AI-Assisted Counterexample Refutes Last-Iterate Convergence for Definable Mini-Batch Optimization

In a preprint first uploaded on August 19, 2026, Weiwei Kong gives a counterexample to the last-iterate convergence conjecture in Remark 12 of Bolte and Pauwels (2021). The example uses two convex piecewise-affine, hence semialgebraic, functions on $\mathbb{R}$ and a deterministic nonincreasing block stepsize sequence with $\alpha_k=o(1/\log k)$.

Within successive blocks, the mini-batch iterates form lazy reflected random walks on nested dyadic lattices. Kong proves that almost surely every sufficiently late block visits its entire lattice. The iterates stay in $[-1,1]$ but fail to converge, and their accumulation set is exactly that interval, on which the averaged objective is constant. The construction has $\sum_k\alpha_k^2=\infty$, so it does not address convergence under the stronger square-summability condition.

The paper states that Chat-GPT 5.6 (Sol) and Gemini Pro 3.1 (DeepThink) were used in developing and drafting the result; it does not divide the mathematical contributions between the two systems.