In-context learning and Occam's razor (2410.14086v2)

Abstract: A central goal of machine learning is generalization. While the No Free Lunch Theorem states that we cannot obtain theoretical guarantees for generalization without further assumptions, in practice we observe that simple models which explain the training data generalize best: a principle called Occam's razor. Despite the need for simple models, most current approaches in machine learning only minimize the training error, and at best indirectly promote simplicity through regularization or architecture design. Here, we draw a connection between Occam's razor and in-context learning: an emergent ability of certain sequence models like Transformers to learn at inference time from past observations in a sequence. In particular, we show that the next-token prediction loss used to train in-context learners is directly equivalent to a data compression technique called prequential coding, and that minimizing this loss amounts to jointly minimizing both the training error and the complexity of the model that was implicitly learned from context. Our theory and the empirical experiments we use to support it not only provide a normative account of in-context learning, but also elucidate the shortcomings of current in-context learning methods, suggesting ways in which they can be improved. We make our code available at https://github.com/3rdCore/PrequentialCode.

In-Context Learning and Occam's Razor

The paper, "In-context learning and Occam's razor," presents a compelling theoretical framework that connects in-context learning (ICL) with the principle of Occam's razor in machine learning. The authors propose that ICL can be likened to a meta-learning process that implicitly employs data compression, specifically through prequential coding, to achieve model generalization.

Core Conceptual Framework

The central thesis revolves around the equivalence of the next-token prediction loss used in ICL with prequential coding. This equivalence is significant because it suggests that ICL can minimize both the training error and model complexity, adhering to Occam's principle which favors simpler models for better generalization. The paper meticulously details how the meta-learning objective for ICL models aligns with minimizing prequential code length, thereby linking model simplicity and compression together.

Theoretical Insights

The authors ground their arguments in Kolmogorov complexity, presenting it as a measure of information quantity. They outline how model complexity and likelihood are interrelated, thereby forming the basis for generalization. By defining a parameterized learner, they illustrate the prequential code length as an upper-bound yet efficient mechanism for estimating joint model and data complexity.

The paper of prequential coding illustrates that optimal data compression can lead to effective generalization, aligning with the Occam's razor principle that simpler explanations are favored. The authors leverage this foundational insight to argue that sequence models trained with traditional ICL methods achieve this by design, using next-token prediction to continuously refine model simplicity.

Empirical Evaluation

The paper reports extensive experiments comparing prequential ICL and train-risk ICL (predicting past tokens rather than future ones), in addition to standard gradient-based learners like SGD. Across linear and sinusoidal regression tasks, as well as a more complex Mastermind task, prequential ICL demonstrates superior performance in low-data regimes. This highlights the method's ability to favor simpler models that generalize well from limited data.

A notable aspect of this paper is its exploration of different architecture designs for ICL learners, emphasizing how effective architecture design, such as bottlenecked Transformers, contributes significantly to minimizing prequential code length. Another key finding is the limited generalization of models, including large pretrained ones, when tasked with novel and complex prediction problems. This underscores the necessity of specializing in-context learners for problem-specific distributions to fully leverage their generalization capabilities.

Implications and Future Directions

The implications of these findings are significant for both theoretical and practical applications. Theoretically, the link with Occam's razor provides a powerful lens through which one can better understand model complexity in contemporary machine learning paradigms. Practically, this framework can guide the development of in-context learners and LLMs, potentially leading to more efficient and versatile models.

Future directions propose further exploration into refining ICL architectures, including integrating optimization primitives for adaptable compute budgets. Experiments with distributing data by context length suggest promising directions for augmenting existing ICL training paradigms, especially in complex, language-like tasks.

Conclusion

Overall, this paper offers substantial contributions to our understanding of ICL by demonstrating its connection to both Occam's razor and optimal compression strategies. The insights drawn will undeniably guide future research and methodological innovations aimed at building more robust, efficient, and generalizable machine learning models.