The Query Complexity of Local Search and Brouwer in Rounds

TL;DR

This paper investigates the number of queries needed to find local minima and Brouwer fixed points on a grid with limited interaction rounds, providing algorithms and lower bounds for constant and polynomial rounds.

Contribution

It introduces new algorithms and lower bounds for the round-limited query complexity of local search and Brouwer fixed points on grids, with novel proof techniques for lower bounds.

Findings

Characterizes query complexity for constant rounds.

Provides bounds for polynomial rounds.

Introduces a new lower bound proof technique.

Abstract

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.