*Can David Beat Goliath?* On Multi-Hop Reasoning with Resource-Constrained Agents

Reinforcement learning (RL) trains small language model agents to answer multi-hop questions by retrieving evidence over multiple turns, but reported gains typically rely on thousands of on-policy rollouts per update. We study RL for such agents under the budget constraint of commodity GPUs, where each update samples only a few rollouts per question. Under this constraint, most sampled trajectories retrieve none of the required evidence, so the outcome reward gives the policy little to learn from and small agents settle for answering without retrieval, a failure we call retrieval collapse. David-GRPO addresses this with two mechanisms: (1) Expert trajectory seeding places a handful of off-policy expert trajectories into the GRPO groups of the early updates, and (2) evidence-guided continuation rewards evidence coverage and resumes the most promising partial trajectory. The evidence for each training question is constructed from the corpus link graph, so no annotated evidence is required. On six multi-hop QA benchmarks, David-GRPO trained on four RTX 3090 GPUs with 144 rollouts per step brings Qwen2.5-1.5B to 22.6 average EM against 11.9 for the best baseline under the same budget, matches Tree-GRPO trained with 20 times more rollouts, and, unlike the baselines that stop after at most one search, learns to retrieve across turns. The implementation is available at: https://anonymous.4open.science/r/David-GRPO-5261/