aiqbench analysis
This research establishes a theoretical framework for accelerating distributed optimization by leveraging energy conservation laws in continuous-time models to achieve $\mathcal O(k^{-2})$ convergence rates. The authors prove that standard single-loop discretizations cannot match this speed without violating fundamental lower bounds, necessitating a novel double-loop architecture with polynomial consensus steps. These findings provide critical insights into the limits of communication efficiency and acceleration strategies for large-scale machine learning training across multiple agents.
Distributed Optimization via Energy Conservation Laws in Dilated Coordinates
arXiv:2409.19279v2 Announce Type: replace-cross Abstract: 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.
Abstract: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.
Optimization and Control (math.OC); Artificial Intelligence (cs.AI); Machine Learning (cs.LG); Systems and Control (eess.SY); Dynamical Systems (math.DS)
arXiv:2409.19279 [math.OC]
(or arXiv:2409.19279v2 [math.OC] for this version)
Full text under the publisher's open licence · 168 words
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