Exponential Speedups by Rerooting Levin Tree Search

2024-12-06

arXiv.org

AI Summary

To address the limitations of Levin Tree Search (LTS)—namely, its reliance on hand-crafted policies and suboptimal search efficiency in deterministic environments—this paper proposes √LTS. The method implicitly reroots the search tree at each node, automatically decomposing global search into *q* cooperative subtasks. A learnable *rerooter* module dynamically assigns weights to subtasks, enabling weighted parallel search and uncertainty-aware resource allocation. This work introduces the first dynamic rerooting mechanism, with theoretical analysis proving a time complexity of *O(q·T^(1/q))*—exhibiting exponential acceleration potential in *q*. Empirical results demonstrate that √LTS significantly reduces the number of search steps under the optimal *q*, matching the performance of manually designed optimal subtask decomposition, while incurring only a bounded, controllable uncertainty cost from the rerooter.

Technology Category

• Search and Optimization: Learning to Search
• Planning, Routing, and Scheduling: Learning for Planning and Scheduling
• Constraint Satisfaction and Optimization: Search

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Responsible Web: Machine-in-the-loop, human agency and autonomy

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 for finding a solution node that depends on the quality of the policy. In this paper, we introduce a new algorithm, called $sqrt{ ext{LTS}}$ (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 search steps that $sqrt{ ext{LTS}}$ 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, $sqrt{ ext{LTS}}$ only takes time $O(qsqrt[q]{T})$. Like the policy, the rerooter can be learnt from data, and we expect $sqrt{ ext{LTS}}$ to be applicable to a wide range of domains.

Problem

• Improves search efficiency in deterministic environments.
• Introduces rerooting mechanism for exponential speedups.
• Competes with best subtask decomposition in node visits.

Innovation

• Implicit LTS search at every node
• Rerooting weights guide search effort
• Decomposes search space into subtasks