prompt embedding space to bridge the gap between the order sensitivity of ICL with inner workings of decoder-only LLMs, uncovering the clustering property: prompts sharing the first and last demonstrations have closer embeddings, with first-demonstration clustering usually being stronger in practice. We explain this property through extensive theoretical analyses and empirical evidences. Our finding suggests that the positional encoding and the causal attention mask are key contributors to the clustering phenomenon. Leveraging this clustering insight, we introduce Cluster-based Search, a novel method that accelerates the selection and ordering of demonstrations in self-adaptive ICL settings. Our approach substantially decreases the time complexity from factorial to quadratic, saving 92% to nearly 100% execution time while maintaining comparable performance to exhaustive search.

The capacity to achieve out-of-distribution (OOD) generalization is a hallmark of human intelligence and yet remains out of reach for machines. This remarkable capability has been attributed to our abilities to make conceptual abstraction and analogy, and to a mechanism known as indirection, which binds two representations and uses one representation to refer to the other. Inspired by these mechanisms, we hypothesize that OOD generalization may be achieved by performing analogy-making and indirection in the functional space instead of the data space as in current methods. To realize this, we design FINE (Functional Indirection Neural Estimator), a neural framework that learns to compose functions that map data input to output on-the-fly. FINE consists of a backbone network and a trainable semantic memory of basis weight matrices. Upon seeing a new input-output data pair, FINE dynamically constructs the backbone weights by mixing the basis weights. The mixing coefficients are indirectly computed through querying a separate corresponding semantic memory using the data pair. We demonstrate empirically that FINE can strongly improve out-of-distribution generalization on IQ tasks that involve geometric transformations. In particular, we train FINE and competing models on IQ tasks using images from the MNIST, Omniglot and CIFAR100 datasets and test on tasks with unseen image classes from one or different datasets and unseen transformation rules. FINE not only achieves the best performance on all tasks but also is able to adapt to small-scale data scenarios.