Counter-examples in first-order optimization: a constructive approach
Baptiste Goujaud, Aymeric Dieuleveut, Adrien Taylor
TL;DR
This work tackles gaps in worst-case analysis for first-order optimization by automating the search for cyclic trajectories that witness non-convergence. It builds a cycle-detection framework using performance estimation problems (PEP) and SDP lifting to search over a function class $\mathcal{F}$ and algorithm parameters. The authors apply the method to four stationary first-order methods—Heavy-ball, Nesterov accelerated gradient, inexact gradient descent, and three-operator splitting—and show cycles for parameter regions lacking a Lyapunov certificate, thereby complementing traditional convergence proofs. Overall, the approach provides a constructive, scalable tool for certifying the non-existence of universal convergence guarantees and informs the design of robust optimization algorithms.
Abstract
While many approaches were developed for obtaining worst-case complexity bounds for first-order optimization methods in the last years, there remain theoretical gaps in cases where no such bound can be found. In such cases, it is often unclear whether no such bound exists (e.g., because the algorithm might fail to systematically converge) or simply if the current techniques do not allow finding them. In this work, we propose an approach to automate the search for cyclic trajectories generated by first-order methods. This provides a constructive approach to show that no appropriate complexity bound exists, thereby complementing the approaches providing sufficient conditions for convergence. Using this tool, we provide ranges of parameters for which some of the famous heavy-ball, Nesterov accelerated gradient, inexact gradient descent, and three-operator splitting algorithms fail to systematically converge, and show that it nicely complements existing tools searching for Lyapunov functions.