The Query Complexity of Local Search in Rounds on General Graphs

TL;DR

This paper investigates the query complexity of finding local minima in general graphs within a limited number of rounds, providing new bounds and insights that extend understanding beyond grid graphs, with implications for optimization tasks.

Contribution

It introduces new deterministic upper bounds and randomized lower bounds for local minimum search in general graphs, expanding the theoretical understanding of query complexity in this context.

Findings

Deterministic upper bound of O(t n^{1/t} (sΔ)^{1-1/t}) for local minimum search

Randomized lower bound of Ω(t n^{1/t}-t) for connected graphs

Parallel steepest descent improves bounds on graphs with high separation number and bounded degree

Abstract

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 $O(t n^{1/t} (s\Delta)^{1-1/t})$, where $s$ is the separation number and $\Delta$ is the maximum degree of the graph. We complement this result with a randomized lower bound of $\Omega(t n^{1/t}-t)$ 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. To obtain our results, we utilized an advanced version of Gemini at various stages of our research. We discuss our experience in a methodology section.