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# The Query Complexity of Local Search and Brouwer in Rounds

*Proceedings of Thirty Fifth Conference on Learning Theory*, PMLR 178:5128-5145, 2022.

#### Abstract

We consider the query complexity of finding a local minimum of a function defined on a graph, where at most $k$ rounds of interaction (aka adaptivity) with the oracle are allowed. Adaptivity is a fundamental concept studied due to the need to parallelize computation and understand the speedups attainable. The query complexity of local search is tightly related to the complexity of computing stationary points of a function, thus bounds for local search can give insights into the performance of algorithms such as gradient descent. 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 trade-off between the number of rounds of adaptivity and the query complexity of local search, when the number of rounds is constant and polynomial in $n$, respectively. The local search analysis also enables us to characterize the query complexity of computing a Brouwer fixed point in rounds. 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.