Efficient Extraction of Higher-Order Information in Neural Networks

In this paper, we delve into the realm of optimization, where the goal is to find the best solution to a problem by minimizing or maximizing a certain function. We explore various techniques for achieving this goal, with a focus on practicality and efficiency.
Firstly, we examine the concept of higher-order information, which refers to the additional details that can be extracted from the loss function to improve optimization. To address this, we propose a novel technique for summarizing the derivative of the loss into a smaller tensor, making it easier to compute and more efficient overall.
However, the assumption behind these methods is that we need access to the Hessian matrix or its inverse, which may not always be the case. In particular, when dealing with non-convex losses, the eigenvalues of the Hessian tend to concentrate around zero, making it challenging to apply Newton’s method. Therefore, we question the need to compute the full Hessian in such situations and propose alternative approaches that are more practical and efficient.
Throughout the paper, we strive to demystify complex concepts by using everyday language and engaging analogies or metaphors. We aim to find a balance between simplicity and thoroughness to provide a comprehensive overview of the topic without oversimplifying it. By the end of this summary, readers should have a better understanding of the practical considerations involved in optimization and the techniques that can be employed to tackle these challenges.