A Concise Lyapunov Analysis of Nesterov's Accelerated Gradient Method

Abstract: Convergence analysis of Nesterov's accelerated gradient method has attracted significant attention over the past decades. While extensive work has explored its theoretical properties and elucidated the intuition behind its acceleration, a simple and direct proof of its convergence rates is still lacking. We provide a concise Lyapunov analysis of the convergence rates of Nesterov's accelerated gradient method for both general convex and strongly convex functions.

• The paper introduces a more accessible Lyapunov function approach to prove the convergence of Nesterov's Accelerated Gradient for both general and strongly convex functions.
• A simplified analysis yields a 1/k² rate for general convex functions and an accelerated (1 - 1/√κ)ᵏ rate for strongly convex cases.
• The study highlights potential extensions to stochastic optimization, paving the way for future research in accelerated gradient methods.

A Concise Lyapunov Analysis of Nesterov's Accelerated Gradient Method

Introduction

The paper "A Concise Lyapunov Analysis of Nesterov's Accelerated Gradient Method" (2502.17373) offers a streamlined approach to understanding the convergence behavior of the Nesterov's Accelerated Gradient (NAG) method using Lyapunov functions. While NAG is well-regarded for its acceleration capabilities in first-order optimization, traditional convergence proofs rely on the complex technique of estimating sequences. This work proposes a more accessible approach through Lyapunov analysis, applicable to both general convex and strongly convex function classes.

General Convex Functions

The analysis of NAG for general convex functions involves reformulating the method's dynamics in terms of Lyapunov functions. The NAG method is specified by iteratively updating sequences $x_k$ and $y_k$, with parameters $\alpha_k$ and $\beta_k$ governing the update rules. The convergence rate for general convex functions is derived as:

$$f(x_k) - f^* \le \frac{(r-1)^2 \| x_0 - x^* \|^2}{2\alpha(k+r-2)^2}$$

Theorem \ref{thm:weakly_convex} provides a rigorous yet more straightforward proof than traditional methods, offering both clarity and insight. A notable special case with optimal parameter settings reduces to a convergence rate of:

$$f(x_k) - f^* \le \frac{2L \| x_0 - x^* \|^2}{(k+1)^2}$$

This $1/k^2$ rate serves as an improvement over standard gradient descent, showcasing Nesterov's acceleration.

Strongly Convex Functions

The paper extends its Lyapunov approach to strongly convex functions, where the optimization landscape is more constrained. Here, the Lyapunov function framework reveals:

$$f(x_k) - f^* \le \left(1-\frac{1}{\sqrt{\kappa}}\right)^k \left(f(x_0) - f^* + \frac{\mu}{2}\| x_0 - x^* \|^2\right)$$

This translates to an accelerated convergence rate of $(1 - 1/\sqrt{\kappa})^k$ compared to traditional approaches, thus offering a significant performance gain.

Practical Implications and Speculations

The concise Lyapunov analysis provides a robust, unified framework for analyzing NAG that can extend beyond deterministic settings. In particular, this analysis has potential applications in stochastic settings, offering a pathway to proving acceleration for non-quadratic functions. Given the burgeoning interest in the stochastic applications of NAG, these insights could lead to advances in areas requiring efficient optimization under uncertainty.

Conclusions

This paper presents a refined and accessible Lyapunov analysis of Nesterov's accelerated gradient method, applicable to both general and strongly convex scenarios. By eschewing the complexity of traditional estimating sequences, it provides both pedagogical clarity and practical insight, potentially influencing future analyses of optimization algorithms within the broader machine learning framework. The implications extend to stochastic optimization, suggesting avenues for future research that could explore diverse application contexts or introduce innovations in convergence analysis techniques.