We consider the query complexity of finding a local minimum of a function defined on a graph. This abstract problem is fundamental to many optimization tasks, such as finding a local minimum of the loss function when training deep neural networks. In such applications, each query is an expensive loss evaluation, making it crucial to parallelize computations. This motivates our study of local search where at most $k$ rounds of interaction (aka adaptivity) with the oracle are allowed. We focus on the $d$-dimensional grid $\{1, 2, \ldots, n \}^d$, where the dimension $d \geq 2$ is a constant. Our main contribution is to give algorithms and lower bounds that characterize the query complexity of finding a local minimum in $k$ rounds, when $k$ is constant and polynomial in $n$, respectively. Our proof technique for lower bounding the query complexity in rounds may be of independent interest as an alternative to the classical relational adversary method of Aaronson from the fully adaptive setting. The local search analysis also enables us to characterize the query complexity of computing a Brouwer fixed point in rounds.

翻译：我们研究在图上定义的函数寻找局部最小值的查询复杂度。这一抽象问题对许多优化任务至关重要，例如在训练深度神经网络时寻找损失函数的局部最小值。在此类应用中，每次查询都是一次昂贵的损失评估，因此并行化计算至关重要。这促使我们研究局部搜索问题，其中至多允许与预言机进行$k$轮交互（亦称自适应性）。我们聚焦于$d$维网格$\\{1, 2, \\ldots, n \\}^d$，其中维度$d \\geq 2$为常数。我们的主要贡献在于给出算法和下界，分别刻画了当$k$为常数和$n$的多项式时，在$k$轮内寻找局部最小值的查询复杂度。我们用于下界轮次查询复杂度的证明技术可能具有独立意义，可作为Aaronson经典关系对抗方法在完全自适应场景下的替代方案。局部搜索的分析也使我们能够刻画在轮次内计算布劳威尔不动点的查询复杂度。