# Counter-examples in first-order optimization: a constructive approach

arXiv: https://arxiv.org/abs/2303.10503 · HTML: https://mlsanity.com/paper/2303.10503/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 18 Mar 2023.

## Built on by (most cited first, 2 of 2)

- [Tight analyses of first-order methods with error feedback](https://mlsanity.com/paper/2506.05271/) (2025)
- [AI-Assisted Discovery and Construction of a Counterexample to the Convergence of Three-Block ADMM with the Identity Matrix as its Third Constraint Block](https://mlsanity.com/paper/2608.14396/) (2026)

## Abstract (arXiv)

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.