# Distributed Optimization via Energy Conservation Laws in Dilated Coordinates

arXiv: https://arxiv.org/abs/2409.19279 · HTML: https://mlsanity.com/paper/2409.19279/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 19 of its 36 references that are in the map.
- First posted on arXiv 28 Sep 2024.

## Builds on (most cited first, 19 of 19)

- [A Differential Equation for Modeling Nesterov's Accelerated Gradient Method: Theory and Insights](https://mlsanity.com/paper/1503.01243/) (2015)
- [EXTRA: An Exact First-Order Algorithm for Decentralized Consensus Optimization](https://mlsanity.com/paper/1404.6264/) (2014)
- [On the Convergence of Decentralized Gradient Descent](https://mlsanity.com/paper/1310.7063/) (2013)
- [Achieving Geometric Convergence for Distributed Optimization over Time-Varying Graphs](https://mlsanity.com/paper/1607.03218/) (2016)
- [Distributed optimization over time-varying directed graphs](https://mlsanity.com/paper/1303.2289/) (2013)
- [An Improved Analysis of Gradient Tracking for Decentralized Machine Learning](https://mlsanity.com/paper/2202.03836/) (2022)
- [Accelerated Distributed Nesterov Gradient Descent](https://mlsanity.com/paper/1705.07176/) (2017)
- [Direct Runge-Kutta Discretization Achieves Acceleration](https://mlsanity.com/paper/1805.00521/) (2018)
- [Multi-consensus Decentralized Accelerated Gradient Descent](https://mlsanity.com/paper/2005.00797/) (2020)
- [Acceleration in Distributed Optimization under Similarity](https://mlsanity.com/paper/2110.12347/) (2021)
- [Accelerated Gradient Tracking over Time-varying Graphs for Decentralized Optimization](https://mlsanity.com/paper/2104.02596/) (2021)
- [Towards accelerated rates for distributed optimization over time-varying networks](https://mlsanity.com/paper/2009.11069/) (2020)
- [Decentralized Accelerated Gradient Methods With Increasing Penalty Parameters](https://mlsanity.com/paper/1810.01053/) (2018)
- [Accelerated Primal-Dual Algorithms for Distributed Smooth Convex Optimization over Networks](https://mlsanity.com/paper/1910.10666/) (2019)
- [Exponential Convergence for Distributed Smooth Optimization Under the Restricted Secant Inequality Condition](https://mlsanity.com/paper/1909.03282/) (2019)
- [Continuous-Time Analysis of Accelerated Gradient Methods via Conservation Laws in Dilated Coordinate Systems](https://mlsanity.com/paper/2202.05501/) (2022)
- [Distributed Gradient Methods for Nonconvex Optimization: Local and Global Convergence Guarantees](https://mlsanity.com/paper/2003.10309/) (2020)
- [Breaking the Convergence Barrier: Optimization via Fixed-Time Convergent Flows](https://mlsanity.com/paper/2112.01363/) (2021)
- [Robust Distributed Fixed-Time Economic Dispatch under Time-Varying Topology](https://mlsanity.com/paper/1910.14214/) (2019)

## Abstract (arXiv)

Continuous-time models can reveal accelerated structures in distributed optimization, but their rates need not survive direct discretization. We introduce a second-order primal--dual flow for smooth convex distributed optimization and construct an exactly conserved energy that yields an mathcal O(t-2) rate for both the aggregate objective gap and the squared consensus error. We then prove a horizon-wise Omega(k-1) lower bound for a broad class of single-loop finite-memory primal--dual discretizations, ruling out a mathcal O(k-2) aggregate-objective guarantee within this class. Motivated by this barrier, we develop a double-loop method that combines finite-step polynomial consensus with an accelerated outer update. It uses one gradient evaluation and at most m-1 communication rounds per outer iteration, m being the number of agents, maintains exact consensus and achieves an mathcal O(k-2) aggregate-objective rate. Numerical comparisons with representative distributed methods support the theory and quantify the communication cost of acceleration.