Spectral Lower Bounds for Local Search

TL;DR

This theorem formally establishes a connection between the query complexity of local search and the mixing time of the fastest mixing Markov chain for the given graph.

Abstract

Local search is a powerful heuristic in optimization and computer science, the complexity of which has been studied in the white box and black box models. In the black box model, we are given a graph $G = (V,E)$ and oracle access to a function $f : V \to \mathbb{R}$. The local search problem is to find a vertex $v$ that is a local minimum, i.e. with $f(v) \leq f(u)$ for all $(u,v) \in E$, using as few queries to the oracle as possible. We show that if a graph $G$ admits a lazy, irreducible, and reversible Markov chain with stationary distribution $π$, then the randomized query complexity of local search on $G$ is $Ω\left( \frac{\sqrt{n}}{t_{mix} \cdot \exp(3σ)}\right)$, where $t_{mix}$ is the mixing time of the chain and $σ= \max_{u,v \in V(G)} \frac{π(v)}{π(u)}.$ This theorem formally establishes a connection between the query complexity of local search and the mixing time of the fastest mixing Markov chain for the given graph. We also get several corollaries that lower bound the complexity as a function of the spectral gap, one of which slightly improves a lower bound based on spectral gaps from prior work.

Figures (3)

• Figure 1: Consider a graph $G$ with a lazy, irreducible, and reversible Markov chain $\mathcal{P}$ with stationary distribution $\pi$ and mixing time $T$. The proof fixes a walk $\mathbf{x} = [x_0, \ldots, x_L]$, where $L = \lfloor \sqrt{n} \rfloor \cdot T$. The walk $\mathbf{x}$ is illustrated as a solid line, where every $T$-th node is highlighted. Sample a random walk $\mathbf{y} = [y_0, \ldots, y_L]$ according to $\mathcal{P}$, conditioned on $\mathbf{y}$ and $\mathbf{x}$ having a shared prefix of length $jT$, where $j \sim U(0, \lfloor \sqrt{n} \rfloor )$. We say that $x_{jT} = y_{jT}$ is the "divergence point". In the figure, the shared prefix of $\mathbf{x}$ and $\mathbf{y}$ is $[x_0, \ldots, x_T]$, so $j=1$. A critical step of the proof is to show that no vertex $v$ is too likely to lie on $\mathbf{y}$ after its divergence from $\mathbf{x}$. To show this, we divide the walk $\mathbf{y}$ in two regions. Vertices in the region $R_1 = [y_{jT}, \ldots, y_{jT+1}]$ are collectively close to being distributed according to $\pi$. This is because the divergence point from $\mathbf{x}$ is chosen randomly. In the region $R_2 = [ y_{jT+1}, ..., y_{L}]$, the walk $\mathbf{y}$ has mixed, so the vertices in $R_2$ are close to being distributed according to $\pi$. In either case, no vertex $v$ is too likely to lie on $\mathbf{y}$ after diverging from $\mathbf{x}$.
• Figure 2: The milestone $z_{(j+1)T}$, shown in purple, may not match any of the orange milestones: $x_0$ through $x_{jT}$ because it would make $\mathbf{z}$ bad, and $x_{(j+1)T}$ because it would make $J(\mathbf{x},\mathbf{z}) > j$.
• Figure 3: The milestone $z_{(j+3)T}$, shown in purple, may not match any of the orange milestones because it would make $\mathbf{z}$ bad.

Theorems & Definitions (40)

• Theorem 0
• Remark 1
• Corollary 0
• Corollary 0
• Proposition 0
• Lemma 1: BCR23, Theorem 3
• Definition 1: Set of walks $\mathcal{W}$ and parameter $T$
• Definition 2: Milestones
• Definition 3: Good/bad walk
• Definition 4: Heads and Tails.