One Step of Gradient Descent is Provably the Optimal In-Context Learner with One Layer of Linear Self-Attention

• URL: http://arxiv.org/abs/2307.03576v1
• Date: Fri, 7 Jul 2023 13:09:18 GMT
• Title: One Step of Gradient Descent is Provably the Optimal In-Context Learner with One Layer of Linear Self-Attention
• Authors: Arvind Mahankali, Tatsunori B. Hashimoto, Tengyu Ma
• Abstract summary: Recent works have empirically analyzed in-context learning. One-layer transformers with linear self-attention learn to implement one step of gradient descent.
• Score: 31.522320487765878
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Recent works have empirically analyzed in-context learning and shown that transformers trained on synthetic linear regression tasks can learn to implement ridge regression, which is the Bayes-optimal predictor, given sufficient capacity [Aky\"urek et al., 2023], while one-layer transformers with linear self-attention and no MLP layer will learn to implement one step of gradient descent (GD) on a least-squares linear regression objective [von Oswald et al., 2022]. However, the theory behind these observations remains poorly understood. We theoretically study transformers with a single layer of linear self-attention, trained on synthetic noisy linear regression data. First, we mathematically show that when the covariates are drawn from a standard Gaussian distribution, the one-layer transformer which minimizes the pre-training loss will implement a single step of GD on the least-squares linear regression objective. Then, we find that changing the distribution of the covariates and weight vector to a non-isotropic Gaussian distribution has a strong impact on the learned algorithm: the global minimizer of the pre-training loss now implements a single step of $\textit{pre-conditioned}$ GD. However, if only the distribution of the responses is changed, then this does not have a large effect on the learned algorithm: even when the response comes from a more general family of $\textit{nonlinear}$ functions, the global minimizer of the pre-training loss still implements a single step of GD on a least-squares linear regression objective.

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