# Dissipativity Theory for Nesterov's Accelerated Method

arXiv: https://arxiv.org/abs/1706.04381 · HTML: https://mlsanity.com/paper/1706.04381/

## Key facts

- Cited by 22 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 14 Jun 2017.

## Built on by (most cited first, 22 of 22)

- [No-Regret Dynamics in the Fenchel Game: A Unified Framework for Algorithmic Convex Optimization](https://mlsanity.com/paper/2111.11309/) (2021)
- [DiSK: Differentially Private Optimizer with Simplified Kalman Filter for Noise Reduction](https://mlsanity.com/paper/2410.03883/) (2024)
- [An Accelerated Algorithm for Stochastic Bilevel Optimization under Unbounded Smoothness](https://mlsanity.com/paper/2409.19212/) (2024)
- [Convergence and Stability of the Stochastic Proximal Point Algorithm with Momentum](https://mlsanity.com/paper/2111.06171/) (2021)
- [Revisiting the Role of Euler Numerical Integration on Acceleration and Stability in Convex Optimization](https://mlsanity.com/paper/2102.11537/) (2021)
- [Continuized Acceleration for Quasar Convex Functions in Non-Convex Optimization](https://mlsanity.com/paper/2302.07851/) (2023)
- [Memory Augmented Optimizers for Deep Learning](https://mlsanity.com/paper/2106.10708/) (2021)
- [Dissipative Gradient Descent Ascent Method: A Control Theory Inspired Algorithm for Min-max Optimization](https://mlsanity.com/paper/2403.09090/) (2024)
- [Convex Programs and Lyapunov Functions for Reinforcement Learning: A Unified Perspective on the Analysis of Value-Based Methods](https://mlsanity.com/paper/2202.06922/) (2022)
- [Transient growth of accelerated optimization algorithms](https://mlsanity.com/paper/2103.08017/) (2021)
- [On Centralized and Distributed Mirror Descent: Convergence Analysis Using Quadratic Constraints](https://mlsanity.com/paper/2105.14385/) (2021)
- [A Variational Perspective on High-Resolution ODEs](https://mlsanity.com/paper/2311.02002/) (2023)
- [Fast symplectic integrator for Nesterov-type acceleration method](https://mlsanity.com/paper/2106.07620/) (2021)
- [Tradeoffs between convergence rate and noise amplification for momentum-based accelerated optimization algorithms](https://mlsanity.com/paper/2209.11920/) (2022)
- [Accelerated Gradient Methods with Biased Gradient Estimates: Risk Sensitivity, High-Probability Guarantees, and Large Deviation Bounds](https://mlsanity.com/paper/2509.13628/) (2025)
- [Online Optimization and Ambiguity-based Learning of Distributionally Uncertain Dynamic Systems](https://mlsanity.com/paper/2102.09111/) (2021)
- [Exact Formulas for Finite-Time Estimation Errors of Decentralized Temporal Difference Learning with Linear Function Approximation](https://mlsanity.com/paper/2204.09801/) (2022)
- [A Passivity-Based Method for Accelerated Convex Optimisation](https://mlsanity.com/paper/2306.11474/) (2023)
- [Generalized Continuous-Time Models for Nesterov's Accelerated Gradient Methods](https://mlsanity.com/paper/2409.00913/) (2024)
- [A Concise Lyapunov Analysis of Nesterov's Accelerated Gradient Method](https://mlsanity.com/paper/2502.17373/) (2025)
- [Pseudospectral Bounds for Transient Amplification in Coupled Gradient Descent](https://mlsanity.com/paper/2606.04031/) (2026)
- [When Rates Are Geometric: Rate-Certificate Transfer for Contact Splittings in Optimization](https://mlsanity.com/paper/2607.23642/) (2026)

## Abstract (arXiv)

In this paper, we adapt the control theoretic concept of dissipativity theory to provide a natural understanding of Nesterov's accelerated method. Our theory ties rigorous convergence rate analysis to the physically intuitive notion of energy dissipation. Moreover, dissipativity allows one to efficiently construct Lyapunov functions (either numerically or analytically) by solving a small semidefinite program. Using novel supply rate functions, we show how to recover known rate bounds for Nesterov's method and we generalize the approach to certify both linear and sublinear rates in a variety of settings. Finally, we link the continuous-time version of dissipativity to recent works on algorithm analysis that use discretizations of ordinary differential equations.