This paper investigates whether Transformers can emulate higher-order optimization methods, such as Newton's method, for in-context learning tasks. The authors demonstrate that linear attention Transformers with ReLU layers can approximate second-order optimization algorithms for logistic regression and achieve ε error with only logarithmic layers. They also show that even linear attention-only Transformers can implement a single step of Newton's iteration for matrix inversion with just two layers. These results suggest that Transformers can implement complex algorithms beyond gradient descent. The study focuses on linear and logistic regression tasks. For linear regression, the authors show that a linear Transformer with 3 + T layers can solve the task by performing T steps of Newton's iteration for matrix inversion. For logistic regression, they show that a linear Transformer can approximate T iterations of Newton's method with a width bounded by O(d(1 + μ)^6/ε^4μ^8) and depth bounded by T(11 + 2k), where k is a function of the condition number of the Hessian. The paper also provides empirical evidence showing that trained Transformers outperform Newton's method for initial layers/steps. The results suggest that Transformers can efficiently implement higher-order optimization methods, achieving quadratic convergence rates and better performance than gradient descent. The study contributes to the understanding of how Transformers can perform complex optimization tasks through in-context learning.This paper investigates whether Transformers can emulate higher-order optimization methods, such as Newton's method, for in-context learning tasks. The authors demonstrate that linear attention Transformers with ReLU layers can approximate second-order optimization algorithms for logistic regression and achieve ε error with only logarithmic layers. They also show that even linear attention-only Transformers can implement a single step of Newton's iteration for matrix inversion with just two layers. These results suggest that Transformers can implement complex algorithms beyond gradient descent. The study focuses on linear and logistic regression tasks. For linear regression, the authors show that a linear Transformer with 3 + T layers can solve the task by performing T steps of Newton's iteration for matrix inversion. For logistic regression, they show that a linear Transformer can approximate T iterations of Newton's method with a width bounded by O(d(1 + μ)^6/ε^4μ^8) and depth bounded by T(11 + 2k), where k is a function of the condition number of the Hessian. The paper also provides empirical evidence showing that trained Transformers outperform Newton's method for initial layers/steps. The results suggest that Transformers can efficiently implement higher-order optimization methods, achieving quadratic convergence rates and better performance than gradient descent. The study contributes to the understanding of how Transformers can perform complex optimization tasks through in-context learning.