Second order splitting dynamics with vanishing damping for additively structured monotone inclusions

Boţ, Radu Ioan; Hulett, David Alexander

doi:10.48550/arxiv.2201.01017

Cited by 1 publication

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“…Specifically, Su et al (2014) studied the dynamics of 0 = Ẍ + r t Ẋ +∇f (X) and proved A wide range of variations of the ODE with vanishing damping were also studied (Attouch & Chbani, 2015;May, 2017;Attouch et al, 2018b;Attouch & Cabot, 2018a;Attouch et al, 2019b;Attouch & Peypouquet, 2019;Attouch & László, 2020;Attouch et al, 2020a;2021a;Attouch & László, 2021;Attouch & Cabot, 2017;Attouch & Laszlo, 2021;Attouch et al, 2022;2021b). Similar analyses were extended to differential inclusions for non-differentiable functions (Attouch & Maingé, 2011;Attouch & Peypouquet, 2016;Aujol & Dossal, 2017b;Apidopoulos et al, 2017;, monotone inclusions (Bot ¸& Csetnek, 2016;Bot & Hulett, 2022), primal-dual methods (Bot ¸& Nguyen, 2021) This intense study of ODEs modeling optimization algorithms motivated the development of tools utilizing the following ideas: variational principle and Lagrangian mechanics (Wibisono et al, 2016;Jordan, 2018;Wilson et al, 2021); duality gap and convex-analytical techniques (Diakonikolas & Orecchia, 2019); Hamiltonian mechanics (Diakonikolas & Jordan, 2021); control theory (Hu & Lessard, 2017); continuous-time complexity lower bounds (Muehlebach & Jordan, 2020); and perturbation analysis of physics, leading to the high-resolution ODE (Shi et al, 2021).…”

Section: Prior Workmentioning

confidence: 97%

Continuous-Time Analysis of Accelerated Gradient Methods via Conservation Laws in Dilated Coordinate Systems

Suh¹,

Roh²,

Ryu³

2022

Preprint

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We analyze continuous-time models of accelerated gradient methods through deriving conservation laws in dilated coordinate systems. Namely, instead of analyzing the dynamics of X(t), we analyze the dynamics of W (t) = t α (X(t) − X c ) for some α and X c and derive a conserved quantity, analogous to physical energy, in this dilated coordinate system. Through this methodology, we recover many known continuous-time analyses in a streamlined manner and obtain novel continuous-time analyses for OGM-G, an acceleration mechanism for efficiently reducing gradient magnitude that is distinct from that of Nesterov. Finally, we show that a semi-second-order symplectic Euler discretization in the dilated coordinate system leads to an O(1/k 2 ) rate on the standard setup of smooth convex minimization, without any further assumptions such as infinite differentiability.

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