arXiv:2610.14730v2 Announce Type: replace Abstract: AI agents require specialised retrieval across domains and tasks, but adapting embedding models is constrained by costly manual training-data design. Here we introduce SYNTRA, a self-improving agentic framework that turns a few retrieval examples into reusable synthesis instructions. Given an unlabelled corpus and a few relevant query-document pairs, SYNTRA infers implicit information needs and relevance criteria, then uses feedback on trial outputs to refine instructions for query generation and relevance assessment. Experiments across 24 datasets spanning 15 domains demonstrate that the same synthesis workflow produces effective training data, benefiting even strong embedding models with and without existing labelled training sets. Self-improvement strengthens this supervision on all six evaluated tasks, yielding better retrieval than human calibration on four. The training value persists at larger scales: on MS MARCO, retrieval gains continue across a 32-fold range of synthetic training-data sizes. SYNTRA thus enables scalable embedding-model adaptation with minimal human intervention.