Improving an agent through successive revisions requires balancing exploration and evaluation under a limited budget. A narrow search can miss useful sequences of edits, while fully evaluating every intermediate version often exhausts the budget before those sequences yield a verified improvement. Inspired by the draft--verify principle of speculative decoding, we propose Speculative Recursive Self-Improvement (SpecRSI), a two-level framework coupling inner-loop agent search with outer-loop improver learning. SpecRSI drafts successors before their predecessors are fully evaluated, while adopting only fully evaluated, admissible versions. In the inner loop, Verification-Aware Tree Search (VATS) allocates the budget between further edits and outstanding checks using the predicted verified return of a continuation. In the outer loop, Group Verification Optimization (GVO) updates the improver from completed searches while keeping the real evaluator fixed. We evaluate SpecRSI on scientific optimization (including active queue management (AQM) for network infrastructure), coding, and general agent-system benchmarks. Under matched evaluation-call budgets, SpecRSI improves scientific terminal gain over Janus by 60.0%--96.3% and area under the budget curve by 32.1%--91.6%, while achieving higher means across all 50 scientific backbone--task pairs and outperforming MetaRSI-v1 by 1.30--2.70 percentage points (pp) in system score across twenty backbones. On instance-level comparisons, it further improves PyBaMM terminal gain by 142.6% and SWE-Pro success by 6.0 pp at 7.0% and 9.2% lower token cost, respectively. These results demonstrate the efficacy of the proposed approach.

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