Organizations: Center for Cognitive Interaction Technology CITEC, Universität Bielefeld, Germany · Department of Electrical and Electronics Engineering, Birla Institute of Technology and Science Pilani, India · Department of Computer Science, Royal Holloway, University of London, United Kingdom
In-context learning lets a sequence model adapt to a new task from examples in its input. A prominent line of work shows how self-attention can be constructed to implement gradient descent on a linear predictor fit to the in-context examples during the forward pass. State-space models (SSMs) and other linear recurrent networks (LRNNs) model sequences at linear time cost, but it is unclear how their recurrent update could carry out the same in-context gradient descent. We introduce Gradient-based Recurrent In-context Learner (GRIL), a diagonal LRNN that factorizes a supervised gradient step into a short-window cross-product write and a multiplicative readout of the next query. For linear regression, this construction accumulates the context gradient in a matrix state and applies it in a single forward pass, with O(f2) learned degrees of freedom. The same design extends to multi-step updates and cross-entropy classification, with a limited MLP-based extension to non-linear regression. We show empirically that trained GRILs recover the behavior and parameters analytically predicted by the construction on synthetic ICL tasks. Furthermore, the same architecture can be extended and trained on general-purpose benchmarks, including Long Range Arena, language modeling and associative recall. Together, these results establish windowed cross-product self-attention as a concrete inductive bias that lets LRNNs learn in context through gradient-descent-like updates, while remaining trainable on general-purpose tasks.
In-context learning has recently been linked to implicit gradient descent in linear self-attention models, suggesting that context can induce a forward-pass update. Retrieval-augmented generation (RAG) also relies on context, but retrieved documents are usually treated as static evidence rather than signals for adaptation. We study RAG as an in-context optimization process. First, we show that one linear self-attention layer can implement one gradient-descent step on a unified linearized RAG objective covering both projection-based and dot-product retrieval interfaces. This gives an exact regime where retrieval-augmented prediction and in-context optimization coincide. We use this result not as a literal model of LLM computation, but as a guide for adapting the interaction between queries and retrieved evidence. We then test the boundary of this correspondence: it remains stable under controlled linear extensions, but becomes feature-distribution dependent under nonlinear architectures. Finally, we turn this view into a lightweight method for frozen RAG LLMs. The method keeps the retriever and backbone fixed, and predicts a context-conditioned update to a generator-side evidence-use interface. Across seven QA benchmarks, two retrievers, and two frozen LLM backbones, this forward-only update improves a shared-interface baseline, transfers to held-out tasks, and approaches test-time gradient adaptation at much lower per-query cost.
Mingchen Li, Jiatan Huang, Chuxu Zhang +2
University of Massachusetts, Amherst · University of Connecticut · Emory University
Recent work has shown that large language models (LLMs) can iteratively improve their outputs by incorporating generated samples and their corresponding evaluation scores as in-context examples. Despite these empirical findings, the theoretical foundations underlying this phenomenon remain poorly understood. In this paper, we show that score-conditioned In-Context Learning (ICL) admits a structural correspondence to policy gradient optimization. We first provide a constructive proof that self-attention mechanisms can implement reward-weighted aggregation analogous to the REINFORCE algorithm under specific weight matrix configurations, and discuss the relationship between this construction and the behavior of pretrained transformers. The correspondence is directional in hidden-state space and holds exactly only under the stated simplifying conditions; we quantify its strength empirically. Within our simplified hidden-state model, we furthermore derive an exact upper bound on the distribution shift induced by a bounded attention update, yielding a trust-region-like analogy to KL-constrained policy optimization. We validate our theory through extensive experiments across multiple LLMs, demonstrating that LLMs effectively utilize score information to shift output distributions toward high-scoring exemplars, and that attention weights exhibit a strong correlation with example scores.
Large language models (LLMs) exhibit two striking and ostensibly unrelated behaviours: in-context learning (ICL) and repetitive generation. In both, the model behaves as though it had summarised the context into a population-level statistic and discarded token-level detail. We ask whether this ``summarisation and forgetting'' can be derived from the attention mechanism itself, and answer in the affirmative. Under stationary, ergodic and elliptical inputs, the softmax attention output converges almost surely to ΘVΣΘK⊤ΘQxt, where Σ is the input covariance; the long-context limit is therefore a linear readout of the input's second-order statistics. Two consequences follow. (i) For in-context linear regression, a single softmax head can implement one step of population gradient descent. Stacking such heads with residual connections iterates this update and implements multiple gradient descent steps. (ii) Propagated across an L-layer transformer, this readout drives the terminal hidden state at the parametric 1/t rate to a deterministic function of the current token alone, so that autoregressive generation collapses asymptotically to a first-order Markov chain whose attracting orbits furnish a structural account of repetition and mode collapse. The two phenomena thus emerge as facets of a single covariance-readout principle.