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Implicit Optimization Bias of Next-Token Prediction in Linear Models (2402.18551v2)

Published 28 Feb 2024 in cs.LG, cs.CL, and stat.ML

Abstract: We initiate an investigation into the optimization properties of next-token prediction (NTP), the dominant training paradigm for modern LLMs. Specifically, we study the structural properties of the solutions selected by gradient-based optimizers among the many possible minimizers of the NTP objective. By framing NTP as cross-entropy minimization across distinct contexts, each tied with a sparse conditional probability distribution across a finite vocabulary of tokens, we introduce "NTP-separability conditions" that enable reaching the data-entropy lower bound. With this setup, and focusing on linear models with fixed context embeddings, we characterize the optimization bias of gradient descent (GD): Within the data subspace defined by the sparsity patterns of distinct contexts, GD selects parameters that equate the logits' differences of in-support tokens to their log-odds. In the orthogonal subspace, the GD parameters diverge in norm and select the direction that maximizes a margin specific to NTP. These findings extend previous research on implicit bias in one-hot classification to the NTP setting, highlighting key differences and prompting further research into the optimization and generalization properties of NTP, irrespective of the specific architecture used to generate the context embeddings.

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Citations (12)
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Summary

  • The paper demonstrates that gradient descent in overparameterized linear NTP models converges to a max-margin solution in specific data subspaces.
  • It reveals that the implicit bias aligns the model with solving a quadratic programming problem analogous to an SVM classification in NTP setups.
  • The findings provide theoretical insights that can inform new training strategies for improving model robustness, generalization, and interpretability in NLP tasks.

Exploring the Implicit Bias of Next-Token Prediction in LLMs

Introduction to Implicit Bias in NTP

Next-token prediction (NTP) is a cornerstone of modern NLP, underlying the success of LLMs across a spectrum of applications from text summarization to machine translation. While the empirical advancements in NTP are undisputed, a theoretical understanding of the optimization and generalization behaviors of models trained under this paradigm remains nascent. This gap in our knowledge introduces challenges in robustness, interpretability, and bias of models, particularly as they become deeply integrated into societal systems.

The Study of Implicit Bias in NTP

A paper addresses the fundamental question of whether gradient-based optimizers display an implicit bias towards particular solutions during the training of linear NTP models. This question is crucial as understanding this bias can lead to insights into how models generalize to unseen data and potentially how they can be made more robust and interpretable.

The paper demonstrates that for linear models trained using gradient descent under overparameterization, iterates converge in a specific direction within the parameter space. This direction aligns with the unique solution of a system of linear equations when projected onto a particular data subspace, and towards the solution of a max-margin quadratic programming problem in the orthogonal data subspace.

Insights from the Paper

NTP-SVM and Implicit Bias

The investigation uncovers a max-margin classifier (termed the NTP-SVM) within the NTP training setup, revealing that gradient descent's implicit bias in this context steers the model parameters towards maximizing the margin between in-support and out-of-support tokens. This result is analogous to findings in traditional one-hot prediction scenarios but is novel in the context of NTP.

Practical Implications

From a practical standpoint, these findings offer pathways to enhancing model generalization and providing a theoretical foundation for regularization techniques in NTP settings. For instance, understanding the role of the NTP-SVM direction can guide the development of new training strategies that inherently promote robustness and better generalization.

Looking Forward

Future research directions are ripe for exploration, including identifying exact conditions under which NTP linear-separability is guaranteed, leveraging the theoretical insights for soft-label classification, and extending the analysis beyond linear models to encompass deep learning architectures inherent in modern LLMs.

Conclusion

This paper makes significant strides towards demystifying the implicit bias in next-token prediction training, thereby contributing to the broader quest of understanding deep learning optimization and generalization. As the field strides forward, marrying empirical successes with theoretical insights will be pivotal in crafting models that are not only powerful but also robust, fair, and interpretable.

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