In-context learning (ICL) is one of the most powerful and most unexpected capabilities to emerge in recent transformer-based large language models (LLMs).

In-context learning (ICL) of large language models has proven to be a surprisingly effective method of learning a new task from only a few demonstrative examples. In this paper, we shed light on the efficacy of ICL from the viewpoint of statistical learning theory. We develop approximation and generalization error analyses for a transformer model composed of a deep neural network and one linear attention layer, pretrained on nonparametric regression tasks sampled from general function spaces including the Besov space and piecewise $\gamma$-smooth class. In particular, we show that sufficiently trained transformers can achieve -- and even improve upon -- the minimax optimal estimation risk in context by encoding the most relevant basis representations during pretraining. Our analysis extends to high-dimensional or sequential data and distinguishes the \emph{pretraining} and \emph{in-context} generalization gaps, establishing upper and lower bounds w.r.t.

Recent research has shown that Transformers with linear attention are capable of in-context learning (ICL) by implementing a linear estimator through gradient descent steps. However, the existing results on the optimization landscape apply under stylized settings where task and feature vectors are assumed to be IID and the attention weights are fully parameterized. In this work, we develop a stronger characterization of the optimization and generalization landscape of ICL through contributions on architectures, low-rank parameterization, and correlated designs: (1) We study the landscape of 1-layer linear attention and 1-layer H3, a state-space model. Under a suitable correlated design assumption, we prove that both implement 1-step preconditioned gradient descent. We show that thanks to its native convolution filters, H3 also has the advantage of implementing sample weighting and outperforming linear attention in suitable settings.

Transformers excel at *in-context learning* (ICL)---learning from demonstrations without parameter updates---but how they do so remains a mystery. Recent work suggests that Transformers may internally run Gradient Descent (GD), a first-order optimization method, to perform ICL. In this paper, we instead demonstrate that Transformers learn to approximate second-order optimization methods for ICL. For in-context linear regression, Transformers share a similar convergence rate as *Iterative Newton's Method*, both *exponentially* faster than GD. Empirically, predictions from successive Transformer layers closely match different iterations of Newton's Method linearly, with each middle layer roughly computing 3 iterations; thus, Transformers and Newton's method converge at roughly the same rate.

In-Context Learning (ICL) empowers Large Language Models (LLMs) to tackle various tasks by providing input-output examples as additional inputs, referred to as demonstrations. Nevertheless, the performance of ICL could be easily impacted by the quality of selected demonstrations. Existing efforts generally learn a retriever model to score each demonstration for selecting suitable demonstrations, however, the effect is suboptimal due to the large search space and the noise from unhelpful demonstrations. In this study, we introduce MoD, which partitions the demonstration pool into groups, each governed by an expert to reduce search space. We further design an expert-wise training strategy to alleviate the impact of unhelpful demonstrations when optimizing the retriever model. During inference, experts collaboratively retrieve demonstrations for the input query to enhance the ICL performance.

Transformers can efficiently learn in-context from example demonstrations. Most existing theoretical analyses studied the in-context learning (ICL) ability of transformers for linear function classes, where it is typically shown that the minimizer of the pretraining loss implements one gradient descent step on the least squares objective. However, this simplified linear setting arguably does not demonstrate the statistical efficiency of ICL, since the pretrained transformer does not outperform directly solving linear regression on the test prompt.

Large language models (LLMs) excel at few-shot in-context learning (ICL) -- learning from a few examples provided in context at inference, without any weight updates. Newly expanded context windows allow us to investigate ICL with hundreds or thousands of examples - the many-shot regime. Going from few-shot to many-shot, we observe significant performance gains across a wide variety of generative and discriminative tasks. While promising, many-shot ICL can be bottlenecked by the available amount of human-generated outputs. To mitigate this limitation, we explore two new settings: (1) Reinforced ICL that uses model-generated chain-of-thought rationales in place of human rationales, and (2) Unsupervised ICL where we remove rationales from the prompt altogether, and prompts the model only with domain-specific inputs. We find that both Reinforced and Unsupervised ICL can be quite effective in the many-shot regime, particularly on complex reasoning tasks. We demonstrate that, unlike few-shot learning, many-shot learning is effective at overriding pretraining biases, can learn high-dimensional functions with numerical inputs, and performs comparably to supervised fine-tuning. Finally, we reveal the limitations of next-token prediction loss as an indicator of downstream ICL performance.

In-context learning (ICL) is a cornerstone of large language model (LLM) functionality, yet its theoretical foundations remain elusive due to the complexity of transformer architectures. In particular, most existing work only theoretically explains how the attention mechanism facilitates ICL under certain data models. It remains unclear how the other building blocks of the transformer contribute to ICL. To address this question, we study how a two-attention-layer transformer is trained to perform ICL on $n$-gram Markov chain data, where each token in the Markov chain statistically depends on the previous n tokens. We analyze a sophisticated transformer model featuring relative positional embedding, multi-head softmax attention, and a feed-forward layer with normalization. We prove that the gradient flow with respect to a cross-entropy ICL loss converges to a limiting model that performs a generalized version of the induction head mechanism with a learned feature, resulting from the congruous contribution of all the building blocks. Specifically, the first attention layer acts as a copier, copying past tokens within a given window to each position, and the feed-forward network with normalization acts as a selector that generates a feature vector by only looking at informationally relevant parents from the window. Finally, the second attention layer is a classifier thatcompares these features with the feature at the output position, and uses the resulting similarity scores to generate the desired output. Our theory is further validated by simulation experiments.

In-context learning (ICL) is one of the most powerful and most unexpected capabilities to emerge in recent transformer-based large language models (LLMs). Yet the mechanisms that underlie it are poorly understood. In this paper, we demonstrate that comparable ICL capabilities can be acquired by an alternative sequence prediction learning method using clone-structured causal graphs (CSCGs). Moreover, a key property of CSCGs is that, unlike transformer-based LLMs, they are {\em interpretable}, which considerably simplifies the task of explaining how ICL works. Specifically, we show that it uses a combination of (a) learning template (schema) circuits for pattern completion, (b) retrieving relevant templates in a context-sensitive manner, and (c) rebinding of novel tokens to appropriate slots in the templates. We go on to marshall evidence for the hypothesis that similar mechanisms underlie ICL in LLMs. For example, we find that, with CSCGs as with LLMs, different capabilities emerge at different levels of overparameterization, suggesting that overparameterization helps in learning more complex template (schema) circuits. By showing how ICL can be achieved with small models and datasets, we open up a path to novel architectures, and take a vital step towards a more general understanding of the mechanics behind this important capability.