How Well Can Transformers Emulate In-context Newton's Method?

TL;DR

This work investigates whether Transformers can perform higher-order optimization in-context, extending beyond gradient-descent behavior to Newton-type methods. It provides constructive proofs showing linear-attention Transformers with ReLU can implement Newton steps for matrix inversion, linear regression, and Newton-based optimization for regularized logistic regression, with explicit depth and width bounds and convergence guarantees for inexact updates. Theoretical results are complemented by experiments demonstrating that trained linear Transformers can approximate Newton iterations and can outperform naive baselines in early steps, highlighting the architecture's capacity to embody complex algorithms. Overall, the paper advances our understanding of in-context algorithmic reasoning in Transformers and outlines concrete design principles for implementing advanced optimization routines within this architecture.

Abstract

Transformer-based models have demonstrated remarkable in-context learning capabilities, prompting extensive research into its underlying mechanisms. Recent studies have suggested that Transformers can implement first-order optimization algorithms for in-context learning and even second order ones for the case of linear regression. In this work, we study whether Transformers can perform higher order optimization methods, beyond the case of linear regression. We establish that linear attention Transformers with ReLU layers can approximate second order optimization algorithms for the task of logistic regression and achieve $ε$ error with only a logarithmic to the error more layers. As a by-product we demonstrate the ability of even linear attention-only Transformers in implementing a single step of Newton's iteration for matrix inversion with merely two layers. These results suggest the ability of the Transformer architecture to implement complex algorithms, beyond gradient descent.

Figures (8)

• Figure 1: The logistic regression loss of a trained Transformer with 1-40 layers and corresponding steps of Newton's method.
• Figure 2: Loss of LSA, LSA with layernorm and different order Newton iteration for linear regression error.
• Figure 3: Performance of Transformer on logistic regression tasks.(Left) The logistic regression loss for the Transformer (TF) and the Newton Method, with a regularization $\mu=0.1$. According to our theoretical construction, a single step of the Newton Method can be implemented by at least $11$ layers, therefore, we have scaled the Newton Method plot to $13$ layers per step. The Transformer is shown to approximate the method more effectively within a few layers. (Right) The training error of the Transformer when it is trained to predict the solution derived from Newton's Method.
• Figure 4: Loss of LSA and different order Newton iteration for linear regression with different input conditions.
• Figure 5: Loss of LSA w/ layernorm and different order Newton iteration for linear regression with different input conditions.

Theorems & Definitions (33)

• Theorem 1.1: Informal
• Definition 3.0
• Proposition 3.2: Lemma 2, zhang2015communicationefficient
• Lemma 4.0
• Theorem 4.1: Linear regression
• Remark 4.2
• Remark 4.3
• Remark 4.4
• Theorem 5.1
• Theorem 5.2