Transformers as Algorithms: Generalization and Stability in In-context Learning (2301.07067v2)

Abstract: In-context learning (ICL) is a type of prompting where a transformer model operates on a sequence of (input, output) examples and performs inference on-the-fly. In this work, we formalize in-context learning as an algorithm learning problem where a transformer model implicitly constructs a hypothesis function at inference-time. We first explore the statistical aspects of this abstraction through the lens of multitask learning: We obtain generalization bounds for ICL when the input prompt is (1) a sequence of i.i.d. (input, label) pairs or (2) a trajectory arising from a dynamical system. The crux of our analysis is relating the excess risk to the stability of the algorithm implemented by the transformer. We characterize when transformer/attention architecture provably obeys the stability condition and also provide empirical verification. For generalization on unseen tasks, we identify an inductive bias phenomenon in which the transfer learning risk is governed by the task complexity and the number of MTL tasks in a highly predictable manner. Finally, we provide numerical evaluations that (1) demonstrate transformers can indeed implement near-optimal algorithms on classical regression problems with i.i.d. and dynamic data, (2) provide insights on stability, and (3) verify our theoretical predictions.

An Analysis of Transformers for In-Context Learning: Implications for Generalization and Algorithm Stability

The paper "Transformers as Algorithms: Generalization and Stability in In-context Learning" addresses the concept of in-context learning (ICL) as implemented in transformer models, such as those used in NLP. In ICL, a transformer interprets a sequence of (input, output) pairs in its prompt to make inferences at test time without needing to update its weights. The authors formalize ICL as an algorithm learning problem where the transformer constructs a hypothesis function during inference. The paper explores various aspects of generalization and algorithmic stability in the context of ICL, providing theoretical and empirical insights.

Statistical Learning and Generalization Bounds

The primary analytical contribution of the paper is the development of generalization bounds in the context of multitask learning (MTL) for ICL. The transformer is considered as an algorithm-learning entity, which generates prediction functions f^m based on its input prompt, comprising m examples. The MTL problem involves training the transformer with multiple tasks, each task consisting of an independent sequence of data points. The authors demonstrate that generalization bounds for ICL can achieve $\mathcal{O}(1/\sqrt{Tn})$ rates with respect to the number of tasks (T) and the number of samples per task (n). This is significant because it offers a rigorous approach to quantify the learning capabilities of transformers when exposed to different lengths of input sequences.

The bounds rely on a well-established stability condition known as algorithmic stability—a framework that measures how alterations in input sequences impact a learning algorithm's outputs. The authors substantiate that these stability conditions are naturally satisfied by certain architectures and training regimes of transformer models. Specifically, stability relates the excess risk in predictions to the architectural features of the transformer, such as the attention mechanism's weights.

Algorithmic Stability and Transformer Architecture

A key part of the paper is how the particularities of transformer architecture—self-attention and dense layers—contribute to stability. Notably, the authors derive conditions under which the transformer's prediction process is stable with respect to changes in the input prompts. By considering the transformer's multi-layered, attention-based structure, they show that careful normalization and control of attention weights ensure stability and, consequently, reliable generalization performance.

The authors empirically verify the theoretical stability arguments using numerical evaluations across varied regression and dynamical system tasks. These results show that attention architectures can faithfully implement stable learning algorithms, capable of making accurate predictions by leveraging patterns seen across different tasks and input prompts.

Transfer Learning and Inductive Bias

Beyond the bounds for MTL, the research also touches on the implications of these bounds for transfer learning—how well an algorithm learned on one set of tasks can generalize to unseen tasks. Through empirical studies, the paper reveals an intriguing inductive bias phenomenon: the transfer learning risk is significantly influenced not by the complexity of the transformers but more so by the intrinsic complexity of the tasks and the number of tasks used in training (indicating a 'task complexity' that interacts with model selection).

Empirical evidence shows that ICL can match or even surpass conventional regression techniques, making it a promising avenue for advanced tasks without extensive retraining or parameter tuning. This capability of implicit model selection suggests the potential of transformers to adapt dynamically to a range of contexts presented to them as input, owing to their pre-trained knowledge acquired during the MTL phase.

Implications and Future Directions

The paper argues that the insights gained from studying ICL through this lens could inform the design of more efficient learning-based systems, where learning occurs without explicit weight updates—a valuable property when dealing with vast amounts of data in real time.

The implications for AI research are broad, presenting an opportunity to explore the development of transformer models that balance stability, complexity, and task adaptability further. This could lead to better understanding of the inductive biases that pre-trained models exhibit and their potential for various applications beyond NLP, such as autonomous systems and control-based tasks involving dynamic data.

Overall, this work presents evidence and methodologies that advance our understanding of how transformer models generalize and adapt, paving the way for their application in broader automated decision-making settings.