The Query Complexity of Local Search in Rounds on General Graphs

TL;DR

This work studies the query complexity of finding a local minimum in $t$ rounds on general graphs with oracle access to an unknown function $f:V\to\mathbb{R}$. It provides a deterministic upper bound of $O(t\,n^{1/t}\,(s\Delta)^{1-1/t})$ and a randomized lower bound of $\Omega(t\,n^{1/t}-t)$, where $s$ is the graph separation number and $\Delta$ the maximum degree; it also shows that parallel steepest descent with a warm start improves performance for graphs with large $s$ and bounded degree. The paper introduces a hierarchical separator decomposition and a shattering lemma to drive the deterministic analysis, and a staircase-based adversarial construction to establish lower bounds via spanning-tree arguments. It unifies local-search-round results from grids to general graphs and informs parallel optimization when loss evaluations are expensive. Overall, the results illuminate how graph structure, via separation number and degree, governs the benefits of adaptivity and parallelization in local search tasks with practical optimization implications.

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 $O(t n^{1/t} (sΔ)^{1-1/t})$, 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.

Figures (6)

• Figure 1: Left: In the Grid graph, a neighbor of $X$ differs from $X$ in a single coordinate. Right: In the Matrix graph, a neighbor of $X$ differs from $X$ in multiple coordinates (specifically, an entire row). This dense connectivity results in a high-expansion graph, distinct from a grid. Bottom: To mitigate high probe latency, the algorithm queries batches in parallel.
• Figure 2: Visual representation of the two-round algorithm. In Round 1, the separator $S$ is queried to find $v_{min}$. In Round 2, the algorithm only queries the components ($C_2, C_3$) containing neighbors of $v_{min}$. Components ($C_1, C_4$) are not connected to $v_{min}$, so they are ignored.
• Figure 3: Parallel search from $v^{(0)}$.
• Figure 4: The left figure shows an example of a graph $G$. A spanning tree of $G$ rooted at vertex 1 is shown on the right, with the set of vertices queried by an algorithm in round 1 circled in red.
• Figure 5: Spanning tree with the round 1 queries circled in red. The value at each node is also shown, for the case where the target local minimum is vertex $4$ (i.e. the input function is the function $f_4 \in \mathcal{F}$).

Theorems & Definitions (25)

• Theorem 1
• Proposition 1
• Theorem 2
• Lemma 1: Shattering Lemma
• proof
• Lemma 2: Non-Exploration
• Lemma 3
• Lemma 4
• Definition 1: The Family $\mathcal{F}$ and Distribution $\mathcal{D}$.
• Remark 1