Bootstrap Flow-Map Tree Sampling Enables Online Feedback Driven Search

Abstract

In many scientific and engineering domains, maximizing discovery within a limited sampling budget demands strategic, observation-guided exploration. While generative models have enabled training-free reward alignment, current methods typically excel in local searches within narrow regions of the underlying distribution. These approaches struggle when preferences are unknown a priori and only revealed through sequential feedback-a scenario demanding broad exploration to uncover high-utility regions. To address this, we introduce Bootstrap Flow-Map-Tree (a.k.a BFMT), a novel computationally efficient sampling framework designed for history-aware global search and alignment under sampling budget constraints. BFMT enables full tree-path construction from any tree depth using a single function evaluation, drastically reducing computational overhead while providing critical foresight for sequential sampling. By enabling dynamic transition time steps scheduling, BFMT efficiently allocates its sampling budget, smoothly transitioning from broad global exploration to fine-grained local refinement of high-utility modes discovered through exploration. Extensive experiments and ablations across diverse search and alignment tasks demonstrate that BFMT substantially outperforms baseline approaches.