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Generalized Continuous-Time Models for Nesterov's Accelerated Gradient Methods

Chanwoong Park, Youngchae Cho, Insoon Yang

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.

Generalized Continuous-Time Models for Nesterov's Accelerated Gradient Methods

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.
Paper Structure (32 sections, 7 theorems, 130 equations, 12 figures, 1 algorithm)

This paper contains 32 sections, 7 theorems, 130 equations, 12 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Contributions
  3. Related work
  4. Preliminaries
  5. Convex optimization
  6. Bregman divergences
  7. Nesterov's method
  8. Generalized Continuous-Time Models for Nesterov's Methods
  9. Relationships with Existing ODE Models
  10. ODE-C and the ODE model of su2016differential
  11. ODE-SC and the ODE model of wilson2021lyapunov
  12. G-ODE and the Bregman Lagrangian flow of wilson2021lyapunov
  13. G-ODE and the dynamical system model of muehlebach2019dynamical
  14. G-ODE and the high-resolution ODEs of shi2021understandingchen2022revisiting
  15. Restart Schemes for Nesterov's Methods
  16. ...and 17 more sections

Key Result

Theorem 1

Let $X(t), Y(t), Z(t): [0,\infty) \rightarrow \mathbb{R}^n$ be a solution to eqn:G-ODE-C when $f$ is convex, and a solution to eqn:G-ODE-SC when $f$ is $\mu$-uniformly convex with respect to $g$. Then, the energy function $\mathcal{E} (t) = V(X(t), Z(t), t)$ is monotonically non-increasing for both

Figures (12)

  • Figure 1: Relationships between our models and the six existing models. In the case of shi2021understanding, our generalized models contain an approximation of it.
  • Figure 2: Comparison of \ref{['eqn:ODE-C']} and the existing ODEs su2016differentialshi2021understanding for (NAG-C-C).
  • Figure 3: Comparison of \ref{['eqn:ODE-C']} and the existing ODEs su2016differentialshi2021understanding for (NAG-C).
  • Figure 4: Comparison of \ref{['eqn:ODE-SC']} and existing ODEs wilson2021lyapunovshi2021understanding for (NAG-SC-C).
  • Figure 5: Comparison of \ref{['eqn:ODE-SC']} and existing models wilson2021lyapunovshi2021understanding for (NAG-SC)
  • ...and 7 more figures

Theorems & Definitions (7)

  • Theorem 1
  • Proposition 2
  • Theorem 3
  • Theorem 4
  • Corollary 5
  • Lemma 6
  • Lemma 7