We analyze the query complexity of finding a local minimum in $t$ rounds on general graphs. More precisely, given a graph $G = (V,E)$ and oracle access to an unknown function $f : V \to \mathbb{R}$, the goal is to find a local minimum--a vertex $v$ such that $f(v) \leq f(u)$ for all $(u,v) \in E$--using at most $t$ rounds of interaction with the oracle. The query complexity is well understood on grids, but much less is known beyond. This abstract problem captures many optimization tasks, such as finding a local minimum of a loss function during neural network training. For each graph with $n$ vertices, we prove a deterministic upper bound of $O(t n^{1/t} (sΔ)^{1-1/t})$, where $s$ is the separation number and $Δ$ is the maximum degree of the graph. We complement this result with a randomized lower bound of $Ω(t n^{1/t}-t)$ that holds for any connected graph. We also find that parallel steepest descent with a warm start provides improved bounds for graphs with high separation number and bounded degree.

翻译：本文分析了在通用图上通过$t$轮查询寻找局部最小值的查询复杂度。具体而言，给定图$G = (V,E)$及对未知函数$f : V \to \mathbb{R}$的预言机访问权限，目标是在最多$t$轮与预言机的交互中找到一个局部最小值——即满足对所有$(u,v) \in E$有$f(v) \leq f(u)$的顶点$v$。该问题在网格图上的查询复杂度已得到充分研究，但在更广泛图类中的认知仍较为有限。这一抽象问题涵盖了许多优化任务，例如在神经网络训练过程中寻找损失函数的局部最小值。针对具有$n$个顶点的任意图，我们证明了$O(t n^{1/t} (sΔ)^{1-1/t})$的确定性上界，其中$s$为分离数，$Δ$为图的最大度数。我们进一步给出了适用于任意连通图的随机下界$Ω(t n^{1/t}-t)$作为补充。研究还发现，采用预热启动的并行最速下降法能在高分离数且有界度数的图上获得更优的界。