可定义优化中最后迭代收敛的小批量反例
A Mini-Batch Counterexample to Last-Iterate Convergence in Definable Optimization
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中文总结 AI 辅助
本文针对带可定义势函数的小批量随机近似的收敛猜想，构造出迭代不收敛的小批量反例，明确其聚点集为[-1,1]，且构造过程使用了Chat-GPT 5.6和Gemini Pro 3.1。
中文摘要 AI 辅助
针对[Bolte & Pauwels, 2021]中注12关于带可定义势函数的小批量随机近似的收敛猜想，本文给出一个反例。构造使用了ℝ上的两个凸分段仿射（即半代数）加项；选取满足αₖ=o(1/log k)的确定性非递增块步长序列，以及每个聚合批量场的容许最小范数选择。在连续块上，迭代在嵌套二进格上形成惰性反射随机游走。显式端点覆盖时间估计、马尔可夫不等式及第一波莱尔-坎泰利引理几乎必然地表明，每个足够晚的块的迭代都会访问其整个格。因此，迭代保持在[-1,1]内但不收敛，其聚点集恰好为[-1,1]，而平均目标在此集上为常数。此外，该构造满足∑ₖαₖ²=∞。本结果的开发与撰写过程中使用了Chat-GPT 5.6（Sol）和Gemini Pro 3.1（DeepThink）。
英文摘要
We give a counterexample to the convergence conjecture in Remark 12 of [Bolte & Pauwels, 2021] for mini-batch stochastic approximation with definable potentials. The construction uses two convex piecewise-affine, hence semialgebraic, summands on $\mathbb{R}$. We choose a deterministic nonincreasing block stepsize sequence satisfying $α_k = o(1/\log k)$ and an admissible minimum-norm selection from each aggregate batch field. On successive blocks, the iterates form lazy reflected random walks on nested dyadic lattices. An explicit endpoint-cover-time estimate, Markov's inequality, and the first Borel-Cantelli lemma imply that almost surely every sufficiently late block's iterates visit their entire lattice. Consequently, the iterates remain in $[-1,1]$ but do not converge, and their accumulation set is exactly $[-1,1]$, on which the averaged objective is constant. Finally, the construction has $\sum_k α_k^2 =\infty$. Both Chat-GPT 5.6 (Sol) and Gemini Pro 3.1 (DeepThink) were used in the development and drafting of this result.
