arXiv:2409.00913v2 Announce Type: replace-cross 
Abstract: Recent research has indicated a substantial rise in interest in understanding Nesterov's accelerated gradient methods via their continuous-time models. However, most existing studies focus on specific classes of Nesterov's methods, which hinders the attainment of an in-depth understanding and a unified perspective. To address this deficit, we present generalized continuous-time models that cover a broad range of Nesterov's methods, including those previously studied under existing continuous-time frameworks. Our key contributions are as follows. First, we identify the convergence rates of the generalized models, eliminating the need to determine the convergence rate for any specific continuous-time model derived from them. Second, we show that six existing continuous-time models are special cases of our generalized models, thereby positioning our framework as a unifying tool for analyzing and understanding these models. Third, we design a restart scheme for Nesterov's methods based on our generalized models and show that it ensures a monotonic decrease in objective function values. Owing to the broad applicability of our models, this scheme can be used to a broader class of Nesterov's methods compared to the original restart scheme. Fourth, we uncover a connection between our generalized models and gradient flow in continuous time, showing that the accelerated convergence rates of our generalized models can be attributed to a time reparametrization in gradient flow. Numerical experiment results are provided to support our theoretical analyses and results.

أظهرت الأبحاث الأخيرة زيادة في الاهتمام بفهم طرق تدرج نسترون المتسارعة من خلال نماذجها الزمنية المستمرة. تقدم دراسة جديدة نماذج زمنية مستمرة عامة تغطي مجموعة واسعة من طرق نسترون، مما يعالج العوائق السابقة في فهم معدلات التقارب ويوحد الأطر الموجودة.

Investigaciones recientes han destacado un creciente interés en los métodos de gradiente acelerado de Nesterov a través de sus modelos en tiempo continuo. Un nuevo estudio presenta modelos en tiempo continuo generalizados que abarcan una amplia gama de estos métodos, abordando limitaciones previas en la comprensión de sus tasas de convergencia y unificando marcos existentes.

Des recherches récentes ont mis en lumière un intérêt croissant pour les méthodes de gradient accéléré de Nesterov à travers leurs modèles en temps continu. Une nouvelle étude présente des modèles en temps continu généralisés qui englobent un large éventail de ces méthodes, abordant les limitations précédentes dans la compréhension de leurs taux de convergence et unifiant les cadres existants.

Recent research has highlighted a growing interest in Nesterov's accelerated gradient methods through their continuous-time models. A new study presents generalized continuous-time models that encompass a wide range of these methods, addressing previous limitations in understanding their convergence rates and unifying existing frameworks.