The Iterates of Nesterov's Accelerated Algorithm Converge in The Critical Regimes

TL;DR

This paper proves that Nesterov's accelerated algorithm iterates converge weakly to a global minimizer in the critical regime, confirming a long-standing conjecture and connecting discrete algorithms with continuous-time systems.

Contribution

It establishes the convergence of Nesterov's algorithm in the critical regime within a Hilbert space, resolving a conjecture from a decade ago.

Findings

Weak convergence of iterates proven in the critical regime

Connection between discrete Nesterov's algorithm and continuous-time systems

Resolution of a long-standing conjecture in optimization

Abstract

In this paper, we prove that the iterates of the accelerated Nesterov's algorithm in the critical regime do converge in the weak topology to a global minimizer of an $L$-smooth function in a real Hilbert space, hence answering positively a conjecture posed by H. Attouch and co-authors a decade ago. This result is the algorithmic case of a very recent result on the continuous-time system posted by E. Ryu on X, with assistance from ChatGPT.