Exponential Speedups by Rerooting Levin Tree Search

TL;DR

The paper introduces $ ootLTS$, an algorithm that reroots Levin Tree Search at multiple nodes to decompose the search space, resulting in exponential speedups by sharing effort across subtasks, with theoretical guarantees.

Contribution

It presents $ ootLTS$, a novel rerooting approach for Levin Tree Search that achieves significant speedups through subtask decomposition and shared effort, with formal performance bounds.

Findings

$ ootLTS$ achieves exponential speedups over standard LTS.

The algorithm's performance depends on the quality of the rerooter.

Theoretical bounds relate search effort to subtask decomposition and rerooting quality.

Abstract

Levin Tree Search (LTS) (Orseau et al., 2018) is a search algorithm for deterministic environments that uses a user-specified policy to guide the search. It comes with a formal guarantee on the number of search steps (node visits) for finding a solution node that depends on the quality of the policy. In this paper, we introduce a new algorithm, called $\sqrt{\text{LTS}}$ (pronounce root-LTS), which implicitly starts an LTS search rooted at every node of the search tree. Each LTS search is assigned a rerooting weight by a (user-defined or learnt) rerooter, and the search effort is shared between all LTS searches proportionally to their weights. The rerooting mechanism implicitly decomposes the search space into subtasks, leading to significant speedups. We prove that the number of node visits that $\sqrt{\text{LTS}}$ takes is competitive with the best decomposition into subtasks, at the price of a factor that relates to the uncertainty of the rerooter. If LTS takes time $T$, in the best case with $q$ rerooting points, $\sqrt{\text{LTS}}$ only takes time $O(q\sqrt[q]{T})$. Like the policy, the rerooter can be learnt from data, and we expect $\sqrt{\text{LTS}}$ to be applicable to a wide range of domains.